Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations.

Transcription

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

2 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 2/81 Hans-Jürgen ENGELBERT A great probabilist: a pionner on stochastic differential equations with singular drift. Many souvenirs. One in particular: the conference of Eisenach (DDR) 1986.

3 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 3/81 Outline 1. About a robust representation problem for random variables. 2. Finite dimensional calculus via regularization. 3. Stochastic calculus via regularizations in Banach spaces. 4. Window processes. 5. Towards a robust Clark-Ocone type formula. 6. Kolmogorov path dependent PDEs. The window of diffusion processes. 7. Path-dependent semilinear Kolmogorov equation.

4 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 4/81 Basic survey reference A. Cosso, C. Di Girolami, F. Russo (2014) Calculus via regularizations in Banach spaces and Kolmogorov-type path-dependent equations.

5 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 5/81 Some related references to our work C. Di Girolami, F. Russo (2010). Infinite dimensional calculus via regularizations and applications. Technical Report. C. Di Girolami, F. Russo (2011). Clark-Ocone type formula for non-semimartingales with finite quadratic variation. Comptes Rendus de l Académie des Sciences, Section Mathématiques. Number 3-4, pp Vol C. Di Girolami, F. Russo (2010). Generalized covariation for Banach space valued processes, Itô formula and applications. Osaka Journal of Mathematics 51 (3), 2014 (forthcoming).

6 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 6/81 C. Di Girolami, F. Russo (2011). Generalized covariation and extended Fukushima decompositions for Banach space valued processes. Application to windows of Dirichlet processes. Infinite dimensional analysis, Quantum probability and related topics (IDA-QP) 15(2): , 50, G. Fabbri, F. Russo (2012). Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control. Preprint HAL-INRIA,

7 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 7/81 C. Di Girolami, G. Fabbri and F. Russo (2014). The covariation for Banach space valued processes and applications. Metrika 77: DOI /s A. Cosso and F. Russo (2014). A regularization approach to functional Itô calculus and strong viscosity solutions to path-dependent SDEs. Preprint HAL-INRIA

8 Available preprints: russo/ Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 8/81

9 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 9/81 1 About a robust representation problem for random variables. 1.1 Window processes Let X be a continuous process with quadratic variation [X].

10 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 10/81 Definition 1 Let T > 0 and X = (X t ) t [0,T] be a real continuous process prolongated by continuity. Process X( ) defined by X( ) = {X t (u) := X t+u ;u [ T,0]} will be called window process.

11 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 11/81 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(X T ( )) admits a representation of the type

12 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 12/81 and h = V 0 + T 0 Z s d X s, V 0 R, Z adapted process with respect to the canonical filtration of X. Possibly we look for explicit expressions of V 0 and Z.

13 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 13/81 Idea: Representation ofh = H(X T ( )) The idea consists in finding functions u,v : [0,T] C R such that h = H(X T ( )) as In particular V t = u(t,x t ( )), Z t = v(t,x t ( )),t [0,T], V t = h T t Z s d X s,t [0,T]. h = u(0,x 0 ( ))+ T 0 v(s,x s ( ))d X s.

14 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 14/81 u, v are related to a deterministic purely analytical tool as a PDE, in order to keep separated probability and analysis. Previous procedure make Clark-Ocone formula robust with respect to the quadratic variation.

15 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 15/81 Natural extensions. V t = h T t Z s d X s + T t F(s,X s ( ),Y s,z s )d[x] s,t [0,T], for some F : [0,T] C([ T,0] R R). [X] = 0 σ2 (s,x s ( ))ds, σ : [0,T] R R.

16 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 16/81 2 Finite dimensional calculus via regularization Definition 2 Let X (resp. Y ) be a continuous (resp. locally integrable) process. Suppose that the random variables t 0 t Y s d X s := lim ǫ 0 0 Y s X s+ǫ X s ǫ ds exists in probability for every t [0,T].

