Curriculum Vitae of Yaozhong HU Yaozhong Hu Department of Mathematics University of Kansas Lawrence, KS 66045-2142 Office Tel: (785) 864-3565 Email: [email protected]http://www.math.ku.edu/˜hu Education Louis Pasteur University, France Ph.D 1992 Advisor: Paul Andr´ e Meyer Chinese Academy of Science, China M.S. 1984 Advisor: Guoping Li University of Jiangxi, China B.S. 1981 Appointments 2007-Present Professor University of Kansas. 2002-2007 Associate Professor University of Kansas. 1997-2002 Assistant Professor University of Kansas. 1996-1997 Visiting Assistant Professor University of California at Irvine. 93, 94-95 Post-Doc and Research Associate Norwegian Research Council, Norway (Advisor: B. Øksendal) 1994 Post-Doc and Research Associate University of North Carolina at Chapel Hill (Advisor: G. Kallianpur) 1994 Alexander von Humboldt Ruhr-University-Bochum, Germany (Advisor: S. Albeverio) 1992-1993 Maˆ ıtre de Conf´ erences Louis Pasteur University, France. 1984-1991 Lecturer Chinese Academy of Science, China Honors Fellow of Institute of Mathematical Statistics Wells Morrison Teaching Award, Department of Mathematics, University of Kansas. Research Grants 1. Some problems in stochastic differential equations, Simons Collaboration Grants for Mathemati- cians 7/1/2011 - 6/30/ 2016. 2. Nonlinear Functionals of fractional Brownian motion, National Science Foundation, 7/1/2005 - 6/30/2008. 3. Stochastic Differential Systems Driven by fractional Brownian motion, National Science Founda- tion, 7/15/2002 - 6/30/2005. 4. Keeler University Professorship 08/2003 - 12/2003 5. Brox diffusions, Amount: $9,935.00, Dates: 07/01/14 - 06/30/15 6. Statistics of stochastic differential equations driven by stable noises, Amount $8,331.00, 07/01/12 - 06/30/13 7. Probabilistic Methods in Fractal Markets, General Research Fund, University of Kansas, 7/1/01- 6/30/02. 8. Stochastic Calculus for General Gaussian Processes and Applications, General Research Fund, University of Kansas, 7/1/00-6/30/01. 1
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Curriculum Vitae of Yaozhong HU
Yaozhong HuDepartment of MathematicsUniversity of KansasLawrence, KS 66045-2142
EducationLouis Pasteur University, France Ph.D 1992 Advisor: Paul Andre MeyerChinese Academy of Science, China M.S. 1984 Advisor: Guoping LiUniversity of Jiangxi, China B.S. 1981
Appointments
2007-Present Professor University of Kansas.2002-2007 Associate Professor University of Kansas.1997-2002 Assistant Professor University of Kansas.1996-1997 Visiting Assistant Professor University of California at Irvine.93, 94-95 Post-Doc and Research Associate Norwegian Research Council, Norway
(Advisor: B. Øksendal)1994 Post-Doc and Research Associate University of North Carolina at Chapel Hill
(Advisor: G. Kallianpur)1994 Alexander von Humboldt Ruhr-University-Bochum, Germany
(Advisor: S. Albeverio)1992-1993 Maıtre de Conferences Louis Pasteur University, France.1984-1991 Lecturer Chinese Academy of Science, China
Honors
Fellow of Institute of Mathematical StatisticsWells Morrison Teaching Award, Department of Mathematics, University of Kansas.
Research Grants
1. Some problems in stochastic differential equations, Simons Collaboration Grants for Mathemati-cians 7/1/2011 - 6/30/ 2016.
2. Nonlinear Functionals of fractional Brownian motion, National Science Foundation, 7/1/2005 -6/30/2008.
3. Stochastic Differential Systems Driven by fractional Brownian motion, National Science Founda-tion, 7/15/2002 - 6/30/2005.
4. Keeler University Professorship 08/2003 - 12/2003
6. Statistics of stochastic differential equations driven by stable noises, Amount $8,331.00, 07/01/12- 06/30/13
7. Probabilistic Methods in Fractal Markets, General Research Fund, University of Kansas, 7/1/01-6/30/02.
8. Stochastic Calculus for General Gaussian Processes and Applications, General Research Fund,University of Kansas, 7/1/00-6/30/01.
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9. Stochastic Calculus for General Gaussian Processes and Applications to Finance and NonlinearFiltering, First Awards EPSCoR, National Science Foundation, 7/1/99-6/30/00.
10. Ito Calculus for FBM and Applications, General Research Fund, University of Kansas, 7/1/99-6/30/00.
11. Stochastic Differential Systems with Fractional White Noise, General Research Fund, Universityof Kansas, 7/1/98-6/30/99.
12. Stochastic Differential Systems with Fractional White Noise, New Faculty General Research Fund,University of Kansas, 7/1/98-6/30/99.
13. Big 12 Faculty Fellowship, University of Kansas, 1999
14. Alexander von Humboldt Research Fellow, Ruhr University-Bochum, Germany, 7/1994-8/1994
Professional Service
Associate Editor of the following three journals
1. Stochastics and Stochastics Report
2. Acta Mathematica Scientia
3. Journal of Applied Mathematics and Stochastic Analysis
Reviewer of Mathematical Review
Referee for large amount of journals and proceedings.
Conferences organizations
1. Organized an invited session on “Stochastic partial differential equations driven by Gaussiannoise” in the third IMS-China international conference on Statistics and Probability, Kuming,China, July 1 - July 3, 2015
2. Co-organizer “Mini Workshop on Probability and Statistics”, Wuhan, China, June 19, 2013.
3. Co-organizer “AMS sectional meeting”, March 29-April 1, 2012, University of Kansas.
4. Co-organizer “Seminar on Stochastic Processes 2012”, March 22-24, 2012, University of Kansas.
5. Co-organizer of the “International Conference on Malliavin Calculus and Stochastic Analysis”,March 19-21, 2011, University of Kansas.
6. Scientific committee of international symposium “optimal stopping with applications”, June, 23-26, 2009, Abo/Turku, Finland.
Some Service at KU
University Academic Assessment Committee, January 2017 -
College of Liberal Arts & Sciences Committee on Appointments, Promotion and Tenure, Universityof Kansas, Fall 2015 - Spring 2018
Graduate Admission Director of the Mathematics Department, University of Kansas, Fall, 2009
College of Liberal Arts & Sciences Committee on Appointments, Promotion and Tenure, Universityof Kansas, Fall 2011, Spring 2011.
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Faculty Senate Research Committee, The University of Kansas, 2003-2006
University Senate Library Committee, 2002 to 2003.
Postdoctoral students co-advised:
Fangjun Xu (Fall 2010 - Spring, 2013, co-adviser: David Nualart)Le Chen (Spring , 2015 - present, co-adviser: David Nualart)
Ph.D students advised (advising):
1. Xiong Wang (current, co-advisor: David Nualart)
2. Cheng Yiying (current)
3. Xia Panqiu (current, co-advisor: David Nualart)
4. Zhou Hongjuan (current, co-advisor: David Nualart)
5. Cui Yanhao (current)
6. Zhang Junxi (current)
Ph.D students advised (advising):
1. Su Chen
Graduated May 4, 2016 (co-advisor: Xuemin Tu). Wells Fargo.
Thesis title: Some studies on parameter estimations.
2. Liu Yanghui
Graduated April 2016 (co-advisor: David Nualart). Postdoc: Purdue University.
Thesis title: Numerical solutions of rough differential equations and stochastic differential equa-tions.
3. Han Zheng
Graduated Summer 2015. University of Kansas, Business School
Thesis title: Controlling reflected diffusions and applications to finance and operation management
4. Hu Guannan
Graduated Summer 2015. University of Oklahoma, Business School.
Thesis title: Fractional Diffusion in Gaussian Noisy Environment
5. Le Khoa
Graduated Spring 2015. Postdoc at University of California at Berkeley.
Thesis title: Nonlinear Integrals, Diffusion in Random Environments and Stochastic Partial Dif-ferential Equations
6. Huang Jingyu
Graduated Spring 2015. (co-advisor: David Nualart), Postdoc at University of California atBerkeley.
Thesis title: Stochastic Partial Differential Equations Driven by Colored Noise
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7. Lu Fei (co-advisor: David Nualart)
Graduated 2013. Postdoc at University of California at Berkeley.
Thesis title: Applications of Malliavin calculus to SPDE and convergence of densities
8. Song Xiaoming (co-advisor: David Nualart)
Graduated 2011. Postdoc at University of North Carolina at Chapel Hill and Assistant professorat Drexel University.
Thesis title: Backward stochastic differential equations and stochastic differential equations drivenby fractional Brownian motion and their numerical solutions.
9. Song Jian (co-advisor: David Nualart)
Graduated 2010. Postdoc at Rutgers University and Assistant professor at Hong Kong University.
Thesis title: Some topics on the fractional Brownian motion and stochastic partial differentialequations.
10. Wang Baobin
Graduated 2008. Associate Professor at Central China University of nationalities, China.
Thesis title: Rate of convergence to approximate the multiple Stratonovich integrals for fractionalBrownian motions.
11. Dalkir Elif
Graduated 2008. Assistant Professor at University of New Brunswick, Canada.
Thesis title: Uniqueness of responsive voting equilibrium.
Master degree students:
1. Yin Yu (current)
2. Chu, David, December 2016
3. Wang, Yu, September 2016
4. Huang, Tianyu , May 2, 2016
5. Yang, Liu, April 28, 2015
Almost sure behavior of U-statistics and related inequalities.
6. Cui Qingqing, January 27, 2014
1/e law for classical secretary problem and related topics.
7. Li Han, August 31, 2012
Frequency domain analysis of cyclical components of US GDP.
8. Whittaker Timothy, January 24, 2012
Generalized autoregressive conditional heteroskedasticity with long memory and time varyingparameters.
9. Yi Lin, 05/05/2009
The application of genetic algorithm in return-risk optimization.
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10. Yao Kan, 04/24/2009
Time series models with garch errors: a review.
11. Zhang Ming, 04/06/2007
Stochastic volatility models.
12. Zhang Qinghua, 04/18/2007
Parameter estimation for Ornstein-Uhlenbeck processes driven by alpha-stable Levy motions.
13. Takavingofa, Bob, completed 8/2006
Long memory phenomena and weather derivatives, University of Zimbabwe.
Visiting scholars or students hosted:
1. Framstad, Nils (August 1, 2001 - December 31, 2001)
Department of Economics, University of Oslo, Norway
2. Xiao, Weilin (September 1, 2008 - August 31, 2010)
School of Management, Zhejing University
3. Han, Yuecai (August 1, 2010 - July 31, 2011)
Department of Mathematics, Jilin University
4. Tian, Ping (August 1, 2010 - July 31, 2011)
Department of Mathematics, Jilin University
5. Rang, Guanglin (August 1, 2012 - July 31, 2013)
Department of Mathematics, Wuhan University
6. Tong, Jinying (December 1, 2013 - November 30, 2014)
College of Science, Donghua University
7. Gong, Yicheng (August 1, 2014 - February 28, 2015)
College of Science, Wuhan University of Science and Technology
8. Wang, Baobin (September 1, 2014 - August 31, 2015)
College of Science, Central South China University of Nationalities.
9. He, Kun (July 1, 2014 - June 30, 2015)
College of Science, Donghua University
10. Zang Qingpei (July 2015 - February 2016)
ZhengJiang University
11. Tu Lilan (July 2015 - July 2016)
Wuhan University of Science and Technology
12. Sun Lin (August 27, 2015 - February 26, 2016)
Guanzhou University of Technology
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13. Cui Qing (August 30, 2015 - February 28, 2016)
Anhui Normal University
14. Jiang Guo (Feb 2016 - Feb 2017)
Hubei Normal University
15. Chen Yong
Hunan University of Science and Technology
16. Chen Guici
Wuhan University of Science and Technology
17. Wang Zhi
College of Science, Ningbo University of Technology
18. Guo Jingjun
Lanzhou University of Finance and Economics.
19. You Zouwei
School of Economics and Management, Beihang University, Beijing, China
Recent PhD student committees at KU served
Quanz Brian (Department of Electrical Engineering & Computer Science)
Lan Yi (Department of Civil, Environmental, & Architectural Engineering )
Yan Kaige ( (Department of Civil, Environmental, & Architectural Engineering)
Kumar Jitendra Thakur (Department of Civil, Environmental, & Architectural Engineering)
Ayyalapu Nishitha (Department of Electrical Engineering & Computer Science)
Kumar Varatharajan Sarvesh (Department of Electrical Engineering & Computer Science)
Wang Wenhao (School of Education)
Rina Na (Department of Economics, Oral Comp March 2, 2017)
Su Liting (Department of Economics, graduated March 13, 2017)
Teney, Alexander Clark (Department of Economics, graduated May 6, 2015)
Corey Ryan (Department of Civil, Environmental, & Architectural Engineering, September 2015)
Li Xi (Department of Mathematics, August 2015)
Kacaribu Febrio (Department of Economics, February 4, 2014)
Nguyen Huy Quoc (Department of Economics, August 27, 2014)
Peng Chen (Department of Economics, April 19, 2013)
Liu Jia (Department of Economics, May 15, 2013)
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Lugovskyy, Josephine Cruz (Department of Economics, November 30, 2012)
Lim Sungjin (Department of Economics, May 15, 2012)
Huang Yilei (Dapartment of Civil, Environmental, and Architectural engineering, May 7, 2012)
Li Han (Department of Economics, May 10, 2012)
Lei Pedro (Math department, September 10, 2012)
Chen Lili (Department of Economics, May 7, 2012)
Zheng Mingming (Department of Economics, 2011)
Carvajal-Espinoza, Jorge E. (School of Education, 2011)
Li Yue (Department of Civil, Environmental, & Architectural Engineering, 2011)
Ikuyasu, Usui (Department of Economics, 2010)
Chui, Calvin (Department of Civil, Environmental, & Architectural Engineering, 2010)
Yi Juan (Department of Civil, Environmental, & Architectural Engineering, 2010)
Kim Seonghoon (Department of Civil, Environmental, and Architectural engineering, 2008)
Chen Fei See (Deapartment of Civil, Environmental, and Architectural engineering, 2008)
Kitaoka Hisaya (Department of Economics, 2008)
Li Yingfeng (Eric) (Dapartment of Civil, Environmental, and Architectural engineering, 2007)
Zimmer Peter (Oral committee)
Matache Dora (Oral committee)
Peng Yan (Department of Economics, 1999)
Wei Heng (Dapartment of Civil, Environmental, and Architectural engineering, 1999)
Student committees outside KU served
Dr. Gille Harge’s habiltation committee to direct research, the Universite d’Evry-Val d’Essonne,France, 2000.
Dr. Laurent Decreusefond’s habilitation committee to direct research, the Universite d’Evry-Vald’Essonne, France, 2001.
Mr. Alberto Lanconelli’s Ph.D defense committee, University of Oslo, Norway, 2004.
Mr. F. Durrell’s PhD thesis examiner, University of Cape Town, South Africa, 2007.
Miss Hassilah Binti Salleh’s Ph.D defense committee, University of Oslo, Norway, 2009.
Mr. Winston Buckley,s Ph.D dissertation committee, Florida Atlantic University, 2009.
Mr. Farai Julius Mhlanga’s PhD thesis examiner, University of Cape Town, South Africa, 2010.
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Mr. Shuai Jing’s Ph.D defense committee, Universite de Bretagne Occidentale, France, 2011.
Mr. Tianyang Nie ’s Ph.D defense committee, Universite de Bretagne Occidentale, France, September20, 2012.
Mr. Wang Qingfeng’s Ph.D defense committee, Loughborough University, England, October 19, 2012.
Mr. Haidar Al-Talibi’s Ph.D thesis opponent, Linnaeus University, Sweden, November 22, 2012.
Mr. Sven Haadem’s Ph.D defense committee, opponent, University of Oslo, March 2014.
