Date post: | 04-Jun-2018 |
Category: |
Documents |
Upload: | spin-fotonio |
View: | 221 times |
Download: | 0 times |
of 268
8/13/2019 Mathemat i Kun Do Eko No Mie
1/268
Mathematics and Economics
Frank Riedel
Institute for Mathematical EconomicsBielefeld University
Mathematics Colloquium Bielefeld, January 2011
http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
2/268
Three Leading Questions
Mathematics and Economics: Big Successes in History
Leon Walras,Elements deconomie politique pure 1874
Francis Edgeworth,Mathematical Psychics, 1881
John von Neumann, Oskar Morgenstern,
Theory of Games and Economic Behavior, 1944
Paul Samuelson,Foundations of Economic Analysis, 1947
Kenneth Arrow, Gerard Debreu,
Competitive Equilibrium 1954
John Nash 1950, Reinhard Selten, 1965,Noncoperative Game Theory
Fischer Black, Myron Scholes, Robert Merton, 1973,
Mathematical Finance
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
3/268
Three Leading Questions
Mathematics and Economics: Big Successes in History
Leon Walras,Elements deconomie politique pure 1874
Francis Edgeworth,Mathematical Psychics, 1881
John von Neumann, Oskar Morgenstern,
Theory of Games and Economic Behavior, 1944
Paul Samuelson,Foundations of Economic Analysis, 1947
Kenneth Arrow, Gerard Debreu,
Competitive Equilibrium 1954
John Nash 1950, Reinhard Selten, 1965,Noncoperative Game Theory
Fischer Black, Myron Scholes, Robert Merton, 1973,
Mathematical Finance
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
4/268
Three Leading Questions
Mathematics and Economics: Big Successes in History
Leon Walras,Elements deconomie politique pure 1874
Francis Edgeworth,Mathematical Psychics, 1881
John von Neumann, Oskar Morgenstern,
Theory of Games and Economic Behavior, 1944
Paul Samuelson,Foundations of Economic Analysis, 1947
Kenneth Arrow, Gerard Debreu,
Competitive Equilibrium 1954John Nash 1950, Reinhard Selten, 1965,Noncoperative Game Theory
Fischer Black, Myron Scholes, Robert Merton, 1973,
Mathematical Finance
Th L di Q i
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
5/268
Three Leading Questions
Mathematics and Economics: Big Successes in History
Leon Walras,Elements deconomie politique pure 1874
Francis Edgeworth,Mathematical Psychics, 1881
John von Neumann, Oskar Morgenstern,
Theory of Games and Economic Behavior, 1944
Paul Samuelson,Foundations of Economic Analysis, 1947
Kenneth Arrow, Gerard Debreu,
Competitive Equilibrium 1954John Nash 1950, Reinhard Selten, 1965,Noncoperative Game Theory
Fischer Black, Myron Scholes, Robert Merton, 1973,
Mathematical Finance
Th L di Q ti
http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
6/268
Three Leading Questions
Mathematics and Economics: Big Successes in History
Leon Walras,Elements deconomie politique pure 1874
Francis Edgeworth,Mathematical Psychics, 1881
John von Neumann, Oskar Morgenstern,
Theory of Games and Economic Behavior, 1944
Paul Samuelson,Foundations of Economic Analysis, 1947
Kenneth Arrow, Gerard Debreu,
Competitive Equilibrium 1954John Nash 1950, Reinhard Selten, 1965,Noncoperative Game Theory
Fischer Black, Myron Scholes, Robert Merton, 1973,
Mathematical Finance
Three Leading Questions
http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
7/268
Three Leading Questions
Mathematics and Economics: Big Successes in History
Leon Walras,Elements deconomie politique pure 1874
Francis Edgeworth,Mathematical Psychics, 1881
John von Neumann, Oskar Morgenstern,
Theory of Games and Economic Behavior, 1944
Paul Samuelson,Foundations of Economic Analysis, 1947
Kenneth Arrow, Gerard Debreu,
Competitive Equilibrium 1954John Nash 1950, Reinhard Selten, 1965,Noncoperative Game Theory
Fischer Black, Myron Scholes, Robert Merton, 1973,
Mathematical Finance
Three Leading Questions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
8/268
Three Leading Questions
Mathematics and Economics: Big Successes in History
Leon Walras,Elements deconomie politique pure 1874
Francis Edgeworth,Mathematical Psychics, 1881
John von Neumann, Oskar Morgenstern,
Theory of Games and Economic Behavior, 1944
Paul Samuelson,Foundations of Economic Analysis, 1947
Kenneth Arrow, Gerard Debreu,
Competitive Equilibrium 1954John Nash 1950, Reinhard Selten, 1965,Noncoperative Game Theory
Fischer Black, Myron Scholes, Robert Merton, 1973,Mathematical Finance
Three Leading Questions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
9/268
Three Leading Questions
Three Leading Questions
1 Rationality ?Isnt it simply wrong to impose heroic foresight and
intellectual abilities to describe humans?
2 Egoism ?Humans show altruism, envy, passions etc.
3 Probability ?Doesnt the crisis show that mathematics is useless, even
dangerous in markets?
Three Leading Questions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
10/268
Three Leading Questions
Three Leading Questions
1 Rationality ?Isnt it simply wrong to impose heroic foresight and
intellectual abilities to describe humans?
2 Egoism ?Humans show altruism, envy, passions etc.
3 Probability ?Doesnt the crisis show that mathematics is useless, even
dangerous in markets?
Three Leading Questions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
11/268
g Q
Three Leading Questions
1 Rationality ?Isnt it simply wrong to impose heroic foresight and
intellectual abilities to describe humans?
2 Egoism ?Humans show altruism, envy, passions etc.
3 Probability ?Doesnt the crisis show that mathematics is useless, even
dangerous in markets?
Three Leading Questions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
12/268
g Q
Three Leading Questions: Details
Rationality ? Egoism ?These assumptions are frequently justified
Aufklarung! . . . answers Kants Was soll ich tun?
design of institutions: good regulation must be robust against
rational, egoistic agents (Basel II was not, e.g.)Doubts remain . . .; Poincare to Walras:
Par exemple, en mechanique, on neglige souvent lefrottement et on regarde les corps comme infiniment polis.Vous, vous regardez les hommes comme infiniment egoisteset infiniment clairvoyants. La premiere hypothese peut etreadmise dans une premiere approximation, mais la deuxiemenecessiterait peut-etre quelques reserves
Three Leading Questions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
13/268
g
Three Leading Questions: Details
Rationality ? Egoism ?These assumptions are frequently justified
Aufklarung! . . . answers Kants Was soll ich tun?
design of institutions: good regulation must be robust against
rational, egoistic agents (Basel II was not, e.g.)Doubts remain . . .; Poincare to Walras:
Par exemple, en mechanique, on neglige souvent lefrottement et on regarde les corps comme infiniment polis.Vous, vous regardez les hommes comme infiniment egoisteset infiniment clairvoyants. La premiere hypothese peut etreadmise dans une premiere approximation, mais la deuxiemenecessiterait peut-etre quelques reserves
Three Leading Questions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
14/268
Three Leading Questions: Details
Rationality ? Egoism ?These assumptions are frequently justified
Aufklarung! . . . answers Kants Was soll ich tun?
design of institutions: good regulation must be robust against
rational, egoistic agents (Basel II was not, e.g.)Doubts remain . . .; Poincare to Walras:
Par exemple, en mechanique, on neglige souvent lefrottement et on regarde les corps comme infiniment polis.Vous, vous regardez les hommes comme infiniment egoisteset infiniment clairvoyants. La premiere hypothese peut etreadmise dans une premiere approximation, mais la deuxiemenecessiterait peut-etre quelques reserves
Three Leading Questions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
15/268
Three Leading Questions: Details
Rationality ? Egoism ?These assumptions are frequently justified
Aufklarung! . . . answers Kants Was soll ich tun?
design of institutions: good regulation must be robust against
rational, egoistic agents (Basel II was not, e.g.)Doubts remain . . .; Poincare to Walras:
Par exemple, en mechanique, on neglige souvent lefrottement et on regarde les corps comme infiniment polis.Vous, vous regardez les hommes comme infiniment egoisteset infiniment clairvoyants. La premiere hypothese peut etreadmise dans une premiere approximation, mais la deuxiemenecessiterait peut-etre quelques reserves
Three Leading Questions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
16/268
Three Leading Questions: Details
Rationality ? Egoism ?These assumptions are frequently justified
Aufklarung! . . . answers Kants Was soll ich tun?
design of institutions: good regulation must be robust against
rational, egoistic agents (Basel II was not, e.g.)Doubts remain . . .; Poincare to Walras:
Par exemple, en mechanique, on neglige souvent lefrottement et on regarde les corps comme infiniment polis.Vous, vous regardez les hommes comme infiniment egoisteset infiniment clairvoyants. La premiere hypothese peut etreadmise dans une premiere approximation, mais la deuxiemenecessiterait peut-etre quelques reserves
Three Leading Questions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
17/268
Three Leading Questions: Details
Rationality ? Egoism ?These assumptions are frequently justified
Aufklarung! . . . answers Kants Was soll ich tun?
design of institutions: good regulation must be robust against
rational, egoistic agents (Basel II was not, e.g.)Doubts remain . . .; Poincare to Walras:
Par exemple, en mechanique, on neglige souvent lefrottement et on regarde les corps comme infiniment polis.Vous, vous regardez les hommes comme infiniment egoisteset infiniment clairvoyants. La premiere hypothese peut etreadmise dans une premiere approximation, mais la deuxiemenecessiterait peut-etre quelques reserves
Three Leading Questions
http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
18/268
Three Leading Doubts: Details ctd.
