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

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.

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: 𝑋^𝛾𝑡(𝑠)=𝛾_𝑡(𝑡)+^𝑠_𝑡𝑏𝑋^𝛾𝑡_𝑟𝑑𝑟+^𝑠_𝑡𝜎𝑋^𝛾𝑡_𝑟(𝑟),𝑋𝑑𝑊_𝛾𝑡(𝑟)=𝛾_𝑡(𝑟)𝑌,0≤𝑟≤𝑡,_𝛾𝑡(𝑠)=𝑔𝑋^𝛾𝑡_𝑇−^𝑇_𝑠ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)𝑑𝑟−^𝑇_𝑠𝑍^𝛾𝑡(𝑟)𝑑𝑊(𝑟),𝑠∈[𝑡,𝑇]. (1)

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: 𝐷_𝑡𝑢𝛾_𝑡+ℒ𝑢𝛾_𝑡=ℎ𝛾_𝑡,𝑢𝛾_𝑡,𝐷_𝑥𝑢𝛾_𝑡𝜎𝛾_𝑡,𝑢𝛾_𝑇=𝑔𝛾_𝑇,𝛾_𝑇∈Λ, (2) where 1ℒ𝑢=2tr𝜎𝜎^𝑇𝐷_𝑥𝑥𝑢+𝑏,𝐷_𝑥𝑢. (3)

The paper is organized as follows: in Section 2, 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

Let Ω=𝐶([0,𝑇];ℝ^𝑑) 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 ℱ_𝑡 the 𝑃-completion of 𝜎(𝑊(𝑠),𝑠∈[0,𝑡]).

For any 𝑡∈[0,𝑇], we denote by 𝐿²(Ω,ℱ_𝑡;ℝ^𝑛) the set of all square integrable ℱ_𝑡-measurable random variables, 𝑀²(0,𝑇;ℝ^𝑛) the set of all ℝ^𝑛-valued ℱ_𝑡-adapted processes 𝜗(⋅) such that 𝐸^𝑇₀|𝜗(𝑠)|²𝑑𝑠<+∞. (4)

Let 𝑡∈[0,𝑇] and 𝛾_𝑡∈Λ. For every 𝑠∈[𝑡,𝑇], we consider the following functional forward-backward SDEs: 𝑋^𝛾𝑡(𝑠)=𝛾_𝑡(𝑡)+^𝑠_𝑡𝑏𝑋^𝛾𝑡_𝑟𝑑𝑟+^𝑠_𝑡𝜎𝑋^𝛾𝑡_𝑟(𝑟),𝑌𝑑𝑊_𝛾𝑡(𝑠)=𝑔𝑋^𝛾𝑡_𝑇−^𝑇_𝑠ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟−)𝑑𝑟_𝑇𝑠𝑍^𝛾𝑡(𝑟)𝑑𝑊(𝑟), (5)(6) where 𝑋^𝛾𝑡(𝑠)=𝛾_𝑡(𝑠),𝑠∈[0,𝑡]. (7) The processes 𝑋,𝑌,𝑍 take values in ℝ^𝑛, ℝ^𝑛, ℝ^𝑛×𝑑; 𝑏, ℎ, 𝜎, and 𝑔 take values in ℝ^𝑛, ℝ^𝑛, ℝ^𝑛×𝑑, and ℝ^𝑛. Equations (5) and (6) can be rewritten as 𝑑𝑋^𝛾𝑡(𝑠)=𝑏𝑋^𝛾𝑡_𝑠𝑑𝑠+𝜎𝑋^𝛾𝑡_𝑠𝑑𝑊(𝑠),𝑑𝑌^𝛾𝑡(𝑠)=ℎ𝑋^𝛾𝑡_𝑠,𝑌^𝛾𝑡(𝑠),𝑍^𝛾𝑡(𝑠)𝑑𝑠+𝑍^𝛾𝑡(𝑠)(𝑠),𝑋𝑑𝑊_𝛾𝑡(𝑡)=𝛾_𝑡(𝑡),𝑌^𝛾𝑡(𝑇)=𝑔𝑋^𝛾𝑡_𝑇. (8) For 𝑧∈ℝ^𝑛×𝑑, we define |𝑧|={tr(𝑧𝑧^𝑇)}^1/2. For 𝑧¹∈ℝ^𝑛×𝑑, 𝑧²∈ℝ^𝑛×𝑑, 𝑧¹,𝑧²=tr𝑧¹𝑧^2𝑇, (9) and for 𝑢¹=(𝑦¹,𝑧¹)∈ℝ^𝑛×ℝ^𝑛×𝑑, 𝑢²=(𝑦²,𝑧²)∈ℝ^𝑛×ℝ^𝑛×𝑑𝑢¹,𝑢²=𝑦¹,𝑦²+𝑧¹,𝑧². (10)

