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

Authors: Shaolin Ji; Shuzhen Yang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

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


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## Body

## 1. Introduction

It is well known that quasilinear parabolic partial differential equations are related to Markovian forward-backward stochastic differential equations (see [1–3]), which generalizes the classical Feynman-Kac formula. Recently in the framework of functional Itô calculus, a path-dependent PDE was introduced by Dupire [4] 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 [5–11].In this paper, we study a functional forward-backward system and its relation to a quasilinear parabolic path-dependent PDE. In more details, the functional forward-backward system is described by the following forward-backward SDE:(1)Xγt(s)=γt(t)+∫ts‍b(Xrγt)dr+∫ts‍σ(Xrγt)dW(r),Xγt(r)=γt(r),0≤r≤t,Yγt(s)=g(XTγt)-∫sT‍h(Xrγt,Yγt(r),Zγt(r))dr=-∫sT‍Zγt(r)dW(r),s∈[t,T].Equation (1) is an uncoupled functional forward-backward system, its general the results of [8], and there are many applications of the uncoupled functional forward-backward system in optimal control problem. The main difference is that we give a weaker requirement of g and h about X, and we also establish some estimates and regularity results for the solution with respect to paths. Then, we prove that the solution of (1) is the unique classical solution of the following path-dependent PDE:
(2)Dtu(γt)+ℒu(γt)=h(γt,u(γt),Dxu(γt)σ(γt)),u(γT)=g(γT),γT∈Λ,
where
(3)ℒu=12tr[σσTDxxu]+〈b,Dxu〉.The paper is organized as follows: in Section2, we give the notations and results on functional FBSDEs and functional Itô calculus. Some estimates and regularity results for the solution of FBSDEs are established in Section 3. Finally, we prove the relationship between functional FBSDEs and path-dependent PDEs in Section 4.


## 2. Preliminaries

### 2.1. Functional FBSDEs

LetΩ=C([0,T];ℝd) and let P be the Wiener measure on (Ω,𝔹(Ω)). We denote by W=(W(t)t∈[0,T]) the cannonical Wiener process, with W(t,ω)=ω(t), t∈[0,T], ω∈Ω. For any t∈[0,T] we denote by ℱt the P-completion of σ(W(s),s∈[0,t]).For anyt∈[0,T], we denote by L2(Ω,ℱt;ℝn) the set of all square integrable ℱt-measurable random variables, M2(0,T;ℝn) the set of all ℝn-valued ℱt-adapted processes ϑ(·) such that
(4)E∫0T‍|ϑ(s)|2ds<+∞.Lett∈[0,T] and γt∈Λ. For every s∈[t,T], we consider the following functional forward-backward SDEs:
(5)Xγt(s)=γt(t)+∫ts‍b(Xrγt)dr+∫ts‍σ(Xrγt)dW(r),(6)Yγt(s)=g(XTγt)-∫sT‍h(Xrγt,Yγt(r),Zγt(r))dr==-∫sT‍Zγt(r)dW(r),
where
(7)Xγt(s)=γt(s),s∈[0,t].
The processes X,Y,Z take values in ℝn, ℝn, ℝn×d; b, h, σ, and g take values in ℝn, ℝn, ℝn×d, and ℝn. Equations (5) and (6) can be rewritten as
(8)dXγt(s)=b(Xsγt)ds+σ(Xsγt)dW(s),dYγt(s)=h(Xsγt,Yγt(s),Zγt(s))ds+Zγt(s)dW(s),Xγt(t)=γt(t),Yγt(T)=g(XTγt).
For z∈ℝn×d, we define |z|={tr(zzT)}1/2.  For z1∈ℝn×d, z2∈ℝn×d,
(9)((z1,z2))=tr(z1(z2)T),
and for u1=(y1,z1)∈ℝn×ℝn×d, u2=(y2,z2)∈ℝn×ℝn×d(10)[u1,u2]=〈y1,y2〉+((z1,z2)).We give the following assumption.Assumption 1.
For allxT1,xT2∈Λ, b(x·1), σ(x·1)∈M2, and t∈[0,T], there exists a constant c1>0, such that
(11)|b(xt1)-b(xt2)|+|σ(xt1)-σ(xt2)|≤c1∥xt1-xt2∥,a.e.
and for all xt∈Λ,
(12)|b(xt)|+|σ(xt)|≤c1(1+∥xt∥),a.e.Definition 2.
X
:
[
0
,
T
]
×
Ω
→
ℝ
n is called an adapted solution of (5), if X∈M2(0,T;ℝn), and it satisfies (5) P-a.s.Then we have the following theorem (see [12]).Theorem 3.
Let Assumption1 hold, then there exists a unique adapted solution X for (5).


### 2.2. Functional Itô Calculus

The following notations and tools are mainly from Dupire [4]. Let T>0 be fixed. For each t∈[0,T], we denote by Λt the set of càdlàg ℝd-valued functions on [0,t]. For each γ∈ΛT 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 γt, that is, γt=γ(s)0≤s≤t∈Λt. We denote Λ=⋃t∈[0,T]‍Λt. For each γt∈Λ and x∈ℝd we denote by γt(s) the value of γt at s∈[0,t] and γtx:=(γt(s)0≤s<t, γt(t)+x) which is also an element in Λt.Let〈·,·〉 and |·| denote the inner product and norm in ℝn. We now define a distance on Λ. For each 0≤t,t-≤T and γt,γ-t-∈Λ, we denote
(13)∥γt∥:=sups∈[0,t]|γt(s)|,∥γt-γ-t-∥:=sups∈[0,t∨t-]|γt(s∧t)-γ-t-(s∧t-)|,d∞(γt,γ-t-):=sup0≤s≤t∨t-|γt(s∧t)-γ-t-(s∧t-)|+|t-t-|.
It is obvious that Λt is a Banach space with respect to ∥·∥ and d∞ is not a norm.Definition 4.
A functionu:Λ↦ℝ is said to be Λ-continuous at γt∈Λ, if for any ε>0 there exists δ>0 such that for each γ-t-∈Λ with d∞(γt,γ-t-)<δ, we have |u(γt)-u(γ-t-)|<ε. u is said to be Λ-continuous if it is Λ-continuous at each γt∈Λ.Definition 5.
Letu:Λ↦ℝ and γt∈Λ be given. If there exists p∈ℝd, such that
(14)u(γtx)=u(γt)+〈p,x〉+o(|x|)asx⟶0,x∈ℝd,
then we say that u is vertically differentiable at γt and denote the gradient of Dxu(γt)=p. If Dxu(γt) exists for each γt∈Λ, u is said to be vertically differentiable in Λ.
We can similarly define the HessianDxxu(γt). It is an 𝕊(d)-valued function defined on Λ, where 𝕊(d) is the space of all d×d symmetric matrices.For eachγt∈Λ we denote
(15)γt,s(r)=γt(r)1[0,t)(r)+γt(t)1[t,s](r),r∈[0,s].
It is clear that γt,s∈Λs.Definition 6.
For a givenγt∈Λ if we have
(16)u(γt,s)=u(γt)+a(s-t)+o(|s-t|)ass⟶t,s≥t,
then we say that u(γt) is (horizontally) differentiable in t at γt and Dtu(γt)=a. u is said to be horizontally differentiable in Λ if Dtu(γt) exists for each γt∈Λ.Definition 7.
Defineℂj,k(Λ) as the set of function u:=(u(γt))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.The following Itô formula was firstly obtained by Dupire [4] and then generalized by Cont and Fournié [5–7].Theorem 8 (functional Itô’s formula).
Let(Ω,ℱ,(ℱt)t∈[0,T],P) be a probability space, if X is a continuous semi-martingale and u is in ℂ1,2(Λ), then for any t∈[0,T),
(17)u(Xt)-u(X0)=∫0t‍Dsu(Xs)ds+∫0t‍Dxu(Xs)dX(s)=+12∫0t‍Dxxu(Xs)d〈X〉(s),P-a.s.