Probability ?
Option Pricing based on probability theory assumptions isextremelysuccessful
some blame mathematicians for financial crisis nonsense, butdoes probability theory apply to single events like
Greece is going bankrupt in 2012SF Giants win the World SeriesmedinForm in Bielefeld wird profitabel
Ellsberg experiments
Three Leading Questions
http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
19/268
Three Leading Doubts: Details ctd.
Probability ?
Option Pricing based on probability theory assumptions isextremelysuccessful
some blame mathematicians for financial crisis nonsense, butdoes probability theory apply to single events like
Greece is going bankrupt in 2012SF Giants win the World SeriesmedinForm in Bielefeld wird profitabel
Ellsberg experiments
Three Leading Questions
http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
20/268
Three Leading Doubts: Details ctd.
Probability ?
Option Pricing based on probability theory assumptions isextremelysuccessful
some blame mathematicians for financial crisis nonsense, butdoes probability theory apply to single events like
Greece is going bankrupt in 2012SF Giants win the World SeriesmedinForm in Bielefeld wird profitabel
Ellsberg experiments
Three Leading Questions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
21/268
Three Leading Doubts: Details ctd.
Probability ?
Option Pricing based on probability theory assumptions isextremelysuccessful
some blame mathematicians for financial crisis nonsense, butdoes probability theory apply to single events like
Greece is going bankrupt in 2012SF Giants win the World SeriesmedinForm in Bielefeld wird profitabel
Ellsberg experiments
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
22/268
Three Leading Questions
8/13/2019 Mathemat i Kun Do Eko No Mie
23/268
Three Leading Doubts: Details ctd.
Probability ?
Option Pricing based on probability theory assumptions isextremelysuccessful
some blame mathematicians for financial crisis nonsense, butdoes probability theory apply to single events like
Greece is going bankrupt in 2012SF Giants win the World SeriesmedinForm in Bielefeld wird profitabel
Ellsberg experiments
Egoism
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
24/268
Egoism
based on Dufwenberg, Heidhues, Kirchsteiger, R., Sobel, Review ofEconomic Studies 2010
Empirical Evidence
Humans react on their environment
relative concerns, in particular with peers, are important
especially in situations with few players
not in anonymous situations
FehrSchmidt Otherregarding preferences matter in games, but not inmarkets
Egoism
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
25/268
Egoism
based on Dufwenberg, Heidhues, Kirchsteiger, R., Sobel, Review ofEconomic Studies 2010
Empirical Evidence
Humans react on their environment
relative concerns, in particular with peers, are important
especially in situations with few players
not in anonymous situations
FehrSchmidt Otherregarding preferences matter in games, but not inmarkets
Egoism
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
26/268
Egoism
based on Dufwenberg, Heidhues, Kirchsteiger, R., Sobel, Review ofEconomic Studies 2010
Empirical Evidence
Humans react on their environment
relative concerns, in particular with peers, are important
especially in situations with few players
not in anonymous situations
FehrSchmidt Otherregarding preferences matter in games, but not inmarkets
Egoism
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
27/268
Egoism
based on Dufwenberg, Heidhues, Kirchsteiger, R., Sobel, Review ofEconomic Studies 2010
Empirical Evidence
Humans react on their environment
relative concerns, in particular with peers, are important
especially in situations with few players
not in anonymous situations
FehrSchmidt Otherregarding preferences matter in games, but not inmarkets
Egoism
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
28/268
Egoism
based on Dufwenberg, Heidhues, Kirchsteiger, R., Sobel, Review ofEconomic Studies 2010
Empirical Evidence
Humans react on their environment
relative concerns, in particular with peers, are important
especially in situations with few players
not in anonymous situations
FehrSchmidt Otherregarding preferences matter in games, but not inmarkets
Egoism
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
29/268
Egoism
based on Dufwenberg, Heidhues, Kirchsteiger, R., Sobel, Review ofEconomic Studies 2010
Empirical Evidence
Humans react on their environment
relative concerns, in particular with peers, are important
especially in situations with few players
not in anonymous situations
FehrSchmidt Otherregarding preferences matter in games, but not inmarkets
Egoism
E i
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
30/268
Egoism
General Equilibrium is the general theory of free, competitive markets withrational,selfinterestedagents
The Big Theorems
ExistenceFirst Welfare Theorem: Equilibrium Allocations are efficient
. . . in the core, even
Second Welfare Theorem: efficient allocations can be implemented
via free markets and lumpsum transfersCoreEquivalence: in large economies, the outcome of rationalcooperation (core) is close to market outcomes
Egoism
E i
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
31/268
Egoism
General Equilibrium is the general theory of free, competitive markets withrational,selfinterestedagents
The Big Theorems
ExistenceFirst Welfare Theorem: Equilibrium Allocations are efficient
. . . in the core, even
Second Welfare Theorem: efficient allocations can be implemented
via free markets and lumpsum transfersCoreEquivalence: in large economies, the outcome of rationalcooperation (core) is close to market outcomes
Egoism
E i
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
32/268
Egoism
General Equilibrium is the general theory of free, competitive markets withrational,selfinterestedagents
The Big Theorems
ExistenceFirst Welfare Theorem: Equilibrium Allocations are efficient
. . . in the core, even
Second Welfare Theorem: efficient allocations can be implemented
via free markets and lumpsum transfersCoreEquivalence: in large economies, the outcome of rationalcooperation (core) is close to market outcomes
Egoism
E i
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
33/268
Egoism
General Equilibrium is the general theory of free, competitive markets withrational,selfinterestedagents
The Big Theorems
ExistenceFirst Welfare Theorem: Equilibrium Allocations are efficient
. . . in the core, even
Second Welfare Theorem: efficient allocations can be implemented
via free markets and lumpsum transfersCoreEquivalence: in large economies, the outcome of rationalcooperation (core) is close to market outcomes
Egoism
Eg is
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
34/268
Egoism
General Equilibrium is the general theory of free, competitive markets withrational,selfinterestedagents
The Big Theorems
ExistenceFirst Welfare Theorem: Equilibrium Allocations are efficient
. . . in the core, even
Second Welfare Theorem: efficient allocations can be implemented
via free markets and lumpsum transfersCoreEquivalence: in large economies, the outcome of rationalcooperation (core) is close to market outcomes
Egoism
Egoism
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
35/268
Egoism
General Equilibrium is the general theory of free, competitive markets withrational,selfinterestedagents
The Big Theorems
ExistenceFirst Welfare Theorem: Equilibrium Allocations are efficient
. . . in the core, even
Second Welfare Theorem: efficient allocations can be implemented
via free markets and lumpsum transfersCoreEquivalence: in large economies, the outcome of rationalcooperation (core) is close to market outcomes
Egoism
Egoism
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
36/268
Egoism
General Equilibrium is the general theory of free, competitive markets withrational,selfinterestedagents
The Big Theorems
ExistenceFirst Welfare Theorem: Equilibrium Allocations are efficient
. . . in the core, even
Second Welfare Theorem: efficient allocations can be implemented
via free markets and lumpsum transfersCoreEquivalence: in large economies, the outcome of rationalcooperation (core) is close to market outcomes
Egoism
Simple yet Famous Models
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
37/268
Simple, yet Famous Models
OtherRegarding Utility functions used to explain experimental data
FehrSchmidt (BoltonOckenfels) introduce fairness and envy:Ui=mi
iI1 kmax{(mkmi), 0}
iI1 kmax{(mimk), 0}
CharnessRabin: mi+ iI1
imin{m1, . . . , mI} + (1 i)
Ij=1mj
Edgeworth already has looked at mi +mj
Shaked: such ad hoc models are not science (and Poincare would
agree)
Egoism
Simple yet Famous Models
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
38/268
Simple, yet Famous Models
OtherRegarding Utility functions used to explain experimental data
FehrSchmidt (BoltonOckenfels) introduce fairness and envy:Ui=mi
iI1 kmax{(mkmi), 0}
iI1 kmax{(mimk), 0}
CharnessRabin: mi+ iI1
imin{m1, . . . , mI} + (1 i)
Ij=1mj
Edgeworth already has looked at mi +mj
Shaked: such ad hoc models are not science (and Poincare would
agree)
Egoism
Simple yet Famous Models
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
39/268
Simple, yet Famous Models
OtherRegarding Utility functions used to explain experimental data
FehrSchmidt (BoltonOckenfels) introduce fairness and envy:Ui=mi
iI1 kmax{(mkmi), 0}
iI1 kmax{(mimk), 0}
CharnessRabin: mi+ iI1
imin{m1, . . . , mI} + (1 i)
Ij=1mj
Edgeworth already has looked at mi +mj
Shaked: such ad hoc models are not science (and Poincare would
agree)
Egoism
Simple yet Famous Models
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
40/268
Simple, yet Famous Models
OtherRegarding Utility functions used to explain experimental data
FehrSchmidt (BoltonOckenfels) introduce fairness and envy:Ui=mi
iI1 kmax{(mkmi), 0}
iI1 kmax{(mimk), 0}
CharnessRabin: mi+ iI1
imin{m1, . . . , mI} + (1 i)
Ij=1mj
Edgeworth already has looked at mi +mj
Shaked: such ad hoc models are not science (and Poincare would
agree)
Egoism
Simple yet Famous Models
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
41/268
Simple, yet Famous Models
OtherRegarding Utility functions used to explain experimental data
FehrSchmidt (BoltonOckenfels) introduce fairness and envy:Ui=mi
iI1 kmax{(mkmi), 0}
iI1 kmax{(mimk), 0}
CharnessRabin: mi+ iI1
imin{m1, . . . , mI} + (1 i)
Ij=1mj
Edgeworth already has looked at mi +mj
Shaked: such ad hoc models are not science (and Poincare would
agree)
Egoism
Mathematical Formulation
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
42/268
Mathematical Formulation
in anonymous situations, an agent cannot debate prices or influence
what others consumeown consumption x RL+, others consumption y R
K, pricesp RL+, income w>0
utilityu(x, y), strictly concave and smooth in x
when is the solution d(y, p, w) of
maximize u(x, y) subject to p x=w
independent ofy ?