17 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 17/81 If the limiting random function admits a continuous modification, it is denoted by 0 Yd X and called (proper) forward integral of Y with respect to X. (FR-Vallois 1991) If lim t T t 0 Yd X exists in probability we call that limit improper forward integral of Y with respect to X, again denoted by T 0 Yd X.

18 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 18/81 Covariation of real valued processes Definition 3 The covariation of X and Y is defined by 1 [X,Y] t = lim ǫ 0 + ǫ t 0 (X s+ǫ X s )(Y s+ǫ Y s )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 variation process and [X] := [X,X].

19 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 19/81 Connections with semimartingales Let S 1, S 2 be (F t )-semimartingales with decomposition S i = M i +V i, i = 1,2 where M i (F t )-local continuous martingale and V i continuous bounded variation processes. Then [S i ] classical bracket and [S i ] = M i. [S 1,S 2 ] classical bracket and [S 1,S 2 ] = M 1,M 2. If S semimartingale and Y cadlag and predictable 0 Yd S = 0 YdS (Itô)

20 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 20/81 Itô formula for finite quadratic variation processes Theorem 4 Let F : [0,T] R R such that F C 1,2 ([0,T[ R) and X be a finite quadratic variation process. Then t exists and equals 0 x F(s,X s )d X s t F(t,X t ) F(0,X 0 ) 0 s F(s,X s )ds 1 2 t 0 xx F(s,X s )d[x] s

21 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 21/81 3 Stochastic calculus via regularization in Banach spaces A stochastic integral for B -valued integrand with respect to B-valued integrators, which are not necessarily semimartingales. χ-quadratic variation of X A new concept of quadratic variation which generalizes the tensor quadratic variation and which involves a Banach subspace χ of (Bˆ π B).

22 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 22/81 Definition 5 Let X (resp. Y) be a B-valued (resp. a B -valued) continuous stochastic process. Suppose that the random function defined for every fixed t [0,T] by t 0 t B Y s,d X s B := lim ǫ 0 0 B Y s, X s+ǫ X s ǫ B ds in probability exists and admits a continuous version. Then, the corresponding process will be called forward stochastic integral of Y with respect to X.

23 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 23/81 Connection with Da Prato-Zabczyk integral Let B = H is separable Hilbert space. Theorem 6 Let W be a H-valued Q-Brownian motion with Q L 1 (H) and Y be H -valued process such that t 0 Y s 2 H ds < a.s. Then, for every t [0,T], t 0 H Y s,d W s H = t 0 Y s dw dz s (Da Prato-Zabczyk integral)

24 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 24/81 Notion of Chi-subspace Definition 7 A Banach subspace χ continuously injected into (Bˆ π B) will be called Chi-subspace (of (Bˆ π B) ). In particular it holds χ (Bˆ π B).

25 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 25/81 Chi-quadratic variation Let X be a B-valued continuous process, χ a Chi-subspace of (Bˆ π B), C([0, T]) space of real continuous processes equipped with the ucp topology. Two processes: 1. [X] : χ C([0,T]); 2. [X] : [0,T] Ω χ with bounded variation.

26 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 26/81 They are (loosely speaking) approached by [X] ǫ be the applications [X] ǫ : χ C([0,T]) defined by φ ( t 0 χ φ, (X s+ǫ X s ) 2 ǫ ) χ ds t [0,T],

27 Definition 8 We say that X admits a global quadratic variation (g.q.v.) if it admits a χ-quadratic variation with χ = (Bˆ π B). Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 27/81

28 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 28/81 When χ = (Bˆ π B) H2 is related to weak convergence in (Bˆ π B). If Ω were a singleton then (H2) would imply [X] ǫ t (Φ) ǫ 0 [X] t (Φ), Φ (Bˆ π B),t [0,T]. The g.q.v. [X] is (Bˆ π B) -valued.