We give the following assumption.

Assumption 1. For all 𝑥¹_𝑇,𝑥²_𝑇∈Λ, 𝑏(𝑥¹_⋅), 𝜎(𝑥¹_⋅)∈𝑀², and 𝑡∈[0,𝑇], there exists a constant 𝑐₁>0, such that 𝑏𝑥¹_𝑡−𝑏𝑥²_𝑡+𝜎𝑥¹_𝑡−𝜎𝑥²_𝑡≤𝑐₁𝑥¹_𝑡−𝑥²_𝑡,a.e. (11) and for all 𝑥_𝑡∈Λ, 𝑏𝑥_𝑡+𝜎𝑥_𝑡≤𝑐₁1+𝑥_𝑡,a.e. (12)

Definition 2. 𝑋∶[0,𝑇]×Ω→ℝ^𝑛 is called an adapted solution of (5), if 𝑋∈𝑀²(0,𝑇;ℝ^𝑛), and it satisfies (5) 𝑃-a.s.

Then we have the following theorem (see [12]).

Theorem 3. Let Assumption 1 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 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, that is, 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,𝛾_𝑡∈Λ, we denote 𝛾_𝑡∶=sup_{[]𝑠∈0,𝑡}𝛾_𝑡(𝑠,𝛾_𝑡)−𝛾_𝑡∶=sup_{𝑠∈0,𝑡∨𝑡}𝛾_𝑡(𝑠∧𝑡)−𝛾_𝑡𝑠∧𝑡,𝑑_∞𝛾_𝑡,𝛾_𝑡∶=sup_{0≤𝑠≤𝑡∨𝑡}𝛾_𝑡(𝑠∧𝑡)−𝛾_𝑡𝑠∧𝑡+𝑡−𝑡. (13) It is obvious that Λ_𝑡 is a Banach space with respect to ‖⋅‖ and 𝑑_∞ is not a norm.

Definition 4. A function 𝑢∶Λ↦ℝ is said to be Λ-continuous at 𝛾_𝑡∈Λ, if for any 𝜀>0 there exists 𝛿>0 such that for each 𝛾_𝑡∈Λ with 𝑑_∞(𝛾_𝑡,𝛾_𝑡)<𝛿, we have |𝑢(𝛾_𝑡)−𝑢(𝛾_𝑡)|<𝜀. 𝑢 is said to be Λ-continuous if it is Λ-continuous at each 𝛾_𝑡∈Λ.

Definition 5. Let 𝑢∶Λ↦ℝ and 𝛾_𝑡∈Λ be given. If there exists 𝑝∈ℝ^𝑑, such that 𝑢𝛾^𝑥_𝑡=𝑢𝛾_𝑡+𝑝,𝑥+𝑜(|𝑥|)as𝑥⟶0,𝑥∈ℝ^𝑑, (14) then we say that 𝑢 is vertically differentiable at 𝛾_𝑡 and denote the gradient of 𝐷_𝑥𝑢(𝛾_𝑡)=𝑝. If 𝐷_𝑥𝑢(𝛾_𝑡) exists for each 𝛾_𝑡∈Λ, 𝑢 is said to be vertically differentiable in Λ.
We can similarly define the Hessian 𝐷_𝑥𝑥𝑢(𝛾_𝑡). It is an 𝕊(𝑑)-valued function defined on Λ, where 𝕊(𝑑) is the space of all 𝑑×𝑑 symmetric matrices.