Definition
We say that agent ibehaves asif selfishif her demand functiondi
p, w, xi
does not depend on others consumption plans xi.
Egoism
Mathematical Formulation
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
43/268
Mathematical Formulation
in anonymous situations, an agent cannot debate prices or influence
what others consumeown consumption x RL+, others consumption y R
K, pricesp RL+, income w>0
utilityu(x, y), strictly concave and smooth in x
when is the solution d(y, p, w) of
maximize u(x, y) subject to p x=w
independent ofy ?
Definition
We say that agent ibehaves asif selfishif her demand functiondi
p, w, xi
does not depend on others consumption plans xi.
Egoism
Mathematical Formulation
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
44/268
in anonymous situations, an agent cannot debate prices or influence
what others consumeown consumption x RL+, others consumption y R
K, pricesp RL+, income w>0
utilityu(x, y), strictly concave and smooth in x
when is the solution d(y, p, w) of
maximize u(x, y) subject to p x=w
independent ofy ?
Definition
We say that agent ibehaves asif selfishif her demand functiondi
p, w, xi
does not depend on others consumption plans xi.
Egoism
Mathematical Formulation
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
45/268
in anonymous situations, an agent cannot debate prices or influence
what others consumeown consumption x RL+, others consumption y R
K, pricesp RL+, income w>0
utilityu(x, y), strictly concave and smooth in x
when is the solution d(y, p, w) of
maximize u(x, y) subject to p x=w
independent ofy ?
Definition
We say that agent ibehaves asif selfishif her demand functiondi
p, w, xi
does not depend on others consumption plans xi.
Egoism
Mathematical Formulation
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
46/268
in anonymous situations, an agent cannot debate prices or influence
what others consumeown consumption x RL+, others consumption y R
K, pricesp RL+, income w>0
utilityu(x, y), strictly concave and smooth in x
when is the solution d(y, p, w) of
maximize u(x, y) subject to p x=w
independent ofy ?
Definition
We say that agent ibehaves asif selfishif her demand functiondi
p, w, xi
does not depend on others consumption plans xi.
Egoism
AsIf Selfish Demand Examples
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
47/268
p
Clearly, standard egoistic utility functions vi(xi) =vi(xi1, . . . , viL)lead to as-if selfish behavior
Additive social preferences: let Ui(xi, xj) =vi(xi) +vj(xj). Then
marginal utilities are independent ofxj,Product Preferences:Ui(xi) =vi(xi)vj(xj) =vi(xi1, . . . , viL)vj(xj1, . . . , vjL)
marginal utilities do depend on others consumption bundlesbut marginal rates of substitution do not!
asif selfish behavior
Egoism
AsIf Selfish Demand Examples
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
48/268
Clearly, standard egoistic utility functions vi(xi) =vi(xi1, . . . , viL)lead to as-if selfish behavior
Additive social preferences: let Ui(xi, xj) =vi(xi) +vj(xj). Then
marginal utilities are independent ofxj,Product Preferences:Ui(xi) =vi(xi)vj(xj) =vi(xi1, . . . , viL)vj(xj1, . . . , vjL)
marginal utilities do depend on others consumption bundlesbut marginal rates of substitution do not!
asif selfish behavior
Egoism
AsIf Selfish Demand Examples
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
49/268
Clearly, standard egoistic utility functions vi(xi) =vi(xi1, . . . , viL)lead to as-if selfish behavior
Additive social preferences: let Ui(xi, xj) =vi(xi) +vj(xj). Then
marginal utilities are independent ofxj,Product Preferences:Ui(xi) =vi(xi)vj(xj) =vi(xi1, . . . , viL)vj(xj1, . . . , vjL)
marginal utilities do depend on others consumption bundlesbut marginal rates of substitution do not!
asif selfish behavior
Egoism
AsIf Selfish Demand Examples
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
50/268
Clearly, standard egoistic utility functions vi(xi) =vi(xi1, . . . , viL)lead to as-if selfish behavior
Additive social preferences: let Ui(xi, xj) =vi(xi) +vj(xj). Then
marginal utilities are independent ofxj,Product Preferences:Ui(xi) =vi(xi)vj(xj) =vi(xi1, . . . , viL)vj(xj1, . . . , vjL)
marginal utilities do depend on others consumption bundlesbut marginal rates of substitution do not!
asif selfish behavior
Egoism
AsIf Selfish Demand Examples
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
51/268
Clearly, standard egoistic utility functions vi(xi) =vi(xi1, . . . , viL)lead to as-if selfish behavior
Additive social preferences: let Ui(xi, xj) =vi(xi) +vj(xj). Then
marginal utilities are independent ofxj,Product Preferences:Ui(xi) =vi(xi)vj(xj) =vi(xi1, . . . , viL)vj(xj1, . . . , vjL)
marginal utilities do depend on others consumption bundlesbut marginal rates of substitution do not!
asif selfish behavior
Egoism
AsIf Selfish Demand Examples
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
52/268
Clearly, standard egoistic utility functions vi(xi) =vi(xi1, . . . , viL)lead to as-if selfish behavior
Additive social preferences: let Ui(xi, xj) =vi(xi) +vj(xj). Then
marginal utilities are independent ofxj,Product Preferences:Ui(xi) =vi(xi)vj(xj) =vi(xi1, . . . , viL)vj(xj1, . . . , vjL)
marginal utilities do depend on others consumption bundlesbut marginal rates of substitution do not!
asif selfish behavior
Egoism
AsIf Selfish Preferences
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
53/268
Theorem
Agent i behaves as if selfish if and only if her preferences can berepresented by a separable utility function
Vi(mi(xi), xi)
where mi :Xi R is theinternal utility function, continuous, strictly
monotone, strictly quasiconcave, and Vi :D R R(I1)L+ R is an
aggregator, increasing in own utility mi.
Technical AssumptionPreferences are smooth enough such that demand is continuouslydifferentiable. Needed for the only if.
Egoism
AsIf Selfish Preferences
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
54/268
Theorem
Agent i behaves as if selfish if and only if her preferences can berepresented by a separable utility function
Vi(mi(xi), xi)
where mi :Xi R is theinternal utility function, continuous, strictly
monotone, strictly quasiconcave, and Vi :D R R(I1)L+ R is an
aggregator, increasing in own utility mi.
Technical AssumptionPreferences are smooth enough such that demand is continuouslydifferentiable. Needed for the only if.
Egoism
General Consequences
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
55/268
Free Markets are a good institution in the sense that they maximizematerialefficiency (in terms ofmi(xi))
but not necessarily good as a social institution, i.e. in terms of realutilityui(xi, xi)
Egoism
General Consequences
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
56/268
Free Markets are a good institution in the sense that they maximizematerialefficiency (in terms ofmi(xi))
but not necessarily good as a social institution, i.e. in terms of realutilityui(xi, xi)
Egoism
Inefficiency
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
57/268
ExampleTake two agents, two commodities with the same internal utility andUi=m1+m2. Take as endowment an internally efficient allocation closeto the edge of the box. Unique Walrasian equilibrium, but not efficient, as
the rich agent would like to give endowment to the poor. Markets cannotmake gifts!
Remark
Public goods are a way to make gifts. Heidhues/R. have an example in
which the rich agent uses a public good to transfer utility to the pooragent (but still inefficient allocation).
Egoism
Inefficiency
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
58/268
ExampleTake two agents, two commodities with the same internal utility andUi=m1+m2. Take as endowment an internally efficient allocation closeto the edge of the box. Unique Walrasian equilibrium, but not efficient, as
the rich agent would like to give endowment to the poor. Markets cannotmake gifts!