29 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 29/81 Infinite dimensional Itô s formula Let B a separable Banach space Theorem 9 Let X a B-valued continuous process admitting a χ-quadratic variation. Let F : [0,T] B R be C 1,2 Fréchet such that D 2 F : [0,T] B χ (Bˆ π B) continuously Then for every t [0,T] the forward integral t 0 B DF(s,X s),d X s B exists and the following formula holds.

30 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 30/81 F(t,X t ) = F(0,X 0 )+ + t 0 t t 0 s F(s,X s )ds+ B DF(s,X s),d X s B χ D 2 F(s,X s ),d [X] s χ

31 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 31/81 4 Window processes. We fix now the attention on B = C = C([ T,0])-valued window processes. X continuous real valued process and X( ) its window process. X = X( )

32 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 32/81 If X has Hölder continuous paths of parameter γ > 1/2, then X( ) has a zero g.q.v. For instance: X = B H fractional Brownian motion with parameter H > 1/2. X = B H,K bifractional Brownian motion with parameters H ]0,1[, K ]0,1] s.t. HK > 1/2. W( ) does not admit a global quadratic variation.

33 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 33/ About a significant Chi-subspace We shall consider the following Chi-subspace of (C([ T,0])ˆ π C([ T,0])) : χ 0 := ( D 0 L 2 ([ T,0] ) ˆ 2 h Diag, with D 0 := {λδ 0 (dx), λ R}, Diag := {µ(dx,dy) = g(x)δ y (dx)dy;g L ([ T,0])}.

34 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 34/81 Evaluations of χ-quadratic variation for window processes Let X be a real finite quadratic variation process.

35 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 35/81 X( ) has a χ 0 -quadratic variation and [X( )] : χ 0 C[0,T] [X( )] t (µ) = dµ(x,y)[x] t+x, D t where D t = {(x,y) t x = y = 0}.

36 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 36/81 In particular, if µ (D 0 L 2 ([ T,0]) ˆ 2 h, [X( )] t (µ) = µ({0,0})[x] t,

37 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 37/81 The particular case of a finite quadratic variation process X such that[x] t = t 0 σ2 (s,x s ( ))ds. If µ χ 0 then [X,X] t (µ) = D t dµ(x,y) If µ (D 0 L 2 ([ T,0])) ˆ 2 h, then t+x 0 σ 2 (s,x s ( ))ds. [X,X] t = µ({(0,0)}) t 0 σ 2 (s,x s ( ))ds.

38 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 38/81 Itô s formula for the corresponding window process. Let F : [0,T] C R of class C 1,2 such that D 2 F : [0,T] C χ 0 continuous. We denote DF(t,η) = D δ 0 F(t,η)δ 0 (dx)+d F(t,η), with D δ 0 F : [0,T] C R. The application of Itô formula gives F(t,X t ) = F(0,X 0 )+ + t 0 t 0 s F(s,X s )ds, D δ 0 F(s,X s )d X s + t 0 L s F(s,X s )ds

39 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 39/81 L t G(η) = I(G)(t,η) + 1 D 2 dxdyg(η)σ 2 2 (t+x,η(x)), D t I(G)(t,η) = D dx G(η)d η(x), provided that [ t,0] I(G)(t,η) = lim ε 0 [ t,0] D dx G(η) η(x+ε) η(x) ε exists and I fulfills some technical assumptions..

40 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 40/81 5 Towards a Robust Clark-Ocone type formula We set B = C = C([ T,0]). X real continuous stochastic process with values in R. X 0 = 0, [X] t = t 0 σ2 (r,x r ( ))dr, where σ : [0,T] C R is continuous.

41 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 41/81 Representation ofh = H(X T ( )) Conformally to what we mentioned at the beginning, we aim at finding functions u,v : [0,T] C R such that In particular V t = u(t,x t ( )), Z t = v(t,x t ( )), V t = h T t Z s d X s,t [0,T]. h = u(0,x 0 ( ))+ T 0 v(s,x s ( ))d X s.

