Intrinsic stochastic differential equations as jetsdbrigo/201610Pisajets.pdf · AgendaI 1...

Intrinsic stochastic differential equations as jets

Theory and applicationsScuola Normale Superiore, Pisa, October 10, 2016

Damiano BrigoDept. of Mathematics, Imperial College, London

wwwf.imperial.ac.uk/∼dbrigo/—

Joint work with John ArmstrongDept. of Mathematics, King’s College, London

—Full paper in http://arxiv.org/abs/1602.03931

I would like to dedicate this talk toGiovanni Battista Di Masi (1944-2016),

who passed away on April 4.

PhD at Brown University, Author in signalprocessing, stochastic control & filtering,probability, stochastic analysis, statistics.

Professor of Probability & Mathematical Statistics, later Head of theDepartment of Mathematics at the University of Padua and Assessorat the Padua Local Administration

Gianni was my Laurea dissertation supervisor (1990) and he waspresent at my PhD viva in Amsterdam. He taught me stochasticcalculus, nonlinear filtering, and much more beyond mere science.

Agenda I1 Stochastic Differential Equations: a simple example2 The traditional view of SDEs: Ito and Stratonovich

Randomness driver: Brownian motionSDEs and stochastic integralsProbability and geometry

3 Ito SDEs on manifolds: 2-JetsDrawing and simulating SDEs as “fields of curves”Coordinate-free converging difference scheme as SDECoordinate free Ito SDE as 2-jet scheme limitCoordinate-free Ito formula and stochastic analysisThe case of vector Brownian motion as driver

4 Applications: Optimal approximation of SDEs on submanifoldsStratonovich projectionIto-vector projectionIto jet projection

J. Armstrong (KCL) & D. Brigo (ICL) Intrinsic SDEs as jets SNS Colloquium 3

Agenda II5 Signal processing: from∞-dimensional SPDE to Rn SDE

6 Interest rate dynamics and term structure parameterization

7 Conclusions and References


Stochastic Differential Equations: a simple example

Stochastic vs deterministic differential equations

Randomness in motion: ExamplesThe future evolution of a financial asset, spacecraft re-entry trajectory,population, submarine probe position, tides levels...

A Stochastic Differential Eq. (SDE) looks like this (r = 5% growth rate):

dX (t)︸︷︷︸ = r X (t)︸︷︷︸ dt + σ X (t)︸︷︷︸ dWt︸︷︷︸Change in X function of X Amplitude New

between t and t + dt in t , coefficient random”MEAN of shock shock

CHANGE” (Volatility)

Let us suppose this is the future price of an asset with return 5% andsee how this varies with σ (or a future popolation toy model)



SDE: dXt = 0.05Xtdt + 0.1XtdWt , X0 = 100,ODE dXt = 0.05Xtdt

































SDE: dXt = 0.05Xtdt + 0.1XtdWt , X0 = 100,ODE dXt = 0.05Xtdt . Randomness, Dynamics & Prob



dXt = 0.05Xtdt + σXtdWt , X0 = 100:σ = 0.1 vs σ = 0.04


The traditional view of SDEs: Ito and Stratonovich Randomness driver: Brownian motion

Brownian Motion as the randomness driver

dXt︸︷︷︸ = a(Xt )︸︷︷︸ dt + b(Xt )︸︷︷︸ dWt︸︷︷︸Change in X ”MEAN Amplitude of Random

between t and t + dt CHANGE” random shock shock

W is Brownian motion or Wiener process

Independent stationary increments, Wt+∆1t −Wt indep of Wt −Wt−∆2t ,continous paths, W0 = 0. This implies Gaussian ∆Wt ∼ N (0,∆t).

These properties can coexist but W ’s paths have unbounded variation- rough paths - nowhere differentiable. So what does dW really mean?

Quadratic variation (nested dyadic grids) 0 = tn0 < tn

1 < . . . < tnn = T ,

limn

n−1∑i=0

(Wtni+1−Wtn

i)2 = T , or “dWtdWt = dt ′′ (“dt dWt = 0, dt dt = 0′′)


The traditional view of SDEs: Ito and Stratonovich SDEs and stochastic integrals

Classic theory of Stochastic Differential Equations

dXt = a(Xt )dt + b(Xt )dWt , X0. dW not a real differential. So?

Write it as

Xt = X0 +

∫ t

0a(Xs)ds +

∫ t

0b(Xs)dWs.

Now the matter is defining the stochastic integral driven by dWSince W has unbounded variation, we cannot define this as anordinary Stiltjes integral on the paths.