## 2.1. Functional FBSDEs

LetΩ=C([0,T];ℝd) and let P be the Wiener measure on (Ω,𝔹(Ω)). We denote by W=(W(t)t∈[0,T]) the cannonical Wiener process, with W(t,ω)=ω(t), t∈[0,T], ω∈Ω. For any t∈[0,T] we denote by ℱt the P-completion of σ(W(s),s∈[0,t]).For anyt∈[0,T], we denote by L2(Ω,ℱt;ℝn) the set of all square integrable ℱt-measurable random variables, M2(0,T;ℝn) the set of all ℝn-valued ℱt-adapted processes ϑ(·) such that
(4)E∫0T‍|ϑ(s)|2ds<+∞.Lett∈[0,T] and γt∈Λ. For every s∈[t,T], we consider the following functional forward-backward SDEs:
(5)Xγt(s)=γt(t)+∫ts‍b(Xrγt)dr+∫ts‍σ(Xrγt)dW(r),(6)Yγt(s)=g(XTγt)-∫sT‍h(Xrγt,Yγt(r),Zγt(r))dr==-∫sT‍Zγt(r)dW(r),
where
(7)Xγt(s)=γt(s),s∈[0,t].
The processes X,Y,Z take values in ℝn, ℝn, ℝn×d; b, h, σ, and g take values in ℝn, ℝn, ℝn×d, and ℝn. Equations (5) and (6) can be rewritten as
(8)dXγt(s)=b(Xsγt)ds+σ(Xsγt)dW(s),dYγt(s)=h(Xsγt,Yγt(s),Zγt(s))ds+Zγt(s)dW(s),Xγt(t)=γt(t),Yγt(T)=g(XTγt).
For z∈ℝn×d, we define |z|={tr(zzT)}1/2.  For z1∈ℝn×d, z2∈ℝn×d,
(9)((z1,z2))=tr(z1(z2)T),
and for u1=(y1,z1)∈ℝn×ℝn×d, u2=(y2,z2)∈ℝn×ℝn×d(10)[u1,u2]=〈y1,y2〉+((z1,z2)).We give the following assumption.Assumption 1.
For allxT1,xT2∈Λ, b(x·1), σ(x·1)∈M2, and t∈[0,T], there exists a constant c1>0, such that
(11)|b(xt1)-b(xt2)|+|σ(xt1)-σ(xt2)|≤c1∥xt1-xt2∥,a.e.
and for all xt∈Λ,
(12)|b(xt)|+|σ(xt)|≤c1(1+∥xt∥),a.e.Definition 2.
X
:
[
0
,
T
]
×
Ω
→
ℝ
n is called an adapted solution of (5), if X∈M2(0,T;ℝn), and it satisfies (5) P-a.s.Then we have the following theorem (see [12]).Theorem 3.
Let Assumption1 hold, then there exists a unique adapted solution X for (5).


## 2.2. Functional Itô Calculus

The following notations and tools are mainly from Dupire [4]. Let T>0 be fixed. For each t∈[0,T], we denote by Λt the set of càdlàg ℝd-valued functions on [0,t]. For each γ∈ΛT 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 γt, that is, γt=γ(s)0≤s≤t∈Λt. We denote Λ=⋃t∈[0,T]‍Λt. For each γt∈Λ and x∈ℝd we denote by γt(s) the value of γt at s∈[0,t] and γtx:=(γt(s)0≤s<t, γt(t)+x) which is also an element in Λt.Let〈·,·〉 and |·| denote the inner product and norm in ℝn. We now define a distance on Λ. For each 0≤t,t-≤T and γt,γ-t-∈Λ, we denote
(13)∥γt∥:=sups∈[0,t]|γt(s)|,∥γt-γ-t-∥:=sups∈[0,t∨t-]|γt(s∧t)-γ-t-(s∧t-)|,d∞(γt,γ-t-):=sup0≤s≤t∨t-|γt(s∧t)-γ-t-(s∧t-)|+|t-t-|.
It is obvious that Λt is a Banach space with respect to ∥·∥ and d∞ is not a norm.Definition 4.
A functionu:Λ↦ℝ is said to be Λ-continuous at γt∈Λ, if for any ε>0 there exists δ>0 such that for each γ-t-∈Λ with d∞(γt,γ-t-)<δ, we have |u(γt)-u(γ-t-)|<ε. u is said to be Λ-continuous if it is Λ-continuous at each γt∈Λ.Definition 5.
Letu:Λ↦ℝ and γt∈Λ be given. If there exists p∈ℝd, such that
(14)u(γtx)=u(γt)+〈p,x〉+o(|x|)asx⟶0,x∈ℝd,
then we say that u is vertically differentiable at γt and denote the gradient of Dxu(γt)=p. If Dxu(γt) exists for each γt∈Λ, u is said to be vertically differentiable in Λ.
We can similarly define the HessianDxxu(γt). It is an 𝕊(d)-valued function defined on Λ, where 𝕊(d) is the space of all d×d symmetric matrices.For eachγt∈Λ we denote
(15)γt,s(r)=γt(r)1[0,t)(r)+γt(t)1[t,s](r),r∈[0,s].
It is clear that γt,s∈Λs.Definition 6.
For a givenγt∈Λ if we have
(16)u(γt,s)=u(γt)+a(s-t)+o(|s-t|)ass⟶t,s≥t,
then we say that u(γt) is (horizontally) differentiable in t at γt and Dtu(γt)=a. u is said to be horizontally differentiable in Λ if Dtu(γt) exists for each γt∈Λ.Definition 7.
Defineℂj,k(Λ) as the set of function u:=(u(γt))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.The following Itô formula was firstly obtained by Dupire [4] and then generalized by Cont and Fournié [5–7].Theorem 8 (functional Itô’s formula).
Let(Ω,ℱ,(ℱt)t∈[0,T],P) be a probability space, if X is a continuous semi-martingale and u is in ℂ1,2(Λ), then for any t∈[0,T),
(17)u(Xt)-u(X0)=∫0t‍Dsu(Xs)ds+∫0t‍Dxu(Xs)dX(s)=+12∫0t‍Dxxu(Xs)d〈X〉(s),P-a.s.


## 3. Regularity

We first recall some notions in Pardoux and Peng [2]. Cn(ℝp;ℝq), Cbn(ℝp;ℝq), Cpn(ℝp;ℝq) will denote, respectively, the set of functions of class Cn from ℝp into ℝq, the set of those functions of class Cbn whose partial derivatives of order less than or equal to n are bounded, and the set of those functions of class Cpn 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 Assumption1 we have that
(18)dXγt(s)=b(Xsγt)ds+σ(Xsγt)dW(s),Xγt(t)=γt(t),
has a unique solution. For t≤s≤T, set
(19)Λ~γt,s:={γ-s:γ-(h)=Xγt(h,ω),0≤h≤s,ω∈Ω},Λ~t,s:=⋃γt∈Λt‍Λ~γt,s,Λ~t:=⋃t≤s≤T‍Λ~t,s.
Then the following definition of derivatives will be used frequently in the sequel.Definition 9.
Anℝn-valued function g is said to be in C2(Λ~γt,T), if for γ1∈Λ~γt,T and γ2∈Λ~γty,T, there exist p1∈ℝd and p2∈𝒮d (𝒮d is the set of all d order symmetric matrix) such that
(20)g(γ2)-g(γ1)=〈p1,y〉+12〈p2y,y〉+o(|y|2),x∈ℝd.
We denote gγt′(γ1):=p1, and gγt′′(γ1):=p2. g is said to be in Cl,lip2(Λ~t,T) if gγt′(γ) and gγt′′(γ) exist for each γt∈Λt, and there exist some constants C≥0 and k≥0 depending only on g such that for each γ,γ-∈ΛT,t,s∈[0,T],
(21)|g(γ)-g(γ-)|≤C(∥γ∥k+∥γ-∥k)∥γ-γ-∥,
and for each γ∈Λ~t,T,γ-∈Λ~s,T,t,s∈[0,T],
(22)|Φγt(γ)-Φγs(γ-)|≤C(∥γ∥k+∥γ-∥k)(|t-s|+∥γ-γ-∥)
with Φ=gγt′(γ),gγt′′(γ). We can also define C2(Λ~t,s), Cl,lip2(Λ~t,s), Cl,lip1(Λ~t,s), Cl,lip(Λ~t,s) and C2(Λ~t), Cl,lip2(Λ~t), Cl,lip1(Λ~t), Cl,lip(Λ~t).Now we consider the solvability of (6).Assumption 10.
Letg be an ℝn-valued function on ΛT. Moreover g∈Cl,lip2(Λ~t,T) with the Lipschitz constants C and k.Assumption 11.
Leth(γt,y,z)=h-(t,γ(t),y,z), where h-:[0,T]×ℝn×ℝn×ℝn×d↦ℝn is such that (t,r,y,z)↦Ψ-(t,r,y,z) is of class Cp0,3([0,T]×ℝn×ℝn×ℝn×d;ℝn) 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.
It is obvious under Assumptions1, 10, and 11 the FBSDE (5) and (6) has a unique solution (see [12–14]).