Remark
Public goods are a way to make gifts. Heidhues/R. have an example in
which the rich agent uses a public good to transfer utility to the pooragent (but still inefficient allocation).
Egoism
Is Charity Enough to Restore Efficiency?
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
59/268
in example: charity would lead to efficiency
not true for more than 2 agents!Prisoners Dilemma
Egoism
Is Charity Enough to Restore Efficiency?
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
60/268
in example: charity would lead to efficiency
not true for more than 2 agents!Prisoners Dilemma
Egoism
Is Charity Enough to Restore Efficiency?
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
61/268
in example: charity would lead to efficiency
not true for more than 2 agents!Prisoners Dilemma
Egoism
Second Welfare Theorem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
62/268
Some (unplausible) preferences have to be ruled out:
Example
Hateful Society: Ui=mi 2mjfor two agents i=j. No consumption isefficient.
Social Monotonicity
For z RL+ \ {0} and any allocation x, there exists a redistribution (zi)with
zi =z such that
Ui(x+z)> Ui(x)
Egoism
Second Welfare Theorem
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
63/268
Some (unplausible) preferences have to be ruled out:
Example
Hateful Society: Ui=mi 2mjfor two agents i=j. No consumption isefficient.
Social Monotonicity
For z RL+ \ {0} and any allocation x, there exists a redistribution (zi)with
zi =z such that
Ui(x+z)> Ui(x)
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
64/268
Egoism
Second Welfare Theorem
8/13/2019 Mathemat i Kun Do Eko No Mie
65/268
Theorem
Under social monotonicity, the set of Pareto optima is included in the setof internal Pareto optima.
Corollary
Second Welfare Theorem. The price system does not create inequality.
Uncertainty and Probability
Uncertainty Theory as new Probability Theory
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
66/268
As P is not exactly known, work with a whole class of probability
measures P, (Huber, 1982, Robust Statistics)
Knightian Decision Making
GilboaSchmeidler: U(X) = minPPiEPu(x)
FollmerSchied, Maccheroni, Marinacci, Rustichinigeneralize tovariational preferences
U(X) = minP
EPu(X) +c(P)
for a cost function cdo not trust your model! be aware of sensitivities! do not believe inyour EXCEL sheet!
Uncertainty and Probability
Uncertainty Theory as new Probability Theory
http://goforward/http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
67/268
As P is not exactly known, work with a whole class of probability
measures P, (Huber, 1982, Robust Statistics)
Knightian Decision Making
GilboaSchmeidler: U(X) = minPPiEPu(x)
FollmerSchied, Maccheroni, Marinacci, Rustichinigeneralize tovariational preferences
U(X) = minP
EPu(X) +c(P)
for a cost function cdo not trust your model! be aware of sensitivities! do not believe inyour EXCEL sheet!
Uncertainty and Probability
Uncertainty Theory as new Probability Theory
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
68/268
As P is not exactly known, work with a whole class of probability
measures P, (Huber, 1982, Robust Statistics)
Knightian Decision Making
GilboaSchmeidler: U(X) = minPPiEPu(x)
FollmerSchied, Maccheroni, Marinacci, Rustichinigeneralize tovariational preferences
U(X) = minP
EPu(X) +c(P)
for a cost function cdo not trust your model! be aware of sensitivities! do not believe inyour EXCEL sheet!
Uncertainty and Probability
Uncertainty Theory as new Probability Theory
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
69/268
As P is not exactly known, work with a whole class of probability
measures P, (Huber, 1982, Robust Statistics)
Knightian Decision Making
GilboaSchmeidler: U(X) = minPPiEPu(x)
FollmerSchied, Maccheroni, Marinacci, Rustichinigeneralize tovariational preferences
U(X) = minP
EPu(X) +c(P)
for a cost function cdo not trust your model! be aware of sensitivities! do not believe inyour EXCEL sheet!
Uncertainty and Probability Optimal Stopping
IMW Research on Optimal Stopping and KnightianUncertainty
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
70/268
U y
Dynamic Coherent Risk Measures, Stochastic Processes and TheirApplications 2004
Optimal Stopping with Multiple Priors, Econometrica, 2009
Optimal Stopping under Ambiguity in Continuous Time (with XueCheng), IMW Working Paper 2010
The Best Choice Problem under Ambiguity (with TatjanaChudjakow), IMW Working Paper 2009
Chudjakow, Vorbrink, Exercise Strategies for American Exotic Optionsunder Ambiguity, IMW Working Paper 2009
Vorbrink, Financial Markets with Volatility Uncertainty, IMW WorkingPaper 2010
JanHenrik Steg, Irreversible Investment in Oligopoly, Finance andStochastics 2011
Uncertainty and Probability Optimal Stopping
Optimal Stopping Problems: Classical Version
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
71/268
Let
,F, P, (Ft)t=0,1,2,...
be a filtered probability space.
Given a sequence X0, X1, . . . , XTof random variables
adapted to the filtration (Ft)
choose a stopping time T
that maximizes EX.
classic: Snell, Chow/Robbins/Siegmund: Great Expectations
Uncertainty and Probability Optimal Stopping
Optimal Stopping Problems: Solution, Discrete FiniteTime
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
72/268
based on R., Econometrica 2009
Solution
Define the Snell envelope Uvia backward induction:
UT =XT
Ut= max {Xt,E [Ut+1|Ft]} (t
8/13/2019 Mathemat i Kun Do Eko No Mie
73/268
based on R., Econometrica 2009
Solution
Define the Snell envelope Uvia backward induction:
UT =XT
Ut= max {Xt,E [Ut+1|Ft]} (t
8/13/2019 Mathemat i Kun Do Eko No Mie
74/268
based on R., Econometrica 2009
Solution
Define the Snell envelope Uvia backward induction:
UT =XT
Ut= max {Xt,E [Ut+1|Ft]} (t
8/13/2019 Mathemat i Kun Do Eko No Mie
75/268
Uncertainty and Probability Optimal Stopping
Two classics
The Parking Problem
8/13/2019 Mathemat i Kun Do Eko No Mie
76/268
The Parking Problem
You drive along a road towards a theatre
You want to park as close as possible to the theatre
Parking spaces are freeiid with probability p>0
When is the right time to stop? take the first free after 68%1/pdistance
Secretary Problem = When to Marry?
You see sequentially N applicants
maximize the probability to get the best one
rejected applicants do not come backapplicants come in random (uniform) order
optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants
probability of getting the best one approx 1 e Uncertainty and Probability Optimal Stopping
Two classics
The Parking Problem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
77/268
The Parking Problem
You drive along a road towards a theatre
You want to park as close as possible to the theatre
Parking spaces are freeiid with probability p>0
When is the right time to stop? take the first free after 68%1/pdistance
Secretary Problem = When to Marry?
You see sequentially N applicants
maximize the probability to get the best one
rejected applicants do not come backapplicants come in random (uniform) order
optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants
probability of getting the best one approx 1 e
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
78/268
Uncertainty and Probability Optimal Stopping
Two classics
The Parking Problem
8/13/2019 Mathemat i Kun Do Eko No Mie
79/268
The Parking Problem
You drive along a road towards a theatre
You want to park as close as possible to the theatre
Parking spaces are freeiid with probability p>0
When is the right time to stop? take the first free after 68%1/pdistance
Secretary Problem = When to Marry?
You see sequentially N applicants
maximize the probability to get the best one
rejected applicants do not come backapplicants come in random (uniform) order
optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants
probability of getting the best one approx 1 e Uncertainty and Probability Optimal Stopping
Two classics
The Parking Problem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
80/268
The Parking Problem
You drive along a road towards a theatre
You want to park as close as possible to the theatre
Parking spaces are freeiid with probability p>0
When is the right time to stop? take the first free after 68%1/pdistance
Secretary Problem = When to Marry?
You see sequentially N applicants
maximize the probability to get the best one
rejected applicants do not come backapplicants come in random (uniform) order
optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants
probability of getting the best one approx 1 e Uncertainty and Probability Optimal Stopping
Two classics
The Parking Problem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
81/268
The Parking Problem
You drive along a road towards a theatre
You want to park as close as possible to the theatre
Parking spaces are freeiid with probability p>0
When is the right time to stop? take the first free after 68%1/pdistance
Secretary Problem = When to Marry?
You see sequentially N applicants
maximize the probability to get the best one
rejected applicants do not come backapplicants come in random (uniform) order
optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants
probability of getting the best one approx 1 e Uncertainty and Probability Optimal Stopping
Two classics
The Parking Problem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
82/268
The Parking Problem
You drive along a road towards a theatre
You want to park as close as possible to the theatre
Parking spaces are freeiid with probability p>0
When is the right time to stop? take the first free after 68%1/pdistance
Secretary Problem = When to Marry?
You see sequentially N applicants
maximize the probability to get the best one
rejected applicants do not come backapplicants come in random (uniform) order
optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants
probability of getting the best one approx 1 e Uncertainty and Probability Optimal Stopping
Two classics
The Parking Problem
http://find/http://goback/8/13/2019 Mathemat i Kun Do Eko No Mie
83/268
g
You drive along a road towards a theatre
You want to park as close as possible to the theatre
Parking spaces are freeiid with probability p>0
When is the right time to stop? take the first free after 68%1/pdistance
Secretary Problem = When to Marry?
