calculus via regularizations in banach spaces: path ... · kolmogorov type equations. francesco...

Calculus via regularizations in Banachspaces: path dependent calculus and

Kolmogorov type equations.

Francesco Russo, ENSTA ParisTech

STOCHASTIC ANALYSIS, CONTROLLED DYNAMICAL SYSTEMS AND

APPLICATIONS

Jena, March 9-13th 2015

In honour of Prof. Dr. Hans-Jürgen ENGELBERT

Covers joint work with

Cristina Di Girolami (Pescara)

and Andrea Cosso (Politecnico di Milano and Paris VII).

Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. – p. 1/81

Hans-Jürgen ENGELBERT

A great probabilist: a pionner on stochastic differentialequations with singular drift.

Many souvenirs. One in particular: the conference ofEisenach (DDR) 1986.


Outline

1. About a robust representation problem for randomvariables.

2. Finite dimensional calculus via regularization.

3. Stochastic calculus via regularizations in Banachspaces.

4. Window processes.

5. Towards a robust Clark-Ocone type formula.

6. Kolmogorov path dependent PDEs. The window ofdiffusion processes.

7. Path-dependent semilinear Kolmogorov equation.


Basic survey reference

A. Cosso, C. Di Girolami, F. Russo (2014)

Calculus via regularizations in Banach spaces andKolmogorov-type path-dependent equations.

http://arxiv.org/abs/1411.8000


Some related references to our work

C. Di Girolami, F. Russo (2010). Infinite dimensional calculus viaregularizations and applications. Technical Report.http://hal.archives-ouvertes.fr/inria-00473947/fr/

C. Di Girolami, F. Russo (2011). Clark-Ocone type formula fornon-semimartingales with finite quadratic variation.Comptes Rendus de l’Académie des Sciences, SectionMathématiques. Number 3-4, pp 209–214. Vol. 349.

C. Di Girolami, F. Russo (2010). Generalized covariation forBanach space valued processes, Itô formula and applications.Osaka Journal of Mathematics 51 (3), 2014 (forthcoming).


C. Di Girolami, F. Russo (2011). Generalized covariation andextended Fukushima decompositions for Banach space valuedprocesses. Application to windows of Dirichlet processes.Infinite dimensional analysis, Quantum probability andrelated topics (IDA-QP) 15(2):1250007, 50, 2012.

G. Fabbri, F. Russo (2012).Infinite dimensional weak Dirichlet processes, stochasticPDEs and optimal control.Preprint HAL-INRIA, http://hal.inria.fr/hal-00720490.


C. Di Girolami, G. Fabbri and F. Russo (2014).The covariation for Banach space valued processes andapplications.Metrika 77:51–104 DOI 10.1007/s00184-013-0472-6

A. Cosso and F. Russo (2014).A regularization approach to functional Itô calculus andstrong viscosity solutions to path-dependent SDEs.Preprint HAL-INRIA http://hal.inria.fr/hal-00933678.http://fr.arxiv.org/abs/1401.5034


Available preprints :http://uma.ensta.fr/∼russo/


1 About a robust representation

problem for random variables.

1.1 Window processes

Let X be a continuous process with quadratic variation [X].


Definition 1 Let T > 0 and X = (Xt)t∈[0,T ] be a realcontinuous process prolongated by continuity.Process X(·) defined by

X(·) = Xt(u) := Xt+u; u ∈ [−T, 0]

will be called window process .


X(·) is a C([−T, 0])-valued stochastic process.

C([−T, 0]) is a typical non-reflexive Banach space.

1.2 The robust representation

Is there a reasonnable rich class of functionals

H : C := C([−T, 0]) −→ R

such that the r.v.h := H(XT (·))

admits a representation of the type


h = V0 +

∫ T

0

Zsd−Xs,

and

V0 ∈ R,

Z adapted process with respect to the canonicalfiltration of X.

Possibly we look for explicit expressions of V0 and Z.