### 3.1. Regularity of the Solution of FBSDEs

We assume the Lipschitz constants with respect tob, σ, h are C and k. Then we have the following estimates for the solutions of FBSDE (5) and (6).Lemma 12.
Under Assumptions1, 10, and 11, for all p≥2 there exist C2 and q depending only on C,T,k,x such that
(23)E[sups∈[t,T]|Xγt(s)|p]≤C2(1+∥γt∥p),E[sups∈[t,T]|Yγt(s)|p]≤C2(1+∥γt∥q),E[(∫tT‍|Zγt(s)|2ds)p/2]≤C2(1+∥γt∥q).Proof.
To simplify presentation, we only study the casen=d=1, and p=2.
The application of Itô's formula to(Yγt,x(s))2eβ1s yields that
(24)(Yγt,(s))2eβ1s+∫sT‍eβ1r[(Zγt(r))2+β1(Yγt(r))2]dr==g2(XTγt)eβ1T==-∫sT‍2eβ1rYγt(r)h(Xrγt,Yγt(r),Zγt(r))dr==-∫sT‍2eβ1rYγt(r)Zγt(r)dW(r).
So
(25)(Yγt(s))2+E[∫sT‍eβ1(r-s)[(Zγt(r))2==========+β1(Yγt(r))2(Zγt(r))]dr∣ℱs∫sT‍]==E[g2(XTγt)eβ1(T-s)∣ℱs]==-E[∫sT‍2eβ1(r-s)Yγt(r)======×h(Xrγt,Yγt(r),Zγt(r))dr∣ℱs∫sT‍].
Then we have
(26)Esupt≤s≤T(Yγt(s))2===+E[∫tT‍eβ1(r-t)[(Zγt(r))2+β1(Yγt(T))2]dr]==≤E[g2(XTγt)eβ1(T-t)]===+E[∫tT‍eβ1(r-t)2β1h2(Xrγt,Yγt(r),Zγt(r))dr]===+E[∫tT‍eβ1(r-t)β12(Yγt(r))2d(r)],(27)Esupt≤s≤T(Yγt(s))2===+E[∫tT‍eβ1(r-t)[(Zγt(r))2+β12(Yγt(r))2]dr]==≤E[g2(XTγt)eβ1(T-t)]===+E[∫tT‍eβ1(r-t)2β1h2(Xrγt,Yγt(r),Zγt(r))dr].
Applying Itô's formula to (Xγt,x(s))2 yields that
(28)(Xγt(s))2=γt(t)2+∫ts‍2Xγt(r)b(Xrγt)dr=+∫ts‍2Xγt(r)σ(Xrγt)dW(r)+∫ts‍σ2(Xrγt)dr.
By inequality 2ab≤a2+b2 and Burkholder-Davis-Gundy's inequality, there is a C0 such that
(29)Esupt≤r≤s(Xγt(r))2=≤C0[γt(t)2+E∫ts‍b2(Xrγt)dr======+E∫ts‍(Xγt(r))2dr+E∫ts‍σ2(Xrγt)dr].
By Assumption 1 and Gronwall's inequality, from (29) we have (note that C0 will change line by line)
(30)Esupt≤r≤T(Xγt(r))2≤C0(1+∥γt∥2).
By Assumptions 10 and 11 and taking β1=4C2+1, from (27) we have
(31)Esupt≤s≤T(Yγt(s))2===+E[∫tT‍[(Zγt(r))2+(Yγt(r))2]dr]==≤C0(1+PγtPq),
where q=2(1+k). Similarly we can get the same result for p≥2.
This completes the proof.Now we study the regularity properties of the solution of FBSDE (5), (6) with respect to the “parameter” γt. For 0≤s<t≤T, define Yγt(s)=Yγt(s∨t) and Zγt(s)=0.Theorem 13.
Under Assumptions1, 10 and 11, for allp≥2 there exist C2 and q depending only on C,c2,x such that for any t,t-∈[0,T], γt,γ-t-, and h,h-∈ℝ∖{0}
(i)(32)E[supu∈[t∨t-,T]|Yγt(u)-Yγ-t-(u)|p]==≤C2(1+∥γt∥q+∥γ-t-∥q)(∥γt-γ-t-∥p+|t-t-|p/2),(ii)(33)E[|∫t∨t-T‍|Zγt(u)-Zγ-t-(u)|2du|p/2]==≤C2(1+∥γt∥q+∥γ-t-∥q)(∥γt-γ-t-∥p+|t-t-|p/2),(iii)(34)E[supu∈[t∨t-,T]|ΔhiYγt(u)-ΔhiYγ-t-(u)|p]==≤C2(1+∥γt∥q+∥γ-t-∥q+|h|q+|h-|q)===×(|h-h-|p+∥γt-γ-t-∥p+|t-t-|p/2),(iv)(35)E[|∫t∨t-T‍|ΔhiZγt(u)-ΔhiZγ-t-(u)|2du|p/2]==≤C2(1+∥γt∥q+∥γ-t-∥q+|h|q+|h-|q)===×(|h-h-|p+∥γt-γ-t-∥p+|t-t-|p/2),
where
(36)ΔhiYγt,x(s)=1h(Yγthei(s)-Yγt(s)),ΔhiZγt(s)=1h(Zγthei(s)-Zγt(s))
and (e1,…,en) is an orthonormal basis of  ℝn.Proof.
(
Y
γ
t
-
Y
γ
-
t
-
,
Z
γ
t
-
Z
γ
-
t
-
) can be formed as a linearized BSDE: for each s∈[t∨t-,T],
(37)Yγt(s)-Yγ-t-(s)==g(XTγt)-g(XTγ-t-)==+∫sT‍[(Xrγ-t-,Yγ-t-(r),Zγ-t-(r))h(Xrγt,Yγt(r),Zγt(r))======-h(Xrγ-t-,Yγ-t-(r),Zγ-t-(r))]dr==+∫sT‍(Zγt(r)-Zγ-t-(r))dW(r)==g(XTγt)-g(XTγ-t-,)==-∫sT‍[α^γt,γ-t-(r)+β^γt,γ-t-(Yγt(r)-Yγ-t-(r))=======+δ^γt,γ-t-(Zγt(r)-Zγ-t-(r))]dr==+∫sT‍(Zγt(r)-Zγ-t-(r))dW(r),
where (with Uγt=(Yγt,Zγt))
(38)α^γt,γ-t-(r)=h(Xrγt,Yγ-t-(r),Zγ-t-(r))-h(Xrγ-t-,Yγ-t-(r),Zγ-t-(r)),β^γt,γ-t-(r)=∫01‍∂h∂y(Xrγt,Uγ-t-(r)+θ(Uγt(r)-Uγ-t-(r)))dθ,δ^γt,γ-t-(r)=∫01‍∂h∂z(Xrγt,Uγ-t-(r)+θ(Uγt(r)-Uγ-t-(r)))dθ.
Under Assumptions 10, 11, using the same method as in Lemma 12, we get the first three inequalities.
For the next three inequalities, we write(ΔhiYγt,ΔhiZγt) as the solution of the following linearized BSDE:
(39)ΔhiYγt(s)==1h(g(XTγthei)-g(XTγt))==-∫sT‍[1hα^γt,γthei(r)+β^γt,γthei(r)ΔhiYγt(r)=======+δ^γt,γtheiΔhiZγt(r)1h]dr==-∫tT‍ΔhiZγt(r)dW(r).
Then the same calculus implies that
(40)E[sups∈[t,T]|ΔhiYγt(s)|p+|∫tT‍|ΔhiZγt|2dr|p/2]==≤C2(1+∥γt∥q+|h|q).
Consider
(41)ΔhiYγt(s)-Δh-iYγ-t-(s)==1h(g(XTγthei)-g(XTγt))-1h-(g(XTγ-th-ei)-g(XTγ-t-))==-∫sT‍(ΔhiZγt(r)-Δh-iZγ-t-(r))dW(r)==-{∫sT[1hα^γt,γthei(r)-1h-α^γ-t-,γ-t-h-ei(r)========+β^γt,γthei(r)ΔhiYγt(r)-β^γ-t-,γ-t-h-ei(r)Δh-iYγ-t-(r)========+δ^γt,γtΔhiZγt(r)-δ^γ-t-,γ-t-h-ei(r)Δh-iZγ-t-(r)]dr}.
Set