For each 𝛾_𝑡∈Λ we denote 𝛾_𝑡,𝑠(𝑟)=𝛾_𝑡(𝑟)𝟏_[0,𝑡)(𝑟)+𝛾_𝑡(𝑡)𝟏_[𝑡,𝑠](𝑟),𝑟∈[0,𝑠]. (15) It is clear that 𝛾_t,s∈Λ_s.

Definition 6. For a given 𝛾_𝑡∈Λ if we have 𝑢𝛾_𝑡,𝑠=𝑢𝛾_𝑡+𝑎(𝑠−𝑡)+𝑜(|𝑠−𝑡|)as𝑠⟶𝑡,𝑠≥𝑡, (16) then we say that 𝑢(𝛾_𝑡) is (horizontally) differentiable in 𝑡 at 𝛾_𝑡 and 𝐷_𝑡𝑢(𝛾_𝑡)=𝑎. 𝑢 is said to be horizontally differentiable in Λ if 𝐷_𝑡𝑢(𝛾_𝑡) exists for each 𝛾_𝑡∈Λ.

Definition 7. 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.

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,(ℱ_𝑡)_{𝑡∈[0,𝑇]},𝑃) be a probability space, if 𝑋 is a continuous semi-martingale and 𝑢 is in ℂ^1,2(Λ), then for any 𝑡∈[0,𝑇), 𝑢𝑋_𝑡−𝑢𝑋₀=^𝑡₀𝐷_𝑠𝑢𝑋_𝑠𝑑𝑠+^𝑡₀𝐷_𝑥𝑢𝑋_𝑠(𝑠)+1𝑑𝑋2^𝑡₀𝐷_𝑥𝑥𝑢𝑋_𝑠𝑑⟨𝑋⟩(𝑠),𝑃-𝑎.𝑠. (17)

3. Regularity

We first recall some notions in Pardoux and Peng [2]. 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 Assumption 1 we have that 𝑑𝑋^𝛾𝑡(𝑠)=𝑏𝑋^𝛾𝑡_𝑠𝑑𝑠+𝜎𝑋^𝛾𝑡_𝑠(𝑠),𝑋𝑑𝑊_𝛾𝑡(𝑡)=𝛾_𝑡(𝑡), (18) has a unique solution. For 𝑡≤𝑠≤𝑇, set Λ_{𝛾𝑡,𝑠}∶=𝛾_𝑠∶𝛾(ℎ)=𝑋^𝛾𝑡(),Λℎ,𝜔,0≤ℎ≤𝑠,𝜔∈Ω_𝑡,𝑠∶=_{𝛾𝑡∈Λ𝑡}Λ_{𝛾𝑡,𝑠},Λ_𝑡∶=_{𝑡≤𝑠≤𝑇}Λ_𝑡,𝑠. (19) Then the following definition of derivatives will be used frequently in the sequel.