Idea: Representation of h = H(XT (·))

The idea consists in finding functions u, v : [0, T ]× C → R

such that h = H(XT (·)) as

Vt = u(t,Xt(·)), Zt = v(t,Xt(·)), t ∈ [0, T ],

Vt = h−

∫ T

t

Zsd−Xs, t ∈ [0, T ].

In particular

h = u(0, X0(·)) +

∫ T

0

v(s,Xs(·))d−Xs.


u, v are related to a deterministic purely analytical toolas a PDE, in order to keep separated probability andanalysis.

Previous procedure make Clark-Ocone formula robustwith respect to the quadratic variation.


Natural extensions.

Vt = h−∫ T

tZsd

−Xs +∫ T

tF (s,Xs(·), Ys, Zs)d[X]s, t ∈

[0, T ],for some F : [0, T ]× C([−T, 0]× R× R).

[X] =∫ ·

0σ2(s,Xs(·))ds, σ : [0, T ]× R → R.


2 Finite dimensional calculus via

regularization

Definition 2 Let X (resp. Y ) be a continuous (resp. locallyintegrable) process.Suppose that the random variables

∫ t

0

Ysd−Xs := lim

ǫ→0

∫ t

0

Ys

Xs+ǫ −Xs

ǫds

exists in probability for every t ∈ [0, T ].


If the limiting random function admits a continuousmodification, it is denoted by

∫ ·

0Y d−X and called

(proper) forward integral of Y with respect to X.(FR-Vallois 1991)

If limt→T

∫ t

0Y d−X exists in probability we call that limit

improper forward integral of Y with respect to X, againdenoted by

∫ T

0Y d−X.


Covariation of real valued processes

Definition 3 The covariation of X and Y is defined by

[X, Y ]t = limǫ→0+

1

ǫ

∫ t

0

(Xs+ǫ −Xs)(Ys+ǫ − Ys)ds

if the limit exists in the ucp sense with respect to t.Obviously [X, Y ] = [Y,X].If X = Y, X is said to be finite quadratic variationprocess and [X] := [X,X].


Connections with semimartingales Let S1, S2 be(Ft)-semimartingales with decomposition Si = M i + V i,i = 1, 2 where M i (Ft)-local continuous martingale and V i

continuous bounded variation processes. Then

[Si] classical bracket and [Si] = 〈M i〉.

[S1, S2] classical bracket and [S1, S2] = 〈M1,M2〉.

If S semimartingale and Y cadlag and predictable∫ ·

0

Y d−S =

∫ ·

0

Y dS (Itô)


Itô formula for finite quadratic variation processes

Theorem 4 Let F : [0, T ]× R −→ R such thatF ∈ C1,2 ([0, T [×R) and X be a finite quadratic variationprocess. Then ∫ t

0

∂xF (s,Xs)d−Xs

exists and equals

F (t,Xt)−F (0, X0)−

∫ t

0

∂sF (s,Xs)ds−1

2

∫ t

0

∂xxF (s,Xs)d[X]s


3 Stochastic calculus via

regularization in Banach spaces

A stochastic integral for B∗-valued integrand withrespect to B-valued integrators, which are notnecessarily semimartingales.

χ-quadratic variation of XA new concept of quadratic variation which generalizesthe tensor quadratic variation and which involves aBanach subspace χ of (B⊗πB)∗.


Definition 5 Let X (resp. Y) be a B-valued (resp. aB∗-valued) continuous stochastic process.Suppose that the random function defined for every fixedt ∈ [0, T ] by

∫ t

0B∗〈Ys, d

−Xs〉B := lim

ǫ→0

∫ t

0B∗〈Ys,

Xs+ǫ − Xs

ǫ〉Bds

in probability exists and admits a continuous version.Then, the corresponding process will be called forwardstochastic integral of Y with respect to X.