(42)(Y~(s),Z~(s))==∶=(ΔhiYγt(s)-Δh-iYγ-t-(s),ΔhiZγt(s)-Δh-iZγ-t-(s)).
Then it solves the following BSDE:
(43)Y~(s)==1h(g(XTγthei)-g(XTγt))-1h-(g(XTγ-th-ei)-g(XTγ-t-))==-∫sT‍[β^γt,γthei(r)Y~(r)+δ^γt,γtheiZ~(r)+h~(r)]dr==-∫sT‍Z~(r)dW(r),
where
(44)h~(r)=:=[β^γt,γthei(r)-β^γ-t-,γ-t-h-ei(r)]Δh-iYγ-t-(r)==+[δ^γt,γthei(r)-δ^γ-t-,γ-t-h-ei(r)]Δh-iZγ-t-(r)==+1hα^γt,γthei(r)-1h-α^γ-t-,γ-t-h-ei(r).
Thus, under Assumptions 10, 11, similarly as in Lemma 12, we can get the last three inequalities.Theorem 14.
For eachγt∈Λ, {Yγtz(s),s∈[t,T],z∈ℝn} has a version which is a.e. of class C0,2([0,T]×ℝn).Proof.
We only consider one dimensional case. Applying Lemma12, for each h,h-∈ℝ∖{0} and k,k-∈ℝ,
(45)E[supu∈[t,T]|Yγtk(u)-Yγtk-(u)|p]≤Cp(1+∥γt∥q)|k-k-|p,E[|∫tT‍|Zγtk(u)-Zγtk-|pdu|p/2]≤C2(1+∥γt∥q)|k-k-|p,E[supu∈[t,T]|ΔhiYγtk(u)-Δh-iYγtk-|p]==≤C2(1+∥γt∥q+∥γ-t-∥q+|h|q+|h-|q)===×(|k-k-|p+|h-h-|p),E[|∫tT‍|ΔhiZγtk(u)-Δh-iZγtk-(u)|2du|p/2]==≤C2(1+∥γt∥q+∥γ-t-∥q+|h|q+|h-|q)===×(|k-k-|p+|h-h-|p).
By Kolmogorov's criterion, there exists a continuous derivative of Yγtz(s) with respect to z. There also exists a mean-square derivative of Zγtz(s) with respect to z, which is mean square continuous in z. We denote them by
(46)(DzYγt,DzZγt).
By Theorem 13 and Definition 9, (DzYγt,DzZγt) is the solution of the following BSDE:
(47)DxYγt(s)=gγt′(XTγt)-∫sTDxZγt(r)dW(r)‍=-∫sT[hγt′(Xrγt,Yγt(r),Zγt(r))‍======+hy′(Xrγt,Yγt(r),Zγt(r))DxYγt(r)======+hz′(Xrγt,Yγt(r),Zγt(r))DxZγt(r)]dr.
It is easy to check that the above BSDE has a unique solution. Thus the existence of a continuous second order derivative of Yγtz(s) with respect to z is proved in a similar way.Define(48)u(γt):=Yγt(t),forγt∈Λ.
We have the following results about u(γt).Lemma 15.
For allt1≤t≤T, one has u(Xtγt1)=Yγt1(t).Proof.
For givenγt1, t1<t, set Xt1=γt1. Consider the solutions of FBSDE (5) and (6) on [t,T]:
(49)Xγt1(s)=Xγt1(t)+∫ts‍b(Xrγt1)dr+∫ts‍σ(Xrγt1)dW(r),Yγt1(s)===g(XTγt1)-∫sT‍h(Xrγt1,Yγt1(r),Zγt1(r))dr===-∫sT‍Zγt1(r)dW(r),s∈[t,T].
We need to prove u(Xtγt1)=Yγt1(t).
For each fixedt∈[0,T] and positive integer n, we introduce a mapping γn(γ-s):Λs↦Λs(50)γn(γ-s)(r)=γ-s(r)I[0,t1)=+∑k=0n-1γ-s(tk+1n∧s)I[tkn∧s,tk+1n∧s)(r)+γ-s(s)I{s}(r),======MMMMMMMMMMMs∈[0,t],
where tkn=t1+(k(t-t1)/n), k=0,1,…,n,
Denote(51)γn(Xtγt1)(r)=Xn,γt1(r),t1≤r≤t.
Set
(52)Xtn,N;γt1:=∑i=1NIAixti,
where {Ai}i=1N is a division of ℱt, xti∈Λt, i=1,2,…,N. For any i, (Yxti(s),Zxti(s)) is the solution of the following BSDE:
(53)Yxti(s)=g(XTxti)==-∫sT‍h(Xrxti,Yxti(r),Zxti(r))dr==-∫sT‍Zxti(r)dW(r),s∈[t,T].
Multiplying by IAi and adding the corresponding terms, we obtain
(54)∑i=1N‍IAiYxti(s)==g(∑i=1N‍IAiXTxti)==-∫sT‍h(∑i=1N‍IAiXrxti,∑i=1NIAiYxti(r)‍,∑i=1N‍IAiZxti(r))dr==-∫sT‍∑i=1N‍IAiZxti(r)dW(r),s∈[t,T].
By the uniqueness and existence theorem of BSDE, we have
(55)YXtn,N;γt1(s)=∑i=1NIAiYxti(s),ZXtn,N;γt1(s)=∑i=1NIAiZxti(s),=================================P-a.s.Then, by the definition of u, we get
(56)YXtn,N;γt1(t)=∑i=1N‍IAiYxti(t)=∑i=1N‍IAiu(xti)=u(Xtn,N;γt1).
Note that
(57)limn,N→∞Xtn,N;γt1=Xtγt1,P-a.s.
This completes the proof.By Theorem13 and 14 and the definition of vertical derivative, we have the following corollary.Corollary 16.
u
(
γ
t
) is Λ-continuous and Dxu(γt),Dxxu(γt) exist; moreover they are both Λ-continuous.Proof.
By Theorem14 we know that Dxu(γt) and Dxxu(γt) exist. In the following, we only prove u(γt) is Λ-continuous. The proof for the continuous property of Dxu(γt) and Dxxu(γt) is similar. Taking expectation on both sides of (6),
(58)u(γt)=Eg(XTγt)-E∫tT‍h(Xrγt,Yγt(r),Zγt(r))dr.
For γt,γ-t-∈Λ,t-≥t, we have
(59)|u(γt)-u(γ-t-)|=≤E[|g(XTγt)-g(XTγ-t-)|]==+E[∫tt-‍|h(Xrγt,Yγt(r),Zγt(r))|dr]==+E[∫t-T‍|h(Xrγt,Yγt(r),Zγt(r))======-h(Xrγ-t-,Yγ-t-(r),Zγ-t-(r))|dr∫t-T‍]=≤E[∫tt-‍C1(1+∥XTγt∥k+∥XTγ-t-∥k)∥γt-γ-t-∥=====+3(t--t)1/2(∫tt-‍(|h(Xγ-t-(r),0,0)|2=======MMMMMM+|CYγ-t-(r)|2+|CYγ-t-(r)|2)dr∫tt-‍)1/2=====+C∫t-T‍(|Yγt(r)-Yγ-t-(r)|+|Zγt(r)-Zγ-t-(r)|)dr].
By Theorem 13, for some constant C1 depending on C,k, and T,
(60)|u(γt)-u(γ-t-)|==≤C1(1+∥γt∥k+∥γ-t-∥k)(∥γt-γ-t-∥+|t-t-|1/2).
This completes the proof.