Short visits - One Month or Two Months Visit:
1. University of Oslo, Oslo, Norway. 07/25/2016 - 08/19/2016.
2. East China Normal University, Shanghai, China. 05/20/2016 - 07/19/2016.
3. Wuhan Institute of Physics and Mathematics, Chinese Academy of Science, Wuhan, China.06/05/2015 - 08/03/2015.
4. Universite de Lorraine, Nancy, France. 05/04/2015 - 05/29/2015.
5. Wuhan Institute of Physics and Mathematics, Chinese Academy of Science, China. 06/13/2014 -08/22/2014.
6. University of Oslo, Oslo, Norway. 05/02/14 - 05/31/14.
7. Wuhan Institute of Physics and Mathematics, Chinese Academy of Science, China. 05/25/2013 -8/23/2013.
8. University of British Columbia, Canada. 03/01/2013 - 04/30/2013.
9. Institute of Mathematics and its Applications, University of Minnesota. 01/01/2013 - 02/28/2013.
10. School of Industrial Engineering and Management, Israel Institute of Technology. Israel. 12/1/2012- 12/27/2012.
11. Wuhan University of Science and Technology, Wuhan, China. 06/12/2012 - 08/17/2012.
12. The Chinese Academy of Science, Wuhan, China. 06/12/2012 - 08/17/2012.
40. University of Bielefeld, Bielefeld, Germany. 07/92.
Conferences and titles of the invited presentations:
1. 07/17/2017 - 07/21/2017: 13th International Workshop on Markov Processes and Related Fields.Wuhan, China.
Give a talk on ”Ito stochastic differential equation driven by fractional Brownian motion of Hurstparameter H > 1/2.
2. 08/15/2016 - 08/19/2016: Abel symposium on combinatoric, numerics, stochastics. Rosendal,Norway.
Give a talk on “density bounds of parabolic Anderson field”.
3. 07/13/2016 -07/17/2016: 12-th International Workshop on Markov Processes and Related Fields.Jiangsu Normal University, Xuzhou.
Give a talk on “density bounds of parabolic Anderson field”.
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4. 07/09/2016 - 07/12/2016: Workshop on Stochastic Partial Differential Equations and StochasticDynamics. Shanghai Normal University. Shanghai.
Give a talk on “Fractional diffusions in a Gaussian noisy environment”.
5. 06/27/2016 -06/30/2016: The 4-th Institute of Mathematical Statistics Asia Pacific Rim Meeting(IMS-APRM), The Chinese University of Hong Kong.
Give a talk on “density of parabolic Anderson random variable”.
6. 06/22/2016 - 06 /24/2016: International workshop on dynamical system and stochastic differentialequations. Huazhong University of Science and Technology, Wuhan, China.
Presented a talk on “Feynman-Kac formula for stochastic partial differential equation driven byrough signal”.
7. 06/03/2016 - 06/05/2016: International Workshop on Stochastic Processes and Applied Proba-bility. Jilin University, Changchun, China.
“Intermittency for the stochastic heat equation driven by time-fractional noise with H ∈ (0, 1/2)”.
8. 04/09/2016 - 04/10/2016: AMS regional meeting on Stochastic partial differential equations,University of Utah, Salt Lake City.
“Density of parabolic Anderson random variable”.
9. 08/02/2015 -08/03/2015: Participate International Conference on Stochastic Analysis and RelatedTopics, Wuhan University, China and present a talk on
“Nonlinear Young integrals and differential systems in Holder media”.
10. 07/01/2015 - 07/03/2015: Participate the third IMS-China international conference on Statisticsand Probability, Kuming, China and present a talk on
“Stochastic partial differential equations driven by general Gaussian noise: Holder continuity andintermittency”.
11. 06/26/2015 - 06/30/2015: participate the 11-th international workshop on Markov processes andrelated topics, Shanghai, China and present a talk on
“Brox diffusion and its stochastic differential equations”.
12. 04/20/2015 - 04/24/2015: participate the international Conference on Stochastics in Environmen-tal and Financial Economics. Oslo, Norway and present a talk on
“Brownian in an white noise environment”.
13. 03/27/2015 - 03/29/2015: AMS spring southeastern sectional meeting at University of Alabamaat Huntsville and present a talk on
“Brox diffusion and its stochastic differential equations”.
14. 9/14/2014 - 9/19/2014: participate the international Conference on Stochastics in Environmentaland Financial Economics. Oslo, Norway and present a talk on
“Parametric estimation of long memory Ornstein-Uhlenbeck processes”.
15. 08/13/2014 - 08/18/2014: participate the 10-th international workshop on Markov processes andrelated topics. Xi An, China and present a talk on
“Density convergence for some nonlinear Gaussian stationary sequences”.
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16. 07/21/2014 - 07/25/2014:
participate the 20th international conference on difference equations and applications, WuhanInstitute of Physics and Mathematics, Chinese Academy of Science, China and present a talk on
“Parameter estimation for long memory Ornstein-Uhlenbeck process”.
17. 07/03/2014 - 07/07/2014
participate the workshop on stochastic processes and applied probability, Jilin University, Changchun,China and present a talk
“Stochastic heat equation with general multiplicative Gaussian noise: Holder continuity and in-termittency”.
18. 06/22/2014 - 06/27/2014
participate the seventh international symposium on backward stochastic differential equations,Shandong University, Weihai, China and present a talk on
“Singular mean-field control games with asymmetric information”.
19. 08/04/2013 - 08/07/2013
Participate the second workshop on stochastic analysis and related topics, Beijing, China andpresent a talk on
“Malliavin calculus and convergence in density of some nonlinear Gaussian functionals”.
20. 07/28/2013 - 08/02/2013
Participate the first workshop on stochastic differential equations and stochastic partial differentialequations, Hefei, China and present a talk on
“Modified Euler approximation scheme for stochastic differential equations driven by fractionalBrownian motions”.
21. 07/14/2013 - 07/18/2013
Participate the 3rd international conference on random dynamical system and present a talk on
“Stochastic quantization and ergodic theorem for diffusions”.
22. 07/06/2013 - 07/13/2013
Participate the 9-th workshop on Markov processes and related fields at Emei, China and presenta talk on
“A multiparameter Garsia-Rodemich-Rumsey inequality and some applications”.
23. 06/30/2013 - 07/04/2013
Participate the second IMS-China international conference on Statistics and Probability, Chengdu,China and present a talk on
“Central limit theorem for an additive functional of the fractional Brownian motion”.
24. 06/19/2013 Mini Workshop on Probability and Statistics, Wuhan Institute of Physics and Math-ematics, Chinese Academy of Science, Wuhan, China.
“Feynman-Kac formula for fractional heat equation driven by fractional white noise”.
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25. 05/30/2013 - 06/02/2013
Attend the inaugural conference of the Institute for Advanced Study Honoring Professor ChenJiang Gong in Hangzhou Normal University, Hangzhou, China.
26. 07/27/2012 - 08/11/2012: Give a minicourse on Analysis on Wiener Space. Jilin University,Changchun, China.
27. 7/15/2012 - 7/22/2012: 8th workshop on Markov processes and related topics.
Convergence in Density of Some Nonlinear Gaussian Functionals. Beijing Normal University,Beijing, China.
28. Give a minicourse on Fractional Brownian motions and applications. Wuhan University of Scienceand Technology, Wuhan, China. 6/12/2012 - 7/12/2012.
29. 6/19/2012 - 6/25/2012: Advanced workshop on probability and statistics, Zhejiang University,Hangzhou, China. Parametric estimation for long memory O-U processes with discrete observa-tions.
30. 6/3/2012 - 6/8/2012: NSF/CBMS conference at University of Alabama at Huntsville, Malliavincalculus and convergence in density.
31. 4/1/2012 - 4/6/2012: Banff international research station international conference on ”StochasticAnalysis and stochastic partial differential equations”. Feynman-Kac formula for stochastic partialdifferential equations.
32. 8/18/2011: Mini-workshop in stochastic analysis and applications, Oslo, Norway: Convergence indensity.
33. 7/8/2011-7/11/2011: IMS-China international Conference, Xi’An, China: Holder continuity for aclass of nonlinear SPDE arising from superprocess.
34. 7/4/2011-7/6/2011: International conference on stochastic analysis and its application to mathe-matical finance: Maximum principle of stochastic systems driven by fractional Brownian motions.Chinese Academy of Science, Beijing, China.
35. 11/13/2010-11/15/2010: International Conference on Modern Analysis in memory of 100 anniver-sary of Professor Li Guoping: Rough path analysis via fractional calculus.
36. 07/19/2010-07/23/2010: 7th international conference on Markov processes and related topics,Beijing Normal University, Beijing, China: A central limit theorem of Brownian local time in Lp.
37. 06/05/2010-06/08/2010: International conference on Applied Analysis, Shanghai, China: Feynman-Kac formula for stochastic heat equations driven by fractional noises.
38. 07/19/2009-12/24/2009: Workshop on Stochastic Partial Differential Equations, Weihai, China:Feynman-Kac formula for stochastic partial differential equations.
39. 12/17/2008-12/22/2008: AMS-SMS joint meeting, in Fudan University, Shanghai, China: Malli-avin calculus and numerical solution of backward stochastic differential equations.
40. 12/14/2008-12/16/2008: Suzhou conference on stochastic analysis: Rough path analysis via frac-tional calculus.
41. 02/14/2008-02/15/2008: Kansas-Missouri Winter School on Applied Probability: Backward stochas-tic differential equations driven by fractional Brownian motions.
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42. 11/19/2007-11/24/2007: Workshop on application of stochastic partial differential equations,Mittag-Leffler Institute, Stockholm, Sweden: Stochastic heat equations driven by fractional noiseand local times.
43. 10/3/2007-10/12/2007: Visit University of Oslo and gave a talk in the “Workshop on insidertrading”.
44. 7/5/2007-7/15/2007: Gave a series of talks in Yantai summer seminar on mathematical finance.
45. 10/2006: AMS regional conference on stochastic control and stochastic game, University of Cincin-nati.
46. 7/27/2005-8/5/2005: Abel Symposium on Stochastic Analysis and Application in honor of K. Ito.Oslo, Norway.
47. 7/4/2005-7/8/2005: Conference on Stochastic Partial Differential Equations, Vancouver, Canada.
48. 6/8/2005-6/10/2005: Conference on Stochastic Analysis and Application in honor of B. Øksendal,Oslo, Norway.
49. 8/10/2004-8/14/2004: Workshop on markov processes and related topics, Beijing Normal Univer-sity, Beijing, China.
50. 8/3/2004-8/8/2004: Workshop on stochastic analysis, Institute of Applied Mathematics, TheChinese Academy of Science, Beijing, China.
51. 6/22/2004-6/26/2004: Workshop on probability and application, Wuhan University, Wuhan,China.
52. 5/24/2004-5/31/2004: International Workshop on Mathematical Finance and Insurance, YellowMountain, China.
53. 5/20/2004-5/22/2004: Workshop on Markov processes and related fields, Beijing Normal Univer-sity, Beijing, China.
54. 8/4/2003-8/15/2003: Warwick workshop on stochastic partial differential equations, Warwick,UK: Stochastic equations driven by fractional noises.
55. 6/22/03-6/26/03: AMS-IMS-SIAM Conference at Snowbird summer resort, Utah: Optimal port-folio for an insider.
56. 4/4/03-4/6/03: AMS regional meeting on Mathematical Finance, Indiana University, Blooming-ton: Optimal portfolio for an insider.
57. 3/14/03-3/16/03: AMS regional meeting on stochastics, quantization and Segal-Bargman analysis,Louisiana University, Baton Rouge: Rotationary approximation for tangent process.
58. 10/14/02-10/18/02: 9th workshop on Mathematical Fiance (insider trading), University of Oslo,Oslo, Norway: Stochastic analysis of fBm and optimal consumption and portfolio in a stochasticvolatility market.
59. 07/15/02-07/17/02: Workshop on Probability with Applications to Finance and Insurance, HongKong University, Hong Kong, China: Optimal Consumption and Portfolio in a stochastic volatilitymarket.
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60. 06/07/01-06/09/01: 973 meeting on stochastic analysis, Chengde, China: Stochastic calculus forfractional Brownian motions and applications.
61. 05/10/01-05/13/01: International Conference on Mathematical Finance, Shanghai, China: Optionpricing in a fractal markets.
62. 02/15/01-02/16/01: Workshop on Fractional Brownian motion: stochastic calculus and applica-tions, Barcelona, Spain: Probability structure preserving mapping and absolute continuity.
63. 06/27/98-07/02/98: International Workshop on Gaussian Correlation Inequalities, College Sta-tion, Texas: Unification of several inequalities for Gaussian measures.
64. 12/13/97-12/15/97: Symposium on Stochastic Control and Nonlinear Filtering, Los Angeles:Finite Difference Approximation of Zakai equations.
65. 10/17/97-10/18/97: AMS meeting Special Session on Stochastic Inequalities and Their Applica-tions, Atlanta: Exponential Integrability for Diffusions
66. 03/27/95-03/31/95: Workshop on stochastic evolution equations as dynamical systems, Warwick,United Kingdom: stochastic quantization.
67. 10/24/94-10/29/94: International Conference on Stochastic Analysis: Mathematics, FinancialMarkets, Biology, Engineering and Physics, Bielefeld, Germany: Approximation of of stochasticdifferential equations of Ito type.
68. 07/09/95-07/14/95: International Meeting on Stochastic Analysis and Applications, Gregynog,United Kingdom: Continuity of some anticipating integral processes.
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Colloquium, Seminar Talks and Research Cooperation:
1. Nankai University, 07/25/2017: “Density of parabolic Anderson random variable”.
2. HuaZhong University of Science and Technology 07/05/2017: “Feynman-Kac formula for thestochastic heat equation driven by fractional noise in time with H < 1/2”.
3. Wuhan University of Technology, 07/04/2017: Fractional diffusions driven by Gaussian noises”.
4. East China Normal University 06/30/2017: “The Expected Time Approach to Optimal Adjust-ment Problems: Application to Domestic Energy Prices”.
5. East China Normal University 06/27/2017: “Intermittency for the stochastic heat equation drivenby time-fractional noise with H ∈ (0, 1/2)”.
6. Wuhan University, 06/24/2017: “Intermittency for the stochastic heat equation driven by time-fractional noise with H ∈ (0, 1/2)”.
7. University of Oslo, 06/07/2017: “Intermittency for the stochastic heat equation driven by time-fractional noise with H ∈ (0, 1/2)”.
8. University of Alberta at Edmonton, 02/10/2017: “Density of parabolic Anderson random vari-able”.
9. University of Alberta at Edmonton, 02/07/2017: “Generalized moment estimation for -stableOrnstein-Uhlenbeck processes from discrete observations”.
10. Purdue University, 01/17/2017: “Ito stochastic differential equations driven by fractional Brown-ian motions of Hurst parameter H > 1/2.”
11. Stevens Institute of Technology, 09/21/2016: “Some mean field singular stochastic control prob-lems”.
12. University of Oslo, 08/11/2016: “Rate of convergence and asymptotic error distribution of Eulerapproximation schemes for fractional diffusions”.
14. Anhui Normal University, 07/08/2016: Convergence in density of some nonlinear Gaussian func-tionals.
15. East China Normal University, 06/26/2016: Singular mean-field control games with asymmetricinformation
16. Donghua University, 06/23/2016: Rate of convergence and asymptotic error distribution of Eulerapproximation schemes for fractional diffusions.
17. University of Macau, 06/30/2016. Give a talk on “Fractional diffusions in a Gaussian noisyenvironment”.
18. Jiangxi University of Economics and Finance, 06/16/2016.
Option price in a market with long memory.
19. Wuhan University of Science and Technology, 06/13/2016 - 06/15/2016: Give a series of talks on“Fractional diffusions in a Gaussian noisy environment”.
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20. Wuhan University, 06/10/2016: Fractional diffusions in a Gaussian noisy environment.
21. Donghua University, 05/23/2016: Rate of convergence and asymptotic error distribution of Eulerapproximation schemes for fractional diffusions.
22. East China Normal University, 05/26/2016: Singular mean-field control games with asymmetricinformation.
23. University of Tennessee at Knoxville. 03/29/2016. Convergence in density of some nonlinearGaussian functionals.
24. Donghua University, 07/07/2015: Parameter estimation for long memory Ornstein-Uhlenbeckprocesses.
25. Bengbu University, 07/06/2015: Black-Scholes formula in a long memory market.
26. Wuhu Polytechnical University, 07/05/2015: Optimal time to invest for an insider.
27. Anhui Normal University, 07/05/2015: Stochastic heat equation with general multiplicative Gaus-sian noise: Holder continuity and intermittency.
28. Xiamen University, 06/23/2015: Parameter estimation for long memory Ornstein-Uhlenbeck pro-cesses.
29. Wuhan University of Science and and Technology: seminar talk, 06/10/2015: An optimal stoppingtime problem.
30. Universite Lorraine: 05/07/2015: Brownian motion in a white noise environment.
31. Iowa State University: 01/26/2015: Parameter estimation for long memory Ornstein-Uhlenbeckprocesses.
32. University of Tennessee: seminar talk: 10/08/2014: Brox diffusion and its stochastic differentialequations.
33. Huazhong University of Science and Technology: Colloquium talk, 07/29/2014: Stochastic heatequation with general multiplicative Gaussian noise: Holder continuity and intermittency.