Connection with Da Prato-Zabczyk integral

Let B = H is separable Hilbert space.Theorem 6 Let W be a H-valued Q-Brownian motion withQ ∈ L1(H) and Y be H∗-valued process such that∫ t

0‖Ys‖

2H∗ds < ∞ a.s. Then, for every t ∈ [0, T ],

∫ t

0H∗〈Ys, d

−Ws〉H =

∫ t

0

Ys · dWdzs

(Da Prato-Zabczyk integral)


Notion of Chi-subspace

Definition 7 A Banach subspace χ continuously injectedinto (B⊗πB)∗ will be called Chi-subspace (of (B⊗πB)∗).In particular it holds

‖ · ‖χ ≥ ‖ · ‖(B⊗πB)∗.


Chi-quadratic variationLet

X be a B-valued continuous process,

χ a Chi-subspace of (B⊗πB)∗,

C([0, T ]) space of real continuous processes equippedwith the ucp topology.

Two processes:1. [X] : χ → C([0, T ]);

2. [X] : [0, T ]× Ω → χ∗ with bounded variation.


They are (loosely speaking) approached by [X]ǫ be theapplications

[X]ǫ : χ −→ C([0, T ])

defined by

φ 7→

(∫ t

0χ〈φ,

(Xs+ǫ − Xs)⊗2

ǫ〉χ∗ ds

)

t∈[0,T ]

,


Definition 8 We say that X admits a global quadraticvariation (g.q.v.) if it admits a χ-quadratic variation withχ = (B⊗πB)∗.


When χ = (B⊗πB)∗

H2 is related to weak∗ convergence in (B⊗πB)∗∗.If Ω were a singleton then (H2) would imply

[X]ǫ

t(Φ) −−→ǫ→0[X]t(Φ), ∀Φ ∈ (B⊗πB)∗, t ∈ [0, T ].

The g.q.v. [X] is (B⊗πB)∗∗-valued.


Infinite dimensional Itô’s formula

Let B a separable Banach spaceTheorem 9 Let X a B-valued continuous processadmitting a χ-quadratic variation.Let F : [0, T ]× B −→ R be C1,2 Fréchet such that

D2F : [0, T ]× B −→ χ ⊂ (B⊗πB)∗ continuously

Then for every t ∈ [0, T ] the forward integral

∫ t

0B∗〈DF (s,Xs), d

−Xs〉B

exists and the following formula holds.


F (t,Xt) = F (0,X0) +

∫ t

0

∂sF (s,Xs)ds+

+

∫ t

0B∗〈DF (s,Xs), d

−Xs〉B+

+1

2

∫ t

0χ〈D

2F (s,Xs), d[X]s〉χ∗


4 Window processes.

We fix now the attention on B = C = C([−T, 0])-valuedwindow processes.

X continuous real valued process and X(·) its windowprocess.

X = X(·)


If X has Hölder continuous paths of parameterγ > 1/2, then X(·) has a zero g.q.v.For instance:

X = BH fractional Brownian motion with parameterH > 1/2.

X = BH,K bifractional Brownian motion withparameters H ∈]0, 1[, K ∈]0, 1] s.t. HK > 1/2.

W (·) does not admit a global quadratic variation.


4.1 About a significant Chi-subspace

We shall consider the following Chi-subspace of(C([−T, 0])⊗πC([−T, 0]))∗:

χ0 :=(D0 ⊕ L2([−T, 0]

)⊗

2h ⊕Diag,

withD0 := λ δ0(dx), λ ∈ R,

Diag := µ(dx, dy) = g(x)δy(dx)dy; g ∈ L∞([−T, 0]) .


Evaluations of χ-quadratic variation for window processes

Let X be a real finite quadratic variation process.


X(·) has a χ0-quadratic variation and

[X(·)] : χ0 −→ C[0, T ]

[X(·)]t(µ) =

∫

D−t

dµ(x, y)[X]t+x,

where D−t = (x, y)| − t ≤ x = y = 0.


In particular, if µ ∈ (D0 ⊕ L2([−T, 0]) ⊗2h,

[X(·)]t(µ) = µ(0, 0)[X]t,


The particular case of a finite quadratic variation process X such

that [X]t =∫ t

0σ2(s,Xs(·))ds.