### 3.2. Path Regularity of ProcessZ

In Pardoux and Peng [2], BSDE is only state-dependent, that is, h=h(t,γ(t),y,z) and g=g(γ(T)). Under appropriate assumptions, Y and Z are related in the following sense:
(61)Zγt(s)=∇xu(s,γt(t)+W(s)-W(t)),P-a.s.
Under the conditions (H1) and (H2) in [8], Peng and Wang extend this result to the path-dependent case. The corresponding BSDE is
(62)Yγt(s)==g(WTγt)-∫sT‍h(Wrγt,Yγt(r),Zγt(r))dr==-∫sT‍Zγt(r)dW(r),s∈[t,T],
where
(63)WTγt=Is≤tγt(s)+It<s≤T(γt(t)+W(s)-W(t)).
Then under some assumptions, they obtained
(64)Zγt(s)=Dxu(Wsγt),P-a.s.
In our context, we have the following theorem.Theorem 17.
Under Assumptions1, 10, and 11, for each γt∈Λ, the process (Zγt(s))s∈[t,T] has a continuous version with the form
(65)Zγt(s)=σ(Xsγt)Dxu(Xsγt),fors∈[t,T]P-a.s.To prove the above theorem, we need the following lemma essentially from Pardoux and Peng [2].Lemma 18.
Letγt and some t-∈[t,T] be given. Suppose that
(66)g(γ)=φ(γ(t-),γ(T)-γ(t-)),
where φ is in Cp3(ℝ2d×ℝm;ℝm). For b,σ,h, suppose that
(67)b(γs)=b1(s,γs(s))I[0,t-)(s)=+b2(s,γs(s)-γs(t-))I[t-,T](s),σ(γs)=σ1(s,γs(s))I[0,t-)(s)=+σ2(s,γs(s)-γs(t-))I[t-,T](s),h(γs,y,z)=h1(s,γs(s),y,z)I[0,t-)(s)=+h2(s,γs(s)-γs(t-),y,z)I[t-,T](s),
where bi,σi,hi∈C0,3,i=1,2. Then for each s∈[t,T],
(68)Zγt(s)=σ(Xsγt)Dxu(Xsγt),fors∈[t,T]P-a.s.Proof.
We only consider the one dimensional case.
Fors∈[t-,T], the FBSDE (5), (6) can be rewritten as
(69)Xγs(u)=γs(s)+∫su‍b2(r,Xγs(s)-γs(t-))dr+∫su‍σ2(r,Xγs(s)-γs(t-))dW(r),Yγs(u)==φ(γs(t-),Xγs(T)-γs(t-))-∫uT‍Zγs(r)dW(r)==-∫uT‍h2(r,γs(t-),Xγs(r)-γs(t-),Yγs(r),Zγs(r))dr,mmmmmmmmmmmmmmmmmmmmmm=u∈[s,T].
For s∈[t,t-],
(70)Xγs(u)=γs(s)+∫su‍b1(r,Xγs(s))dr=+∫su‍σ1(r,Xγs(s))dW(r),MMMMMMM===u∈[s,t-],Xγs(u)=Xγs(t-)+∫t-ub2(r,Xγs(s)-Xγs(t-))dr‍=+∫su‍σ2(r,Xγs(s)-Xγs(t-))dW(r),M=MMMMMMMMM==u∈[t-,T],Yγs(u)=φ(Xγs(t-),Xγs(T)-Xγs(t-))=-∫uT‍Zγs(r)dW(r)=-∫uT‍h2(r,Xγs(t-),Xγs(r)=====-Xγs(t-),Yγs(r),Zγs(r))dr,==================u∈[t-,T],Yγs(u)=Yγs(t-)-∫ut-‍h1(r,Xγs(t-),Yγs(r),Zγs(r))dr-∫ut-‍Zγs(r)dW(r),u∈[s,t-].
Now consider the following system of quasilinear parabolic differential equations, which is defined on[t-,T]×ℝ2 and parameterized by x∈ℝ:
(71)∂su2(s,x,y)+ℒu2(s,x,y)=h2(s,x,y,u2(s,x,y),∂yu2(s,x,y)σ(rs)),u2(T,x,y)=φ(x,y),
where ℒ=(1/2)σ2(∂2/∂xx)+b(∂/∂x). The other one is defined on [t,t-]×ℝ:
(72)∂su1(s,x)+ℒu1(s,x)=h1(r,x,u1(s,x),∂yu1(s,x)σ(r)),u1(t-,x)=u2(t-,x,0),
where ℒ=(1/2)σ2(∂2/∂xx)+b(∂/∂x). By Corollary 16 and Theorems 3.1, 3.2 in Paroux-Peng [2], we have u2∈C1,2([t-,T]×ℝ2;ℝ), u1∈C1,2([t,t-]×ℝ;ℝ), and
(73)u(γs)=u1(s,γs(s))I[t,t-)(s)=+u2(s,γs(t-),γs(s)-γs(t-))I[t-,T](s).
Then we obtain
(74)Yγt(s)=u1(s,Xγt(s)),t≤s<t-,Yγt(s)=u2(s,Xγt(t-),Xγt(s)-Xγt(t-)),t-≤s≤T,Zγt(s)=∂xu1(s,Xγt(s))σ1(Xsγt),t≤s<t-,Zγt(s)=∂xu2(s,Xγt(t-),Xγt(s)-Xγt(t-))σ2(Xsγt),MMMMMMMMMMMMMM====t-≤s≤T.
Finally, for each s∈[t,T],
(75)Zγt(s)=σ(Xsγt)Dxu(Xsγt),P-a.s.
In particular,
(76)Zγt(t)=σ(γt)Dxu(γt),γt∈Λ.
This completes the proof.Now we give the proof of Theorem17.Proof.
For each fixedt∈[0,T] and positive integer n, we introduce a mapping γn(γ-s):Λs↦Λs(77)γn(γ-s)(r)=γ-s(r)I[0,t)+∑k=0n-1γ-s(tk+1n∧s)I[tkn∧s,tk+1n∧s)(r)+γ-s(s)I{s}(r),s∈[0,T],
where tkn=t+(k(T-t)/n), k=0,1,…,n(78)gn(γ-):=g(γn(γ-)),hn(γ-s,y,z):=h(γn(γ-s),y,z).
For each n, there exist some functions φn defined on Λt×ℝn×d and ψn,ϕn,ϕ~n defined on [t,T]×Λt×ℝn×d×ℝm×ℝm×d such that
(79)gn(γ-)=:=φn(γ-t,γ-(t1n)-γ-(t),…,γ-(tnn)-γ-(tn-1n)),bn(γ-s)=:=ϕn(s,γ-t,γ-s(t1n∧s)-γ-s(t),…,γ-s(tnn∧s)======-γ-s(tn-1n∧s)),σn(γ-s)=:=ϕn(s,γ-t,γ-s(t1n∧s)-γ-s(t),…,γ-s(tnn∧s)==-γ-s(tn-1n∧s)),hn(γ-s,y,z)=:=ψn(s,γ-t,γ-s(t1n∧s)-γ-s(t),…,γ-s(tnn∧s)======-γ-s(tn-1n∧s),y,z).
Indeed, if we set
(80)φ-n(γ-t,x1,…,xn)=:=g(∑i=1nxi(∑i=1nxiγ-t(s)I[0,t)(s)========+∑k=1nxkI[tk-1n,tkn)(s)+xnI{T}(s))0≤s≤T),φn(γ-t,x1,…,xn)=≔φ-n(∑i=1nxiγ-t,γ-t+x1,γ-t(t)+x1========+x2,…,γ-t(t)+∑i=1nxi),
then by Assumptions1, 10, and 11, we obtain that, for each fixed γ-t, φn(γ-t,x1,…,xn) is a Cp3-function of x1,…,xn. In particular, for each γ-∈Λ,
(81)∂xiφn(γ-t,γ-(t1n)-γ-(t),…,γ-(tnn)-γ-(tn-1n))=gγti-1n′(γn(γ-)).
For anyt-≥t,γ-t-∈Λt-, we consider the following FBSDE:
(82)Xn,γ-t-(s)=γ-t-(t-)+∫t-s‍bn(Xrn,γ-t-)dr+∫t-s‍σn(Xrn,γ-t-)dW(r),Yn,γ-t-(s)=gn(XTn,γ-t-)-∫sT‍hn(Xrn,γ-t-,Yn,γ-t-(r),Zn,γ-t-(r))dr-∫sT‍Zn,γ-t-(r)dW(r),t-≤s.
We denote
(83)un(γ-t-):=Yn,γ-t-(t),γ-t-∈Λ.
Following the argument as in Lemma 18, for each s∈[t,T], we have
(84)Zn,γ-t-(s)=σn(Xsn,γ-t-)Dxun(γ-t-),P-a.s.
By the definition of gn, bn, σn, hn and (82) we have
(85)limnXTn,γ-t-=XTγ-t-,P-a.s,limn(Yn,γ-t-(s),Zn,γ-t-(s))=(Yγ-t-(s),Zγ-t-(s)),==============a.e.s∈[t,T],P-a.s,(86)limnun(γ-t-)=u(γ-t-),limnDxun(γ-t-)=Dxu(γ-t-),limnDxxun(γ-t-)=Dxxu(γ-t-),limn(un(Xsn,γ-t),Dxun(Xsn,γ-t),Dxxun(Xsn,γ-t))=(u(Xsγ-t),Dxu(Xsγ-t),Dxxu(Xsγ-t)),a.e.s∈[t,T],P-a.s.
Therefore(87)Zγt(s)=σ(Xsγ-t)Dxu(Xsγ-t),a.e.s∈[t,T],P-a.s.
This completes the proof.