You see sequentially N applicants
maximize the probability to get the best one
rejected applicants do not come backapplicants come in random (uniform) order
optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants
probability of getting the best one approx 1 e Uncertainty and Probability Optimal Stopping
Two classics
The Parking Problem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
84/268
g
You drive along a road towards a theatre
You want to park as close as possible to the theatre
Parking spaces are freeiid with probability p>0
When is the right time to stop? take the first free after 68%1/pdistance
Secretary Problem = When to Marry?
You see sequentially N applicants
maximize the probability to get the best one
rejected applicants do not come backapplicants come in random (uniform) order
optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants
probability of getting the best one approx 1 e Uncertainty and Probability Optimal Stopping
Two classics
The Parking Problem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
85/268
g
You drive along a road towards a theatre
You want to park as close as possible to the theatre
Parking spaces are freeiid with probability p>0
When is the right time to stop? take the first free after 68%1/pdistance
Secretary Problem = When to Marry?
You see sequentially N applicants
maximize the probability to get the best one
rejected applicants do not come backapplicants come in random (uniform) order
optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants
probability of getting the best one approx 1 e Uncertainty and Probability Optimal Stopping
Two classics
The Parking Problem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
86/268
You drive along a road towards a theatre
You want to park as close as possible to the theatre
Parking spaces are freeiid with probability p>0
When is the right time to stop? take the first free after 68%1/pdistance
Secretary Problem = When to Marry?
You see sequentially N applicants
maximize the probability to get the best one
rejected applicants do not come backapplicants come in random (uniform) order
optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants
probability of getting the best one approx 1 e Uncertainty and Probability Optimal Stopping
Two classics
The Parking Problem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
87/268
You drive along a road towards a theatre
You want to park as close as possible to the theatre
Parking spaces are freeiid with probability p>0
When is the right time to stop? take the first free after 68%1/pdistance
Secretary Problem = When to Marry?
You see sequentially N applicants
maximize the probability to get the best one
rejected applicants do not come backapplicants come in random (uniform) order
optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants
probability of getting the best one approx 1 e Uncertainty and Probability Optimal Stopping
Optimal Stopping with Multiple Priors: Discrete Time
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
88/268
We choose the following modeling approachLet X0, X1, . . . , XTbe a (finite) sequence of random variables
adapted to a filtration (Ft)
on a measurable space (,F),
let Pbe a set of probability measures
choose a stopping time T
that maximizesinfPP
EPX
Uncertainty and Probability Optimal Stopping
Optimal Stopping with Multiple Priors: Discrete Time
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
89/268
We choose the following modeling approachLet X0, X1, . . . , XTbe a (finite) sequence of random variables
adapted to a filtration (Ft)
on a measurable space (,F),
let Pbe a set of probability measures
choose a stopping time T
that maximizesinfPP
EPX
Uncertainty and Probability Optimal Stopping
Optimal Stopping with Multiple Priors: Discrete Time
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
90/268
We choose the following modeling approachLet X0, X1, . . . , XTbe a (finite) sequence of random variables
adapted to a filtration (Ft)
on a measurable space (,F),
let Pbe a set of probability measures
choose a stopping time T
that maximizesinfPP
EPX
Uncertainty and Probability Optimal Stopping
Optimal Stopping with Multiple Priors: Discrete Time
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
91/268
We choose the following modeling approachLet X0, X1, . . . , XTbe a (finite) sequence of random variables
adapted to a filtration (Ft)
on a measurable space (,F),
let Pbe a set of probability measures
choose a stopping time T
that maximizesinfPP
EPX
Uncertainty and Probability Optimal Stopping
Optimal Stopping with Multiple Priors: Discrete Time
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
92/268
We choose the following modeling approachLet X0, X1, . . . , XTbe a (finite) sequence of random variables
adapted to a filtration (Ft)
on a measurable space (,F),
let Pbe a set of probability measures
choose a stopping time T
that maximizesinfPP
EPX
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
93/268
Uncertainty and Probability Optimal Stopping
Assumptions
8/13/2019 Mathemat i Kun Do Eko No Mie
94/268
(Xt) bounded by a Puniformly integrable random variable
there exists a reference measure P0: all PPare equivalent to P0
(wlog,Tutsch, PhD 07)
agent knows all null sets,Epstein/Marinacci 07
Pweakly compact in L1
,F, P0
inf is always min,Follmer/Schied 04, Chateauneuf, Maccheroni, Marinacci, Tallon 05
Uncertainty and Probability Optimal Stopping
Extending the General Theory to Multiple Priors
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
95/268
Aims
Work as close as possible along the classical lines
Time ConsistencyMultiple Prior Martingale Theory
Backward Induction
Uncertainty and Probability Optimal Stopping
Time Consistency
With general P, one runs easily into inconsistencies in dynamictti (S i /W kk )
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
96/268
settings (Sarin/Wakker)
Time consistency law of iterated expectations:
minQP
EQ
ess infPP
EP [ X |Ft]
= min
PPEPX
Literature on time consistency in decision theory /risk measure theoryEpstein/Schneider, R. , Artzner et al., Detlefsen/Scandolo, Peng,Chen/Epsteintime consistency is equivalent to stability under pasting:
let P,QPand let (pt), (qt) be the density processesfix a stopping time
define a new measure Rvia setting
rt=
pt if tpqt/q else
then RPas well
Uncertainty and Probability Multiple Prior Martingale Theory
Multiple Prior Martingales
Definition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
97/268
Definition
An adapted, bounded process (St) is called a multiple priorsupermartingale iff
St ess infPP
EP [ St+1 |Ft]
holds true for all t 0.
multiple prior martingale: =multiple prior submartingale:
Remark
Nonlinear notion of martingales.Different from Pmartingale (martingale for all PPsimultaneously)
Uncertainty and Probability Multiple Prior Martingale Theory
Multiple Prior Martingales
Definition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
98/268
Definition
An adapted, bounded process (St) is called a multiple priorsupermartingale iff
St ess infPP
EP [ St+1 |Ft]
holds true for all t 0.
multiple prior martingale: =multiple prior submartingale:
Remark
Nonlinear notion of martingales.Different from Pmartingale (martingale for all PPsimultaneously)
Uncertainty and Probability Multiple Prior Martingale Theory
Multiple Prior Martingales
Definition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
99/268
Definition
An adapted, bounded process (St) is called a multiple priorsupermartingale iff
St ess infPP
EP [ St+1 |Ft]
holds true for all t 0.
multiple prior martingale: =multiple prior submartingale:
Remark
Nonlinear notion of martingales.Different from Pmartingale (martingale for all PPsimultaneously)
Uncertainty and Probability Multiple Prior Martingale Theory
Multiple Prior Martingales
Definition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
100/268
An adapted, bounded process (St) is called a multiple priorsupermartingale iff
St ess infPP
EP [ St+1 |Ft]
holds true for all t 0.
multiple prior martingale: =multiple prior submartingale:
Remark
Nonlinear notion of martingales.Different from Pmartingale (martingale for all PPsimultaneously)
Uncertainty and Probability Multiple Prior Martingale Theory
Characterization of Multiple Prior Martingales
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
101/268
Theorem(St) is a multiple prior submartingale iff(St) is a Psubmartingale.
(St) is a multiple prior supermartingale iff there exists a PP suchthat(St) is a Psupermartingale.
(Mt) is a multiple prior martingale iff(Mt) is a Psubmartingale andfor some PPa Psupermartingale.
Remark
For multiple prior supermartingales: holds always true. needs
timeconsistency.
Uncertainty and Probability Multiple Prior Martingale Theory
Characterization of Multiple Prior Martingales
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
102/268
Theorem(St) is a multiple prior submartingale iff(St) is a Psubmartingale.
(St) is a multiple prior supermartingale iff there exists a PP suchthat(St) is a Psupermartingale.
(Mt) is a multiple prior martingale iff(Mt) is a Psubmartingale andfor some PPa Psupermartingale.
Remark
For multiple prior supermartingales: holds always true. needs
timeconsistency.
Uncertainty and Probability Multiple Prior Martingale Theory
Characterization of Multiple Prior Martingales
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
103/268
Theorem(St) is a multiple prior submartingale iff(St) is a Psubmartingale.
(St) is a multiple prior supermartingale iff there exists a PP suchthat(St) is a Psupermartingale.
(Mt) is a multiple prior martingale iff(Mt) is a Psubmartingale andfor some PPa Psupermartingale.
Remark
For multiple prior supermartingales: holds always true. needs
timeconsistency.
Uncertainty and Probability Multiple Prior Martingale Theory
Characterization of Multiple Prior Martingales
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
104/268
Theorem(St) is a multiple prior submartingale iff(St) is a Psubmartingale.
(St) is a multiple prior supermartingale iff there exists a PP suchthat(St) is a Psupermartingale.