If µ ∈ χ0 then

[X,X]t(µ) =

∫

D−t

dµ(x, y)

∫ t+x

0

σ2(s,Xs(·))ds.

If µ ∈ (D0 ⊕ L2([−T, 0])) ⊗2h, then

[X,X]t = µ((0, 0))

∫ t

0

σ2(s,Xs(·))ds.


Itô’s formula for the corresponding window process.

Let F : [0, T ]× C → R of class C1,2 such thatD2F : [0, T ]× C → χ0 continuous. We denote

DF (t, η) = Dδ0F (t, η)δ0(dx) +D⊥F (t, η),

with Dδ0F : [0, T ]× C → R. The application of Itô formulagives

F (t,Xt) = F (0,X0) +

∫ t

0

Dδ0F (s,Xs)d−Xs +

∫ t

0

LsF (s,Xs)ds

+

∫ t

0

∂sF (s,Xs)ds,

where Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. – p. 38/81

LtG(η) = I(G)(t, η)

+1

2

∫

D−t

D2dxdyG(η)σ2(t+ x, η(x)),

I(G)(t, η) =

∫

[−t,0]

DdxG(η)d−η(x),

provided that

I(G)(t, η) = limε→0

∫

[−t,0]

DdxG(η)η(x+ ε)− η(x)

ε.

exists and I fulfills some technical assumptions.


5 Towards a Robust Clark-Ocone

type formula

We set B = C = C([−T, 0]).

X real continuous stochastic process with values in R.

X0 = 0,

[X]t =∫ t

0σ2(r,Xr(·))dr, where σ : [0, T ]× C → R is

continuous.


Representation of h = H(XT (·))

Conformally to what we mentioned at the beginning, weaim at finding functions u, v : [0, T ]× C → R such that

Vt = u(t,Xt(·)), Zt = v(t,Xt(·)),

Vt = h−

∫ T

t

Zsd−Xs, t ∈ [0, T ].

In particular

h = u(0, X0(·)) +

∫ T

0

v(s,Xs(·))d−Xs.


5.1 A toy model (related to the Black-Scholes

formula)

Let (St) be the ”price of a financial asset” of the type

St = exp(σWt −σ2

2t) , σ > 0 .

Let f : R → R be a continuous function and

h = f(ST ) = f(WT ) where f(y) = f(exp(σy − σ2

2T )

).


Let U : [0, T ]× R −→ R solving

∂tU(t, x) +

σ2

2∂xxU(t, x) = 0

U(T, x) = f(x) x ∈ R.

Applying classical Itô formula we obtain

h = U(0, S0) +

∫ T

0

∂xU(s, Ss)dSs

= U(0,W0) +

∫ T

0

∂xU(s,Ws)dWs,

for a suitable U : [0, T ]× R −→ R.


Robustness with respect to the volatility.

Does one have a similar representation if W is replaced by a finitequadratic variation X such that [X]t ≡ t?

The answer is positive! because of Theorem 4.


Proposition 10 Let X such that [X]t = σ2t.

A1 f : R −→ R continuous and polynomial growth.

A2 U ∈ C1,2([0, T [×R) ∩ C0([0, T ]× R) such that

∂tU(t, x) +

σ2

2∂xxU(t, x) = 0

v(T, x) = f(x).


Then

h := f(XT ) = U(0, X0) +

∫ T

0

∂xU(s,Xs)d−Xs

︸︷︷︸improper forward integral

Schoenmakers-Kloeden (1999) Zähle (2002)Coviello-Russo (2006) Bender-Sottinen-Valkeila (2006)


Natural question

Generalization to the case of ”path dependent options”?

As first step we revisit the toy model.