Definition 9. An ℝ^𝑛-valued function 𝑔 is said to be in 𝐶²(Λ_{𝛾𝑡,𝑇}), if for 𝛾₁∈Λ_{𝛾𝑡,𝑇} and 𝛾₂∈Λ_𝛾^𝑦_𝑡,𝑇, there exist 𝑝₁∈ℝ^𝑑 and 𝑝₂∈𝒮^𝑑 (𝒮^𝑑 is the set of all 𝑑 order symmetric matrix) such that 𝑔𝛾₂−𝑔𝛾₁=𝑝₁+1,𝑦2𝑝₂𝑦,𝑦+𝑜𝑦²,𝑥∈ℝ^𝑑. (20) We denote 𝑔^′_𝛾𝑡(𝛾₁)∶=𝑝₁, and 𝑔_𝛾^′′_𝑡(𝛾₁)∶=𝑝₂. 𝑔 is said to be in 𝐶²_{𝑙,𝑙𝑖𝑝}(Λ_𝑡,𝑇) if 𝑔^′_𝛾𝑡(𝛾) and 𝑔_𝛾^′′_𝑡(𝛾) exist for each 𝛾_𝑡∈Λ_𝑡, and there exist some constants 𝐶≥0 and 𝑘≥0 depending only on 𝑔 such that for each 𝛾,𝛾∈Λ_𝑇,𝑡,𝑠∈[0,𝑇], 𝑔𝛾−𝑔𝛾≤𝐶𝛾^𝑘+𝛾^𝑘𝛾−𝛾, (21) and for each 𝛾∈Λ_𝑡,𝑇,𝛾∈Λ_𝑠,𝑇,𝑡,𝑠∈[0,𝑇], Φ_𝛾𝑡𝛾−Φ_𝛾𝑠𝛾≤𝐶𝛾^𝑘+𝛾^𝑘|𝑡−𝑠|+𝛾−𝛾 (22) with Φ=𝑔^′_𝛾𝑡(𝛾),𝑔_𝛾^′′_𝑡(𝛾). We can also define 𝐶²(Λ_𝑡,𝑠), 𝐶²_{𝑙,𝑙𝑖𝑝}(Λ_𝑡,𝑠), 𝐶¹_{𝑙,𝑙𝑖𝑝}(Λ_𝑡,𝑠), 𝐶_{𝑙,𝑙𝑖𝑝}(Λ_𝑡,𝑠) and 𝐶²(Λ_𝑡), 𝐶²_{𝑙,𝑙𝑖𝑝}(Λ_𝑡), 𝐶¹_{𝑙,𝑙𝑖𝑝}(Λ_𝑡), 𝐶_{𝑙,𝑙𝑖𝑝}(Λ_𝑡).

Now we consider the solvability of (6).

Assumption 10. Let 𝑔 be an ℝ^𝑛-valued function on Λ_𝑇. Moreover 𝑔∈𝐶²_{𝑙,𝑙𝑖𝑝}(Λ_𝑡,𝑇) with the Lipschitz constants 𝐶 and 𝑘.

Assumption 11. Let ℎ(𝛾_𝑡,𝑦,𝑧)=ℎ(𝑡,𝛾(𝑡),𝑦,𝑧), where ℎ∶[0,𝑇]×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×𝑑↦ℝ^𝑛 is such that (𝑡,𝑟,𝑦,𝑧)↦Ψ(𝑡,𝑟,𝑦,𝑧) is of class 𝐶_𝑝^0,3([0,𝑇]×ℝ^𝑛×ℝ^𝑛×ℝ^𝑛×𝑑;ℝ^𝑛) 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 Assumptions 1, 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 to 𝑏, 𝜎, ℎ are 𝐶 and 𝑘. Then we have the following estimates for the solutions of FBSDE (5) and (6).

Lemma 12. Under Assumptions 1, 10, and 11, for all 𝑝≥2 there exist 𝐶₂ and 𝑞 depending only on 𝐶,𝑇,𝑘,𝑥 such that 𝐸sup_{[]𝑠∈𝑡,𝑇}𝑋^𝛾𝑡(𝑠)^𝑝≤𝐶₂1+𝛾_𝑡^𝑝,𝐸sup_{[]𝑠∈𝑡,𝑇}𝑌^𝛾𝑡(𝑠)^𝑝≤𝐶₂1+𝛾_𝑡^𝑞,𝐸^𝑇_𝑡𝑍^𝛾𝑡(𝑠)²𝑑𝑠^𝑝/2≤𝐶₂1+𝛾_𝑡^𝑞. (23)