34. Jiangsu Normal University: Colloquium talk, 06/31/2014: Malliavin calculus and convergence indensity of some nonlinear Gaussian functionals.
35. Wuhan University: Colloquium talk, 06/14/2014: Parameter estimation for long memory Ornstein-Uhlenbeck process
36. University of Oslo: Colloquium talk, 05/14/2014: Density convergence for some nonlinear Gaus-sian stationary sequences.
37. University of Oslo: Colloquium talk, 03/19/2014: Stochastic heat equation with general multi-plicative Gaussian noise: Holder continuity and intermittency.
38. University of Illinois at Chicago, 10/24/2013. Malliavin calculus and convergence of densities.
59. Wuhan Institute of Physics and Mathematics, The Chinese Academy of Science: Seminar talk,7/19/2011: Convergence in density.
60. Tianjing University: Colloquium talk, 7/13/2011: Improving the Black-Scholes formula by usinglong memory.
61. Purdue University: Seminar talk, 1/25/2011: Malliavin calculus for backward stochastic differen-tial equations and application to numerical solutions.
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62. University of Delaware: Seminar talk, 12/17/2010: Parameter estimation for Ornstein-Uhlenbeckprocess driven by fractional Brownian motion.
63. University of Kansas: Seminar talk, 12/6/2010: Parameter estimation for Ornstein-Uhlenbeckprocess driven by fractional Brownian motion.
64. Iowa State University : Seminar talk, 11/29/2010: Parameter estimation for Ornstein-Uhlenbeckprocess driven by fractional Brownian motion.
65. Rutgers University: Seminar talk, 11/2/2010: Malliavin calculus for backward stochastic differ-ential equations and application to numerical solutions.
66. The University of Science and Technology of China, Hefei, China: Seminar talk, 7/7/2010: Roughpath analysis via fractional calculus.
67. South East University, Nanjing, China: Seminar talk, 6/25/2010: Rough path analysis via frac-tional calculus.
68. Nanjing University, Nanjing, China: Seminar talk, 6/24/2010: Feynman-Kac formula for stochas-tic heat equation driven by fractional noises.
69. Wuhan Institute of Physics and Mathematics, The Chinese Academy of Science, Wuhan, China:Seminar talk, 6/12/2010: Feynman-Kac formula for stochastic heat equation driven by fractionalnoises.
70. Jilin University, Changchun, China: Seminar talk, 5/26/2010: Improving Black-Scholes formulaby using long memory processes.
71. Jilin University, Changchun, China: Seminar talk, 5/25/2010: Optimal time to invest for aninsider.
72. Universite Strasbourg I, Strasbourg, France: Seminar talk, 3/12/2010: Feynman-Kac formula forstochastic partial differential equations driven by fractional noises.
73. Universite Nancy I, Nancy, France: Seminar talk, 3/11/2010: Feynman-Kac formula for stochasticpartial differential equations driven by fractional noises.
74. African Institute of Mathematical Science, Capetown, South Africa: Colloquium talk, 12/16/2009:Optimal time to invest.
75. African Institute of Mathematical Science, Capetown, South Africa: 12/8/2009 - 12/11/2009:Gave eight two-hour lectures on rough path analysis
76. University of Tennessee: Seminar talk , 09/21/2009: Stochastic Heat Equation Driven by Frac-tional Noise and Local Time.
77. Faculty of Science, Donghua University, Shanghai, China: 06/10/2009 - 06/19/2009: Gave eighttwo-hour lectures on Malliavin calculus.
78. Department of Finance, Xing Jiang University of Economics and Finance, Urumuqi, China: col-loquium talk, 06/29/09: Optimal time to invest.
79. Department of Mathematics, Xing Jiang University, Urumuqi, China: colloquium talk, 06/29/09:Improving Black-Scholes formula by using long memory.
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80. Department of Mathematics, Fudan University, Shanghai, China: seminar talk, 01/9/09: Roughpath analysis via fractional calculus
81. Department of Mathematics, Donghua University, Shanghai, China: seminar talk, 01/6/09: Leastsquares estimator for O-U process driven by fractional noise.
82. Wuhan Institute of Physics and Mathematics, The Chinese Academy of Science, Wuhan, China:seminar talk, 12/31/08: Least squares estimator for O-U process driven by fractional noise.
83. Department of Mathematics, Wuhan University, Wuhan, China: seminar talk, 12/31/08: Stochas-tic partial differential equations and local times.
84. Institute of Applied Mathematics, The Chinese Academy of Science, Beijing, China: seminar talk,12/24/08: Stochastic partial differential equations and local times.
85. Institute of Applied Mathematics, The Chinese Academy of Science, Beijing, China: seminar talk,12/9/08: Anticipative Optimal stopping.
86. Wuhan Institute of Physics and Mathematics, Chinese Academy of Science: seminar talk, 6/8/07:Rough path analysis via fractional calculus.
87. Wuhan University of Science and Technology: seminar talk, 6/10/07: Optimal anticipative stop-ping.
88. HuaZhong University of Science and Technology: seminar talk, 6/8/07: Rough path analysis viafractional calculus and stochastic partial differential equations and local time.
89. University of Oslo: seminar talk, 8/14/06: Rough path analysis via fractional Brownian motion.
90. University of Missouri at Columbia: seminar talk, 2/10/06: Rough Path Analysis via Fractionalcalculus.
91. University of Missouri at Kansas City: seminar talk, 10/7/05: Improving Black-Scholes by usingthe long memory.
92. Loughborough University: seminar talk, 2/18/05: Self-intersection local time of fractional Brow-nian motions.
93. University of Swansea: seminar talk, 2/15/05: Self-intersection local time of fractional Brownianmotions.
94. University of Wisconsin at Madison: seminar talk, 12/09/04: Self-intersection local time of frac-tional Brownian motions.
95. University of British Coloumbia: seminar talk, 11/10/04: Self-intersection local time of fractionalBrownian motions.
96. University of Alberta at Edmonton: seminar talk, 11/15/04: Stochastic Calculus for fractionalBrownian motions.
97. University of Alberta at Edmonton: Open Lecture, 11/04/04: Improving the Black-Scholes for-mula: An Introduction to Pricing and Portfolio Management by Using Long Memory.
98. University of Alberta at Edmonton: Open Lecture, 10/28/04: Some Inequalities for Gaussianmeasures.
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99. University of Alberta at Edmonton: Open Lecture, 10/22/04: Simulation of Random DynamicalSystem
100. University of Alberta at Edmonton: Open Lecture, 10/15/04: Simulation of the Oil Pressure inthe Norwegian Sea.
101. University of Alberta at Edmonton: Open Lecture, 10/04/04: Self-intersection local time offractional Brownian motions.
102. University of Alberta at Edmonton: Open Lecture, 11/05/04: Between log-Sobolev and SpectralGap inequalities.
103. University of Alberta at Edmonton: Open Lecture, 11/19/04: Between log-Sobolev and SpectralGap inequalities, continuation.
104. University of Oslo: 09/17/04: Self-intersection local time of fractional Brownian motions.
105. Purdue University: seminar talk, 02/21/03: Fractional Brownian motion and application to fi-nance,
106. Michigan State University: colloquium talk, 11/02: Girsanov transformation of fractional Brow-nian motion,
107. Faculty of Mathematics and System Science, Chinese Academy of Science, China: colloquiumtalk, 06/01: several inequalities for Gaussian measures.
108. Institute of Applied Mathematics, Chinese Academy of Science, China: seminar talk, 06/01:stochastic calculus for fractional Brownian motions and applications.
109. HuaZhong University of Science and Technology, China: seminar talk, 05/25/01: Girsanov trans-formation for fractional Brownian motion.
110. HuaZhong University of Science and Technology, China: seminar talk, 05/24/01: fractional whitenoise analysis and application.
112. Institute of Physics and Mathematics, Chinese Academy of Science, China: colloquium talk,05/23/01: some inequalities for Gaussian measures.
113. Institute of Physics and Mathematics, Chinese Academy of Science, China: colloquium talk,05/22/01: stochastic quantization.
114. Oxford University, Oxford, United Kingdom: colloquium talk, 04/01: unified treatment of severalinequalities for Gaussian measures.
115. University of Barcelona, Spain: seminar talk, 02/01: fractional Black-Scholes market.
116. University of Barcelona, Spain: seminar talk, 07/95: on Wick approximation of stochastic differ-ential equations.
117. University of Barcelona, Spain: seminar talk, 07/93: on multiple Wiener-Ito integrals.
118. Israel Institute of Technology, Haifa, Israel: seminar talk, 01/01: a simple approach to logarithmicSobolev inequalities.
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119. Israel Institute of Technology, Haifa, Israel: seminar talk, 12/00: Girsanov transformation forfractional Brownian motion.
120. University of Nebraska, Lincoln: seminar talk, 04/99: some inequalities for Gaussian measures.
121. University of Southern California, Los Angeles: seminar talk, 04/97: stochastic quantization.
122. University of California at Irvine: seminar talk, 03/96: on Poincare and log Sobolev inequalities.
123. University of California at Irvine: seminar talk, 05/96: on long range dependence.
124. Georgia Institute of Technology, Atlanta: seminar talk, 09/96: on an interpolation inequality forGaussian measure.
125. Southern Illinois University at Carbondale: colloquium talk, 09/96: on an interpolation inequalityfor Gaussian measure.
126. Southern Illinois University at Carbondale: seminar talk, 09/96: exact rate of convergence forEuler-Maruyama scheme.
127. University of Illinois at Urbana-Champagne, Urbana: seminar talk, 08/96: on an interpolationinequality for Gaussian measure.
128. University of Delaware: seminar talk, 09/96: some inequalities for Gaussian measures.
129. Rutgers University, New Brunswick: seminar talk, 09/96: some inequalities for Gaussian measures.
130. University of North Carolina at Chapel Hill: seminar talk, 09/96: approximation of the heatkernel.
131. University of North Carolina at Chapel Hill: seminar talk, 03/95: stochastic quantization.
132. University of North Carolina at Chapel Hill: seminar talk, 11/94: general idea of numericalsolution of stochastic differential equations.
133. Mittag-Leffler Institute, Stockholm, Sweden: colloquium talk, 01/95: numerical approximation ofstochastic pressure equation.
134. University of Warwick, Coventry, United Kingdom: seminar talk, 07/95 stochastic quantization.
135. Hull University, Hull, United Kingdom: colloquium talk, 04/95: exact rate of convergence of somenumerical schemes for stochastic differential equations.
136. University of Edinburgh, Edinburgh, United Kingdom: colloquium talk, 03/95: numerical ap-proximation of some stochastic partial equations.
137. Ruhr University Bochum, Bochum, Germany: seminar talk, 07/94: on stochastic quantization.
138. Ruhr University Bochum, Bochum, Germany: seminar talk, 08/92: on the value at 0 of someWiener functionals.
139. North Carolina State University: seminar, 05/94: on rate of convergence of Euler-Maruyamascheme.
140. Centro de Investigacion en Matematicas, Guanajuato, Mexico: colloquium talk, 11/93: on aninterpolation inequality for Gaussian measure.
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141. University of Oslo, Norway: seminar talk, 12/92: on the value at 0 of some Wiener functionals.
142. University of Bessancon, France: seminar talk, 07/92: on traces on the Wiener space.
143. University of Paris 6, France: colloquium talk, 05/92: on a work of Carmona and Nualart.
144. University Louis Pasteur, France: seminar talk, 11/91: on stochastic Taylor series.
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A. List of submitted papers
1. (with Chen, L. and Nualart, D.) Regularity and strict positivity of densities for the nonlinearstochastic heat equation. Submitted.
2. Ito type stochastic differential equations driven by fractional Brownian motions of Hurst parameterH > 1/2. Submitted.
3. (with Khoa Le) Asymptotics of the density of parabolic Anderson random fields. Near completion.
4. (with Han, Z. and Lee, C.) On the linear-quadratic control in a regulated market via reflecteddiffusions. Ready for submission.
5. (with Han, Z. and Lee, C.) On the pricing barrier control in a regime-switching regulated market.Submitted.
6. (with Cheng, Y. and Long, H.) Generalized moment estimation for Ornstein-Uhlenbeck processesdriven by α-stable Levy motions from discrete time observations. Near completion.
7. (with Cheng, Y.; Ghoddusi, H. and Lee, C.) Optimal Adjustment of Energy Prices. Near com-pletion.
8. (with Nualart, D. and Zhou, H.) Parameter estimation for fractional Ornstein-Uhlenbeck processesof general Hurst parameter. Ready to submit.
9. (with Chen, X.; Song, J. and Song, X.) Temporal asymptotics for fractional parabolic Andersonmodel. Submitted.
10. (with G. Rang) Parameter Estimation For Stochastic Hamiltonian Systems Driven By FractionalBrownian Motions.
11. (with Chen, X. and Song, J.) Feynman-Kac formula for fractional heat equation driven by frac-tional white noise. Submitted.
12. (with Chen, X.; Nualart, D. and Tindel, S.) Spatial asymptotics for the parabolic Anderson modeldriven by a Gaussian rough noise. Submitted.
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B. List of refereed publications
14. (with B. Øksendal and A. Sulem) Singular mean-field control games. Stoch. Anal. Appl. 35(2017), no. 5, 823-851.
15. (with Nualart, D. and Zhang, T.) Large deviations for stochastic heat equation with rough de-pendence in space. To appear in Bernoulli.
16. (with Chen, L.; Kalbasi, K. and Nualart, D.) Intermittency for the stochastic heat equation drivenby time-fractional Gaussian noise with H ∈ (0, 1/2). To appear in Prob. Theory and RelatedFields.
17. (with Chen, L. and Nualart, D.) Two-point correlation function and Feynman-Kac formula forthe stochastic heat equation. Potential Anal. 46 (2017), no. 4, 779-797.
18. (with Le, K. and Mytnik, L.) Stochastic differential equation for Brox diffusion. Stochastic Pro-cess. Appl. 127 (2017), no. 7, 2281-2315.
19. (with J. Huang, Le, K. and Nualart, D. and Tindel, S.) Stochastic heat equation with roughdependence in space. To appear in Annals of Probability.
20. Analysis on Gaussian spaces. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.xi+470 pp.
21. (With Chen, L.; Hu, G. and Huang, J.) Space-time fractional diffusions in Gaussian noisy envi-ronment. Stochastics 89 (2017), no. 1, 171-206.
22. (with Huang, J. and Nualart, D.) On the intermittency front of stochastic heat equation drivenby colored noises. Electron. Commun. Probab. 21 (2016), Paper No. 21, 13 pp.
23. (with Le, K.) Nonlinear Young integrals and differential systems in Holder media. Trans. Amer.Math. Soc. 369 (2017), no. 3, 1935-2002.
24. (with Y. Liu and D. Nualart) Rate of convergence and asymptotic error distribution of Eulerapproximation schemes for fractional diffusions. Ann. Appl. Probab. 26 (2016), no. 2, 1147-1207.
25. (with Le, K.) Nonlinear Young integrals via fractional calculus. Stochastics of environmental andfinancial economics-Centre of Advanced Study, Oslo, Norway, 2014-2015, 81-99, Springer Proc.Math. Stat., 138, Springer, Cham, 2016.
26. (with Liu, Y. and Nualart, D.) Taylor schemes for rough differential equations and fractionaldiffusions. Taylor schemes for rough differential equations and fractional diffusions. DiscreteContin. Dyn. Syst. Ser. B 21 (2016), no. 9, 3115-3162.
27. (with Han, Z. and Lee, C.) Optimal pricing barriers in a regulated market using reflected diffusionprocesses. Quant. Finance 16 (2016), no. 4, 639-647.
28. (with Hu, G.) Fractional diffusion in Gaussian noisy environment. Mathematics 2015, 3, 131-152;doi:10.3390/math3020131.
29. (with Lee, C.; Lee, M. H. and Song, J.) Parameter estimation for reflected Ornstein-Uhlenbeckprocesses with discrete observations. Stat Inference Stoch Process 18 (2015), 279-291.
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30. (with Chen, X., Song, J. and Xing, F.) Exponential asymptotics for time-space Hamiltonians.Annales de l’Institut Henri Poincare - Probabilites et Statistiques Vol. 51 (2015) , No. 4, 1529-1561.