5.2 The toy model revisited

Proposition 11 We set B = C = C([−T, 0]) and η ∈ C andwe define

H : C −→ R, by H(η) := f(η(0))

u : [0, T ]× C −→ R, by u(t, η) := U(t, η(0))

Then

u ∈ C1,2 ([0, T [×C;R) ∩ C0 ([0, T ]× C;R)

and solves


∂tu(t, η) + σ2

2

∫D−t

D2dx dyu (t, η) = 0

u(T, η) = H(η)

(Here D⊥u ≡ 0).


Proof.

u(T, η) = U(T, η(0)) = f(η(0)) = H(η)

∂tU (t, η) = ∂tU (t, η(0))

Du (t, η) = ∂xU (t, η(0)) δ0

D2u (t, η) = ∂2xxU (t, η(0)) δ0 ⊗ δ0

∂tu (t, η) +12D2u (t, η)(0, 0) = 0.


5.3 The general representation

The considerations of previous section bring us to thefollowing.Theorem 12 Let us consider the following.

H : C −→ R continuous.

u ∈ C1,2 ([0, T [×C) ∩ C0 ([0, T ]× C)

For any (t, η) ∈ [0, T ]× C,∫[−t,0]

D⊥dxu(t, η) d

−η(x) is

well-defined (with a technical condition).

(t, η) 7→ D2u (t, η) is continuous from [0, T ]× C to χ0.


Suppose that u solves the Infinite dimensional PDE

∂tu(t, η) + Ltu(t, η) = 0

u(T, η) = H(η).(1)


Then,

h = V0 +

∫ T

0

Zsd−Xs (2)

(in the possibly improper sense) with

V0 = u(0, X0(·))

Zs = Dδ0u(s,Xs(·))


Definition 13 We call (1) path dependent Kolmogorov PDE .related to X.


6 Kolmogorov path dependent PDE.

The window of diffusion process

Let σ : [0, T ]× R → R continuous and Lipschitz. Considerthe stochastic flow (Xs,ξ

t )s≤t≤T , for s ∈ [t, T ], ξ ∈ R, where,setting X = Xs,ξ solves the SDE

Xt = ξ +

∫ t

s

σ(r,Xr)dWr,

where (W,Ft) is a classical Wiener process.


Remark 14 1. X is a.s. continuous in all the threevariables (s, t, ξ).

2. Let σ is of class C0,2(∆× R), such that σ, ∂xσ and ∂2xxσ

are Hölder continuous, with

∆ = (s, t)|0 ≤ s ≤ t ≤ T.

Under previous assumptions we have, for k = 0, 1, 2,

sup0≤s≤T

E( supt∈[s,T ]

|∂(k)ξ Xs,ξ

t |p) ≤ M,

for every p ≥ 1.


Associated with X, we link the following functionalstochastic flow. For η ∈ C, we set

Xs,ηt (x) =

η(t− s+ x) : x ≤ s− t

Xs,η(0)t+x : x ≥ s− t.

(3)

Remark 15 If σ = 1 we denote the corresponding windowBrownian flow by W

s,ηt := X

s,ηt .

From now on we will suppose σ to be as in Remark 14.


Let h = H(XT (·)), where X = X0,x0

T , for some x0 ∈ R. Weset

Vt = E(h|Ft).

Then, there is u : [0, T ]× C → R such that

Vt = u(t,Xt(·)). (4)

It is given byu(t, η) = E(H(Xt,η

T )).


Aim.

Under suitable conditions on H, we aim to show that u isthe unique solution of the path dependent Kolmogorovequation (1), taking values in C(| − T, 0]).

This justifies to say that u defined in (4) is the virtualsolution for (1).


Definition 16 (Strict solution )u : [0, T ]×C → R of class C1,2([0, T [×C) ∩C0([0, T ]×C) issaid to be a solution of (1) if

I(u)(t, η) :=∫]−t,0]

Ddxu(t, η)d−η(x)

is “well-defined”;

(t, η) 7→ D2u(t, η) is continuous into someChi-subspace χ.

It solves effectively (1).


Sufficient conditions to find a strict solution of (1) and σ = 1.

Case σ general in a paper in preparation (Cosso-Russo).