Proof. To simplify presentation, we only study the case 𝑛=𝑑=1, and 𝑝=2.
The application of Itô's formula to (𝑌_{𝛾𝑡,𝑥}(𝑠))²𝑒^𝛽1𝑠 yields that 𝑌^𝛾𝑡,(𝑠)²𝑒^𝛽1𝑠+^𝑇_𝑠𝑒^𝛽1𝑟𝑍^𝛾𝑡(𝑟)²+𝛽₁𝑌^𝛾𝑡(𝑟)²𝑑𝑟=𝑔²𝑋^𝛾𝑡_𝑇𝑒^𝛽1𝑇−^𝑇_𝑠2𝑒^𝛽1𝑟𝑌^𝛾𝑡(𝑟)ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)𝑑𝑟−^𝑇_𝑠2𝑒^𝛽1𝑟𝑌^𝛾𝑡(𝑟)𝑍^𝛾𝑡(𝑟)𝑑𝑊(𝑟). (24) So 𝑌^𝛾𝑡(𝑠)²+𝐸^𝑇_𝑠𝑒^{𝛽1(𝑟−𝑠)}𝑍^𝛾𝑡(𝑟)²+𝛽₁𝑌^𝛾𝑡(𝑟)²𝑑𝑟∣ℱ_𝑠=𝐸𝑔²𝑋^𝛾𝑡_𝑇𝑒^{𝛽1(𝑇−𝑠)}∣ℱ_𝑠−𝐸^𝑇_𝑠2𝑒^{𝛽1(𝑟−𝑠)}𝑌^𝛾𝑡(𝑟)×ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)𝑑𝑟∣ℱ_𝑠. (25) Then we have 𝐸sup_{𝑡≤𝑠≤𝑇}𝑌^𝛾𝑡(𝑠)²+𝐸^𝑇_𝑡𝑒^{𝛽1(𝑟−𝑡)}𝑍^𝛾𝑡(𝑟)²+𝛽₁𝑌^𝛾𝑡(𝑇)²𝑔𝑑𝑟≤𝐸²𝑋^𝛾𝑡_𝑇𝑒^{𝛽1(𝑇−𝑡)}+𝐸^𝑇_𝑡𝑒^{𝛽1(𝑟−𝑡)}2𝛽₁ℎ²𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)𝑑𝑟+𝐸^𝑇_𝑡𝑒^{𝛽1(𝑟−𝑡)}𝛽₁2𝑌^𝛾𝑡(𝑟)²𝑑(𝑟),𝐸sup_{𝑡≤𝑠≤𝑇}𝑌^𝛾𝑡(𝑠)²+𝐸^𝑇_𝑡𝑒^{𝛽1(𝑟−𝑡)}𝑍^𝛾𝑡(𝑟)²+𝛽₁2𝑌^𝛾𝑡(𝑟)²𝑔𝑑𝑟≤𝐸²𝑋^𝛾𝑡_𝑇𝑒^{𝛽1(𝑇−𝑡)}+𝐸^𝑇_𝑡𝑒^{𝛽1(𝑟−𝑡)}2𝛽₁ℎ²𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)𝑑𝑟. (26)(27) Applying Itô's formula to (𝑋^{𝛾𝑡,𝑥}(𝑠))² yields that 𝑋^𝛾𝑡(𝑠)²=𝛾_𝑡(𝑡)²+^𝑠_𝑡2𝑋^𝛾𝑡(𝑟)𝑏𝑋^𝛾𝑡_𝑟+𝑑𝑟_𝑠𝑡2𝑋^𝛾𝑡(𝑟)𝜎𝑋^𝛾𝑡_𝑟𝑑𝑊(𝑟)+^𝑠_𝑡𝜎²𝑋^𝛾𝑡_𝑟𝑑𝑟. (28) By inequality 2𝑎𝑏≤𝑎²+𝑏² and Burkholder-Davis-Gundy's inequality, there is a 𝐶₀ such that 𝐸sup_{𝑡≤𝑟≤𝑠}𝑋^𝛾𝑡(𝑟)²≤𝐶₀𝛾_𝑡(𝑡)²+𝐸^𝑠_𝑡𝑏²𝑋^𝛾𝑡_𝑟𝑑𝑟+𝐸^𝑠_𝑡𝑋^𝛾𝑡(𝑟)²𝑑𝑟+𝐸^𝑠_𝑡𝜎²𝑋^𝛾𝑡_𝑟𝑑𝑟. (29) By Assumption 1 and Gronwall's inequality, from (29) we have (note that 𝐶₀ will change line by line) 𝐸sup_{𝑡≤𝑟≤𝑇}𝑋^𝛾𝑡(𝑟)²≤𝐶₀1+𝛾_𝑡². (30) By Assumptions 10 and 11 and taking 𝛽₁=4𝐶²+1, from (27) we have 𝐸sup_{𝑡≤𝑠≤𝑇}𝑌^𝛾𝑡(𝑠)²+𝐸^𝑇_𝑡𝑍^𝛾𝑡(𝑟)²+𝑌^𝛾𝑡(𝑟)²𝑑𝑟≤𝐶₀1+𝑃𝛾_𝑡𝑃^𝑞, (31) 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” 𝛾_𝑡. For 0≤𝑠<𝑡≤𝑇, define 𝑌^𝛾𝑡(𝑠)=𝑌^𝛾𝑡(𝑠∨𝑡) and 𝑍^𝛾𝑡(𝑠)=0.