31. (With Huang, J.; Nualart, D. and Sun, X.) Smoothness of the joint density for spatially homoge-neous SPDEs. J. Math. Soc. Japan 67 (2015), no. 4, 1605-1630.
32. (with Huang, J.; Nualart, D. and Tindel, S.) Stochastic heat equations with general multiplicativeGaussian noises: Holder continuity and intermittency. Electron. J. Probab. 20 (2015), no. 55, 50pp.
33. (with J. Huang and D. Nualart) On Holder continuity of the solution of stochastic wave equationsin dimension three. Stoch. Partial Differ. Equ. Anal. Comput. 2 (2014), no. 3, 353-407.
34. (with Nualart, D.; Tindel, S. and Xu, F.) Density convergence in the Breuer-Major theorem forGaussian stationary sequences. Bernoulli 21 (2015), no. 4, 2336-2350.
35. (with F. Lu and D. Nualart) Convergence of densities of some functionals of Gaussian processes.J. Funct. Anal. 266 (2014), no. 2, 814-875.
36. (with D. Nualart and F. Xu) Central limit theorem for an additive functional of the fractionalBrownian motion. Ann. Probab. 42 (2014), no. 1, 168-203.
37. (with Nualart, D. and Song, J.) The 43 -variation of the derivative of the self-intersection Brownian
local time and related processes. J. Theoret. Probab. 27 (2014), no. 3, 789-825.
38. (with G. Rang) Identification of the point sources in some stochastic wave equations. Abstr.Appl. Anal. 2014, Art. ID 219876.
39. (with Le, Khoa) A multiparameter Garsia-Rodemich-Rumsey inequality and some applications.Stochastic Process. Appl. 123 (2013), no. 9, 3359-3377.
40. (with F. Lu and D. Nualart) Non-degeneracy of some Sobolev pseudo-norms of fractional Brownianmotion. Electron. Commun. Probab. 18 (2013), no. 84, 8 pp.
41. (with Y. Han and J. Song) Maximum Principle for General Controlled Systems Driven by Frac-tional Brownian Motions. Appl. Math. Optim. 67 (2013), no. 2, 279-322.
42. (with C. Lee) Drift parameter estimation for a reflected fractional Brownian motion based on itslocal time. J. Appl. Probab. 50 (2013), no. 2, 592-597.
43. (with J. Song) Parameter estimation for fractional Ornstein-Uhlenbeck processes with discreteobservations. In Malliavin calculus and stochastic analysis, 427-442, Springer Proc. Math. Stat.,34, Springer, New York, 2013.
44. (with D. Nualart and J. Song) A nonlinear stochastic heat equation: Holder continuity andsmoothness of the density of the solution. Stochastic Process. Appl. 123 (2013), no. 3, 1083-1103.
45. (with F. Biagini, T. Meyer-Brandis and B. Øksendal) Insider trading equilibrium in a market withmemory. Math. Financ. Econ. 6 (2012), no. 3, 229-247.
46. Multiple integrals and expansion of solution to differential equations driven by rough path and byfractional Brownian motion. Stochastics 85 (2013), no. 5, 859-916.
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47. (with F. Lu and D. Nualart) Holder continuity of the solutions for a class of nonlinear SPDE’sarising from one dimensional superprocesses. Probab. Theory Related Fields 156 (2013), no. 1-2,27-49.
48. (with Jolis M. and Tindel S.) On Stratonovich and Skorohod stochastic calculus for Gaussianprocesses. Annals of Probability 41 (2013), no. 3A, 1656-1693.
49. (with S. Tindel) Smooth Density for Some Nilpotent Rough Differential Equations. J. Theoret.Probab. 26 (2013), no. 3, 722-749.
50. Stochastic quantization and ergodic theorem for diffusion processes. Sci. China Math. 55 (2012),no. 11, 2285-2296.
51. (with D. Ocone and J. Song) Some results on backward stochastic differential equations drivenby fractional Brownian motions. in “Stochastic Analysis and Applications to Finance”. Essays inhonor of Jia-An Yan. World Scientific Publishing Co. 2012, 225-242.
52. (with C. Yang) Optimal tracking for bilinear stochastic system driven by fractional Brownianmotions. J. Syst. Sci. Complex. 25 (2012), no. 2, 238-248.
53. (with D. Nualart and F. Lu) Feynman-Kac formula for the heat equation driven by fractionalnoise with Hurst parameter H < 1/2. Ann. Probab. 40 (2012), no. 3, 1041-1068.
54. (with Nualart, D.; Xiao, W. and Zhang, W.) Exact maximum likelihood estimator for drift frac-tional Brownian motion at discrete observation. Acta Math. Sci. Ser. B Engl. Ed. 31 (2011),no. 5, 1851-1859.
55. An enlargement of filtration for Brownian motion. Acta Math. Sci. Ser. B Engl. Ed. 31 (2011),no. 5, 1671-1678.
56. (with D. Nualart and X. Song) Malliavin calculus for backward stochastic differential equationsand application to numerical schemes. The Annals of Applied Probability Vol. 21 (2011), 2379-2423.
57. (with D. Nualart and J. Song) Feynman-Kac formula for heat equation driven by fractional whitenoise. The Annals of Probability 39 (2011), no. 1, 291-326.
58. (with D. Nualart) Central limit theorem for the third moment in space of the Brownian local timeincrements. Electron. Commun. Probab. 15 (2010), 396-410.
59. (with D. Nualart) Parameter estimation for fractional Ornstein-Uhlenbeck processes. Statist.Probab. Lett. 80 (2010), no. 11-12, 1030-1038.
60. (with B. Wang) Convergence rate of an approximation to multiple integral of fractional Brownianmotion. Acta Math. Sci. Volume 30 (2010), 975-992.
61. A random transport-diffusion equation. Acta Math. Scientia. Vol 30 (2010), 2033-2050.
62. (with D. Nualart and J. Song ) Fractional martingales and characterization of the fractionalBrownian motion. Ann. Probab. 37 (2009), no. 6, 2404–2430.
63. (with D. Nualart) Stochastic integral representation of the L2 modulus of Brownian local timeand a central limit theorem. Electron. Commun. Probab. 14 (2009), 529–539.
64. (with J.A. Yan) Wick calculus for nonlinear Gaussian functionals. Acta Math. Appl. Sin. Engl.Ser. 25 (2009), no. 3, 399-414.
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65. (with Long H. W.) Least squares estimator for Ornstein-Uhlenbeck processes driven by α-stablemotions. Stochastic Process. Appl. 119 (2009), no. 8, 2465–2480.
66. (with Long H. W.) On the singularity of least squares estimator for mean-reverting α-stablemotions. Acta Math. Sci. Ser. B Engl. Ed. 29 (2009), no. 3, 599–608.
67. (with S. Peng) Backward stochastic differential equations driven by fractional Brownian motion.SIAM Journal of Control and Optimization, 48 (2009), 1675-1700.
68. (with D. Nualart and J. Song ) Integral representation of renormalized self-intersection local times.J. Funct. Anal. 255 (2008), no. 9, 2507–2532.
69. (with D. Nualart and X. Song ) A singular stochastic differential equation driven by fractionalBrownian motion. Statist. Probab. Lett. 78 (2008), no. 14, 2075–2085.
70. (with B. Øksendal) Partial information linear quadratic control for jump diffusions. SIAM Journalof Control and Optimization, 47 (2008), 1744-1761.
71. (with B. Øksendal) Optimal anticipative stopping. Advances in Math. of Finance (L. Stettnered.). Banach Center Publications vol. 83 (2008), 107-116.
72. (with D. Nualart) Rough path analysis via fractional calculus. Trans. Amer. Math. Soc. 361(2009), no. 5, 2689-2718.
73. (with D. Nualart) Stochastic heat equation driven by fractional noise and local time. Prob.Theory and Related Fields, 143 (2009), 285-328.
74. (with B. Øksendal) Optimal smooth portfolio selection for an insider. J. Appl. Probab. 44 (2007),no. 3, 742–752.
75. (with Biagini, F. B. Øksendal and Zhang, T.S.) Stochastic calculus for fractional Brownian motionand applications. Probability and its Applications (New York). Springer-Verlag London, Ltd.,London, 2008.
76. (with D. Nualart) Regularity of renormalized self-intersection local time for fractional Brownianmotion. Communications in Information and Systems, 7 (2007), 21-30.
77. (with D. Nualart) Differential equation driven by Holder continuous functions of order greaterthan 1/2. in The Abel Symposium on Stochastic Analysis, 399-423. Springer, 2007.
78. (with H. Long) Parameter estimation for Ornstein-Uhlenbeck processes driven by α-stable Levymotions. Communications on Stochastic Analysis, 1 (2007), 175-192.
79. (with Mohammed, S., Arritojas, M. and Pap, G.) A Delayed Black and Scholes Formula, Stoch.Anal. Appl. 25 (2007), no. 2, 471-492.
80. Integral transformations and anticipative calculus for fractional Brownian motions. Mem. Amer.Math. Soc. 175 (2005), no. 825, viii+127 pp.
81. (with Øksendal, B. and Salopek, D. M.) Weighted local time for fractional Brownian motion andapplications to finance. Stoch. Appl. Anal. 23 (2005), no. 1, 15–30.
82. (with Zhou X.Y.) Stochastic control for linear systems driven by fractional noises. SIAM J.Control Optim. 43 (2005), no. 6, 2245–2277
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83. Optimization of portfolio and consumption and minimization of volatility. Mathematics of finance,199–206, Contemp. Math., 351, Amer. Math. Soc., Providence, RI, 2004.
84. (with D. Nualart) Renormalized self-intersection local time for fractional Brownian motion. Ann.Probab. 33 (2005), no. 3, 948–983.
85. (with D. Nualart) Some processes associated with fractional Bessel processes. J. Theoret. Probab.18 (2005), no. 2, 377–397.
86. (with B. Øksendal and A. Sulem) Optimal portfolio in a fractional Black-Scholes market drivenby fractional Brownian motion. Infinite Dimensional Analysis, Quantum Probability and RelatedTopics, Vol. 6 (2004), 519-536.
87. (with S. E. A. Mohammed and F. Yan) Numerical Solution of Stochastic Differential Systemswith Memory. Annals of Probability, 32 (2004), 265-314.
88. Optimal consumption and portfolio in a market where the volatility is driven by fractional Brown-ian motion. Probability, Finance and Insurance. Ed. Lai, T.L. et al. World Scientific Publishing.164-173.
89. (with Øksendal and T. S. Zhang) General fractional multiparameter white noise theory andstochastic partial differential equations, Communications in partial differential equations 29 (2004),1-23.
90. (with B. Øksendal) Fractional white noise calculus and applications to finance. Infinite Dimen-sional Analysis, Quantum Probability and Related Topics, Vol. 6 (2003), 1-32.
91. (with F. Biagini, B. Øksendal and A. Sulem) A Stochastic maximum principle for processes drivenby fractional Brownian motion, Stochastic Processes and Applications, 100 (2002), 233-253.
92. (with G. Kallianpur, J. Xiong) An approximation for Zakai equation, Applied Mathematics andoptimization 45 (2002), no. 1, 23–44.
93. Probability structure preserving and absolute continuity, Annales de l’Institut Henri Poincare, 38(2002), no. 4, 557–580.
94. (with Ustnel, A. S.; Zakai, M.) Tangent processes on Wiener space. J. Funct. Anal. 192 (2002),no. 1, 234–270
95. Chaos expansion of heat equation with white noise potentials, Potential Anal. 16 (2002), no. 1,45–66.
96. (with Øksendal, B.) Chaos expansion of local time of fractional Brownian motions. StochasticAnal. Appl. 20 (2002), no. 4, 815–837.
97. Self-intersection local time of fractional Brownian motions - via chaos expansion, Journal of Math-ematics of Kyoto University, 41 (2001), no. 2, 233–250.
98. Heat equation with fractional white noise potentials, Applied Mathematics and Optimization, 43(2001), 221-243.
99. (with Øksendal and T. S. Zhang) Stochastic fractional potential theory, Papers in Analysis, Re-port. Univ. Jyvaskyla, 83 (2001), 169-180.
100. Option pricing in a market where the volatility is driven by fractional Brownian motions, RecentDevelopment in Mathematical Finance. Ed. J.M. Yong. World Scientific. 2002, 49-59.
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101. Prediction and translation of fractional Brownian motions, Stochastics in Finite and Infinite Di-mensions, Ed. T. Hida et al., Trends Math., Birkhauser, Boston, MA, 2001, 153–171.
102. (with Duncan, T. E. and Pasik-Duncan, B.) Stochastic calculus for fractional Brownian motion.I. Theory. SIAM J. Control Optim. 38 (2000), no. 2, 582-612.
103. (with G. Kallianpur) Schrodinger equations with fractional Laplacians, Applied Mathematics andOptimization, 42 (2000), 281-290.
104. Optimal times to observe in the Kalman-Bucy models, Stochastics and Stochastics Report, 69(2000), 123-140.
106. (with B. Øksendal and T. Zhang) Stochastic partial differential equations driven by multi-parameterfractional white noise, Stochastic Processes, Physics and Geometry: New Interplays. II, Ed. F.Gesztesy et al., AMS 2000, 327-337.
107. A unified approach to several inequalities for Gaussian and diffusion measures, Seminaire XXXIV,Lecture Notes 1729, Springer-Verlag, 2000, 329–335.
108. A Class of stochastic partial differential equations driven by fractional white noises, StochasticProcesses, Physics and Geometry: New Interplays. II, Ed. F. Gesztesy et al., AMS 2000, 317-325.
109. (with B. Øksendal and A. Sulem) Optimal portfolio in a fractional Black-Scholes market, Mathe-matical Physics and Stochastic Analysis, Ed. S. Albeverio et al., World Scientific 2000, 267-279.
110. (with R.J. de Figuereido) On non-linear filtering of non-Gaussian processes through Volterraseries, Volterra Equations and Applications, Arlington, TX, 1996, 197–202, Stability ControlTheory Methods Appl., 10, Gordon and Breach, Amsterdam, 2000.
111. (with S. Albeverio, M. Rockner and X.Y. Zhou) Stochastic quantization of the two-dimensionalpolymer measure, Applied Mathematics and Optimization, 40 (1999), 341-354.
112. Exponential integrability of diffusion processes, Advances in Stochastic Inequalities, Ed. T.P. Hilland C. Houdre, Contemporary Mathematics, 234 (1999), American Mathematical Society, 1999,75-84.
113. On the positivity of the solution of a class of stochastic pressure equations, Stochastics andStochastics Reports, 63 (1998), 27-40.
114. (with B. Øksendal) Optimal time to invest when the price processes are geometric Brownianmotions, Finance and Stochastics, 2 (1998), 295-310.
115. Ito-Wiener chaos expansion with exact residual and correlation, variance inequalities, Journal ofTheoretical Probability, 10 (1997), 835-848.
116. (with Z. Q. Chen, Z. M. Qian and W. A. Zheng) Stability and approximations of symmetricdiffusion semigroups and kernels, Journal of Functional Analysis, 152 (1998), 255-280.
117. (with D. Nualart) Continuity of some anticipating integral processes, Statistics and ProbabilityLetters, 37 (1998), 203-211.
118. (with G. Kallianpur) Exponential integrability and application to stochastic quantization, AppliedMathematics and Optimization, 37 (1998), 295-353.
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119. (with S.Albeverio and X.Y. Zhou) A remark on non smoothness of self-intersection local time ofplanar Brownian motion, Statistics and Probability Letter, 32 (1997), 57-65.
120. (with H. Holden) Finite difference approximation of the pressure equation for fluid flow in astochastic medium–a probabilistic approach, Comm. Partial Differential Equation, 21 (1996),1367-1388.
121. On the self-intersection local time of Brownian motion-via chaos expansion, Publ. Mat., 40 (1996),337-350.
122. (with Bruce K. Driver) On heat kernel logarithmic Sobolev inequalities, Stochastic Analysis andApplications (Powys, 1995), World Sci. Publishing, River Edge, NJ, 1996, 189–200.
123. Strong and weak order of time discretization schemes of stochastic differential equations, Seminairede Probabilites XXX, Ed. J. Azema, P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1626,Springer-Verlag, 1996, 218-227.
124. (with S. Watanabe) Donsker’s delta functions and approximation of heat kernels by time dis-cretization methods, J. Math. Kyoto University, 36 (1996), 499-518.