The stochastic integral as a Stiltjes integral?

In a Stiltjes integral one has∫ T0 b(Xs)dWs =

= limn

n∑i=1

b(X (ti))(Wti+1−Wti )

for ANY choice ti ∈ [ti , ti+1).

However, for Brownian mo-tion this does not work sinceW has unbounded variation.

Add an extra specification:we need to explicitly decidewhich point ti is considered.

In a standard Stiltjes integral one has that the following limit converges∫ T

0σ(Xs)dWs = lim

n

n∑i=1

σ(X (ti))(Wti+1 −Wti )

for ANY possible choice of ti ∈ [ti , ti+1).

However, for Brownian motion this does not work since W hasunbounded variation and is not differentiable.

It turns out that one can still define the stochastic integral in a Stiltjesway adding an extra specification: we need to explicitly decide at whichpoint ti in each limit interval [ti , ti+1) the integrand σ(Xt ) is evaluated.



Xt = X0 +∫ t

0 a(Xs)ds +

∫ t

0b(Xs)dWs ??

Traditionally, 2 main definitions of stochastic integrals withL2(P)-convergence: Initial point ti = ti vs mid point ti =

ti +ti+12∫ T

0b(Xs)dWs = lim

n

n∑i=1

b(X (ti))(Wti+1 −Wti ) (Ito)

∫ T

0b(Xs)◦dWs = lim

n

n∑i=1

b(

X(

ti + ti+1

2

))(Wti+1−Wti )(Stratonovich)

(Str more general def. has [b(X (ti)) + b(X (ti+1))]/2 in front of dW )where it is understood that as n tends to infinity the mesh size ofthe partition {[0, t1), [t1, t2), . . . , [tn−1, tn = T ]} of [0,T ] tends to 0.Stratonovich integral looks into the future, Ito does not.



Battle of the integrals: Ito or Stratonovich?

Ito (-Doeblin) integral:The good: “Does not look into the future” (social sciences)Ito integral is martingale: split in “local mean” & “volatility” aboveGood probabilistically, many important results in probability theory.The bad: due to dWdW = dt 6= 0 does not satisfy chain rule!

dXt = a(Xt )dt + b(Xt )dWt ,

Ito’s formula: df (Xt ) = ((∇f )(Xt ))T dXt +12

(dXt )T (Hf (Xt ))(dXt )

What does it mean as a change of coordinates/variables?The ugly: Given finite variation noises W n →W a.s. uniformly int-bounded intervals, solutions in dW n do not converge to Ito SDEsol. Bad for engineering / physical systems with external noise



Battle of the integrals: Ito or Stratonovich?

Fisk ([12])-Stratonovich ([29]) (-McShane [23]) Integral.The good: satisfies chain rule (same as ordinary differentialeq./vector fields), good for basic geometry

dXt = a(Xt )dt + b(Xt ) ◦ dWt , df (Xt ) = ((∇f )(Xt ))T ◦ dXt

E.g. the above SDE dX stays in a manifold M if a(X ) & b(X ) arein the tangent space of the manifold. If not project on tangentspace and you have approximated original SDE with SDE on M.Now if W n →W , the solution uder W n converges to theStratonovich SDE solution (Wong Zakai)The bad: Looks into the future.The ugly: Cannot interpret SDE terms as local mean and volatility(no martingale property). Not good probabilistically.


The traditional view of SDEs: Ito and Stratonovich Probability and geometry

In a nutshell: Ito ok probabilistically, Stratonovich geometrically.This talk: Let’s make Ito SDEs good for geometry too.Natural question. Ito-Stratonovich Transformation: given ItoSDE, by suitably changing a(Xt ) one obtains a Strat. SDE with thesame solution X . Why not use that back & forth?Because e.g. optimality of projection on submanifolds fordimension reduction depends on choice of calculus (later)History: Ito integral in the 40’s-50’s. Ito dominates amongmathematicians, except for geometry. Stratonovich fared betterwith physicists & engineers, due to Wong Zakai & symmetry.Difficult infancy for symmetric integral. Donald Fisk paper rejectedby Annals in mid 60’s. In 1967 Skorokhod (1930-2011) [28]reviewed Stratonovich’s 1966 book quite critically (euphemism).We now introduce Ito calculus on manifolds using jets. Previousapproaches: Schwartz morphism [11] & Ito bundle [15].