Theorem 13. Under Assumptions 1, 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], 𝛾_𝑡,𝛾_𝑡, and ℎ,ℎ∈ℝ⧵{0}
   (i)𝐸sup_{𝑢∈𝑡∨𝑡,𝑇}𝑌^𝛾𝑡(𝑢)−𝑌^𝛾𝑡(𝑢)^𝑝≤𝐶₂1+𝛾_𝑡^𝑞+𝛾_𝑡^𝑞𝛾_𝑡−𝛾_𝑡^𝑝+𝑡−𝑡^𝑝/2, (32)(ii)𝐸^𝑇_𝑡∨𝑡𝑍^𝛾𝑡(𝑢)−𝑍^𝛾𝑡(𝑢)²𝑑𝑢^𝑝/2≤𝐶₂1+𝛾_𝑡^𝑞+𝛾_𝑡^𝑞𝛾_𝑡−𝛾_𝑡^𝑝+𝑡−𝑡^𝑝/2, (33)(iii)𝐸sup_{𝑢∈𝑡∨𝑡,𝑇}Δ^𝑖_ℎ𝑌^𝛾𝑡(𝑢)−Δ^𝑖_ℎ𝑌^𝛾𝑡(𝑢)^𝑝≤𝐶₂1+𝛾_𝑡^𝑞+𝛾_𝑡^𝑞+|ℎ|^𝑞+ℎ^𝑞×ℎ−ℎ^𝑝+𝛾_𝑡−𝛾_𝑡^𝑝+𝑡−𝑡^𝑝/2, (34)(iv)𝐸^𝑇_𝑡∨𝑡Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑢)−Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑢)²𝑑𝑢^𝑝/2≤𝐶₂1+𝛾_𝑡^𝑞+𝛾_𝑡^𝑞+|ℎ|^𝑞+ℎ^𝑞×ℎ−ℎ^𝑝+𝛾_𝑡−𝛾_𝑡^𝑝+𝑡−𝑡^𝑝/2, (35) where Δ^𝑖_ℎ𝑌^{𝛾𝑡,𝑥}(𝑠)=1ℎ𝑌^{𝛾ℎ𝑒𝑖𝑡}(𝑠)−𝑌^𝛾𝑡(𝑠),Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑠)=1ℎ𝑍^{𝛾ℎ𝑒𝑖𝑡}(𝑠)−𝑍^𝛾𝑡(𝑠) (36) and (𝑒₁,…,𝑒_𝑛) is an orthonormal basis of  ℝ^𝑛.