125. (with S. Cambanis) The exact convergence rate of Euler-Maruyama scheme and application tosample design, Stochastics and Stochastics Reports, 59 (1996), 211-240.
126. Semi-implicit Euler-Maruyama scheme for stiff stochastic equations, Stochastic Analysis and Re-lated Topics, V (Silivri, 1994), Progr. Probab., 38, Birkhauser Boston, Boston, MA, 1996, 183–202.
127. (with B. Øksendal) Wick approximation of quasilinear linear stochastic differential equations,Stochastic Analysis and Related Topics, Progr. Prob. 38, Birkhauser, Boston, 1996, 203-231.
128. (with T. Lindstrøm, B. Øksendal, J. Ubøe and T.S. Zhang) Inverse power of white noise, Proc.Symp. Pure Math. 57 (1995), 439-456.
129. (with V. Peres-Abreu) On the continuous extension of Wiener chaos, Bol. Soc. Mat. Mexicana,1, (1995), 127-135.
130. On the differentiability of functions of an operator, Seminaire de Probabilites XXIX, Ed. J.Azema, P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1613, Springer-Verlag, 1995,218-219.
131. The pathwise solution for a class of quasilinear stochastic equations of evolution in Banach spaceII, Acta Mathematica Scientia, 15 (1995), 264-274.
132. Some operator inequalities, Seminaire de Probabilites XXVIII, Ed. by J. Azema, P.A. Meyer andM. Yor, Lecture Notes in Mathematics 1583, Springer-Verlag, 1994, 316-333.
133. The pathwise solution for a class of quasilinear stochastic equation of evolutions in Banach spacesI, Acta Mathematica Scientia, 14 (1994), 461-474.
134. (with P.A.Meyer) On the approximation of Stratonovitch multiple integrals, Stochastic Processes,a festschrift in honor of G. Kallianpur, Ed. S. Cambanis et al., Springer, 1993, 141-147.
135. ( with Long Hongwei) Symmetric integral and the approximation theorem of stochastic integralin the plane, Acta Mathematica Scientia, 13 (1993), 153-166.
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136. A remark on the value on zero of a Brownian functional, Stochastic Analysis and Related Topics,Proc. Fourth Oslo-Silivri Workshop on Stochastic Analysis, Ed. T. Lindstrøm, B. Øksendal andA.S. Ustunel, Gordon and Breach Science Publishers, 1993, 173-176.
137. (with L. Decreusefond, A.S. Ustunel) Une inegalite d’interpolation sur l’espace de Wiener, CompteRendus Acad. Sci. Paris, 317 (1993), 1065-1067.
138. Hypercontractivite pour les fermions, d’apres Carlen-Lieb, Seminaire de Probabilites XXVII, Ed.J. Azema, P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1557, Springer-Verlag, 1993,86-96.
139. The pathwise solutions for a class of quasi-linear stochastic equations of evolution in Banachspaces III, Acta Mathematica Scientia, 13 (1993), 13-22.
140. Sur un travail de R. Carmona et D. Nualart, Seminaire de Probabilites XXVI, Ed. J. Azema,P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1526, Springer-Verlag, 1992, 587-594.
141. Une formule d’Ito pour le mouvement Brownien fermionique, Seminaire de Probabilites XXVI,Ed. J. Azema, P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1526, Springer-Verlag,1992, 575-578.
142. Une remarque sur l’inegalite de Holder non-commutative, Seminaire de Probabilites XXVI, ed. J.Azema, P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1526, Springer-Verlag, 1992, 595.
143. Calculation of Feynman path integral for certain central forces, Stochastic Analysis and RelatedTopics (Oslo, 1992), Stochastics Monogr., 8, Gordon and Breach, Montreux, 1993, 161-171.
144. Serie de Taylor stochastique et formule de Campbell-Hausdorff - d’apes Ben Arous, Seminaire deProbabilites XXVI, Ed. J. Azema, P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1526,Springer-Verlag, 1992, 579-586.
145. Existence de traces dans les developpements en chaos de Wiener. Dissertation, Universite LouisPasteur, Strasbourg, 1992. Publication de l’Institut de Recherche Mathematique Avancee, 480.Universite Louis Pasteur, Departement de Mathematique, Institut de Recherche MathematiqueAvancee, Strasbourg, 1992. 77 pp.
146. Calculs formels sur les e.d.s. de Stratonovitch, Seminaire de Probabilites XXVI, Ed. J. Azema,P.A. Meyer and M. Yor, Lect. Notes in Math. 1426, Springer-Verlag, 1990, 453-460.
147. Symmetric integral and canonical extension for jump process - some combinatorial results, ActaMath. Sci. 10, (1990), 448-458.
148. Some notes on multiple Stratonovitch integrals, Acta Math. Sci. 9 (1989), 453-462.
149. Un nouvel exemple de distribution de Hida, Seminaire de Probabilites XXVI, Ed. J. Azema, P.A.Meyer and M. Yor, Lecture Notes in Mathematics 1321, Springer-Verlag, 1988, 82-84.
150. (with P.A.Meyer) Chaos de Wiener et integrales de Feynman, Seminaire de Probabilites XXII,Ed. J. Azema, P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1321, Springer-Verlag,1988, 51-71.
151. (with P.A.Meyer) Sur les integrales multiples de Stratonovitch, Seminaire de Probabilites XXVI,Ed. J. Azema, P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1321, Springer-Verlag,1988, 72-81.
152. Stochastic analysis of the stochastic functional on the basic space, Acta Math. Sci. 6 (1986), pp67-74.
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Teaching Statement of Yaozhong HuI taught mathematics in all university levels at the Chinese Academy of Science (China), University
of Strasbourg (France), University of Oslo (Norway), University of California at Irvine and mostly atthe University of Kansas.
Undergraduate Teaching
Since joining KU in 1997 I have taught two courses (six credit hours) almost every semester. Perrequest by students, I also offered some special topics courses such as Math 696, “Statistics with SASprogramming” a number of times to advanced undergraduate students and beginning graduate students.Some of these students found jobs because of their SAS skill. In addition to the mandatory classroomteaching required by the university, I have also offered each semester several direct reading (individual)courses at various levels (to both upper division undergraduate students and to graduate students) sothat students can acquire some knowledge which are not taught by the university on regular basis. Alist of courses I taught is listed at the end of this teaching statement.
At the undergraduate level I taught most of the courses offered by our department, including Math105, “introduction to finite mathematics” (about 600 students), Math 115, “calculus I ” (about 160students), Math 365, Elementary Statistics (about 200 students).
I believe that the first thing a teacher should do is to try all means to make students love the coursethey are taking. This is very difficult in particular at the undergraduate level. Most mathematicscourses taught at KU are designed for non-mathematics major students and most students are “forced”to learn mathematics (at least most students feel so). To make these students like the math course theyare taking, I think the instructor should do the following.
(i) Prepare the lectures with some background material (history, funny stories, potential applicationsand so on). For example, when I taught exponential in Math 105, I told the student of the lordKrishna’s “rice and chessboard problem”.
(ii) More practical and interesting examples should be used to motivate students and to illustratethe main ideas.
(iii) In order for students to build confidence in their learning of the course the lectures should becarried out gradually and homework problems should be chosen very carefully both to increasethe students’ interest in the course and to strengthen their understanding of the material.
(iv) To inspire students in loving the subject, teacher should present his/her lectures with greatpassion and enthusiasm, with great interest and imagination.
Secondly, in order to develop students’ ability, teacher should prepare his/her lectures very carefullyand with a profound depth. I have been writing my lecture notes carefully and add some more additionalexamples and exercises. Before each class, I carefully prepare my lecture. I try to explore the maintextbook I used for the class but also read several other relevant textbooks and materials relevant tothe contents that I plan to lecture. I choose very carefully the examples to present in the class so thatthe examples can help explain the course concepts and results well. I also choose very carefully theexercises to assign to make sure that they can be completed by the students and they also can help thestudents to better understand the course material.
For example when I taught math 365 (basic statistics) course, the intellectual goal that I have forstudents is to let them know how to apply some basic statistical methods to solve problems, knowthe general idea and basic mathematical background for these statistical methods, and how to findmore advanced methods if a different situation occurs. So the contents, examples that I present in theclassroom are carefully chosen and exercises are carefully selected. I also use transparencies, computer
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(projector) and so on in my lectures. But I prepare my powerpoint slides very carefully so that studentscan follow and can take notes. Sometimes students are some kind of understanding the material butwith confusion. So I spend a lot effort to make sure they really understand and when to apply whatand how.
Third, in order to respect the students’ individual freedom of learning and accommodate theirdiversified need, I stimulate students to communicate with me by encouraging them to ask questions inthe class and by encouraging them to come to my office to discuss problems with me. This way, I canunderstand students’ each individual need and concerns and then I can adapt my teaching accordingly.I also asked students about the feedback of my classroom teaching. Sometimes, I was surprised by theiranswers. For example, one student from Math 105 told me that she and her classmates wish to havemore exams (I have already designed two midterm exams and one final for the class). At the beginningI was surprised that students like to take more examinations. But after a discussion with her I foundout that they like to take more frequent exams but with each exam covering less material. I think thisalso makes sense. So, in subsequent teaching I designed more in-class quizzes. I am happy that myeffort was also noticed by my colleagues who pointed out to me the fact that there are always studentsin my office.
Fourth, I often help students to review the materials to gain new insights, to interrelate differentideas. Almost for each course I have typed for students, before each midterm examination and beforefinal, a few pages of material to summarize the important concepts, theorems, examples and so on.Some students expressed to me their appreciation of my effort.
Fifth, in order to be successful in teaching cooperatively, teacher should build a good friendship andmutual trust with students. Teacher should have the attitude to learn from students as well.
I have been constantly making a great deal of effort to use new technologies in my teaching. When Iteach large classes, the department usually assigns some (undergraduate) student teaching assistants toassist me to maintain the class order, to grade homework and to hold help room hours. Most of theseassistants are very good but some of them need a lot of attention: they often do not go to the help roomas scheduled or they just don’t grade the homework in time. It is also much complained that some ofthe student assistants are incompetent and they don’t know how to help to do exercise problems atall. In addition to discuss with the concerned TA or to have department administration to discuss withthe concerned TA, I also prepared each of my lectures using powerpoint and I post these lecture notematerials in the web (including lecture notes, review materials, homework assignments, examinationsolution keys, and lot of other materials). Students said that they liked this way better than the use ofchalk on blackboard. I also use “webassign” and “blackboard” in my course teaching.
I have enjoyed a great deal in teaching at KU and am very much encouraged by the appreciationfrom students: the average of my overall teaching effectiveness is nearly 4.0 of 5.0 although I have taughttough classes such as Math 105 (“introduction to finite mathematics”), Math 365 (basic statistics forbusiness majors).
Graduate Teaching
I taught all graduate level probability courses and some graduate level statistics and algebra coursesat the University of Kansas. In many of those courses there are graduate students from outside of thedepartment. They kept coming to discuss with me the mathematics problems they encountered in theirown research and asked me to be an outside member of their dissertation committee. I helped themas much as I can. Some of these Ph.D graduate students from outside of my department finally got aMaster degree in Mathematics under my supervision in my department in addition to their Ph.D degreein their home department.
In addition to teaching the existing university courses, I also introduced a number of new graduatetopic courses to the university such as “fractional calculus”, “advanced time series I”, “advanced time
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series II”, “stochastic differential equations”, “statistics of stochastic differential equations”, “appliedstochastic control for jump diffusions”, “rough path analysis”, and “multiple stochastic integrals andU -statistics”, “Introduction to random matrices”. I also team-taught an advanced economics coursetogether with Professor Bill Barnett (Oswald Distinguished Professor of Macroeconomics) in the eco-nomics department.
I dedicate enormous time and effort to the training of graduate students both at Master and Ph.Dlevels. I have spent a lot of effort in contributing to recruiting and retaining graduate students atUniversity of Kansas. Eleven Ph.D Students and thirteen master degree students have successfullycompleted their programs at University of Kansas under my supervision. Currently, I am supervising 6Ph.D students (three of them are jointly with Professor David Nualart). I also hosted over 19 visitingscholars or students of duration of 6 months to 2 years. Most of my graduate students are very successfuland continue to excel. For example, Dr. Lu Fei (graduated 2013) has a tenure track position at JohnHopkins University. Dr. Jian Song has a tenure track position at Hong Kong university.
To advise my graduate students (including both Ph.D students and Master students) and visitors,I usually hold individual meetings with them on regular basis (once or twice a week). I also holddiscussion seminars on the topics of their interests on weekly basis. For example, while I was teachinga special topic course on “multiple stochastic integrals and U-statistics”. Associated with course I washolding a weekly study seminar on “U -statistics” and “V -statistics”.
In summary, I believe that everybody deserves the best education possible and I try my best both inmy classroom teaching and individual supervision. I constantly learn and study new topics and acquirenew knowledge with my students.
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The courses that Yaozhong Hu has taught at the University of Kansas are listed as follows. Theenrollment information is from the KU official website.