## 3.1. Regularity of the Solution of FBSDEs

We assume the Lipschitz constants with respect tob, σ, h are C and k. Then we have the following estimates for the solutions of FBSDE (5) and (6).Lemma 12.
Under Assumptions1, 10, and 11, for all p≥2 there exist C2 and q depending only on C,T,k,x such that
(23)E[sups∈[t,T]|Xγt(s)|p]≤C2(1+∥γt∥p),E[sups∈[t,T]|Yγt(s)|p]≤C2(1+∥γt∥q),E[(∫tT‍|Zγt(s)|2ds)p/2]≤C2(1+∥γt∥q).Proof.
To simplify presentation, we only study the casen=d=1, and p=2.
The application of Itô's formula to(Yγt,x(s))2eβ1s yields that
(24)(Yγt,(s))2eβ1s+∫sT‍eβ1r[(Zγt(r))2+β1(Yγt(r))2]dr==g2(XTγt)eβ1T==-∫sT‍2eβ1rYγt(r)h(Xrγt,Yγt(r),Zγt(r))dr==-∫sT‍2eβ1rYγt(r)Zγt(r)dW(r).
So
(25)(Yγt(s))2+E[∫sT‍eβ1(r-s)[(Zγt(r))2==========+β1(Yγt(r))2(Zγt(r))]dr∣ℱs∫sT‍]==E[g2(XTγt)eβ1(T-s)∣ℱs]==-E[∫sT‍2eβ1(r-s)Yγt(r)======×h(Xrγt,Yγt(r),Zγt(r))dr∣ℱs∫sT‍].
Then we have
(26)Esupt≤s≤T(Yγt(s))2===+E[∫tT‍eβ1(r-t)[(Zγt(r))2+β1(Yγt(T))2]dr]==≤E[g2(XTγt)eβ1(T-t)]===+E[∫tT‍eβ1(r-t)2β1h2(Xrγt,Yγt(r),Zγt(r))dr]===+E[∫tT‍eβ1(r-t)β12(Yγt(r))2d(r)],(27)Esupt≤s≤T(Yγt(s))2===+E[∫tT‍eβ1(r-t)[(Zγt(r))2+β12(Yγt(r))2]dr]==≤E[g2(XTγt)eβ1(T-t)]===+E[∫tT‍eβ1(r-t)2β1h2(Xrγt,Yγt(r),Zγt(r))dr].
Applying Itô's formula to (Xγt,x(s))2 yields that
(28)(Xγt(s))2=γt(t)2+∫ts‍2Xγt(r)b(Xrγt)dr=+∫ts‍2Xγt(r)σ(Xrγt)dW(r)+∫ts‍σ2(Xrγt)dr.
By inequality 2ab≤a2+b2 and Burkholder-Davis-Gundy's inequality, there is a C0 such that
(29)Esupt≤r≤s(Xγt(r))2=≤C0[γt(t)2+E∫ts‍b2(Xrγt)dr======+E∫ts‍(Xγt(r))2dr+E∫ts‍σ2(Xrγt)dr].
By Assumption 1 and Gronwall's inequality, from (29) we have (note that C0 will change line by line)
(30)Esupt≤r≤T(Xγt(r))2≤C0(1+∥γt∥2).
By Assumptions 10 and 11 and taking β1=4C2+1, from (27) we have
(31)Esupt≤s≤T(Yγt(s))2===+E[∫tT‍[(Zγt(r))2+(Yγt(r))2]dr]==≤C0(1+PγtPq),
where q=2(1+k). Similarly we can get the same result for p≥2.
This completes the proof.Now we study the regularity properties of the solution of FBSDE (5), (6) with respect to the “parameter” γt. For 0≤s<t≤T, define Yγt(s)=Yγt(s∨t) and Zγt(s)=0.Theorem 13.
Under Assumptions1, 10 and 11, for allp≥2 there exist C2 and q depending only on C,c2,x such that for any t,t-∈[0,T], γt,γ-t-, and h,h-∈ℝ∖{0}
(i)(32)E[supu∈[t∨t-,T]|Yγt(u)-Yγ-t-(u)|p]==≤C2(1+∥γt∥q+∥γ-t-∥q)(∥γt-γ-t-∥p+|t-t-|p/2),(ii)(33)E[|∫t∨t-T‍|Zγt(u)-Zγ-t-(u)|2du|p/2]==≤C2(1+∥γt∥q+∥γ-t-∥q)(∥γt-γ-t-∥p+|t-t-|p/2),(iii)(34)E[supu∈[t∨t-,T]|ΔhiYγt(u)-ΔhiYγ-t-(u)|p]==≤C2(1+∥γt∥q+∥γ-t-∥q+|h|q+|h-|q)===×(|h-h-|p+∥γt-γ-t-∥p+|t-t-|p/2),(iv)(35)E[|∫t∨t-T‍|ΔhiZγt(u)-ΔhiZγ-t-(u)|2du|p/2]==≤C2(1+∥γt∥q+∥γ-t-∥q+|h|q+|h-|q)===×(|h-h-|p+∥γt-γ-t-∥p+|t-t-|p/2),
where
(36)ΔhiYγt,x(s)=1h(Yγthei(s)-Yγt(s)),ΔhiZγt(s)=1h(Zγthei(s)-Zγt(s))
and (e1,…,en) is an orthonormal basis of  ℝn.Proof.
(
Y
γ
t
-
Y
γ
-
t
-
,
Z
γ
t
-
Z
γ
-
t
-
) can be formed as a linearized BSDE: for each s∈[t∨t-,T],
(37)Yγt(s)-Yγ-t-(s)==g(XTγt)-g(XTγ-t-)==+∫sT‍[(Xrγ-t-,Yγ-t-(r),Zγ-t-(r))h(Xrγt,Yγt(r),Zγt(r))======-h(Xrγ-t-,Yγ-t-(r),Zγ-t-(r))]dr==+∫sT‍(Zγt(r)-Zγ-t-(r))dW(r)==g(XTγt)-g(XTγ-t-,)==-∫sT‍[α^γt,γ-t-(r)+β^γt,γ-t-(Yγt(r)-Yγ-t-(r))=======+δ^γt,γ-t-(Zγt(r)-Zγ-t-(r))]dr==+∫sT‍(Zγt(r)-Zγ-t-(r))dW(r),
where (with Uγt=(Yγt,Zγt))
(38)α^γt,γ-t-(r)=h(Xrγt,Yγ-t-(r),Zγ-t-(r))-h(Xrγ-t-,Yγ-t-(r),Zγ-t-(r)),β^γt,γ-t-(r)=∫01‍∂h∂y(Xrγt,Uγ-t-(r)+θ(Uγt(r)-Uγ-t-(r)))dθ,δ^γt,γ-t-(r)=∫01‍∂h∂z(Xrγt,Uγ-t-(r)+θ(Uγt(r)-Uγ-t-(r)))dθ.
Under Assumptions 10, 11, using the same method as in Lemma 12, we get the first three inequalities.
For the next three inequalities, we write(ΔhiYγt,ΔhiZγt) as the solution of the following linearized BSDE:
(39)ΔhiYγt(s)==1h(g(XTγthei)-g(XTγt))==-∫sT‍[1hα^γt,γthei(r)+β^γt,γthei(r)ΔhiYγt(r)=======+δ^γt,γtheiΔhiZγt(r)1h]dr==-∫tT‍ΔhiZγt(r)dW(r).
Then the same calculus implies that
(40)E[sups∈[t,T]|ΔhiYγt(s)|p+|∫tT‍|ΔhiZγt|2dr|p/2]==≤C2(1+∥γt∥q+|h|q).
Consider
(41)ΔhiYγt(s)-Δh-iYγ-t-(s)==1h(g(XTγthei)-g(XTγt))-1h-(g(XTγ-th-ei)-g(XTγ-t-))==-∫sT‍(ΔhiZγt(r)-Δh-iZγ-t-(r))dW(r)==-{∫sT[1hα^γt,γthei(r)-1h-α^γ-t-,γ-t-h-ei(r)========+β^γt,γthei(r)ΔhiYγt(r)-β^γ-t-,γ-t-h-ei(r)Δh-iYγ-t-(r)========+δ^γt,γtΔhiZγt(r)-δ^γ-t-,γ-t-h-ei(r)Δh-iZγ-t-(r)]dr}.
Set