Ito SDEs on manifolds: 2-Jets Drawing and simulating SDEs as “fields of curves”

Jets and SDEs

For all x ∈ Rn consider smooth curve γx : R→ Rn, γx (0) = x

Example: γEx on R2 as follows

(zero 3d-on derivatives):

γE(x1,x2)(t) = (x1, x2)+ t(−x2, x1)︸︷︷︸

circular counterclock

+ 3t2(x1, x2)︸︷︷︸radially outward

-2 -1 0 1 2 3

-2

-1

0

1

2

3



Jets and SDEs

Given such a γ, a starting X0(X0 = (1,0) in our example), a Wt& time step δt define discrete timestochastic process:

X0 := x0, Xt+δt := γEXt

(Wt+δt −Wt )

We have connected the pointsusing the curves in γE

Xt: follow

s 7→ γXt (s) from s = 0 tos = N (0, δt), all N indepedent



Jets and SDEs

δt = 0.2× 2−5 δt = 0.2× 2−7

δt = 0.2× 2−9 δt = 0.2× 2−11

Figure: Discrete time trajectories for γE for a fixed Wt and X0 with differentvalues for δt


Ito SDEs on manifolds: 2-Jets Coordinate-free converging difference scheme as SDE

Jets and SDEs

Xt+δt := γXt (δWt ) reads:

“Follow the curve γ starting from X for a parameter increment ofδWt := Wt+δt −Wt ”. In this description,

Not using the Rn vector space structure. Intrinsic.

These discrete time stochastic processes converge in some sense to alimit as the time step tends to zero for γ such as γE with sufficientlygood regularity. Write the limit equation as

Coordinate free SDE: Xt γXt (dWt ), X0 = x0. (1)

How can the scheme limit be made precise and how does it relate toclassic stochastic calculus?


Ito SDEs on manifolds: 2-Jets Coordinate-free converging difference scheme as SDE

Jets and SDEs

In a coordinate system, consider the Taylor expansion of γx .

γx (t) = x + γ′x (0)t +12γ′′x (0)t2 + Rx t3, Rx =

16γ′′′x (ξ), ξ ∈ [0, t ],

where Rx t3 is the remainder term in Lagrange form. Substituting thisTaylor expansion in our scheme Xt+δt = γXt (Wt+δt −Wt ) we obtain

δXt = γ′Xt(0)δWt +

12γ′′Xt

(0)(δWt )2 + RXt (δWt )

3, X0 = x0. (2)

Properties of Brownian motion such as “(dW )2 = dt” and “(dW )3 = 0”suggest we replace (δWt )

2 with δt and (δWt )3 with 0. We obtain:

δXt = γ′Xt(0)︸︷︷︸

=:b(Xt )

δWt +12γ′′Xt

(0)︸︷︷︸=:a(Xt )

δt , X0 = x0.


Ito SDEs on manifolds: 2-Jets Coordinate free Ito SDE as 2-jet scheme limit

Jets and SDEs

δXt = a(Xt )δt + b(Xt )δWt . (3)

This is the Euler scheme & under suitable assumptions converges inL2(P) to the solution to the Ito stochastic differential equation:

dXt = a(Xt ) dt + b(Xt )dWt , X0 = x0. (4)

More precisely, assume in the given coordinate system γx (t) issmoothly varying in x with first & second t derivatives at 0 satisfyingLipschitz conditions in x . Assume that the third t-derivative at t = 0 isuniformly bounded in x . Theorem: (Armstrong & B. 2016). Thefollowing 3 schemes have as same L2(P) limit the classic Ito SDE X .

Coordinate free γx scheme: Xt+δt := γXt (Wt+δt −Wt ), X0

2-jet scheme: δXt = γ′Xt

(0)δWt + 12γ′′Xt

(0)(δWt )2, X0

The classic Euler scheme: δXt = γ′Xt(0)δWt + 1

2γ′′Xt

(0)δt , X0


Ito SDEs on manifolds: 2-Jets Coordinate free Ito SDE as 2-jet scheme limit

Jets and SDEs

Are the 2-jet scheme and its limit coordinate free?

δXt = b(Xt )δWt + a(Xt )(δWt )2, x0 → dXt = a(Xt )︸︷︷︸

12γ′′Xt

(0)

dt + b(Xt )︸︷︷︸γ′

Xt(0)

dWt , x0

Coefficients a & b of Ito SDE only depend on first two derivatives of γ.

Curves γ1 ∼ γ2 have the same k -jet if their Taylor expansions areequal up to order tk in one (all) coordinate system. k -jet can bedefined as equivalence class j2(γ1) := γ1.