Semester Course Number and Name Enrollement
Spring, 2017 Math 796, Special Topics (Random Matrices) 18MATH 993, Readings in Mathematics 1MATH 699, Direct Reading 2
Fall, 2016 Math 727, Probability Theory 27MATH 624, Discrete Probability 18MATH 896, Master’s Research Component 2MATH 993, Direct Reading 2
Summer, 2016 Math 899, Master’s Thesis 1MATH 896, Master’s Research Component 1MATH 993, Direct Reading 2
Spring, 2016 Math 526, Applied Math. Stat. 57MATH 865, Stochastic Processes 5MATH 896, Master’s Research Component 2MATH 899, Master’s Thesis 1MATH 993, Direct Reading 5MATH 999, Doctoral Dissertation 1
Fall, 2015 Math 526, Applied Mathematical Statistics I 43MATH 990, Seminar 1MATH 993, Readings in Mathematics 4
Summer, 2015 MATH 899, Master’s Thesis 1MATH 993, Direct Reading 3MATH 999, Doctoral Dissertation 2
Spring, 2015 Math 365, Elementary Statistics 112MATH 993, Readings in Mathematics 5MATH 999, Doctoral Dissertation 2
Fall 2014 Math 996, Special Topic: U-statistics 10Math 699, Directing reading 1Math 896, Master’s research component 1Math 990, Seminar 1Math 993, Readings in Mathematics 4
Summer 2014 Math 993, Readings in Mathematics 7Spring 2014 Math 728, Statistical Theory 23
Math 799, Directing reading 1Math 993, Readings in Mathematics 6
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Semester Course Number and Name Enrollement
Fall 2013 Math 526, Introduction to Mathematical Statistics 61Math 727, Probability Theory 39Math 799, Directing reading 1Math 899, Master’s Thesis 1Math 990, Seminar 1Math 993, Readings in Mathematics 3
Summer 2013 Math 993, Readings in Mathematics 4Spring 2013 SabbaticalFall 2012 Math 105, Introduction to Topics of Mathematics 581
MATH 799-2400(13782), Directed Readings 2MATH 896-2200 (13805), Master’s research component 1MATH 993-2200 (13859), Readings in Mathematics 5MATH 999-1900 (18180) Doctoral Dissertation 1
Summer, 2012 MATH 799, Direct =Readings 1MATH 896, Master’s Research Component 1MATH 993, Direct Reading 3
Spring 2012 Math 728, Statistical Theory 32Math 799, Directing reading 1Math 896, Master’s research component 1Math 899, Master’s Thesis 1Math 990, Seminar 1Math 993, Readings in Mathematics 3
Fall 2011 Math 866, Stochastic Processes II 7Math 727, Probability Theory 47Math 699, Directing reading 1Math 799, Directing reading 1Math 896, Master’s research component 2Math 993, Readings in Mathematics 1
Summer 2011 Math 799, Directing reading 1Math 896, Master’s research component 1Math 899, Master’s thesis 1Math 993, Readings in Mathematics 1
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Semester Course Number and Name Enrollement
Spring 2011: Math 865, Stochastic Processes I 18Math 624, Discrete Probability 24Math 699, Directing reading 1Math 799, Directing reading 2Math 990, Seminar 1Math 993, Readings in Mathematics 1
Fall 2010 Math 796, Special Topics: Statistics of StochasticDifferential equations 12Math 590, Linear Algebra 42Math 699, Directing reading 1
Summer 2010 Math 799, Directing reading 1Spring 2010 Math 500, Intermediate Analysis 47
Math 896, Master’s research component 1Math 993, Readings in Mathematics 1
Fall 2009 Math 115, Calculus I 156Math 996, Special Topics: Rough Path Analysis 6Math 699, Directing reading 1Math 799, Directing reading 1Math 896, Master’s research component 1Math 990, Seminar 2
Spring 2009 Math 728, Statistics Theory 22Math 896, Master’s research component 3Math 990, seminar 1Math 993, Readings in Mathematics 1
Fall 2008 Math 365, Elementary Statistics 63Math 727, Probability Theory 29Math 896, Master’s research component 3Math 990, seminar 2Math 993, Readings in Mathematics 1Math 999, Doctoral Dissertation 1
Summer 2008 Math 799, Directing reading 1Math 896, Master’s research component 2Math 999, Doctoral Dissertation 1
Spring 2008 Math 605, Applied Regression Analysis 18Math 796, Advanced Time Series II 12Math 699, Directing reading 2Math 896, Master’s research component 2Math 899, Master’s thesis 1Math 990, seminar 1Math 999, Doctoral Dissertation 1
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Semester Course Number and Name Enrollement
Fall 2007 Math 115, Calculus I 163Math 796, Advanced Time Series I 14Math 896, Master’s research component 2Math 999, Doctoral Dissertation 1
Summer 2007 Math 999, Doctoral Dissertation 1Spring 2007 Math 611, Time Series 18
Math 611, Time Series 2Math 526, Applied Math Stat I 37Math 526, Applied Math Stat I 2Math 699, Directing reading 1Math 699, Directing reading 1Math 896, Master’s research component 1Math 899, Master’s thesis 3Math 990, seminar 2Math 999, Doctoral Dissertation 1
Fall 2006 Math 290, Elementary Linear Algebra 35Math 365, Elementary Statistics 67Math 799, Directing reading 1Math 896, Master’s research component 1Math 899, Master’s thesis 3Math 990, seminar 2Math 993, Readings in Mathematics 3
Summer 2006 Math 899, Master’s thesis 1Math 990, seminar 3
Spring 2006 Math 500, Intermediate Analysis 35Math 796, Special Topics: AppliedStochastic Control 6Math 799, Directing reading 2Math 896, Master’s research component 1Math 899, Master’s thesis 4Math 990, seminar 1Math 990, seminar 2Math 999, Doctoral Dissertation 1
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Semester Course Number and Name Enrollement
Fall 2005 Math 106, Introdctn to Finite Mathematcs 160Math 116, Calculus II 34Math 799, Directing reading 1Math 896, Master’s research component 3Math 899, Master’s thesis 2Math 990, seminar 6Math 993, Readings in Mathematics 1Math 999, Doctoral Dissertation 1
Summer 2005 Math 896, Master’s research component 1Math 993, Readings in Mathematics 3
Spring 2005 sabbaticalMath 699, Directing reading 1Math 799, Directing reading 5Math 993, Readings in Mathematics 2
Fall 2004: sabbaticalMath 799, Directing reading 2Math 993, Readings in Mathematics 2
Summer 2004 Math 993, Readings in Mathematics 3Spring 2004 Math 796, Stochastic Differential Equations 14
Math 526, Appl. Math. Stat. I 27Math 526, Appl. Math. Stat. I 4Math 993, Readings in Mathematics 3
Fall 2003 Keeler Professorship (no teaching duty)Math 699, Directing reading 1Math 990, seminar 5Math 993, Readings in Mathematics 3
Spring 2003 Math 628, Math Theory Statistics 15Math 250, Math Engr Systems 102
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Period Course Number and Name ∗Overall The possiblestudent totalevaluation points
Fall 2002: Math 627, Probability 3.94 5Math 365, Elementary Statistics 4.18 5
Spring 2002: Math 500, Intermediate Analysis 4.23 5Math 696, Special Topics: Prob and Statwith SAS programming 3.75 5
Fall 2001: Math 728, Statistics 4.40 5Math 526, Appl. Math. Stat. I 3.58 5
Spring 2001: Math 790, Linear Algebra II 4.57 5Math 727, Probability Theory 3.77 5
Fall 2000: Math 696, Special Topics: Prob and Statwith SAS programming 4.86 5Math 526, Applied Math Stat I 3.50 5
Spring 2000: Math 365, Elem Statistics 4.03 5Math 320, Elem Diff Equations 3.53 5
Fall 1999: Math 365, Elem Statistics 3.70 5Math 796, Special Topics: Fractional Calculus 4.67 5
Spring 1999: Math 628, Mathematical Statistics 4.21 5Math 796, Special Topics: StochasticDifferential Equations 4.43 5
Fall 1998: Math 627, Probability 4.29 5Math 115 Calculus I 3.87 5
Spring 1998: Math 365, Elem Statistics 3.45 5Math 526, Applied Math Stat I 3.36 5
Fall 1997: Math 123, Linear Algebraand Multivariable Calculus 3.68 5
The KU website does not provide teaching enrollment information for courses before 2003. Sostudent evaluations are listed instead.∗ Response to the question “Overall he is an effective teacher.”
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Some courses that Yaozhong Hu taught at the University of California at Irvine are listed as follows.
Period Course Number and Name ∗Overall The possiblestudent totalevaluation points
Winter 1997: Math 2B, Calculus II 5.9 7Winter 1997: Math 7, Elem Statistics 5.3 7Spring 1997: Math 3A, Linear Algebra
and Multivariable Calculus 6.1 7Spring 1997: Math 3A, Linear Algebra
and Multivariable Calculus 5.3 7Fall 1996: Math 2C, Calculus III 5.0 7Fall 1996: Math 2D, Calculus IV 4.8 7Spring 1996: Math 7, Elem Statistics 5.5 7Spring 1996: Math 7, Elem Statistics 5.8 7Winter 1996: Math 2B, Calculus II 5.2 7Winter 1996: Math 2D, Calculus IV 5.6 7
Some courses that Yaozhong Hu taught at the University of Oslo, Norway, Louis Pasteur University,France, and the Chinese Academy of Science are listed as follows. There were no student evaluationsat those places.
University of Oslo, Norway
Fall 1995: Stochastic Differential EquationSpring 1993: Numerical Solutions of Stochastic
Differential Equation
Louis Pasteur University, France
Fall 1992: Calculus I and Linear Algebra
Chinese Academy of Science, Wuhan, China
Fall 1989: Nonlinear System TheorySpring 1989: Linear Multivariable ControlFall 1988: Differential GeometrySpring 1988: Multidimensional Diffusion Processes
2. Professor Arriojas Mercedes, Department of Mathematics, University Central of Venezuela, Cara-cas, VENEZUELA
3. Professor Biagini, Francesca, Department of Mathematics, University of Bologna, Piazza di PortaS. Donato, 5, I–40127 Bologna, ITALY
4. Professor Cambanis Stamatis, Department of Statistics, University of North Carolina, ChapelHill, NC 27514
5. Chen Le, Department of Mathematics, University of Kansas, Lawrence, KS 66045
6. Professor Chen Zhenqing, Department of Mathematics, University of Washington, Seattle, WA98195
7. Professor Chen Xia, Department of Mathematics, University of Tennessee at Knoxville.
8. Professor Decreusefond Laurent, Departement Reseaux (Network), Ecole Nationale Superieure deTelecommunications, 75634 Paris, FRANCE
9. Professor de Figuereido Rui J. , Department of Electrical and Computer Engineering, Universityof California, Irvine, CA 92717
10. Professor Driver Bruce K., Department of Mathematics, University of California, San Diego, LaJolla, CA 92093
11. Professor Duncan Tyrone E., Department of Mathematics, University of Kansas, Lawrence, KS66045
12. Han, Zheng, Department of Mathematics, University of Kansas, Lawrence, KS 66045
13. Han, Yuecai, School of Mathematics, Jilin University, Changchun 130012, JILIN, CHINA.
14. Professor Holden Helge, Department of Mathematical Sciences, Norwegian Institute of Technology(NTH), Norwegian University of Science and Technology (NTNU), 7034 Trondheim, NORWAY
15. Hu, Guannan, Department of Mathematics, University of Kansas, Lawrence, KS 66045
16. Mr. Huang Jingyu, Department of Mathematics, University of Kansas, Lawrence, KS 66045
17. Professor Jolis, Maria, Departament de Matematiques, Facultat de Ciencies, Universitat Autonomade Barcelona, 08193 Bellaterra, Spain.
18. Professor Kallianpur Gopinath, Department of Statistics, University of North Carolina, ChapelHill, NC 27514
19. Mr. Le Khoa, Department of Mathematics, University of Kansas, Lawrence, KS 66045
20. Professor Lindstrøm Tom, Department of Mathematics, University of Oslo, 0316 Oslo, NORWAY
21. Lee, Chihoon, Department of Statistics, Colorado State University, Fort Collins, CO 80523.
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22. Lee, Myung Hee, Department of Statistics, Colorado State University, Fort Collins, CO 80523.
23. Professor Liu Yanghui, Department of Mathematics, University of Kansas, Lawrence, KS 66045
24. Professor Long Hongwei, Wuhan Institute of Mathematical Sciences, Academia Sinica, Wuhan430071, PEOPLES REPUBLIC OF CHINA
25. Dr. Lu Fei, University of California at Berkeley.
26. Professor Meyer Paul Andre (thesis advisor), Departement de Mathematique, Institut de RechercheMathematique Avancee (IRMA)(CNRS), Universite de Strasbourg I (Louis Pasteur), 67084 Stras-bourg, FRANCE
27. Professor Meyer-Brandis, Thilo, Fachbereich Mathematik Ludwig-Maximilians-Universitat MunchenD-80333, Munich, GERMANY
28. Professor Mohammed Salah-Eldin A. ,Department of Mathematics, Southern Illinois Universityat Carbondale, Carbondale, Illinois 62901
29. Professor Nualart David, Faculty of Mathematics, University of Barcelona, 08071 Barcelona,SPAIN
30. Professor Ocone Daniel, Department of Mathematics Rutgers University, Piscataway, NJ 08854.
31. Professor Øksendal Bernt, Department of Mathematics, University of Oslo, 0316 Oslo, NORWAY
32. Professor Pap Gyula, Institute of Mathematics and Informatics, University of Debrecen, Pf. 12,H-4010 Debrecen, HUNGARY
33. Professor Pasik-Duncan Bozenna, Department of Mathematics, University of Kansas, Lawrence,KS 66045
34. Professor Peng Shige, Department of Mathematics, Shandong University, Jinan, China
35. Professor Perez-Abreu Victor, Center of Investigations in Mathematics (CIMAT), 36000 Guana-juato Gto., MEXICO
36. Professor Qian Zhongmin, Department of Mathematics, Oxford University, UK
37. Professor Rang Guanglin, School of Mathematics, Wuhan University.
39. Professor Song Jian, Department of Mathematics, University of Kansas, Snow Hall 405, Lawrence,KS 66045
40. Dr. Song Xaioming, Department of Mathematics, University of Kansas, Snow Hall 405, Lawrence,KS 66045
41. Professor Sulem Agnes, INRIA, Domaine de Voluceau, Rocquencourt, B.P. 105, F-78153 Le Ches-nay Cedex, FRANCE
42. Dr. Sun, Xiaobin,
43. Professor Tindel, Samy, Institut Elie Cartan Nancy, Universite de Nancy 1, B.P. 239, 54506Vandœuvre-les-Nancy Cedex, France.
43
44. Professor Ubøe Jan, Department of Mathematics, Agder College (Høgskole i Agder–HiA), 4601Kristiansand, NORWAY
45. Professor Ustunel Ali Suleyman, Departement Reseaux (Network), Ecole Nationale Superieure deTelecommunications, 75634 Paris, FRANCE
46. Professor Wang Baobin, Institute of Physics and Mathematics, Chinese Academy of Science,Wuhan 430071, China
47. Professor Watanabe Shinzo, Department of Mathematics, Kyoto University, Sakyo, Kyoto, JAPAN
48. Dr. Xing Fei, Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300
49. Professor Xiao Weilin, Department of Accounting and Finance, School of Management, ZhejiangUniversity, Hangzhou 310006, Zhejiang, CHINA.
50. Professor Xiong Jie, Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300
51. Professor Xu, Fangjun, Department of Mathematics, University of Kansas, Snow Hall 405, Lawrence,KS 66045.
52. Professor Yan Jia An, Institute of Applied Mathematics, Academy of Mathematics and SystemsScience, Chinese Academy of Sciences, Beijing, China.
53. Professor Yang Changli, School of Information Science and Technology, East China Normal Uni-versity, Shanghai 200062, CHINA.
54. Professor Zakai Moshe, Department of Electrical Engineering, Israel Institute of Technology, Haifa,ISRAEL
55. Professor Zhang Weiguo, School of Business and Administration South China (Huanan) Universityof Technology, Guangzhou 510641, Guangdiong, CHINA.
56. Professor Zhang Tusheng, School of Mathematics, University of Manchester, Oxford Road, Manch-ester M13 9PL, ENGLAND
57. Professor Zheng Weian, Department of Mathematics, University of California, Irvine, CA 92664
58. Professor Zhou Xianyin, Mathematisches Institut, Ruhr-Universitat Bochum, D-44780 Bochum,GERMANY
59. Professor Zhou XunYu, Department of Systems Engineering and Engineering Management, TheChinese University of Hong Kong, Shatin, Hong Kong
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Research Statement of Yaozhong Hu
My research focuses on the theoretical, numerical and applied aspects of probability theory, statistics,and its applications to quantum physics, engineering, finance, and other sciences.
To understand and explore natural and social phenomena, researchers in various fields use mathe-matical models. Quite often the phenomena under investigation appears to be random in character.To describe such phenomena mathematically one needs to apply probability and statistics theory. Myresearch is on these fields. In probability theory one of the most fundamental stochastic processes isBrownian motion. Many quantities can be expressed functionals of Brownian motions. The famousWiener-Ito’s chaos expansion theorem states every square integrable functional of Brownian motioncan be expressed as sum of multiple Wiener-Ito integrals. However, when I studied Feynman integralunder the supervision of P.A. Meyer, I found that the multiple Wiener-Ito integrals are no longer conve-nient. We then introduced multiple Stratonovich integrals and established the relation between multipleStratonovich integrals and Wiener-Ito integrals. This formula is now called the “Hu-Meyer formula”.It is in the titles of 9 articles in top journals (see[48]-[56]) and is a section title in a graduate studenttext book ([47]). Some of my research on these topics is included in my recent book [1].
My recent concentration is stochastic calculus of fractional Brownian motions and applications. As iswell-known, fractional Brownian motions are neither semimartingale nor Markov process. The stochasticanalysis developed for semimartingales cannot no longer applicable to fractional Brownian motions. Onthe other hand, fractional Brownian motions have long memory and other desirable properties that fit tothe reality and they can be applied to describe phenomena that cannot be described by semimartingalesor Markov processes. However, they had been very limited tools for fractional Brownian motions. I andcollaborators developed a stochastic analysis similar to the Ito calculus for fractional Brownian motions.This includes stochastic integral; Ito formula; stochastic differential equations; maximum principle forstochastic control problems driven by fractional Brownian motions; parameter estimation for stochasticdifferential systems driven by fractional Brownian motions; Numerical simulations. Part of my researchon this aspect is summarized in my monograph in American Mathematical Society Memoir ([9]), andmy monograph book by Springer ([3]). Another topic of my recent concentration is stochastic heatequations. For the simplest stochastic heat equations, I and collaborators established a Feynman-Kacformula for the solution of the equation, and Feynman-Kac formula for the moments of the solution byusing weighted local times. These formulas are very useful to obtain the exact moment bounds for thesolution which play important role in intermittancy.
Up to the present, I have co-authored with nearly 60 experts in the field and have published ap-proximately 150 articles in referred journals and proceedings. My research has been supported by theGeneral Research Fund of the University of Kansas, National Science Foundation, and Simons Collabo-ration Grants for Mathematicians (through American Mathematical Society), Alexander von Humboldtfoundation (from Germany), and so on. I was elected to be the fellow of Institute of Mathematical In-stitute in 2015 because of “fundamental research on stochastic calculus of fractional Brownian motionsand influential work on stochastic partial differential equations”.
In the following I will briefly describe some parts of my research work. A complete list of mypublications is in my vita.