(Mt) is a multiple prior martingale iff(Mt) is a Psubmartingale andfor some PPa Psupermartingale.
Remark
For multiple prior supermartingales: holds always true. needs
timeconsistency.
Uncertainty and Probability Multiple Prior Martingale Theory
Characterization of Multiple Prior Martingales
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
105/268
Theorem(St) is a multiple prior submartingale iff(St) is a Psubmartingale.
(St) is a multiple prior supermartingale iff there exists a PP suchthat(St) is a Psupermartingale.
(Mt) is a multiple prior martingale iff(Mt) is aP
submartingale andfor some PPa Psupermartingale.
Remark
For multiple prior supermartingales: holds always true. needs
timeconsistency.
Uncertainty and Probability Multiple Prior Martingale Theory
Doob Decomposition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
106/268
Theorem
Let(St) be a multiple prior supermartingale.Then there exists a multiple prior martingale M and a predictable,nondecreasing process A with A0 = 0such that S=M A. Such a
decomposition is unique.
Remark
Standard proof goes through.
Uncertainty and Probability Multiple Prior Martingale Theory
Doob Decomposition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
107/268
Theorem
Let(St) be a multiple prior supermartingale.Then there exists a multiple prior martingale M and a predictable,nondecreasing process A with A0 = 0such that S=M A. Such a
decomposition is unique.
Remark
Standard proof goes through.
Uncertainty and Probability Multiple Prior Martingale Theory
Optional Sampling Theorem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
108/268
Theorem
Let(St)0tTbe a multiple prior supermartingale. Let T bestopping times. Then
ess infPP EP
[S|F] S.
Remark
Not true without time consistency.
Uncertainty and Probability Multiple Prior Martingale Theory
Optional Sampling Theorem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
109/268
Theorem
Let(St)0tTbe a multiple prior supermartingale. Let T bestopping times. Then
ess infPP EP
[S|F] S.
Remark
Not true without time consistency.
Uncertainty and Probability Optimal Stopping Rules
Optimal Stopping under Ambiguity
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
110/268
With the concepts developed, one can proceed as in the classical case!Solution
Define the multiple priorSnell envelope Uvia backward induction:
UT =XT
Ut= max
Xt, ess inf
PPEP [Ut+1|Ft]
(t
8/13/2019 Mathemat i Kun Do Eko No Mie
111/268
With the concepts developed, one can proceed as in the classical case!Solution
Define the multiple priorSnell envelope Uvia backward induction:
UT =XT
Ut= max
Xt, ess inf
PPEP [Ut+1|Ft]
(t
8/13/2019 Mathemat i Kun Do Eko No Mie
112/268
With the concepts developed, one can proceed as in the classical case!Solution
Define the multiple priorSnell envelope Uvia backward induction:
UT =XT
Ut= max
Xt, ess inf
PPEP [Ut+1|Ft]
(t
8/13/2019 Mathemat i Kun Do Eko No Mie
113/268
With the concepts developed, one can proceed as in the classical case!
Solution
Define the multiple priorSnell envelope Uvia backward induction:
UT =XT
Ut= max
Xt, ess inf
PPEP [Ut+1|Ft]
(t
8/13/2019 Mathemat i Kun Do Eko No Mie
114/268
Question: what is the relation between the Snell envelopes U for fixedPPand the multiple prior Snell envelope U?
Theorem
U= ess infPP
UP .
Corollary
Under our assumptions, there exists a measure P P such thatU=UP
. The optimal stopping rule corresponds to the optimal stopping
rule under P.
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
115/268
Uncertainty and Probability Optimal Stopping Rules
Monotonicity and Stochastic Dominance
P
8/13/2019 Mathemat i Kun Do Eko No Mie
116/268
Suppose that (Yt) are iid under P P
andfor all PP
P[Yt x] P[Yt x] (x R)
and suppose that the payoffXt=g(t, Yt) for a function g that isisotone in y,
then P is for all optimal stopping problems(Xt) the worstcasemeasure,
i.e. the robust optimal stopping rule is the optimal stopping ruleunder P.
Uncertainty and Probability Optimal Stopping Rules
Monotonicity and Stochastic Dominance
P
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
117/268
Suppose that (Yt) are iid under P P
andfor all PP
P[Yt x] P[Yt x] (x R)
and suppose that the payoffXt=g(t, Yt) for a function g that isisotone in y,
then P is for all optimal stopping problems(Xt) the worstcasemeasure,
i.e. the robust optimal stopping rule is the optimal stopping ruleunder P.
Uncertainty and Probability Optimal Stopping Rules
Monotonicity and Stochastic Dominance
P
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
118/268
Suppose that (Yt) are iid under P P
andfor all PP
P[Yt x] P[Yt x] (x R)
and suppose that the payoffXt=g(t, Yt) for a function g that isisotone in y,
then P is for all optimal stopping problems(Xt) the worstcasemeasure,
i.e. the robust optimal stopping rule is the optimal stopping ruleunder P.
Uncertainty and Probability Optimal Stopping Rules
Monotonicity and Stochastic Dominance
P
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
119/268
Suppose that (Yt) are iid under P P
andfor all PP
P[Yt x] P[Yt x] (x R)
and suppose that the payoffXt=g(t, Yt) for a function g that isisotone in y,
then P is for all optimal stopping problems(Xt) the worstcasemeasure,
i.e. the robust optimal stopping rule is the optimal stopping ruleunder P.
Uncertainty and Probability Optimal Stopping Rules
Monotonicity and Stochastic Dominance
P
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
120/268
Suppose that (Yt) are iid under P andfor all PP
P[Yt x] P[Yt x] (x R)
and suppose that the payoffXt=g(t, Yt) for a function g that isisotone in y,
then P is for all optimal stopping problems(Xt) the worstcasemeasure,
i.e. the robust optimal stopping rule is the optimal stopping ruleunder P.
Uncertainty and Probability Optimal Stopping Rules
Easy Examples
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
121/268
Parking problem: choose the smallest pfor open lots
House sale: presume the least favorable distribution of bids in the
sense of firstorder stochastic dominanceAmerican Put: just presume the most positive possible drift
Uncertainty and Probability Continuous Time: gexpectations
Diffusion Models
based on Cheng, R., IMW Working Paper 429Framework now: Brownian motion Won a filtered probability space(,F, P , (Ft)) with the usual conditions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
122/268
0
Typical Example: Ambiguous Drift t() [, ]
P= {P :Wis Brownian motion with drift t() [, ]}
(for timeconsistency: stochastic drift important!)
worst case: either + or, depending on the stateLet EtX= minPPE
P[X|Ft]
we have the representation
EtX = Z
tdt+Z
tdW
t
for some predictable process Z
Knightian expectations solve a backward stochastic differentialequation
Uncertainty and Probability Continuous Time: gexpectations
Diffusion Models
based on Cheng, R., IMW Working Paper 429Framework now: Brownian motion Won a filtered probability space(,F, P , (Ft)) with the usual conditions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
123/268
0
Typical Example: Ambiguous Drift t() [, ]
P= {P :Wis Brownian motion with drift t() [, ]}
(for timeconsistency: stochastic drift important!)
worst case: either + or, depending on the stateLet EtX= minPPE
P[X|Ft]
we have the representation
EtX = Z
tdt+Z
tdW
t
for some predictable process Z
Knightian expectations solve a backward stochastic differentialequation
Uncertainty and Probability Continuous Time: gexpectations
Diffusion Models
based on Cheng, R., IMW Working Paper 429Framework now: Brownian motion Won a filtered probability space(,F, P0, (Ft)) with the usual conditions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
124/268
Typical Example: Ambiguous Drift t() [, ]
P= {P :Wis Brownian motion with drift t() [, ]}
(for timeconsistency: stochastic drift important!)
worst case: either + or, depending on the stateLet EtX= minPPE
P[X|Ft]
we have the representation
EtX = Ztdt+ZtdWt
for some predictable process Z
Knightian expectations solve a backward stochastic differentialequation
Uncertainty and Probability Continuous Time: gexpectations
Diffusion Models
based on Cheng, R., IMW Working Paper 429Framework now: Brownian motion Won a filtered probability space(,F, P0, (Ft)) with the usual conditions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
125/268
Typical Example: Ambiguous Drift t() [, ]
P= {P :Wis Brownian motion with drift t() [, ]}
(for timeconsistency: stochastic drift important!)
worst case: either + or, depending on the stateLet EtX= minPPE
P[X|Ft]
we have the representation
EtX = Ztdt+ZtdWt
for some predictable process Z
Knightian expectations solve a backward stochastic differentialequation
Uncertainty and Probability Continuous Time: gexpectations
Diffusion Models
based on Cheng, R., IMW Working Paper 429Framework now: Brownian motion Won a filtered probability space(,F, P0, (Ft)) with the usual conditions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
126/268
Typical Example: Ambiguous Drift t() [, ]
P= {P :Wis Brownian motion with drift t() [, ]}
(for timeconsistency: stochastic drift important!)