Given our convergence results, showing that the limit of our schemedepends only on the two-jet, we may rewrite Xt γXt (dWt ), X0 as:

Coordinate-free 2-jet SDE: Xt j2(γXt )(dWt ), X0 = x0. (5)


Ito SDEs on manifolds: 2-Jets Coordinate-free Ito formula and stochastic analysis

Ito’s formula via 2-jetsLemma (Ito’s lemma — coordinate free formulation)

If the process Xt satisfies Xt j2(γXt )(dWt )then f (Xt ) satisfies f (X )t j2(f ◦ γXt )(dWt ).

Ito’s formula: the transformation rule for jets under a change ofcoordinates is the composition of functions.

We have illustrated a way of drawing an SDE on a rubber sheet suchthat if sheet is stretched, diagram transforms as per Ito’s lemma.

Or: the following diagram commutes



Ito’s formula via 2-jets

Since we now understand the geometric content of Ito’s lemma, wecan draw a picture to illustrate it. Consider the transformation

(θ, s) = φ(x1, x2) =

(arctan(x2/x1), log(

√x2

1 + x22 )

)(φ(z) = i log(z))

applied to our γE (left) process.

1st apply φ to each point (stretchthe rubber sheet).

d(θ, s) =

(0,

72

)dt + (1,0) dWt .



Ito’s formula via 2-jets

The process j2(φ ◦ γE ) plotted using image manipulation software

The process j2(φ ◦ γE ) plotted by applying Ito’s lemma

Figure: Two plots of the process j2(φ ◦ γE ) in the plane (θ, s).J. Armstrong (KCL) & D. Brigo (ICL) Intrinsic SDEs as jets SNS Colloquium 35

Ito SDEs on manifolds: 2-Jets The case of vector Brownian motion as driver

Generalizations and other results

Can generalize to SDE driven by vector-Brownian motion usingjets driven by Rm parameters.Can give jet-based definition of backward and fwd diffusionoperatorsOnly part of the jet information is used for SDE; weak and strongequivalence of SDEs.Fan diagrams and Stratonovich drift a(X ) as median.Ito - Stratonovich transformation interpreted geometrically asfollows: a 2-jet (Ito) can be equivalently represented bysubsequent application of two vector flows (Stratonovich).

And now... applications!


Applications: Optimal approximation of SDEs on submanifolds

Optimal approximation of SDEs on submanifolds

dX = a(X )dt+bα(X )dWα in Rr , r > n, dY = A(Y )dt+Bα(Y )dWα in Rn

X0 = φ(Y0) ∈ M, n-dimensional manifold;We wish to approximate X with φ(Y ) on M.


Applications: Optimal approximation of SDEs on submanifolds Stratonovich projection

Stratonovich projection. Write the Ito SDE for X in Stratonovich form

dX = a dt + bα ◦ dWα in Rr , X0 ∈ M

Apply the tangent space projection to obtain SDE on M for Z = φ(Y ),

dZ = ΠZ [a] dt + ΠZ [bα] ◦ dWα in M, Z0 = X0 ∈ M

Justification: for b = 0 it coincides with optimal ODE projectionminimizing leading term of Taylor expansion for |φ(Y )− X |2.

No obvious optimality for SDE as a whole, rough paths & a,b togetherJ. Armstrong (KCL) & D. Brigo (ICL) Intrinsic SDEs as jets SNS Colloquium 38

Applications: Optimal approximation of SDEs on submanifolds Stratonovich projection

Stratonovich projection via tangent space projection Π


Applications: Optimal approximation of SDEs on submanifolds Ito-vector projection

Ito vector projection. Minimize leading term Ito-Taylor expansions

E[|Xt − φ(Yt )|2], |E[Xt − φ(Yt )]|2 for small t

This is achieved up to order 1 in t by choosing Y SDE with

Bα(Yt , t) = (ψ∗)φ(Yt )Πφ(Yt )bα(·, t)

A(Yt , t) = (ψ∗)φ(Yt )Πφ(Yt )

(a(·, t)− 1

2(∇Bα(Yt ,t)φ∗)Bβ(Yt , t)g

αβE

).


Applications: Optimal approximation of SDEs on submanifolds Ito-vector projection

Metric projection π vs tangent space projection Π


Applications: Optimal approximation of SDEs on submanifolds Ito jet projection

Now π is the metric projection π : Rr → M, of which the earlier linearprojection Π is the first order component. π = ψ ◦ π.