(42)(Y~(s),Z~(s))==∶=(ΔhiYγt(s)-Δh-iYγ-t-(s),ΔhiZγt(s)-Δh-iZγ-t-(s)).
Then it solves the following BSDE:
(43)Y~(s)==1h(g(XTγthei)-g(XTγt))-1h-(g(XTγ-th-ei)-g(XTγ-t-))==-∫sT‍[β^γt,γthei(r)Y~(r)+δ^γt,γtheiZ~(r)+h~(r)]dr==-∫sT‍Z~(r)dW(r),
where
(44)h~(r)=:=[β^γt,γthei(r)-β^γ-t-,γ-t-h-ei(r)]Δh-iYγ-t-(r)==+[δ^γt,γthei(r)-δ^γ-t-,γ-t-h-ei(r)]Δh-iZγ-t-(r)==+1hα^γt,γthei(r)-1h-α^γ-t-,γ-t-h-ei(r).
Thus, under Assumptions 10, 11, similarly as in Lemma 12, we can get the last three inequalities.Theorem 14.
For eachγt∈Λ, {Yγtz(s),s∈[t,T],z∈ℝn} has a version which is a.e. of class C0,2([0,T]×ℝn).Proof.
We only consider one dimensional case. Applying Lemma12, for each h,h-∈ℝ∖{0} and k,k-∈ℝ,
(45)E[supu∈[t,T]|Yγtk(u)-Yγtk-(u)|p]≤Cp(1+∥γt∥q)|k-k-|p,E[|∫tT‍|Zγtk(u)-Zγtk-|pdu|p/2]≤C2(1+∥γt∥q)|k-k-|p,E[supu∈[t,T]|ΔhiYγtk(u)-Δh-iYγtk-|p]==≤C2(1+∥γt∥q+∥γ-t-∥q+|h|q+|h-|q)===×(|k-k-|p+|h-h-|p),E[|∫tT‍|ΔhiZγtk(u)-Δh-iZγtk-(u)|2du|p/2]==≤C2(1+∥γt∥q+∥γ-t-∥q+|h|q+|h-|q)===×(|k-k-|p+|h-h-|p).
By Kolmogorov's criterion, there exists a continuous derivative of Yγtz(s) with respect to z. There also exists a mean-square derivative of Zγtz(s) with respect to z, which is mean square continuous in z. We denote them by
(46)(DzYγt,DzZγt).
By Theorem 13 and Definition 9, (DzYγt,DzZγt) is the solution of the following BSDE:
(47)DxYγt(s)=gγt′(XTγt)-∫sTDxZγt(r)dW(r)‍=-∫sT[hγt′(Xrγt,Yγt(r),Zγt(r))‍======+hy′(Xrγt,Yγt(r),Zγt(r))DxYγt(r)======+hz′(Xrγt,Yγt(r),Zγt(r))DxZγt(r)]dr.
It is easy to check that the above BSDE has a unique solution. Thus the existence of a continuous second order derivative of Yγtz(s) with respect to z is proved in a similar way.Define(48)u(γt):=Yγt(t),forγt∈Λ.
We have the following results about u(γt).Lemma 15.
For allt1≤t≤T, one has u(Xtγt1)=Yγt1(t).Proof.
For givenγt1, t1<t, set Xt1=γt1. Consider the solutions of FBSDE (5) and (6) on [t,T]:
(49)Xγt1(s)=Xγt1(t)+∫ts‍b(Xrγt1)dr+∫ts‍σ(Xrγt1)dW(r),Yγt1(s)===g(XTγt1)-∫sT‍h(Xrγt1,Yγt1(r),Zγt1(r))dr===-∫sT‍Zγt1(r)dW(r),s∈[t,T].
We need to prove u(Xtγt1)=Yγt1(t).
For each fixedt∈[0,T] and positive integer n, we introduce a mapping γn(γ-s):Λs↦Λs(50)γn(γ-s)(r)=γ-s(r)I[0,t1)=+∑k=0n-1γ-s(tk+1n∧s)I[tkn∧s,tk+1n∧s)(r)+γ-s(s)I{s}(r),======MMMMMMMMMMMs∈[0,t],
where tkn=t1+(k(t-t1)/n), k=0,1,…,n,
Denote(51)γn(Xtγt1)(r)=Xn,γt1(r),t1≤r≤t.
Set
(52)Xtn,N;γt1:=∑i=1NIAixti,
where {Ai}i=1N is a division of ℱt, xti∈Λt, i=1,2,…,N. For any i, (Yxti(s),Zxti(s)) is the solution of the following BSDE:
(53)Yxti(s)=g(XTxti)==-∫sT‍h(Xrxti,Yxti(r),Zxti(r))dr==-∫sT‍Zxti(r)dW(r),s∈[t,T].
Multiplying by IAi and adding the corresponding terms, we obtain
(54)∑i=1N‍IAiYxti(s)==g(∑i=1N‍IAiXTxti)==-∫sT‍h(∑i=1N‍IAiXrxti,∑i=1NIAiYxti(r)‍,∑i=1N‍IAiZxti(r))dr==-∫sT‍∑i=1N‍IAiZxti(r)dW(r),s∈[t,T].
By the uniqueness and existence theorem of BSDE, we have
(55)YXtn,N;γt1(s)=∑i=1NIAiYxti(s),ZXtn,N;γt1(s)=∑i=1NIAiZxti(s),=================================P-a.s.Then, by the definition of u, we get
(56)YXtn,N;γt1(t)=∑i=1N‍IAiYxti(t)=∑i=1N‍IAiu(xti)=u(Xtn,N;γt1).
Note that
(57)limn,N→∞Xtn,N;γt1=Xtγt1,P-a.s.
This completes the proof.By Theorem13 and 14 and the definition of vertical derivative, we have the following corollary.Corollary 16.
u
(
γ
t
) is Λ-continuous and Dxu(γt),Dxxu(γt) exist; moreover they are both Λ-continuous.Proof.
By Theorem14 we know that Dxu(γt) and Dxxu(γt) exist. In the following, we only prove u(γt) is Λ-continuous. The proof for the continuous property of Dxu(γt) and Dxxu(γt) is similar. Taking expectation on both sides of (6),
(58)u(γt)=Eg(XTγt)-E∫tT‍h(Xrγt,Yγt(r),Zγt(r))dr.
For γt,γ-t-∈Λ,t-≥t, we have
(59)|u(γt)-u(γ-t-)|=≤E[|g(XTγt)-g(XTγ-t-)|]==+E[∫tt-‍|h(Xrγt,Yγt(r),Zγt(r))|dr]==+E[∫t-T‍|h(Xrγt,Yγt(r),Zγt(r))======-h(Xrγ-t-,Yγ-t-(r),Zγ-t-(r))|dr∫t-T‍]=≤E[∫tt-‍C1(1+∥XTγt∥k+∥XTγ-t-∥k)∥γt-γ-t-∥=====+3(t--t)1/2(∫tt-‍(|h(Xγ-t-(r),0,0)|2=======MMMMMM+|CYγ-t-(r)|2+|CYγ-t-(r)|2)dr∫tt-‍)1/2=====+C∫t-T‍(|Yγt(r)-Yγ-t-(r)|+|Zγt(r)-Zγ-t-(r)|)dr].
By Theorem 13, for some constant C1 depending on C,k, and T,
(60)|u(γt)-u(γ-t-)|==≤C1(1+∥γt∥k+∥γ-t-∥k)(∥γt-γ-t-∥+|t-t-|1/2).
This completes the proof.