worst case: either + or, depending on the stateLet EtX= minPPE
P[X|Ft]
we have the representation
EtX = Ztdt+ZtdWt
for some predictable process Z
Knightian expectations solve a backward stochastic differentialequation
Uncertainty and Probability Continuous Time: gexpectations
Diffusion Models
based on Cheng, R., IMW Working Paper 429Framework now: Brownian motion Won a filtered probability space(,F, P0, (Ft)) with the usual conditions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
127/268
Typical Example: Ambiguous Drift t() [, ]
P= {P :Wis Brownian motion with drift t() [, ]}
(for timeconsistency: stochastic drift important!)
worst case: either + or, depending on the stateLet EtX= minPPE
P[X|Ft]
we have the representation
EtX = Ztdt+ZtdWt
for some predictable process Z
Knightian expectations solve a backward stochastic differentialequation
Uncertainty and Probability Continuous Time: gexpectations
Diffusion Models
based on Cheng, R., IMW Working Paper 429Framework now: Brownian motion Won a filtered probability space(,F, P0, (Ft)) with the usual conditions
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
128/268
Typical Example: Ambiguous Drift t() [, ]
P= {P :Wis Brownian motion with drift t() [, ]}
(for timeconsistency: stochastic drift important!)
worst case: either + or, depending on the stateLet EtX= minPPE
P[X|Ft]
we have the representation
EtX = Ztdt+ZtdWt
for some predictable process Z
Knightian expectations solve a backward stochastic differentialequation
Uncertainty and Probability Continuous Time: gexpectations
Timeconsistent multiple priors are gexpectations
gexpectations
F
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
129/268
Conditional gexpectation of an FTmeasurable random variable Xat time t is Et(X) :=Yt
where (Y, Z) solves the backward stochastic differential equation
YT =X, ,dYt=g(t, Yt, Zt) ZtdWt
the probability theory for gexpectations has been mainly developedbyShige Peng
Theorem (Delbaen, Peng, Rosazza Giannin)
All timeconsistent multiple prior expectations are gexpectations.
Uncertainty and Probability Continuous Time: gexpectations
Timeconsistent multiple priors are gexpectations
gexpectations
C di i l i f F bl d i bl X
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
130/268
Conditional gexpectation of an FTmeasurable random variable Xat time t is Et(X) :=Yt
where (Y, Z) solves the backward stochastic differential equation
YT =X, ,dYt=g(t, Yt, Zt) ZtdWt
the probability theory for gexpectations has been mainly developedbyShige Peng
Theorem (Delbaen, Peng, Rosazza Giannin)
All timeconsistent multiple prior expectations are gexpectations.
Uncertainty and Probability Continuous Time: gexpectations
Timeconsistent multiple priors are gexpectations
gexpectations
C di i l i f F bl d i bl X
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
131/268
Conditional gexpectation of an FTmeasurable random variable Xat time t is Et(X) :=Yt
where (Y, Z) solves the backward stochastic differential equation
YT =X, ,dYt=g(t, Yt, Zt) ZtdWt
the probability theory for gexpectations has been mainly developedbyShige Peng
Theorem (Delbaen, Peng, Rosazza Giannin)
All timeconsistent multiple prior expectations are gexpectations.
Uncertainty and Probability Continuous Time: gexpectations
Timeconsistent multiple priors are gexpectations
gexpectations
C diti l t ti f F bl d i bl X
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
132/268
Conditional gexpectation of an FTmeasurable random variable Xat time t is Et(X) :=Yt
where (Y, Z) solves the backward stochastic differential equation
YT =X, ,dYt=g(t, Yt, Zt) ZtdWt
the probability theory for gexpectations has been mainly developedbyShige Peng
Theorem (Delbaen, Peng, Rosazza Giannin)
All timeconsistent multiple prior expectations are gexpectations.
Uncertainty and Probability Continuous Time: gexpectations
Timeconsistent multiple priors are gexpectations
gexpectations
Co ditio al g e ectatio of a F eas able a do a iable X
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
133/268
Conditional gexpectation of an FTmeasurable random variable Xat time t is Et(X) :=Yt
where (Y, Z) solves the backward stochastic differential equation
YT =X, ,dYt=g(t, Yt, Zt) ZtdWt
the probability theory for gexpectations has been mainly developedbyShige Peng
Theorem (Delbaen, Peng, Rosazza Giannin)
All timeconsistent multiple prior expectations are gexpectations.
Uncertainty and Probability Continuous Time: gexpectations
Timeconsistent multiple priors are gexpectations
gexpectations
Conditional g expectation of an F measurable random variable X
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
134/268
Conditional gexpectation of an FTmeasurable random variable Xat time t is Et(X) :=Yt
where (Y, Z) solves the backward stochastic differential equation
YT =X, ,dYt=g(t, Yt, Zt) ZtdWt
the probability theory for gexpectations has been mainly developedbyShige Peng
Theorem (Delbaen, Peng, Rosazza Giannin)
All timeconsistent multiple prior expectations are gexpectations.
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
Optimal Stopping under gexpectations: Theory
Our Problem recall
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
135/268
Our Problem - recall
Let (Xt) be continuous, adapted, nonnegative process withsuptT |Xt| L
2 (P0).Let g=g(, t, z) be a standard concave driver (in particular,
Lipschitzcontinuous).Find a stopping time T that maximizes
E0(X) .
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
Optimal Stopping under gexpectations: General Structure
LetVt= ess sup
t
Et(X) .
be the value function of our problem
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
136/268
be the value function of our problem.
Theorem
1 (Vt) is the smallest rightcontinuous gsupermartingale dominating(Xt);
2 = inf{t 0 :Vt=Xt}is an optimal stopping time;3 the value function stopped at, (Vt) is a gmartingale.
Proof.
Our proof uses the properties ofgexpectations like regularity,timeconsistency, Fatou, etc. to mimic directly the classical proof (as, e.g.,inPeskir,Shiryaev) with one additional topping: rightcontinous versions ofgsupermartingales
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
Optimal Stopping under gexpectations: General Structure
LetVt= ess sup
t
Et(X) .
be the value function of our problem.
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
137/268
be the value function of our problem.
Theorem
1 (Vt) is the smallest rightcontinuous gsupermartingale dominating(Xt);
2 = inf{t 0 :Vt=Xt}is an optimal stopping time;3 the value function stopped at, (Vt) is a gmartingale.
Proof.
Our proof uses the properties ofgexpectations like regularity,timeconsistency, Fatou, etc. to mimic directly the classical proof (as, e.g.,inPeskir,Shiryaev) with one additional topping: rightcontinous versions ofgsupermartingales
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
Optimal Stopping under gexpectations: General Structure
LetVt= ess sup
t
Et(X) .
be the value function of our problem.
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
138/268
be the value function of our problem.
Theorem
1 (Vt) is the smallest rightcontinuous gsupermartingale dominating(Xt);
2 = inf{t 0 :Vt=Xt}is an optimal stopping time;3 the value function stopped at, (Vt) is a gmartingale.
Proof.
Our proof uses the properties ofgexpectations like regularity,timeconsistency, Fatou, etc. to mimic directly the classical proof (as, e.g.,inPeskir,Shiryaev) with one additional topping: rightcontinous versions ofgsupermartingales
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
Optimal Stopping under gexpectations: General Structure
LetVt= ess sup
t
Et(X) .
be the value function of our problem.
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
139/268
p
Theorem
1 (Vt) is the smallest rightcontinuous gsupermartingale dominating(Xt);
2 = inf{t 0 :Vt=Xt}is an optimal stopping time;3 the value function stopped at, (Vt) is a gmartingale.
Proof.
Our proof uses the properties ofgexpectations like regularity,timeconsistency, Fatou, etc. to mimic directly the classical proof (as, e.g.,inPeskir,Shiryaev) with one additional topping: rightcontinous versions ofgsupermartingales
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
Optimal Stopping under gexpectations: General Structure
LetVt= ess sup
t
Et(X) .
be the value function of our problem.
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
140/268
p
Theorem
1 (Vt) is the smallest rightcontinuous gsupermartingale dominating(Xt);
2 = inf{t 0 :Vt=Xt}is an optimal stopping time;3 the value function stopped at, (Vt) is a gmartingale.
Proof.