Ito jet projection. Minimize leading term Ito-Taylor expansions

E[dM(π(Xt ), φ(Yt ))2], for small t

This is achieved up to order 2 in t with Y SDE: B as before and A

A(Yt , t) = π∗(a(φ(Yt ), t)) +12

(∇bα(φ(Yt ),t)π∗)bβ(φ(Yt ), t)g

αβE .


Applications: Optimal approximation of SDEs on submanifolds Ito jet projection

SDE Xt j2(γXt (dWt )) has Ito jet projection Zt j2(π ◦ γZt (dWt ))

Best probabilistic (mean square) optimality of 3 projectionsJ. Armstrong (KCL) & D. Brigo (ICL) Intrinsic SDEs as jets SNS Colloquium 43

Signal processing: from∞-dimensional SPDE to Rn SDE

Filtering problem (e.g. Apollo 11)

Signal and observation equations:

dSt = µ(St , t)dt + σ(St , t)dBt , dRt = h(St , t)dt + εdVt

where B,V are independent Brownian motions (noises).

Given observations R from 0 to t , estimate St . Full solution:pt (ξ)dξ = P{St ∈ dξ|σ(Rs, s ∈ [0, t ])}. Point estimate: mean pt .J. Armstrong (KCL) & D. Brigo (ICL) Intrinsic SDEs as jets SNS Colloquium 44


Filtering problem and SPDE projection to SDE

pt is our previous X but now in an infinite dimensional function space(typically

√p or p in L2) that plays the role of our former Rr .

pt follows a SPDE (Kushner-Stratonovich or Zakai) and we can use ourthree above projections to estimate an optimal finite dimensionalapproximation Y of p = X according to different criteria.

In B., Hanzon & LeGland [7, 8], Armstrong & B. [3] we studyprojections on M = Gaussians (here), exponential families, mixtures.



Filtering numerical example: Cubic sensor

dSt = dBt , dRt = (St + αS3t )dt + dVt

Hellinger rel. residuals: ‖√

p(t)−√

pN (t)‖2

‖√

p(t)−√

pN ,Ito−jet (t)‖2; L2 resid.: ‖p(t)− pN (t)‖2

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 0.2 0.4 0.6 0.8 1

Time

Extended Kalman FilterStratonovich Projection

Ito-vector ProjectionIto ADF

Ito-jet Projection

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.2 0.4 0.6 0.8 1

Time

Ito ADFExtended Kalman FilterStratonovich Projection

Ito-vector ProjectionIto-jet Projection


Interest rate dynamics and term structure parameterization

Consistency rate dynamics - curve parameterizationIn finance, we use short rate interest ratemodel rt = ϕ(t ,Xt ), where X follows adriving SDE in Rr . X is chosen based onhistory and derivatives prices (calibration).Spot rate at t for maturity T is

R(t ,T ) =1

T − tln

(Et

[exp

(−∫ T

trsds

)])

Practitoners wish curve T 7→ R(t ,T )to have a particular parametric shape,R(t ,T ) = R(T ; θ(t)), θ ∈ Rn.

Use the projection framework to try and optimally approximate thecorrect dynamics of dR coming from dX with one on “manifold” R(θ).Related work was done in the 90’s by Bjork [6] but looking for exactresults rather than optimal approximations.J. Armstrong (KCL) & D. Brigo (ICL) Intrinsic SDEs as jets SNS Colloquium 47

Conclusions and References

Conclusions

SDEs: Ito and Stratonovich. Pros and Cons.Make Ito SDEs good for geometry: Jet interpretationJet formulation of Ito’s formula and other classicsClear relationship with Schwartz Morphism [11] & BelopolskajaDalecky Ito bundle [5, 15] (see paper), but jets more standardOptimal SDEs on submanifolds: dimensionality reduction3 types of projections on submanifolds, the best one based on jetsApplications to signal processing and finance



Thanks

With thanks to Scuola Normale Superiore for the invitation.

Thank you for your attention.

Questions and comments welcomeJ. Armstrong (KCL) & D. Brigo (ICL) Intrinsic SDEs as jets SNS Colloquium 49


References I

[1] J. Armstrong and D. Brigo.Dimensionality reduction for SDEs with differential geometricprojection methods.Working paper, forthcoming on arxiv.org.

[2] John Armstrong and Damiano Brigo.Extrinsic projection of Ito SDEs on submanifolds with applicationsto non-linear filtering.To appear in: Nielsen, F., Critchley, F., & Dodson, K. (Eds),Computational Information Geometry for Image and SignalProcessing, Springer Verlag, 2016.