Proof. (𝑌^𝛾𝑡−𝑌^𝛾𝑡,𝑍^𝛾𝑡−𝑍^𝛾𝑡) can be formed as a linearized BSDE: for each 𝑠∈[𝑡∨𝑡,𝑇], 𝑌^𝛾𝑡(𝑠)−𝑌^𝛾𝑡(𝑠)=𝑔𝑋^𝛾𝑡_𝑇−𝑔𝑋^𝛾𝑡_𝑇+^𝑇_𝑠ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)−ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)+𝑑𝑟_𝑇𝑠𝑍^𝛾𝑡(𝑟)−𝑍^𝛾𝑡(𝑟)(𝑟)𝑋𝑑𝑊=𝑔^𝛾𝑡_𝑇−𝑔𝑋^𝛾𝑡_𝑇,−^𝑇_𝑠𝛼_{𝛾𝑡,𝛾𝑡}(𝑟)+𝛽_{𝛾𝑡,𝛾𝑡}𝑌^𝛾𝑡(𝑟)−𝑌^𝛾𝑡(𝑟)+𝛿_{𝛾𝑡,𝛾𝑡}𝑍^𝛾𝑡(𝑟)−𝑍^𝛾𝑡(𝑟)+𝑑𝑟_𝑇𝑠𝑍^𝛾𝑡(𝑟)−𝑍^𝛾𝑡(𝑟)𝑑𝑊(𝑟), (37) where (with 𝑈^𝛾𝑡=(𝑌^𝛾𝑡,𝑍^𝛾𝑡)) 𝛼_{𝛾𝑡,𝛾𝑡}(𝑟)=ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)−ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟),𝛽_{𝛾𝑡,𝛾𝑡}(𝑟)=¹₀𝜕ℎ𝑋𝜕𝑦^𝛾𝑡𝑟,𝑈^𝛾𝑡(𝑟)+𝜃𝑈^𝛾𝑡(𝑟)−𝑈^𝛾𝑡(𝑟)𝛿𝑑𝜃,_{𝛾𝑡,𝛾𝑡}(𝑟)=¹₀𝜕ℎ𝑋𝜕𝑧^𝛾𝑡𝑟,𝑈^𝛾𝑡(𝑟)+𝜃𝑈^𝛾𝑡(𝑟)−𝑈^𝛾𝑡(𝑟)𝑑𝜃. (38) 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 (Δ^𝑖_ℎ𝑌^𝛾𝑡,Δ^𝑖_ℎ𝑍^𝛾𝑡) as the solution of the following linearized BSDE: Δ^𝑖_ℎ𝑌^𝛾𝑡(𝑠)=1ℎ𝑔𝑋^{𝛾ℎ𝑒𝑖𝑡}_𝑇−𝑔𝑋^𝛾𝑡_𝑇−^𝑇_𝑠1ℎ𝛼_{𝛾𝑡,𝛾𝑡}^ℎ_𝑒𝑖(𝑟)+𝛽_{𝛾𝑡,𝛾𝑡}^ℎ_𝑒𝑖(𝑟)Δ^𝑖_ℎ𝑌^𝛾𝑡(𝑟)+𝛿_{𝛾𝑡,𝛾𝑡}^ℎ_𝑒𝑖Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑟−)𝑑𝑟_𝑇𝑡Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑟)𝑑𝑊(𝑟). (39) Then the same calculus implies that 𝐸sup_{[]𝑠∈𝑡,𝑇}Δ^𝑖_ℎ𝑌^𝛾𝑡(𝑠)^𝑝+^𝑇_𝑡Δ^𝑖_ℎ𝑍^𝛾𝑡2𝑑𝑟^𝑝/2≤𝐶₂1+𝛾_𝑡^𝑞+|ℎ|^𝑞. (40) Consider