H has a smooth Fréchet dependence on C([−T, 0]).

h := H(XT (·)) = f(∫ T

0ϕ1(s)d

−Xs, . . . ,∫ T

0ϕn(s)d

−Xs

);

f : Rn → R continuous and with linear growth;ϕi ∈ C2([0, T ];R), ∀1 ≤ i ≤ n.

The determinant of Σt :=(∫ 0

xϕi(y)ϕj(y)dy

)is

bigger than zero for every x ∈ [−T, 0[.


Uniqueness for solutions of the Kolmogorov

type PDE.

Let u1, u2 : [0, T ]× C → R of classC1,2([0, T [×C)∩C0([0, T ]×C) being strict solutions of (1) ofpolynomial growth. If u1, u2 solve the Kolmogorov typeequation then u1 = u2.


Sketch of the Proof .

We fix (s, η) ∈ [0, T ]× C. We have to prove thatu1(s, η) = u2(s, η).Without restriction of generality we set s = 0.

We apply Itô formula to Xt = X0,ηt .

We take than the expectation and we obtain

ui(0, η) = E(H(X0,ηT )).

This shows the uniqueness.


7 Path-dependent semilinear

Kolmogorov equation

Here we concentrate on the case σ = 1, the general casebeing in Cosso-Russo (in preparation)..

We study the path-dependent semilinear Kolmogorovequation:

∂tU + LtU = F (t, η,U , Dδ0U), ∀ (t, η) ∈ [0, T [×C,

U(T, η) = G(η), ∀ η ∈ C,

where


G : C([−T, 0]) −→ R

F : [0, T ]× C([−T, 0])× R× R −→ R

are Borel measurable functions. We refer to

∂tU + LtU ,

as the path-dependent heat operator.


7.1 Strict solutions

Definition 17 A function U : [0, T ]× C([−T, 0]) → R inC1,2([0, T [×C) ∩ C([0, T ]× C), which solves thepath-dependent semilinear Kolmogorov equation, is calleda strict solution .


Strict solutions: uniqueness

Theorem 18 (Uniqueness) Let G : C([−T, 0]) → R andF : [0, T ]× C([−T, 0])× R× R → R be Borel measurablefunctions satisfying, for some positive constants C and m,

|F (t, η, y, z)− F (t, η, y′, z′)| ≤ C(|y − y′|+ |z − z′|

),

|G(η)|+ |F (t, η, 0, 0)| ≤ C(1 + ‖η‖m∞

),

for all (t, η) ∈ [0, T ]× C([−T, 0]), y, y′ ∈ R, and z, z′ ∈ R.


Let U : [0, T ]× C([−T, 0]) → R be a strict solution to thepath-dependent nonlinear Kolmogorov equation, satisfyingthe polynomial growth condition

|U(t, η)| ≤ C(1 + ‖η‖m∞

), ∀ (t, η) ∈ [0, T ]× C([−T, 0]).


Then, we have

U(t, η) = Y t,ηt , ∀ (t, η) ∈ [0, T ]× C([−T, 0]),

where

(Y t,ηs , Zt,η

s )s∈[t,T ] = (U(s,Wt,ηs ), Dδ0U(s,Wt,η

s )1[t,T [(s))s∈[t,T ]

is the solution to the backward stochastic differentialequation, P -a.s.,


Y t,ηs = G(Wt,η

T ) +

∫ T

s

F (r,Wt,ηr , Y t,η

r , Zt,ηr )dr −

∫ T

s

Zt,ηr dWr,

for all t ≤ s ≤ T . In particular, there exists at most onestrict solution to the path-dependent nonlinear Kolmogorovequation.

Here W is the functional stochastic flow associated withW s,x

t = x+ (Wt −Ws) (classical Wiener process).


Alternative methods to strict solutions for Kolmogorov pathdependent PDEs

Dupire-Cont-Fournié (2010).

Flandoli-Zanco (2014).

Leão-Ohashi-Simas (2014).