## 3.2. Path Regularity of ProcessZ

In Pardoux and Peng [2], BSDE is only state-dependent, that is, h=h(t,γ(t),y,z) and g=g(γ(T)). Under appropriate assumptions, Y and Z are related in the following sense:
(61)Zγt(s)=∇xu(s,γt(t)+W(s)-W(t)),P-a.s.
Under the conditions (H1) and (H2) in [8], Peng and Wang extend this result to the path-dependent case. The corresponding BSDE is
(62)Yγt(s)==g(WTγt)-∫sT‍h(Wrγt,Yγt(r),Zγt(r))dr==-∫sT‍Zγt(r)dW(r),s∈[t,T],
where
(63)WTγt=Is≤tγt(s)+It<s≤T(γt(t)+W(s)-W(t)).
Then under some assumptions, they obtained
(64)Zγt(s)=Dxu(Wsγt),P-a.s.
In our context, we have the following theorem.Theorem 17.
Under Assumptions1, 10, and 11, for each γt∈Λ, the process (Zγt(s))s∈[t,T] has a continuous version with the form
(65)Zγt(s)=σ(Xsγt)Dxu(Xsγt),fors∈[t,T]P-a.s.To prove the above theorem, we need the following lemma essentially from Pardoux and Peng [2].Lemma 18.
Letγt and some t-∈[t,T] be given. Suppose that
(66)g(γ)=φ(γ(t-),γ(T)-γ(t-)),
where φ is in Cp3(ℝ2d×ℝm;ℝm). For b,σ,h, suppose that
(67)b(γs)=b1(s,γs(s))I[0,t-)(s)=+b2(s,γs(s)-γs(t-))I[t-,T](s),σ(γs)=σ1(s,γs(s))I[0,t-)(s)=+σ2(s,γs(s)-γs(t-))I[t-,T](s),h(γs,y,z)=h1(s,γs(s),y,z)I[0,t-)(s)=+h2(s,γs(s)-γs(t-),y,z)I[t-,T](s),
where bi,σi,hi∈C0,3,i=1,2. Then for each s∈[t,T],
(68)Zγt(s)=σ(Xsγt)Dxu(Xsγt),fors∈[t,T]P-a.s.Proof.
We only consider the one dimensional case.
Fors∈[t-,T], the FBSDE (5), (6) can be rewritten as
(69)Xγs(u)=γs(s)+∫su‍b2(r,Xγs(s)-γs(t-))dr+∫su‍σ2(r,Xγs(s)-γs(t-))dW(r),Yγs(u)==φ(γs(t-),Xγs(T)-γs(t-))-∫uT‍Zγs(r)dW(r)==-∫uT‍h2(r,γs(t-),Xγs(r)-γs(t-),Yγs(r),Zγs(r))dr,mmmmmmmmmmmmmmmmmmmmmm=u∈[s,T].
For s∈[t,t-],
(70)Xγs(u)=γs(s)+∫su‍b1(r,Xγs(s))dr=+∫su‍σ1(r,Xγs(s))dW(r),MMMMMMM===u∈[s,t-],Xγs(u)=Xγs(t-)+∫t-ub2(r,Xγs(s)-Xγs(t-))dr‍=+∫su‍σ2(r,Xγs(s)-Xγs(t-))dW(r),M=MMMMMMMMM==u∈[t-,T],Yγs(u)=φ(Xγs(t-),Xγs(T)-Xγs(t-))=-∫uT‍Zγs(r)dW(r)=-∫uT‍h2(r,Xγs(t-),Xγs(r)=====-Xγs(t-),Yγs(r),Zγs(r))dr,==================u∈[t-,T],Yγs(u)=Yγs(t-)-∫ut-‍h1(r,Xγs(t-),Yγs(r),Zγs(r))dr-∫ut-‍Zγs(r)dW(r),u∈[s,t-].
Now consider the following system of quasilinear parabolic differential equations, which is defined on[t-,T]×ℝ2 and parameterized by x∈ℝ:
(71)∂su2(s,x,y)+ℒu2(s,x,y)=h2(s,x,y,u2(s,x,y),∂yu2(s,x,y)σ(rs)),u2(T,x,y)=φ(x,y),
where ℒ=(1/2)σ2(∂2/∂xx)+b(∂/∂x). The other one is defined on [t,t-]×ℝ:
(72)∂su1(s,x)+ℒu1(s,x)=h1(r,x,u1(s,x),∂yu1(s,x)σ(r)),u1(t-,x)=u2(t-,x,0),
where ℒ=(1/2)σ2(∂2/∂xx)+b(∂/∂x). By Corollary 16 and Theorems 3.1, 3.2 in Paroux-Peng [2], we have u2∈C1,2([t-,T]×ℝ2;ℝ), u1∈C1,2([t,t-]×ℝ;ℝ), and
(73)u(γs)=u1(s,γs(s))I[t,t-)(s)=+u2(s,γs(t-),γs(s)-γs(t-))I[t-,T](s).
Then we obtain
(74)Yγt(s)=u1(s,Xγt(s)),t≤s<t-,Yγt(s)=u2(s,Xγt(t-),Xγt(s)-Xγt(t-)),t-≤s≤T,Zγt(s)=∂xu1(s,Xγt(s))σ1(Xsγt),t≤s<t-,Zγt(s)=∂xu2(s,Xγt(t-),Xγt(s)-Xγt(t-))σ2(Xsγt),MMMMMMMMMMMMMM====t-≤s≤T.
Finally, for each s∈[t,T],
(75)Zγt(s)=σ(Xsγt)Dxu(Xsγt),P-a.s.
In particular,
(76)Zγt(t)=σ(γt)Dxu(γt),γt∈Λ.
This completes the proof.Now we give the proof of Theorem17.Proof.
For each fixedt∈[0,T] and positive integer n, we introduce a mapping γn(γ-s):Λs↦Λs(77)γn(γ-s)(r)=γ-s(r)I[0,t)+∑k=0n-1γ-s(tk+1n∧s)I[tkn∧s,tk+1n∧s)(r)+γ-s(s)I{s}(r),s∈[0,T],
where tkn=t+(k(T-t)/n), k=0,1,…,n(78)gn(γ-):=g(γn(γ-)),hn(γ-s,y,z):=h(γn(γ-s),y,z).
For each n, there exist some functions φn defined on Λt×ℝn×d and ψn,ϕn,ϕ~n defined on [t,T]×Λt×ℝn×d×ℝm×ℝm×d such that
(79)gn(γ-)=:=φn(γ-t,γ-(t1n)-γ-(t),…,γ-(tnn)-γ-(tn-1n)),bn(γ-s)=:=ϕn(s,γ-t,γ-s(t1n∧s)-γ-s(t),…,γ-s(tnn∧s)======-γ-s(tn-1n∧s)),σn(γ-s)=:=ϕn(s,γ-t,γ-s(t1n∧s)-γ-s(t),…,γ-s(tnn∧s)==-γ-s(tn-1n∧s)),hn(γ-s,y,z)=:=ψn(s,γ-t,γ-s(t1n∧s)-γ-s(t),…,γ-s(tnn∧s)======-γ-s(tn-1n∧s),y,z).
Indeed, if we set
(80)φ-n(γ-t,x1,…,xn)=:=g(∑i=1nxi(∑i=1nxiγ-t(s)I[0,t)(s)========+∑k=1nxkI[tk-1n,tkn)(s)+xnI{T}(s))0≤s≤T),φn(γ-t,x1,…,xn)=≔φ-n(∑i=1nxiγ-t,γ-t+x1,γ-t(t)+x1========+x2,…,γ-t(t)+∑i=1nxi),
then by Assumptions1, 10, and 11, we obtain that, for each fixed γ-t, φn(γ-t,x1,…,xn) is a Cp3-function of x1,…,xn. In particular, for each γ-∈Λ,
(81)∂xiφn(γ-t,γ-(t1n)-γ-(t),…,γ-(tnn)-γ-(tn-1n))=gγti-1n′(γn(γ-)).
For anyt-≥t,γ-t-∈Λt-, we consider the following FBSDE:
(82)Xn,γ-t-(s)=γ-t-(t-)+∫t-s‍bn(Xrn,γ-t-)dr+∫t-s‍σn(Xrn,γ-t-)dW(r),Yn,γ-t-(s)=gn(XTn,γ-t-)-∫sT‍hn(Xrn,γ-t-,Yn,γ-t-(r),Zn,γ-t-(r))dr-∫sT‍Zn,γ-t-(r)dW(r),t-≤s.
We denote
(83)un(γ-t-):=Yn,γ-t-(t),γ-t-∈Λ.
Following the argument as in Lemma 18, for each s∈[t,T], we have
(84)Zn,γ-t-(s)=σn(Xsn,γ-t-)Dxun(γ-t-),P-a.s.
By the definition of gn, bn, σn, hn and (82) we have
(85)limnXTn,γ-t-=XTγ-t-,P-a.s,limn(Yn,γ-t-(s),Zn,γ-t-(s))=(Yγ-t-(s),Zγ-t-(s)),==============a.e.s∈[t,T],P-a.s,(86)limnun(γ-t-)=u(γ-t-),limnDxun(γ-t-)=Dxu(γ-t-),limnDxxun(γ-t-)=Dxxu(γ-t-),limn(un(Xsn,γ-t),Dxun(Xsn,γ-t),Dxxun(Xsn,γ-t))=(u(Xsγ-t),Dxu(Xsγ-t),Dxxu(Xsγ-t)),a.e.s∈[t,T],P-a.s.
Therefore(87)Zγt(s)=σ(Xsγ-t)Dxu(Xsγ-t),a.e.s∈[t,T],P-a.s.
This completes the proof.


## 4. The Related Parabolic Path-Dependent PDEs

In this section, we relate FBSDE (5), (6) to the following path-dependent partial differential equation:
(88)Dtu(γt)+ℒu(γt)-h(γt,u(γt),σ(γt)Dxu(γt))=0,u(γT)=g(γT),γT∈Λn,
where
(89)ℒu=12tr[(σσT)Dxxu]+〈b,Dxu〉.Theorem 19.
Suppose that Assumptions1, 10, and 11 are fulfilled, and if u∈ℂ1.2(Λ), and that u is the solution of (88), u is uniformly Lipschitz continuous, and bounded by C(1+∥γt∥), then the solution is unique, and for any γt∈Λ, u(γt) is determined by (5) and (6).Proof.
By the assumptions of this theorem, we know thatb(γt) and σ(γt) are uniformly Lipschtiz continuous and the following SDE has a uniqueness solution:
(90)dXγt(s)=b(Xsγt)ds+σ(Xsγt)dW(s),Xt=γt,s∈[t,T].
Set Y(s)=u(Xsγt),t≤s≤T. Applying Itô’s formula to Y(s)=u(Xsγt), we have
(91)dY(s)=-h(Xsγt,Y(s),Z(s))dr-σ(Xsγt)Dxu(Xsγt)dW(s).Y(T)=g(XTγt)s∈[t,T].
Then by the uniqueness and existence theorem of the functional FBSDE, we obtain the result.Now we prove the converse to the above result.Theorem 20.
Under Assumptions1, 10, and 11, the function u(γt)=Yγt(t) is the unique ℂ1,2(Λ)-solution of the path-dependent PDE (88).Proof.
We only study the one dimensional case.u∈ℂ0,2(Λ) follows from Corollary 16. Let δ>0 satisfying t+δ≤T. By Lemma 15 we can get
(92)u(Xt+δγt)=Yγt(t+δ).
Hence
(93)u(γt,t+δ)-u(γt)=u(γt,t+δ)-u(Xt+δγt)+u(Xt+δγt)-u(γt),
By the proof of Theorem 17, we obtain
(94)u(γt,t+δ)-u(γt)=limn→∞[un(γt,t+δ)-un(Xt+δγt)]+∫tt+δ‍h(Xsγt,Yγt(s),Zγt(s))ds+∫tt+δ‍Zγt(s)dW(s).
By Lemma 3.1 and Theorem 3.2 of Pardoux and Peng [2] and Theorem 4.4 of Peng and Wang [8], we deduce that
(95)un(γt,t+δ)-un(Xt+δγt)=∫tt+δ‍Dsun(γt,s)ds-∫tt+δ‍Dsun(Xsγt)ds-∫tt+δ‍Dxun(Xsγt)dXn,γt(s)-12∫tt+δ‍Dxxun(Xsγt)d〈Xn,γt〉(s).
Thus by (86) and the dominated convergence theorem, we have
(96)u(γt,t+δ)-u(γt)=-∫tt+δ‍Dxu(Xsγt)dXγt(s)-12∫tt+δ‍Dxxu(Xsγt)d〈Xγt〉(s)+∫tt+δ‍h(Xsγt,Yγt(s),Zγt(s))ds+∫tt+δ‍Zγt(s)dW(s)+limn→∞Cn,
where
(97)Cn=∫tt+δ‍Dsun(γt,s)ds-∫tt+δ‍Dsun(Xsγt)ds.
Note that un(γt)∈ℂl,lip0,2(Λ). By Lemma 12, we have
(98)|Dsun(γt,s)-Dsun(Xsγt)|≤c∥γt-Xsγt∥,============a.e.s∈[t,T],P-a.s
for some constant c depending on C,T,γt, and k. Hence
(99)|Cn|≤cδsups∈[t,t+δ]|Xγt(s)-γt(t)|,P-a.s.
Taking expectation on both sides of (96), we have
(100)limδ→0u(γt,t+δ)-u(γt)δ=-ℒu(γt)+h(γt,u(γt),Dxu(γt)σ(γt)).
Thus u(γt) belongs to ℂ1,2(Λ) and satisfies (88).



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*Source: 423101-2013-05-16.xml*