1. Analysis of nonlinear Wiener functionals
Feynman path integral is one of the most important tools in quantum physics. However, the mathe-matical aspect of Feynman integral has always been a challenge. An important tool in the mathematicalrigor of Feynman path integrals is the analytic continuation. From stochastic analysis point of viewthis is to study f(σB·), where (Bt, 0 ≤ t ≤ T ) is a Brownian motion (or Wiener process) and σ is a
45
complex number for some functional f on the space Ω of continuous functions with the sup norm. Forfunctional f arising from physical problem usually f(σB·) is well-defined for σ when the real part of σis greater than 0 (right side of the complex plane). The analytic continuation problem is to study if thisfunction of σ can be defined for σ =
√−1 (which lies on the imaginary line). In quantum field theory
or in stochastic analysis the interesting class of f are usually not a continuous functional on Ω. It isdefined only almost surely with respect to a Gaussian measure (such as the standard Wiener measure).Thus f(σB·) may not even well-defined in general. I studied this problem with my Ph.D advisor, P.A. Meyer, one of the world’s best probabilists of his time, for f being given by the multiple Wiener-Itointegrals. I found that it is more natural to use the multiple Stratonovich integrals ([24], [25]). Wehave then established a relationship between multiple Wiener-Ito integrals and multiple Stratonovichintegrals and applied it to the Feynman path integral. These formulas are now called the “Hu-Meyerformulas” and have applications to quantum field theory, analysis on loop space, numerical solutions ofstochastic differential equations. Several refereed papers are entitled “Hu-Meyer formula”, “Hu-Meyertheorem” or so on (see [48]-[56] for the references). In a graduate text book of University of Paris VI([47]), one section is entitled as “Hu-Meyer formula”. I have made substantial contributions to analysisof nonlinear Wiener functionals. Some of these works are presented in my recent book [1].
2. Stochastic quantization
Euclidean quantum field is an important area in mathematical physics. One of the objectives ofthe Euclidean quantum field is to construct an interacting (probability) measure on a certain space of(generalized) functions of several variables. There are several breakthroughs in this field such as theNelson’s hypercontractivity. Two leading physicists Parisi and Wu studied this problem from a new(and dynamic) point of view. They tried to construct a Markov process whose limiting (invariant)measure is the measure that we want to construct. This program attracted a number of physicists andmathematicians. Several books have devoted to this approach. The difficulty to justify this approachmathematically is the infinite dimensional stochastic differential equation satisfied by this Markov pro-cess has a drift term which is not continuous and which is unbounded from both above and below. TheNovikov condition cannot be used for the existence of the solution. Using a less known Kazamaki con-dition for the exponential integrability of some exponential martingale, we give a rigorous mathematicalfoundation for this approach ([23]). This work is well-received.
Originally, the stochastic quantization is to approximate a measure in infinite dimensional space.But when it is used to approximate the measure in finite dimensional space it has also advantagecomparing with the popular Monte Carlo method. For Monte Carlo approximation there are results forconvergence in distribution. But in [10], for stochastic quantization, the rate of convergence in densityis obtained which is sharper than convergence in distribution. The discrete time analogue of stochasticquantization is relevant to the so-called Monte-Carlo Markov chain. I plan to return to this problemfrom the Monte-Carlo Markov chain prospective.
3. Statistics in stochastic differential system
For the simple stochastic Langevin equation dXt = −θXtdt+σdZt, where Zt is a fractional Brownianmotion or a stable process, we studied the estimation problem for the parameter θ. For example whenZt is a fractional Brownian motion, we constructed a new simple least square estimator θT of θ ([30]).
We studied the almost sure convergence of θT to θ. We also obtained the convergence of√T (θT − θ) to
a normal distribution, a central limit type theorem. Since√T (θT −θ) converges to a normal variable in
distribution it is natural to ask if√T (θT −θ) has a density with respect to Lebesgue measure and if the
density converges to the normal density or not. These questions seem impossible to answer. However,they are solved in a recent paper with Lu and Nualart ([21]).
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On the other hand, when Zt is α-stable processes, the rate will be different. In fact, it is proved in
[19] and [20] that(
nlogn
)1/α(θn − θ0) converges to a stable distribution.
When only discrete observations are available, in the literature, one needs to assume the lengthbetween consecutive observations goes to 0 (the high frequency data) to have consistent estimator.With collaborator we constructed a “simple to calculate” estimator which is convergent for any fixedgiven length ([42]). This is a surprising result since we don’t require the length between consecutiveobservations goes to 0. The central limiting type theorem and Berry-Esseen type rate of convergenceare also obtained. This type of estimator has some advantage that it can be used to other systems suchas reflected Brownian motions or reflected OU processes. These are done in some of my other papers([17], [18]).
4. Mathematical finance
I studied some basic mathematical problems such as option pricing, optimal portfolio and consump-tion and other problems for long memory models (fractal markets) and for delay models.
As is well-known, the optimal stopping problem is usually very hard to solve explicitly. Even fora simple model it is hard to obtain explicit solution although it can be easily reduced to a movingboundary problem and one has also some verification theorems available. The first optimal stoppingproblem that Professor Øksendal and I ([39]) considered is the McDonald-Siegel problem which askswhen to sell one asset and to buy another asset. We solved this problem under some quite generalconditions and we also take into account the transaction cost.
When there are insider in the market, one has to consider the optimal stopping problems for aninsider. Or one has to consider anticipative optimal stopping problems. These problems are even muchhard than the nonanticipative optimal stopping problems. One can see this from the following simplefact. If Bt is a Brownian motion and if τ is a (nonanticipative) stopping time, than E (Bτ ) = 0.However, if τ is an anticipative stopping time, then E (Bτ ) may not be 0. And people have no effectiveway to compute this expectation. By using Malliavin calculus, we solve some anticipative stoppingproblems explicitly in [38].
In a recent paper [40] we also studied the optimal singular control problems for mean field systemsby using recent results for Skorohod problems. With collaborators, I also considered Bank-Kyle insiderequilibrium problem, insider trading problems, and some other mathematical finance problems.
5. A simple formula to compute the Clark derivative
In the classical theory of stochastic analysis a basic theorem states that if ξ is a square integrablefunctional of the Brownian motion (Bt, 0 ≤ t ≤ T ), then there is an adapted process (ft, 0 ≤ t ≤ T )such that
ξ = E(ξ) +
∫ T
0
ftdBt .
ft is called the Clark derivative of ξ. There is a famous formula, called the Clark-Ocone-Haussmannformula, to compute the Clark derivative by using the Malliavin derivative. But recently due to theprofound application of stochastic analysis to finance, one has to hedge a given contingent claim. Forthis one needs to more effectively compute the Clark derivative. In the paper [8] in addition to someother results, I found a simple method to compute the Clark derivative ft which uses only the semigroupand Ito formula. This approach is included in a very popular graduate text book [46] (Its sixth editionwas appeared).
6. Several inequalities
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Malliavin Calculus is a very active field in the recent 40 years. A basic inequality that people wishto have is the so-called interpolation inequality which states that if D and D2 are the first and secondorder Malliavin derivatives, then
‖Df‖p ≤ C‖f‖1/2‖D2f‖1/2p .
Professor Suleyman Ustunel introduced this problem to me and told me that Professor D. Stroock askedhim about this problem. Together with Ustunel and his student, Decreusefond, we proved the followingequality
‖Df‖p ≤ Cp[‖f‖p + ‖f‖1/2‖D2f‖1/2p ] .
with a very simple method ([4]). An important consequence of this inequality is that if fn → 0 inLp and D2fn is bounded in Lp, then Dfn → 0 in Lp, which is the original motivation for ProfessorStroock to wish to have such interpolation inequality. So, our inequality is different but serves the samepurpose.
Since we used mainly the Meyer’s inequality, we sent our paper to Paul Andre Meyer and he did notbelieve such a basic and beautiful inequality did not exist in analysis and asked an expert in analysisto review it and it is concluded that this was indeed a new result.
In a similar situation, there is a so-called correlation conjecture: Assume that µ is a standardGaussian measure on the n dimensional Euclidean space Rn. If A and B are two convex and balancedsets in Rn, then
µ(A ∩B) ≥ µ(A)µ(B) .
This inequality was conjectured around the 1960. When n=2 this inequality is proved by L. Pitt. Whenn ≥ 3 many researchers has made great contributions. Motivated by this inequality, I have proved thefollowing inequality. Let f and g be two symmetric convex functions, then
µ(fg) ≥ µ(f)µ(g) , where µ(f) =
∫R
f(x)µ(dx) .
This inequality is some time called Hu’s inequality in some papers and attracted a good amount ofresearchers. In a 1998 international mini-conference on correlation conjecture, the above inequalitybecomes the whole content of the first talk. This inequality is also included in some monograph.
Motivated by the approach to prove the variant of the correlation inequality, Bruce Driver and Iproved a log Sobolev inequality on the Riemanian manifold ([5]). We proved a general inequality undersome mild conditions on Ricci curvature. The Poincare inequality and the log Sobolev inequality areimmediate consequence of our general inequality. Not only the inequality is very general, but also theproof is much simpler. This correlation inequality is very recently solved by Royen [57].
7. Stochastic delay systems
Salah Mohammed, Yan Feng and I studied numerical solution of stochastic delay equations ([26]).We derived the Milstein scheme, a higher order convergence scheme than the Euler scheme. But tostudy the rate of convergence we encounter the non-adapted stochastic integrals although we are stillin the adapted situation. This requires us to develop a new anticipative Ito formula. Subsequently, weapply the stochastic delay equations to option pricing [2], which is supposed to better fit the volatilitysmile.
8. Backward stochastic differential equations and Malliavin calculus
The stochastic control originated from real life applications and has a very rich and sophisticatedmathematical theory. Usually the explicit solution to a stochastic control problem is very hard to obtain
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except in some specific cases. Backward stochastic differential equations appeared first as an importantcomponent to solve some general stochastic control problems. Now they also became an important toolsin mathematical finance. It is interesting to study general backward stochastic differential equationsdriven by long memory processes such as fractional Brownian motions. The difficulty here is the lack ofmartingale representation theorem for fractional Brownian motions since fractional Brownian motionsthemselves are not semimartingales. With Shige Peng [41], we exploit quasi-conditional expectationsand the fractional Clark formula and use them to solve the backward stochastic differential equationsdriven by fractional Brownian motions. Some preliminary work has also been studied when we tried tosolve some stochastic control problems for elementary long memory systems.
The classical backward stochastic differential equations contains two unknown Y and Z. However,in mathematics it is common that one equation determines one unknown. So there must be somerelationship between these two variables Y and Z. In a recent work ([36]) with Nualart and Song, weuse that fact that Z is a Malliavin derivative of Y to study the properties of Z which is hard to obtainotherwise. These results are used to obtain sharp rate of convergence of some numerical schemes forbackward stochastic differential equations. We also developed some new numerical schemes and appliedour results to obtained some rate of convergence of our newly developed schemes.
With Professor Øksendal and other collaborators we solved some stochastic control problems forsystems driven by fractional Brownian motions. In particular, we obtained a sufficient condition thatthe optimal control must satisfy by using backward stochastic differential equations driven by fractionalBrownian motions. However, it is very interesting to know the necessary condition. In stochastic controltheory usually people use duality method to obtain the backward sde, which is hardly applicable tofractional Brownian motion case. In [7] we use a completely different approach to obtain a necessarycondition for general stochastic control problems for systems driven by fractional Brownian motions.The new backward stochastic differential equation is quite interesting itself since it involves both thefractional Brownian motion and the “background” Brownian motion. It is even more interesting topoint out that the new bsde also involves the Malliavin derivative.
9. Numerical stochastic differential equations
Numerical solutions for stochastic differential equations have been widely studied by numerous re-searchers. I contributed to this field in several ways. The first one is that usually people assume thatthe coefficients are globally Lipschitz. It appears that I am the first one to construct a numerical scheme(which is called the semi-Euler scheme) for stiff stochastic differential equations (the coefficients satisfythe one-sided Lipschitz condition) [11]. In the literature, people usually consider the strong convergenceor weak convergence. Namely, if X and Xn are the true and the approximate solutions, people studiedE |Xn −X|p and |E Xn −E X|. These rate are usually different. With Professor Shinzo Watanabe westudied the convergence of the densities [43]. Namely, we studied the problems under what conditionthe law of Xn and X are absolutely continuous with respect to Lebesgues measure. Namely, under
what conditions there are pn(x) and p(x) such that for any a < b, P (a ≤ Xn ≤ b) =∫ bapn(x)dx and
P (a ≤ X ≤ b) =∫ bap(x)dx? An what is the rate of |pn(x) − p(x)|? These problems are solved by
develop some new results in the Malliavin calculus applicable to above problems. These results havenow potential applications to the maximum likelihood estimation in stochastic systems.
I also studied a number of other problems in the numerical solutions of stochastic differential equa-tions such as the optimal sampling points, Wick product approximation, convergence of finite differenceschemes for stochastic pressure equations, numerical solutions of Zakai equations and numerical schemesfor stochastic differential system driven by factional Brownian motions, and so on.
10. Rough Path analysis
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Professor David Nualart and I spent a number of years in the study of rough path analysis usingfractional calculus ([28]). If y(t) is a Holder continuous function of t of order greater than 1/3, then wecan define the integral with respect to y(t) and the differential equation driven by y(t). The advantageof using fractional calculus compared to the professor Terry Lyons [45] original approach is that ourapproach can yield more “quantitative” results. We can obtain precise dependence of solution on theinitial data, the driving rough signals, and so on. As a consequence, we can prove the almost surerate of convergence in terms of Holder norm of the Wong-Zakai type approximation which is new andsurprising. Having completed this work, there left a number of relevant problems. On one hand it isplaned to study more carefully the work of Lyons and to see if it is possible to generalize our work tothe case that y is of Holder continuous of order less than 1/3. On the other hand, it is also planned tostudy the stochastic partial differential equations driven by fractional Brownian fields.
When the Hurst parameter is greater than 1/2, in [13] I systematically studied the multiple integralswith respect to fractional Brownian motion and the expansion of the solution of a stochastic differentialequations driven by fractional Brownian motions into multiple integrals. Quite surprisingly, I alsoobtained the Ito-Wiener chaos expansion for one dimensional stochastic differential equations driven byfractional Brownian motions.
11. Self-intersection local times
Intersection or self-intersection local time of a stochastic process is an very important concept inprobability theory. In a paper jointly with David Nualart on renormalized self-intersection local timefor fractional Brownian motions ([27]), we obtained a detailed account of the self-intersection local timefor fractional Brownian motions. We proved that if dH < 1 (where H is the Hurst parameter and dis the dimension), then the self-intersection local time exists as a square interable random varaible.If 1/d <= H < 3/(2d), then the renormalized self-intersection local time exists. The renormalizedself-intersection local time has been further systematically studied in my joint paper with Nualart andSong [34]. In particular, the problem of exponential integrability is solved. This property is useful inthe study of Euclidean quantum field theory since the work Varadhan in the 1960s and is proved in thetwo dimensional Brownian case by Le Gall in the 1990s.
In a recent work ([29]) Prof. Nualart and I have discovered that a specific weighted self-intersectionlocal time is relevant to the stochastic partial differential equation with multiplicative fractal noise. Theexponential integrability is relevant to the square integrable of solution to the spde. This is an excitingliaison and leads to our further study on the Feynman-Kac formula for the solution of SPDE driven byfractional noises.
12. Feynman-Kac formulas
As we know the Feynman path integral is one of the most important tools in quantum physics.The Feynman-Kac formula (a mathematical rigourization by M. Kac) has become one of the mostimportant formulas in probability theory. Researchers wish to obtain Feynman-Kac formula for thesolution of stochastic partial differential equations driven by multiplicative noises since more than twodecades ago. But the corresponding Feynman-Kac functionals for multiplicative white noises are notwell-defined in the classical sense. In a joint work Nualart and Song ([32]) we studied the functionalscorresponding to fractional multiplicative noises in detail and in particular we studied the exponentialintegrability of these functionals and obtain the Feynman-Kac formula for a reasonably large class ofstochastic partial differential equations. This work mainly dealt with the multiplicative noises whichdisplay long memory property. On the other hand, the persistent noises appear often in the turbulenceand so on. But mathematically the stochastic partial differential equations driven by these noises aremore difficulty to handle. In a joint work with Nualart and Lu [22], we use careful analysis through
50
fractional calculus to deal with this equations and obtained some satisfactory results. The feynman-Kacformula has a broad applications. One application is to obtain some new properties about the solutionof some nonlinear stochastic partial differential equations (see our work [35]).