42 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 42/ A toy model (related to the Black-Scholes formula) Let (S t ) be the price of a financial asset of the type S t = exp(σw t σ2 2 t), σ > 0. Let f : R R be a continuous function and h = f(s T ) = f(w T ) where f(y) = f ( exp(σy ). σ2t) 2

43 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 43/81 Let Ũ : [0,T] R R solving { t Ũ(t,x)+ σ2 2 xxũ(t,x) = 0 Ũ(T,x) = f(x) x R. Applying classical Itô formula we obtain h = Ũ(0,S 0)+ = U(0,W 0 )+ T 0 T for a suitable U : [0,T] R R. 0 x Ũ(s,S s )ds s x U(s,W s )dw s,

44 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 44/81 Robustness with respect to the volatility. Does one have a similar representation ifw is replaced by a finite quadratic variationx such that[x] t t? The answer is positive! because of Theorem 4.

45 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 45/81 Proposition 10 Let X such that [X] t = σ 2 t. A1 f : R R continuous and polynomial growth. A2 U C 1,2 ([0,T[ R) C 0 ([0,T] R) such that { t U(t,x)+ σ2 2 xxu(t,x) = 0 v(t,x) = f(x).

46 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 46/81 Then h := f(x T ) = U(0,X 0 )+ T x U(s,X s )d X s 0 }{{} improper forward integral Schoenmakers-Kloeden (1999) Zähle (2002) Coviello-Russo (2006) Bender-Sottinen-Valkeila (2006)

47 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 47/81 Natural question Generalization to the case of path dependent options? As first step we revisit the toy model.

48 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 48/ The toy model revisited Proposition 11 We set B = C = C([ T,0]) and η C and we define Then H : C R, by H(η) := f(η(0)) u : [0,T] C R, by u(t,η) := U(t,η(0)) and solves u C 1,2 ([0,T[ C;R) C 0 ([0,T] C;R)

49 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 49/81 { t u(t,η) + σ2 2 u(t, η) = D t Ddx 2 dy u(t,η) = 0 H(η) (Here D u 0).

50 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 50/81 Proof. u(t,η) = U(T,η(0)) = f(η(0)) = H(η) t U (t,η) = t U (t,η(0)) Du(t,η) = x U (t,η(0)) δ 0 D 2 u(t,η) = xxu 2 (t,η(0)) δ 0 δ 0 t u(t,η)+ 1 2 D2 u(t,η)({0,0}) = 0.

51 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 51/ The general representation The considerations of previous section bring us to the following. Theorem 12 Let us consider the following. H : C R continuous. u C 1,2 ([0,T[ C) C 0 ([0,T] C) For any (t,η) [0,T] C, [ t,0] D dx u(t,η)d η(x) is well-defined (with a technical condition). (t,η) D 2 u(t,η) is continuous from [0,T] C to χ 0.

52 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 52/81 Suppose that u solves the Infinite dimensional PDE { t u(t,η) + L t u(t,η) = 0 u(t, η) = H(η). (1)

53 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 53/81 Then, h = V 0 + T 0 (in the possibly improper sense) with V 0 = u(0,x 0 ( )) Z s = D δ 0 u(s,x s ( )) Z s d X s (2)

54 Definition 13 We call (1) path dependent Kolmogorov PDE. related to X. Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 54/81

55 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 55/81 6 Kolmogorov path dependent PDE. The window of diffusion process Let σ : [0,T] R R continuous and Lipschitz. Consider the stochastic flow (X s,ξ t ) s t T, for s [t,t],ξ R, where, setting X = X s,ξ solves the SDE X t = ξ + t s σ(r,x r )dw r, where (W,F t ) is a classical Wiener process.