References II

[3] John Armstrong and Damiano Brigo.Nonlinear filtering via stochastic PDE projection on mixturemanifolds in L2 direct metric.Mathematics of Control, Signals and Systems, 28(1):1–33, 2016.

[4] John Armstrong and Damiano Brigo.Coordinate Free Stochastic Differential Equations as Jets.http://arxiv.org/abs/1602.03931

[5] Ya. Belopolskaja and Yu. Dalecky.Stochastic Equations and Differential Geometry.Mathematics and Its Applications, Vol. 30. Dordrecht, KluwerAcademic Publishers, Boston, London, 1990.



References III

[6] Bjork, T.Interest Rate Dynamics and Consistent Forward Rate Curves.Mathematical Finance, Volume 9, Issue 4, 1999.

[7] D Brigo, B Hanzon, and F LeGland.A differential geometric approach to nonlinear filtering: Theprojection filter.IEEE Transactions on automatic control, 43:247–252, 1998.

[8] D. Brigo, B. Hanzon, and F. LeGland.Approximate nonlinear filtering by projection on exponentialmanifolds of densities.Bernoulli, 5(3):495–534, 06 1999.



References IV

[9] K. D. Elworthy.Stochastic differential equations on manifolds.Cambridge University Press, Cambridge, New York, 1982.

[10] K. D. Elworthy.Geometric aspects of diffusions on manifolds.In Ecole d’Ete de Probabilites de Saint-Flour XV–XVII, 1985–87,pages 277–425. Springer, 1988.

[11] M. Emery.Stochastic calculus in manifolds.Springer-Verlag, Heidelberg, 1989.



References V

[12] D. Fisk.Quasi-martingales and stochastic integrals.PhD Dissertation, Michigan State University, Department ofStatistics, 1963.

[13] Avner Friedman.Partial differential equations of parabolic type.Prentice-Hall, reprinted in 2008 by Dover-Publications,Englewood Cliffs N.J, 1964.

[14] Avner Friedman.Stochastic differential equations and applications, vol. I and II.Academic Press, reprinted by Dover Publications, New York,1975.



References VI

[15] Yuri E Gliklikh.Global and Stochastic Analysis with Applications to MathematicalPhysics.Theoretical and Mathematical Physics. Springer, London, 2011.

[16] I. Gyongy.A note on Eulers approximations.Potential Analysis, (8):205–216, 1998.

[17] Steven L Heston.A closed-form solution for options with stochastic volatility withapplications to bond and currency options.Review of financial studies, 6(2):327–343, 1993.



References VII

[18] K. Ito.Stochastic differential equations in a differentiable manifold.Nagoya Math. J., (1):35–47, 1950.

[19] P. E. Kloeden and E. Platen.Higher-order implicit strong numerical schemes for stochasticdifferential equations.Journal of statistical physics, 66(1-2):283–314, 1992.

[20] Peter E. Kloeden and Eckhard Platen.Numerical solution of stochastic differential equations.Applications of mathematics. Springer, Berlin, New York, Thirdprinting, 1999.



References VIII

[21] S. Kobayashi and K. Nomizu.Foundations of differential geometry, volume 1.New York, 1963.

[22] Graham, R.L., Knuth, D.E., and O. Patashnik (1994).Concrete Mathematics, 2nd Ed., ADDISON-WESLEY.

[23] E.J. McShane.Stochastic Calculus and Stochastic Models (2nd Ed.).Academic Press, New York, 1974.

[24] G. A. Pavliotis.Stochastic Processes and Applications: Diffusion Processes, theFokker-Planck and Langevin Equations.Springer, Heidelberg, 2014.



References IX

[25] K. Pearson.Contributions to the mathematical theory of evolution. ii. skewvariation in homogeneous material.Philosophical Transactions of the Royal Society of London. A,186:343–414, 1895.

[26] Protter, P. (2004).Stochastic Integration and Differential Equations. Springer Verlag.

[27] L. C. G. Rogers and D. Williams.Diffusions, markov processes and martingales, vol 2: Ito calculus,1987.



References X

[28] A. V. Skorokhod (1967).R. L. Stratonovich “Conditional Markov processes and theirapplication to the theory of optimal control” (book review), Teor.Veroyatnost. i Primenen., 12:1 (1967), 184187.

[29] R. L. Stratonovich.A new representation for stochastic integrals and equations.SIAM Journal on Control, 4(2):362–371, 1966.

[30] N. G. Van Kampen.Ito vs Stratonovich.Journal of Statistical Physics, 24(1), 1981.


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