7.2 Strong-viscosity solutions: introduction

Various definitions of viscosity-type solutions forpath-dependent PDEs have been given:

(2012) S. PENG.

(2013) S. TANG AND F. ZHANG.

(2014) I. EKREN, C. KELLER, N. TOUZI, AND J. ZHANG.

(2015) R. BUCKDAHN, J. MA AND J. ZHANG.


We propose a notion of solution which is not based on thestandard definition of viscosity solution given in terms oftest functions or jets.

7.3 Strong-viscosity solutions

Idea and origin

Our notion of solution is defined, in a few words, as thepointwise limit of strict solutions to perturbedequations.


Our definition is more similar in spirit to the concept ofgood solution, which turned out to be equivalent to thedefinition of Lp-viscosity solution for certain fullynonlinear partial differential equations.

Our definition is likewise inspired by the notion ofstrong solution, even though strong solutions arerequired to be more regular than simply continuous.


Definition 19 A function U : [0, T ]× C([−T, 0]) → R iscalled a strong-viscosity solution to the path-dependentnonlinear Kolmogorov equation if there exists a sequence(Un, Gn, Fn)n satisfying:

(i) Un : [0, T ]× C([−T, 0]) → R, Gn : C([−T, 0]) → R, andFn : [0, T ]×C([−T, 0])×R×R → R are equicontinuousfunctions such that, for some positive constants C andm, independent of n,

|Fn(t, η, y, z)− Fn(t, η, y′, z′)| ≤ C(|y − y′|+ |z − z′|),

|Un(t, η)|+ |Gn(η)|+ |Fn(t, η, 0, 0)| ≤ C(1 + ‖η‖m∞

),

for all (t, η) ∈ [0, T ]×C([−T, 0]), y, y′ ∈ R, and z, z′ ∈ R.


(ii) Un is a strict solution to

∂tUn + LtUn = Fn(t, η,Un, Dδ0Un),

∀ (t, η) ∈ [0, T )× C([−T, 0]),

Un(T, η) = Gn(η), ∀ η ∈ C([−T, 0]).

(iii) (Un(t, η), Gn(η), Fn(t, η, y, z)) →(U(t, η), G(η), F (t, η, y, z)), as n tends to infinity, for any(t, η, y, z) ∈ [0, T ]× C([−T, 0])× R× R.


7.4 Strong-viscosity solutions: uniqueness

Theorem 20 (Uniqueness) Let U : [0, T ]× C([−T, 0]) → R

be a strong-viscosity solution to the path-dependentnonlinear Kolmogorov equation. Then, we have

U(t, η) = Y t,ηt , ∀ (t, η) ∈ [0, T ]× C([−T, 0]),

where (Y t,ηs , Zt,η

s )s∈[t,T ], with Y t,ηs = U(s,Wt,η

s ), solves thebackward stochastic differential equation, P -a.s.,


Y t,ηs = G(Wt,η

T ) +

∫ T

s

F (r,Wt,ηr , Y t,η

r , Zt,ηr )dr −

∫ T

s

Zt,ηr dWr,

for all t ≤ s ≤ T . In particular, there exists at most onestrong-viscosity solution to the path-dependent nonlinearKolmogorov equation.


Strong-viscosity solutions: existence

Theorem 21 (Existence) Let F ≡ 0 andG : C([−T, 0]) → R be uniformly continuous and satisfyingthe polynomial growth condition

|G(η)| ≤ C(1 + ‖η‖m∞), ∀ η ∈ C([−T, 0]),

for some positive constants C and m. Then, there exists aunique strong-viscosity solution U to the path-dependentheat equation, which is given by

U(t, η) = E[G(Wt,η

T )],

for all (t, η) ∈ [0, T ]× C([−T, 0]).


7.5 Paper in preparation.

A. Cosso, F. Russo.

Existence for functional dependent Kolmogorov typeequation: the strict and strong-viscosity solution case.


Thank you for you attention .


calculus via regularizations in banach spaces: path ... · kolmogorov type equations. francesco...

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