56 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 56/81 Remark X is a.s. continuous in all the three variables (s, t, ξ). 2. Let σ is of class C 0,2 ( R), such that σ, x σ and 2 xxσ are Hölder continuous, with = {(s,t) 0 s t T}. Under previous assumptions we have, for k = 0,1,2, sup 0 s T E( sup (k) ξ X s,ξ t p ) M, t [s,t] for every p 1.

57 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 57/81 Associated with X, we link the following functional stochastic flow. For η C, we set { X s,η η(t s+x) : x s t t (x) = : x s t. X s,η(0) t+x (3) Remark 15 If σ = 1 we denote the corresponding window Brownian flow by W s,η t := X s,η t. From now on we will suppose σ to be as in Remark 14.

58 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 58/81 Let h = H(X T ( )), where X = X 0,x 0 T, for some x 0 R. We set V t = E(h F t ). Then, there is u : [0,T] C R such that V t = u(t,x t ( )). (4) It is given by u(t,η) = E(H(X t,η T )).

59 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 59/81 Aim. Under suitable conditions on H, we aim to show that u is the unique solution of the path dependent Kolmogorov equation (1), taking values in C( T,0]). This justifies to say that u defined in (4) is the virtual solution for (1).

60 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 60/81 Definition 16 (Strict solution) u : [0,T] C R of class C 1,2 ([0,T[ C) C 0 ([0,T] C) is said to be a solution of (1) if I(u)(t,η) := ] t,0] D dxu(t,η)d η(x) is well-defined ; (t,η) D 2 u(t,η) is continuous into some Chi-subspace χ. It solves effectively (1).

61 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 61/81 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]). ( T h := H(X T ( )) = f ϕ 0 1(s)d X s,..., T ϕ 0 n(s)d X s ); f : R n R continuous and with linear growth; ϕ i C 2 ([0,T];R), 1 i n. ( ) 0 The determinant of Σ t := ϕ x i(y)ϕ j (y)dy bigger than zero for every x [ T,0[. is

62 Uniqueness for solutions of the Kolmogorov Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 62/81 type PDE. Let u 1,u 2 : [0,T] C R of class C 1,2 ([0,T[ C) C 0 ([0,T] C) being strict solutions of (1) of polynomial growth. If u 1,u 2 solve the Kolmogorov type equation then u 1 = u 2.

63 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 63/81 Sketch of the Proof. We fix (s,η) [0,T] C. We have to prove that u 1 (s,η) = u 2 (s,η). Without restriction of generality we set s = 0. We apply Itô formula to X t = X 0,η t. We take than the expectation and we obtain u i (0,η) = E(H(X 0,η T )). This shows the uniqueness.

64 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 64/81 7 Path-dependent semilinear Kolmogorov equation Here we concentrate on the case σ = 1, the general case being in Cosso-Russo (in preparation).. We study the path-dependent semilinear Kolmogorov equation: { t U +L t U = F(t,η,U,D δ 0 U), (t,η) [0,T[ C, where U(T,η) = G(η), η C,

65 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 65/81 G: C([ T,0]) R F: [0,T] C([ T,0]) R R R are Borel measurable functions. We refer to t U +L t U, as the path-dependent heat operator.

66 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 66/ Strict solutions Definition 17 A function U: [0,T] C([ T,0]) R in C 1,2 ([0,T[ C) C([0,T] C), which solves the path-dependent semilinear Kolmogorov equation, is called a strict solution.

67 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 67/81 Strict solutions: uniqueness Theorem 18 (Uniqueness) Let G: C([ T,0]) R and F: [0,T] C([ T,0]) R R R be Borel measurable functions 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.

68 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 68/81 Let U: [0,T] C([ T,0]) R be a strict solution to the path-dependent nonlinear Kolmogorov equation, satisfying the polynomial growth condition U(t,η) C ( 1+ η m ), (t,η) [0,T] C([ T,0]).

