Classical Solutions of Path-dependent PDEs and Functional Forward-Backward Stochastic Systems

In this paper we study the relationship between functional forward-backward stochastic systems and path-dependent PDEs. In the framework of functional It\^o calculus, we introduce a path-dependent PDE and prove that its solution is uniquely determined by a functional forward-backward stochastic system.


Introduction
It is well known that quasilinear parabolic partial differential equations are related to Markovian forward-backward stochastic differential equations (see [12], [10] and [9]), which generalizes the classical Feynman-Kac formula. Recently in the framework of functional Itô calculus, a path-dependent PDE was introduced by Dupire [5] and the so-called functional Feynman-Kac formula was also obtained. For a recent account and development of this theory we refer the reader to [1], [2], [3], [14], [13], [6] and [4].
In this paper, we study a functional forword-backward system and its relation to a quasilinear parabolic path-dependent PDE. In more details, the functional forword-backward system is described by the following forword-backward SDE:

Functional Itô calculus
The following notations and tools are mainly from Dupire [5]. Let T > 0 be fixed. For each t ∈ [0, T ], we denote by Λ t the set of càdlàg R d -valued functions on [0, t]. For each γ ∈ Λ T the value of γ at time s ∈ [0, T ] is denoted by γ(s). Thus γ = γ(s) 0≤s≤T is a càdlàg process on [0, T ] and its value at time s is γ(s). The path of γ up to time t is denoted by γ t , i.e., γ t = γ(s) 0≤s≤t ∈ Λ t . We denote Λ = t∈[0,T ] Λ t .
For each γ t ∈ Λ and x ∈ R d we denote by γ t (s) the value of γ t at s ∈ [0, t] and γ x t := (γ t (s) 0≤s<t , γ t (t)+x) which is also an element in Λ t .
Let (·, ·) and | · | denote the inner product and norm in R n . We now define a distance on Λ. For each 0 ≤ t,t ≤ T and γ t ,γt ∈ Λ, we denote It is obvious that Λ t is a Banach space with respect to · and d ∞ is not a norm.
Definition 2.2. Let u : Λ → R and γ t ∈ Λ be given. If there exists p ∈ R d , such that Then we say that u is (vertically) differentiable at γ t and denote the gradient of D x u(γ t ) = p. u is said to be vertically differentiable in Λ if D x u(γ t ) exists for each γ t ∈ Λ. We can similarly define the Hessian For each γ t ∈ Λ we denote It is clear that γ t,s ∈ Λ s .
then we say that u(γ t ) is (horizontally) differentiable in t at γ t and denote D t u(γ t ) = a. u is said to be Definition 2.4. Define C j,k (Λ) as the set of function u := (u(γ t )) γt∈Λ defined on Λ which are j times horizontally and k times vertically differentiable in Λ such that all these derivatives are Λ-continuous.
We give the following assumption: and ∀x t ∈ Λ, Then we have the following theorem (see [7]):

Regularity
We first recall some notions in Pardoux and Peng [10]. C n (R p ; R q ), C n b (R p ; R q ), C n p (R p ; R q ) will denote respectively the set of functions of class C n from R p into R q , the set of those functions of class C n b whose partial derivatives of order less than or equal to n are bounded, and the set of those functions of class C n p which, together with all their partial derivatives of order less than or equal to n, grow at most like a polynomial function of the variable x at infinity. Now we give the definition of derivatives in our context. Under the above Assumption 2.1 we have has a uniqueness solution. For t ≤ s ≤ T , set Then the following definition of derivatives will be used frequently in the sequel.
Definition 3.1. An R n -valued function g is said to be in C 2 (Λ γt,T ), if for γ 1 ∈Λ γt,T and γ 2 ∈Λ γ y t ,T , there exist p 1 ∈ R d and p 2 ∈ R d × R d such that p 2 is symmetric, We denote g ′ γt (γ 1 ) := p 1 , and g ′′ γt (γ 1 ) := p 2 . g is said to be in C 2 l,lip (Λ t,T ) if g ′ γt (γ) and g ′′ γt (γ) exist for each γ t ∈ Λ t , and there exists some constants C ≥ 0 and k ≥ 0 depending only on g such that for Now we consider the solvability of equation (2.2).
and the first order partial derivatives in r,y and z are bounded, as well as their derivatives of up to order two with respect to y,z.