Our proof uses the properties ofgexpectations like regularity,timeconsistency, Fatou, etc. to mimic directly the classical proof (as, e.g.,inPeskir,Shiryaev) with one additional topping: rightcontinous versions ofgsupermartingales
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
WorstCase Priors
Drift ambiguity
V is a gsupermartingale
from the DoobMeyerPeng decomposition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
141/268
y g p
dVt=g(t, Zt)dt ZtdWt+dAt
for some increasing process A
= |Zt|dt ZtdWt+dAt
Girsanov: = ZtdWt +dAtwith kernel sgn(Zt)
Theorem (Duality for ambiguity)
There exists a probability measure P
P
such thatVt= ess suptEt(X) = ess suptE [X|Ft]. In particular:
max
minPP
EP [X|Ft] = minPP
max
EP [X|Ft]
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
WorstCase Priors
Drift ambiguity
V is a gsupermartingale
from the DoobMeyerPeng decomposition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
142/268
y g p
dVt=g(t, Zt)dt ZtdWt+dAt
for some increasing process A
= |Zt|dt ZtdWt+dAt
Girsanov: = ZtdWt +dAtwith kernel sgn(Zt)
Theorem (Duality for ambiguity)
There exists a probability measure P
P
such thatVt= ess suptEt(X) = ess suptE [X|Ft]. In particular:
max
minPP
EP [X|Ft] = minPP
max
EP [X|Ft]
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
WorstCase Priors
Drift ambiguity
V is a gsupermartingale
from the DoobMeyerPeng decomposition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
143/268
dVt=g(t, Zt)dt ZtdWt+dAt
for some increasing process A
= |Zt|dt ZtdWt+dAt
Girsanov: = ZtdWt +dAtwith kernel sgn(Zt)
Theorem (Duality for ambiguity)
There exists a probability measure P
P
such thatVt= ess suptEt(X) = ess suptE [X|Ft]. In particular:
max
minPP
EP [X|Ft] = minPP
max
EP [X|Ft]
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
WorstCase Priors
Drift ambiguity
V is a gsupermartingale
from the DoobMeyerPeng decomposition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
144/268
dVt=g(t, Zt)dt ZtdWt+dAt
for some increasing process A
= |Zt|dt ZtdWt+dAt
Girsanov: = ZtdWt +dAtwith kernel sgn(Zt)
Theorem (Duality for ambiguity)
There exists a probability measure P
P
such thatVt= ess suptEt(X) = ess suptE [X|Ft]. In particular:
max
minPP
EP [X|Ft] = minPP
max
EP [X|Ft]
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
WorstCase Priors
Drift ambiguity
V is a gsupermartingale
from the DoobMeyerPeng decomposition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
145/268
dVt=g(t, Zt)dt ZtdWt+dAt
for some increasing process A
= |Zt|dt ZtdWt+dAt
Girsanov: = ZtdWt +dAtwith kernel sgn(Zt)
Theorem (Duality for ambiguity)
There exists a probability measure P
P
such thatVt= ess suptEt(X) = ess suptE [X|Ft]. In particular:
max
minPP
EP [X|Ft] = minPP
max
EP [X|Ft]
Uncertainty and Probability Optimal Stopping under gexpectations: Theory
WorstCase Priors
Drift ambiguity
V is a gsupermartingale
from the DoobMeyerPeng decomposition
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
146/268
dVt=g(t, Zt)dt ZtdWt+dAt
for some increasing process A
= |Zt|dt ZtdWt+dAtGirsanov: = ZtdW
t +dAtwith kernel sgn(Zt)
Theorem (Duality for ambiguity)
There exists a probability measure P
P
such thatVt= ess suptEt(X) = ess suptE [X|Ft]. In particular:
max
minPP
EP [X|Ft] = minPP
max
EP [X|Ft]
Uncertainty and Probability PDE Approach: Modified HamiltonJacobiBellman Equation
Markov Models
the state variable Ssolves a forward SDE, e.g.
dSt=(St)dt+ (St)dWt, S0 = 1 .
Let
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
147/268
Let
L =(x)
x + 2(x)
2
x2
be the infinitesimal generator ofS.
By Itos formula, v(t, St) is a martingale if
vt(t, x) + Lv(t, x) = 0 (1)
similarly,v(t, St) is a gmartingale if
vt(t, x) + Lv(t, x) +g(t, vx(t, x)(x))= 0 (2)
Uncertainty and Probability PDE Approach: Modified HamiltonJacobiBellman Equation
Markov Models
the state variable Ssolves a forward SDE, e.g.
dSt=(St)dt+ (St)dWt, S0 = 1 .
Let
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
148/268
Let
L =(x)
x + 2(x)
2
x2
be the infinitesimal generator ofS.
By Itos formula, v(t, St) is a martingale if
vt(t, x) + Lv(t, x) = 0 (1)
similarly,v(t, St) is a gmartingale if
vt(t, x) + Lv(t, x) +g(t, vx(t, x)(x))= 0 (2)
Uncertainty and Probability PDE Approach: Modified HamiltonJacobiBellman Equation
Markov Models
the state variable Ssolves a forward SDE, e.g.
dSt=(St)dt+ (St)dWt, S0 = 1 .
Let
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
149/268
Let
L =(x)
x + 2(x)
2
x2
be the infinitesimal generator ofS.
By Itos formula, v(t, St) is a martingale if
vt(t, x) + Lv(t, x) = 0 (1)
similarly,v(t, St) is a gmartingale if
vt(t, x) + Lv(t, x) +g(t, vx(t, x)(x))= 0 (2)
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
150/268
Uncertainty and Probability PDE Approach: Modified HamiltonJacobiBellman Equation
PDE Approach: A Modified HJB Equation
Theorem (Verification Theorem)
Let v be a viscosity solution of the gHJB equation
8/13/2019 Mathemat i Kun Do Eko No Mie
151/268
y g q
max {f(x) v(t, x), vt(t, x) + Lv(t, x) +g(t, vx(t, x)(x))} = 0 . (3)
Then Vt=v(t, St).
nonlinearity only in the firstorder term
numeric analysis feasible
ambiguity introduces an additional nonlinear drift term
Uncertainty and Probability PDE Approach: Modified HamiltonJacobiBellman Equation
PDE Approach: A Modified HJB Equation
Theorem (Verification Theorem)
Let v be a viscosity solution of the gHJB equation
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
152/268
max {f(x) v(t, x), vt(t, x) + Lv(t, x) +g(t, vx(t, x)(x))} = 0 . (3)
Then Vt=v(t, St).
nonlinearity only in the firstorder term
numeric analysis feasible
ambiguity introduces an additional nonlinear drift term
Uncertainty and Probability PDE Approach: Modified HamiltonJacobiBellman Equation
PDE Approach: A Modified HJB Equation
Theorem (Verification Theorem)
Let v be a viscosity solution of the gHJB equation
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
153/268
max {f(x) v(t, x), vt(t, x) + Lv(t, x) +g(t, vx(t, x)(x))} = 0 . (3)
Then Vt=v(t, St).
nonlinearity only in the firstorder term
numeric analysis feasible
ambiguity introduces an additional nonlinear drift term
Uncertainty and Probability PDE Approach: Modified HamiltonJacobiBellman Equation
PDE Approach: A Modified HJB Equation
Theorem (Verification Theorem)
Let v be a viscosity solution of the gHJB equation
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
154/268
max {f(x) v(t, x), vt(t, x) + Lv(t, x) +g(t, vx(t, x)(x))} = 0 . (3)
Then Vt=v(t, St).
nonlinearity only in the firstorder term
numeric analysis feasible
ambiguity introduces an additional nonlinear drift term
Uncertainty and Probability Examples
More general problems
With monotonicity and stochastic dominance, worstcase prior easy
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
155/268
y p yto identify
In general, the worstcase prior is pathdependenteven in iid settings
Barrier OptionsShout Options
Secretary Problem
Uncertainty and Probability Examples
More general problems
With monotonicity and stochastic dominance, worstcase prior easy
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
156/268
to identify
In general, the worstcase prior is pathdependenteven in iid settings
Barrier OptionsShout Options
Secretary Problem
Uncertainty and Probability Examples
More general problems
With monotonicity and stochastic dominance, worstcase prior easy
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
157/268
to identify
In general, the worstcase prior is pathdependenteven in iid settings
Barrier OptionsShout Options
Secretary Problem
Uncertainty and Probability Examples
More general problems
With monotonicity and stochastic dominance, worstcase prior easy
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
158/268
to identify
In general, the worstcase prior is pathdependenteven in iid settings
Barrier OptionsShout Options
Secretary Problem
Uncertainty and Probability Examples
More general problems
With monotonicity and stochastic dominance, worstcase prior easy
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
159/268
to identify
In general, the worstcase prior is pathdependenteven in iid settings
Barrier OptionsShout Options
Secretary Problem
Uncertainty and Probability Secretary Problem
Ambiguous secretaries (with Tatjana Chudjakow)
based on Chudjakow, R., IMW Working Paper 413
Call applicant j a candidate if she is better than all predecessors
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
160/268
Call applicant j a candidateif she is better than all predecessors
We are interested in Xj=Prob[jbest|jcandidate]
Here, the payoffXjis ambiguous assume that this conditional
probability is minimal
If you compare this probability with the probability that latercandidates are best, you presume the maximalprobability for them!
Uncertainty and Probability Secretary Problem
Ambiguous secretaries (with Tatjana Chudjakow)
based on Chudjakow, R., IMW Working Paper 413
Call applicant j a candidate if she is better than all predecessors
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
161/268
Call applicant j a candidateif she is better than all predecessors
We are interested in Xj=Prob[jbest|jcandidate]
Here, the payoffXjis ambiguous assume that this conditional
probability is minimal
If you compare this probability with the probability that latercandidates are best, you presume the maximalprobability for them!
Uncertainty and Probability Secretary Problem
Ambiguous secretaries (with Tatjana Chudjakow)
based on Chudjakow, R., IMW Working Paper 413
Call applicant j a candidate if she is better than all predecessors
http://find/8/13/2019 Mathemat i Kun Do Eko No Mie
162/268
Call applicant j a candidateif she is better than all prede