69 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 69/81 Then, we have U(t,η) = Y t,η t, (t,η) [0,T] C([ T,0]), where (Y t,η s,z t,η s ) s [t,t] = (U(s,W t,η s ),D δ 0 U(s,W t,η s )1 [t,t[ (s)) s [t,t] is the solution to the backward stochastic differential equation, P -a.s.,

70 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 70/81 Y t,η s = G(W t,η T )+ T s F(r,W t,η r,y t,η r,z t,η r )dr T s Z t,η r dw r, for all t s T. In particular, there exists at most one strict solution to the path-dependent nonlinear Kolmogorov equation. Here W is the functional stochastic flow associated with = x+(w t W s ) (classical Wiener process). W s,x t

71 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 71/81 Alternative methods to strict solutions for Kolmogorov path dependent PDEs Dupire-Cont-Fournié (2010). Flandoli-Zanco (2014). Leão-Ohashi-Simas (2014).

72 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 72/ Strong-viscosity solutions: introduction Various definitions of viscosity-type solutions for path-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.

73 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 73/81 We propose a notion of solution which is not based on the standard definition of viscosity solution given in terms of test functions or jets. 7.3 Strong-viscosity solutions Idea and origin Our notion of solution is defined, in a few words, as the pointwise limit of strict solutions to perturbed equations.

74 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 74/81 Our definition is more similar in spirit to the concept of good solution, which turned out to be equivalent to the definition of L p -viscosity solution for certain fully nonlinear partial differential equations. Our definition is likewise inspired by the notion of strong solution, even though strong solutions are required to be more regular than simply continuous.

75 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 75/81 Definition 19 A function U: [0,T] C([ T,0]) R is called a strong-viscosity solution to the path-dependent nonlinear Kolmogorov equation if there exists a sequence (U n,g n,f n ) n satisfying: (i) U n : [0,T] C([ T,0]) R, G n : C([ T,0]) R, and F n : [0,T] C([ T,0]) R R R are equicontinuous functions such that, for some positive constants C and m, independent of n, F n (t,η,y,z) F n (t,η,y,z ) C( y y + z z ), U n (t,η) + G n (η) + F n (t,η,0,0) C ( 1+ η m ), for all (t,η) [0,T] C([ T,0]), y,y R, and z,z R.

76 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 76/81 (ii) U n is a strict solution to t U n +L t U n = F n (t,η,u n,d δ 0 U n ), (t,η) [0,T) C([ T,0]), U n (T,η) = G n (η), η C([ T,0]). (iii) (U n (t,η),g n (η),f n (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.

77 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 77/ Strong-viscosity solutions: uniqueness Theorem 20 (Uniqueness) Let U: [0,T] C([ T,0]) R be a strong-viscosity solution to the path-dependent nonlinear Kolmogorov equation. Then, we have U(t,η) = Y t,η t, (t,η) [0,T] C([ T,0]), where (Ys t,η,zs t,η ) s [t,t], with Ys t,η = U(s,W t,η s ), solves the backward stochastic differential equation, P -a.s.,

78 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 78/81 Y t,η s = G(W t,η T )+ T s F(r,W t,η r,y t,η r,z t,η r )dr T s Z t,η r dw r, for all t s T. In particular, there exists at most one strong-viscosity solution to the path-dependent nonlinear Kolmogorov equation.

79 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 79/81 Strong-viscosity solutions: existence Theorem 21 (Existence) Let F 0 and G: C([ T,0]) R be uniformly continuous and satisfying the polynomial growth condition G(η) C(1+ η m ), η C([ T,0]), for some positive constants C and m. Then, there exists a unique strong-viscosity solution U to the path-dependent heat equation, which is given by U(t,η) = E [ G(W t,η T )], for all (t,η) [0,T] C([ T,0]).

80 Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 80/ Paper in preparation. A. Cosso, F. Russo. Existence for functional dependent Kolmogorov type equation: the strict and strong-viscosity solution case.

81 Thank you for you attention. Calculus via regularizations in Banach spaces: path dependent calculus and Kolmogorov type equations. p. 81/81