The Feynman-Kac formula has been very effectively used to obtain the lower bounds for the momentsof the solution of parabolic Anderson model: If W is a general Gaussian noise and if u(t, x) is the solutionto ∂u
∂t = 12∆u+ uW , then we proved there are positive constants α, β, c1, C1, c2, C2 such that
C1ec1t
αkβ ≤ E (u(t, x)k) ≤ C2ec2t
αkβ
(as t and k go to ∞), where the constants are explicitly given by the covariance structure of W andthe dimension d. In particular, if the space dimension is 1 (x ∈ R) and if W is time independent, spacewhite noise, then
C1 exp(c1t
3k3)≤ E (u(t, x)k) ≤ C2 exp
(c2t
3k3).
The upper bound is easier by using the hypercontractivity. But the lower bound has been known onlyin very few cases. Our result gives a complete solution by using the Feynman-Kac formula.
13. Fractional martingales and characterization of fractional Brownian motion
In the Brownian motion framework, a key result is the the Levy’s characterization theory. It isinteresting to extend this theorem to fractional Brownian motion. In a joint work with Nualart andSong [32], we introduced the notion of fractional martingale as the fractional derivative of order α ofa continuous local martingale, where α ∈ (− 1
2 ,12 ), and we show that it has a nonzero finite variation
of order 21+2α , under some integrability assumptions on the quadratic variation of the local martingale.
based on this theory, we establish an extension of Levy’s characterization theorem for the fractionalBrownian motion.
14. Long memory processes
In the last years one of my main research focuses is on stochastic analysis of long memory processes,in particular stochastic analysis of fractional Brownian motions and its applications to engineering andfinance. Stochastic analysis has a broad and substantial application to many fields including finance.One such applications is the Black and Scholes theory of option pricing. Black and Scholes theoryhas a great impact in today’s finance world. M. Scholes, and R. Merton, as the New York Timesof Wednesday, 15th October 1997 commented, “won the Nobel Memorial Prize in Economics Scienceyesterday for work that enables investors to price accurately their bets on the future, a breakthroughthat has helped power the explosive growth in financial markets since the 1970’s and plays a profoundrole in the economics of everyday life”. One of the fundamental assumptions of their original work isthat the stock price follows a geometric Brownian motion. Due to the spectacular size and dramatic riseand fall of today’s financial market, investors and regulators alike urgently need more accurate models.One of potentially promising models is the geometric fractional Brownian motion. However, this modelfails to meet a basic economic standard: it provides arbitrage opportunities. With Bernt Øksendal wehave introduced a new fractal (Black and Scholes) markets ([37]). We have proved that these marketsare complete without providing any arbitrage opportunity and have derived some basic results such asthe option pricing formula. These works offer an opportunity to better understand the today’s financialmarket. This work has attracted a great of attention. This paper with Bernt Øksendal is based on myprevious joint work with Tyrone Duncan and Bozenna Pasik-Duncan ([6]). In analysis, a basic problem
is to define the integral∫ T0ftdgt. In the deterministic case, if the integral has some desired property,
then g must be a bounded variation function. If ft and gt are stochastic processes, then some good
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properties of the integral require gt to be a semimartingale. Unfortunately, the fractional Brownianmotion is not a semimartingale (except in the case of Brownian motion), so a new theory to integralmust be introduced. In my paper with Tyrone Duncan and Bozenna Pasik-Duncan ([6]) we introduceda new type of stochastic integral for fractional Brownian motion using Wick product. We obtained thefamous Ito formula and so on. These two papers have already had significant impact.
I have continuously made significant contributions to various aspects of fractional Brownian motionsand fractional Brownian fields. My joint work with David Nualart ([27]) studied the self-intersectionlocal time of the fractional Brownian motions of any dimensions. The exponential integrability of theself-intersection local time are also obtained.
I have published numerous papers on the stochastic calculus for fractional Brownian motions. Asystematic study is my memoir [9] where I studied the stochastic integral, Ito formula, conditioning,Girsanov transform and so on for fractional Brownian motions of any Hurst parameter. In [44] I alsoobtained the Riccati equation for finding the optimal Markov control for linear quadratic problemsdriven by fractional Brownian motions which provides the first explicit solution for long memory linearquadratic control problems.
15. Stochastic partial differential equations
For stochastic heat equations with space time white noise, it is well-known that when the spacedimension is one, then the classical solution exists for all time t ≥ 0. But when the space dimension isgreater or equal to two, then the classical solution does not exists for any time. In [12], I consider astochastic heat equation with time independent space white noise. I discovered the following interestingphenomena that when the space dimension is two the classical solutions exists up to some strictlypositive time t0 and when t > t0 the solution has infinite L2 norm. All other dimensions are alsodiscussed.
Recently there has been a great interest on the intermittency problem for parabolic Anderson model.We studied parabolic Anderson model with various general Gaussian noises including fractional Brow-nian noises. In particular, we obtained the asymptotic upper and in particular lower moment boundsfor the solution. The problems become particularly difficult when the Hurst parameter(s) is less thanor equal to 1/2.
With collaborators I also studied the Holder continuity of the solution, existence of density for thestochastic partial differential equations, including stochastic heat and stochastic wave equations ([14],[31] etc). In particular, we obtain the precise Holder continuity of stochastic heat equations by extendingthe famous Garsia-Rodemich-Rumsey inequality to functions of more of several variables with respectto a non-distance gauge ([16]).
16. Convergence in density
The central limit theorem says that a sequence of random variables Fn (for example, the normalizedsample mean of some iid) converges to F in distribution. This means for any a ∈ R, P (Fn ≤ a)converges to P (F ≤ a). Some natural (and yet difficult) questions to ask are the following: Does Fnhave a density pn(x) (namely, P (a < Xn ≤ b) =
∫ bapn(x)dx for any −∞ < a < b < ∞) and does
limn→∞∫R |pn(x)− φ(x)|pdx = 0, where φ(x) = 1√
2πe−x
2/2. In a very recent work [21], we studied the
above problems and we obtain some bounds of |pn(x)− φ(x)|pdx by using the Malliavin derivatives of
Fn and F . We also applied our results to the parametric estimation problem of√T (θT − θ) mentioned
in section 13.
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References
[1] Hu, Y. Analysis on Gaussian spaces. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ,2017. xi+470 pp.
[2] Arritojas, M.; Hu, Y.; Mohammed, S. and Pap, G. A Delayed Black and Scholes Formula, Stoch.Anal. Appl. 25 (2007), 471-492.
[3] Biagini, F.; Hu, Y.; Øksendal, B. and Zhang, T.S. Stochastic calculus of fractional Brownianmotion. Springer, 2008.
[4] Decreusefond, L.; Hu, Y. and Ustunel, A. S. Une inegalite d’interpolation sur l’espace de Wiener,Compte Rendus Acad. Sci. Paris, 317 (1993), 1065-1067.
[5] Hu, Y. and Driver, B. K. On heat kernel logarithmic Sobolev inequalities, Stochastic Analysis andApplications (Powys, 1995), World Sci. Publishing, River Edge, NJ, 1996, 189-200.
[6] Duncan, T. E.; Hu, Y. and Pasik-Duncan, B. Stochastic calculus for fractional Brownian motion.I. Theory. SIAM J. Control Optim. 38 (2000), no. 2, 582-612.
[7] Han, Y.; Hu, Y. and Song, J. Maximum Principle for General Controlled Systems Driven byFractional Brownian Motions. Appl. Math. Optim. 67 (2013), no. 2, 279-322.
[8] Hu, Y. Ito-Wiener chaos expansion with exact residual and correlation, variance inequalities, Jour-nal of Theoretical Probability, 10 (1997), 835-848.
[9] Hu, Y. Integral transformations and anticipative calculus for fractional Brownian motions. Mem.Amer. Math. Soc. 175 (2005), no. 825, viii+127 pp.
[10] Hu, Y. Stochastic quantization and ergodic theorem for diffusion processes. Sci. China Math. 55(2012), no. 11, 2285-2296.
[11] Semi-implicit Euler-Maruyama scheme for stiff stochastic equations, Stochastic Analysis and Re-lated Topics, V (Silivri, 1994), Progr. Probab., 38, Birkhauser Boston, Boston, MA, 1996, 183-202.
[12] Hu, Y. Chaos expansion of heat equation with white noise potentials, Potential Anal. 16 (2002),no. 1, 45–66.
[13] Hu, Y. Multiple integrals and expansion of solution to differential equations driven by rough pathand by fractional Brownian motion. Stochastics 85 (2013), no. 5, 859-916.
[14] Hu, Y.; Huang, J.; Nualart, D. On Holder continuity of the solution of stochastic wave equationsin dimension three. Stoch. Partial Differ. Equ. Anal. Comput. 2 (2014), 353-407.
[15] Hu, Y.; Huang, J.; Nualart, D. and Tindel, S. Stochastic heat equation with general multiplicativeGaussian noises: Holder continuity and intermittency. Submitted.
[16] Hu, Y. and Le, K. A multiparameter Garsia-Rodemich-Rumsey inequality and some applications.Stochastic Process. Appl. 123 (2013), 3359-3377.
[17] Hu, Y. and Lee, C. Drift parameter estimation for a reflected fractional Brownian motion basedon its local time. J. Appl. Probab. 50 (2013), no. 2, 592-597.
[18] Hu, Y.; Lee, C.; Lee, M. H. and Song, J. Parameter estimation for reflected Ornstein-Uhlenbeckprocesses with discrete observations. Statistical Inference for Stochastic Processes. 2014
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[19] Hu, Y. and Long, H.W. Least squares estimator for Ornstein-Uhlenbeck processes driven by α-stable motions. Stochastic Process. Appl. 119 (2009), no. 8, 2465–2480.
[20] Hu, Y. and Long, H.W. On the singularity of least squares estimator for mean-reverting α-stablemotions. Acta Math. Sci. Ser. B Engl. Ed. 29 (2009), no. 3, 599–608.
[21] Hu, Y.; Lu, F. and Nualart, D. Convergence of densities of some functionals of Gaussian processes.J. Funct. Anal. 266 (2014), no. 2, 814-875.
[22] Hu, Y.; Lu, F. and Nualart, D. Feynman-Kac formula for spde driven by fractional Brownian fieldswith Hurst parameter H < 1/2. Ann. Probab. 40 (2012), no. 3, 1041-1068.
[23] Hu, Y. and Kallianpur, G. Exponential integrability and application to stochastic quantization,Applied Mathematics and Optimization, 37 (1998), 295-353.
[24] Hu, Y. and Meyer, P. A. Chaos de Wiener et integrales de Feynman, Seminaire de ProbabilitesXXII, Ed. J. Azema, P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1321, Springer-Verlag,1988, 51-71.
[25] Hu, Y. and Meyer, P. A. Sur les integrales multiples de Stratonovitch, Seminaire de ProbabilitesXXVI, Ed. J. Azema, P.A. Meyer and M. Yor, Lecture Notes in Mathematics 1321, Springer-Verlag, 1988, 72-81.
[26] Hu, Y.; Mohammed, S. E. A. and Yan, F. Numerical Solution of Stochastic Differential Systemswith Memory. Annals of Probability, 32 (2004), 265-314.
[27] Hu, Y. and Nualart, D. Renormalized self-intersection local time for fractional Brownian motion.Ann. Probab. 33 (2005), no. 3, 948-983.
[28] Hu, Y. and Nualart, D. Rough path analysis via fractional calculus. Trans. Amer. Math. Soc. 361(2009), no. 5, 2689-2718.
[29] Hu, Y. and Nualart, D. Stochastic heat equation driven by fractional noise and local time. Prob.Theory and Related Fields, 143 (2009), 285-328.
[30] Hu, Y. and Nualart, D. Parameter estimation for fractional Ornstein-Uhlenbeck processes. Statist.Probab. Lett. 80 (2010), no. 11-12, 1030-1038.
[31] Hu, Y.; Nualart, D. and Song, J. A nonlinear stochastic heat equation: Holder continuity andsmoothness of the density of the solution. Stochastic Process. Appl. 123 (2013), 1083-1103.
[32] Hu, Y.; Nualart, D. and Song, J. Feynman-Kac formula for heat equation driven by fractionalwhite noise. Ann. Probab. 39 (2011), no. 1, 291-326.
[33] Hu, Y.; Nualart, D. and Song, J. Fractional martingales and characterization of the fractionalBrownian motion. Annals of Probability, in press.
[34] Hu, Y.; Nualart, D. and Song, J. Integral representation of renormalized self-intersection localtimes. J. Funct. Anal. 255 (2008), no. 9, 2507-2532.
[35] Hu, Y.; Nualart, D. and Song, J. A nonlinear stochastic heat equation: Holder continuity andsmoothness of the density of the solution. Submitted.
[36] Hu, Y.; Nualart, D. and Song, X. Malliavin calculus for backward stochastic differential equationsand application to numerical schemes. The Annals of Applied Probability Vol. 21 (2011), 2379-2423.
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[37] Hu, Y. and Øksendal, B. Fractional white noise calculus and applications to finance. Infinite Di-mensional Analysis, Quantum Probability and Related Topics, Vol. 6 (2003), 1-32.
[38] Hu, Y. and Øksendal, B. Optimal anticipative stopping. Advances in Math. of Finance. BanachCenter Publications vol. 83 (2008), 107-116.
[39] Hu, Y. and Øksendal, B. Optimal time to invest when the price processes are geometric Brownianmotions, Finance and Stochastics, 2 (1998), 295-310.
[40] Hu, Y.; Øksendal, B. and Sulem, A. Singular mean-field control games with asymmetric informa-tion. Submitted.
[41] Hu, Y. and Peng, S. Backward stochastic differential equations driven by fractional Brownianmotion. SIAM Journal of Control and Optimization, 48 (2009), 1675-1700.
[42] Hu, Y. and Song, J. Parameter estimation for fractional Ornstein-Uhlenbeck processes with discreteobservations. In Malliavin calculus and stochastic analysis, 427-442, Springer Proc. Math. Stat.,34, Springer, New York, 2013.
[43] Hu, Y. and Watanabe, S. Donsker’s delta functions and approximation of heat kernels by timediscretization methods, J. Math. Kyoto University, 36 (1996), 499-518.
[44] Hu, Y. and Zhou X.Y. Stochastic control for linear systems driven by fractional noises. SIAM J.Control Optim. 43 (2005), no. 6, 2245–2277.
[45] Lyons, T. J. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998),no. 2, 215-310.
[46] Øksendal, B. Stochastic differential equations. An introduction with applications. Sixth edition.Universitext. Springer, 2003.
[47] Bouleau, N. and Lepingle, D. Numerical methods for stochastic processes. Wiley Series in Proba-bility and Mathematical Statistics: Applied Probability and Statistics. A Wiley-Interscience Pub-lication. John Wiley & Sons, Inc., New York, 1994. xx+359 pp.
[48] Farre, M.; Jolis, M. and Utzet, F. Multiple Stratonovich integral and Hu-Meyer formula for Levyprocesses. Ann. Probab. 38 (2010), 2136-2169.
[49] Toh, T.-L. and Chew, T.-S. Henstock’s multiple Wiener integral and Henstock’s version of Hu-Meyer theorem. Math. Comput. Modelling 42 (2005), 139-149.
[50] Budhiraja, A.; Kallianpur, G. The generalized Hu-Meyer formula for random kernels. Appl. Math.Optim. 35 (1997), 177-202.
[51] Delgado, R. and Jolis, M. A Hu-Meyer formula for generalized multiple anticipating integrals.Stochastics Stochastics Rep. 58 (1996), 115-138.
[52] Sole, J. L.; Utzet, F. The Hu-Meyer formula. A view of the problem with special attention to thePoisson case. Chaos expansions, multiple Wiener-Ito integrals and their applications (Guanajuato,1992), 35-45, Probab. Stochastics Ser., CRC, Boca Raton, FL, 1994.
[53] Delgado, R. and Sanz, M. The Hu-Meyer formula for nondeterministic kernels. Stochastics Stochas-tics Rep. 38 (1992), 149-158.
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[54] Johnson, G. W.; Kallianpur, G. Some remarks on Y. Z. Hu and P.-A. Meyer’s paper and infinite-dimensional calculus on finitely additive canonical Hilbert space: Teor. Veroyatnost. i Primenen.34 (1989), 742-752; translation in Theory Probab. Appl. 34 (1989), 679-689 (1990).
[55] Sugita, H. Hu-Meyer’s multiple Stratonovich integral and essential continuity of multiple Wienerintegral. Bull. Sci. Math. 113 (1989), 463-474.
[56] Wang, B.; Hu, T. A new proof of fractional Hu-Meyer formula and its applications. J. Inequal.Appl. 2012, 12 pp.
[57] Royen, T. A simple proof of the Gaussian correlation conjecture extended to multivariate gammadistributions. Preprint 2014.
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