Regularity of the solution of FBSDEs
We assume the Lipschitz constants with respect to b, σ, h are C and k. Then we have the following estimates for the solution of FBSDE (2.1) and (2.2).
Proof. To simplify presentation, we only study the case n = d = 1.
Applying Itô's formula to (Y γt,x (s)) 2 e β1s yields that Then we have and Applying Itô's formula to (X γt,x (s)) 2 yields that By inequality 2ab ≤ a 2 + b 2 and Burkholder-Davis-Gundy's inequality, there is a C 0 such that, By Assumption 2.1 and Gronwall's inequality, we have (note that C 0 will change line by line) By Assumptions 3.1 and 3.2 and taking β 1 = 4C 2 + 1, we have where q = 2(1 + k). This completes the proof. Now we study the regularity properties of the solution of FBSDE (2.1), (2.2) with respect to the "parameter" γ t . For 0 ≤ s < t ≤ T , define Y γt (s) = Y γt (s ∨ t) and Z γt (s) = 0. (i) and (e 1 , · · · , e n ) is an orthonormal basis of R n .
Under Assumptions 3.1 , 3.2, using the same method as in Lemma 3.1, we get the first three inequalities.
For the next three inequalities, we write (∆ i h Y γt , ∆ i h Z γt ) as the solution of the following linearized BSDE: Then the same calculus implies that Then it solves the following BSDẼ Thus, under Assumptions 3.1, 3.2, similarly as in Lemma 3.1, we can get the last three inequalities.
T ], z ∈ R n } has a version which is a.e. of class Proof. We only consider one dimensional case. Applying Lemma 3.1, for each h,h ∈ R\{0} and k,k ∈ R, By kolmogorov's criterion, there exists a continuous derivative of Y γ z t (s) with respect to z. There also exists a mean-square derivative of Z γ z t (s) with respect to z, which is mean square continuous in z. We denote them by By Theorem 3.1 and definition 3.1, (D z Y γt , D z Z γt ) is the solution of the following BSDE: It is easy to check that the above BSDE has a uniqueness solutions. Thus the existence of a continuous second order derivative of Y γ z t (s) with respect to z is proved in a similar way.
We have the following results about u(γ t ).
Proof. For given γ t1 , t 1 < t, set X(r) = xI 0≤r≤t1 . Consider the solution of FBSDE (2.1) and (2.2) on [t, T ]: We need to prove u(X is the solution of the following BSDE: Multiplying by I Ai and adding the corresponding terms, we obtain: By the uniqueness and existence theorem of BSDE, we get Y γt a.s. Then, by the definition of u, we get For the general case, following the method in Peng and Wang [14] (Lemma 4.3), we choose a simple adapted process This completes the proof.
By Theorem 3.1 and 3.2 and the definition of vertical derivative, we have the following corollary.
Proof. By Theorem 3.2 we know that D z u(γ t ) and D zz u(γ t ) exist. In the following, we only prove u(γ t ) is Λ-continuous. The proof for the continuous property of D z u(γ t ) and D zz u(γ t ) is similar. Taking expectation on both sides of equation (2.2), By Theorem 3.1, for some constant C 1 depending only in C, k and T , This completes the proof.
Peng and Wang [14] extends this result to the path-dependent case. The corresponding BSDE is where W γt T = I s≤t γ t (s) + I t<s≤T (γ t (t) + W (s) − W (t)). Then under some assumptions, they obtained Z γt (s) = D x u(W γt s ), P − a.s.
In particular, This completes the proof. Now we give the proof of Theorem 3.3.
For anyt ≥ t,γt ∈ Λt, we consider the following BSDE: This completes the proof.
where Lu = 1 2 tr[(σσ T )D xx u] + b, D x u . Proof. By the assumptions of this theorem" we know that b(γ t ) and σ(γ t ) is uniformly Lipschtiz continuous and the following SDE has a uniqueness solution.