Δ^𝑖_ℎ𝑌^𝛾𝑡(𝑠)−Δ^𝑖_ℎ𝑌^𝛾𝑡(𝑠)=1ℎ𝑔𝑋^{𝛾ℎ𝑒𝑖𝑡}_𝑇−𝑔𝑋^𝛾𝑡_𝑇−1ℎ𝑔𝑋^{𝛾ℎ𝑒𝑖𝑡}_𝑇−𝑔𝑋^𝛾𝑡_𝑇−^𝑇_𝑠Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑟)−Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑟)(𝑟)−𝑑𝑊_𝑇𝑠1ℎ𝛼_{𝛾𝑡,𝛾}^ℎ_𝑒𝑖𝑡(𝑟)−1ℎ𝛼_{𝛾𝑡,𝛾ℎ𝑒𝑖𝑡}(𝑟)+𝛽_{𝛾𝑡,𝛾𝑡}^ℎ_𝑒𝑖(𝑟)Δ^𝑖_ℎ𝑌^𝛾𝑡(𝑟)−𝛽_{𝛾𝑡,𝛾ℎ𝑒𝑖𝑡}(𝑟)Δ^𝑖_ℎ𝑌^𝛾𝑡(𝑟)+𝛿_{𝛾𝑡,𝛾𝑡}Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑟)−𝛿_{𝛾𝑡,𝛾ℎ𝑒𝑖𝑡}(𝑟)Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑟)𝑑𝑟. (41) Set 𝑌(𝑠),𝑍(𝑠Δ)∶=^𝑖_ℎ𝑌^𝛾𝑡(𝑠)−Δ^𝑖_ℎ𝑌^𝛾𝑡(𝑠),Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑠)−Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑠). (42) Then it solves the following BSDE: 𝑌(𝑠)=1ℎ𝑔𝑋^{𝛾ℎ𝑒𝑖𝑡}_𝑇−𝑔𝑋^𝛾𝑡_𝑇−1ℎ𝑔𝑋^{𝛾ℎ𝑒𝑖𝑡}_𝑇−𝑔𝑋^𝛾𝑡_𝑇−^𝑇_𝑠𝛽_{𝛾𝑡,𝛾𝑡}^ℎ_𝑒𝑖(𝑟)𝑌(𝑟)+𝛿_{𝛾𝑡,𝛾𝑡}^ℎ_𝑒𝑖𝑍(𝑟)+ℎ(𝑟−)𝑑𝑟_𝑇𝑠𝑍(𝑟)𝑑𝑊(𝑟), (43) where ℎ(𝑟)∶=𝛽_{𝛾𝑡,𝛾𝑡}^ℎ_𝑒𝑖(𝑟)−𝛽_{𝛾𝑡,𝛾ℎ𝑒𝑖𝑡}(𝑟)Δ^𝑖_ℎ𝑌^𝛾𝑡(𝑟)+𝛿_{𝛾𝑡,𝛾𝑡}^ℎ_𝑒𝑖(𝑟)−𝛿_{𝛾𝑡,𝛾ℎ𝑒𝑖𝑡}(𝑟)Δ^𝑖_ℎ𝑍^𝛾𝑡(𝑟)+1ℎ𝛼_{𝛾𝑡,𝛾}^ℎ_𝑒𝑖𝑡(𝑟)−1ℎ𝛼_{𝛾𝑡,𝛾ℎ𝑒𝑖𝑡}(𝑟). (44) Thus, under Assumptions 10, 11, similarly as in Lemma 12, we can get the last three inequalities.

Theorem 14. For each 𝛾_𝑡∈Λ, {𝑌^𝛾𝑧𝑡(𝑠),𝑠∈[𝑡,𝑇],𝑧∈ℝ^𝑛} has a version which is a.e. of class 𝐶^0,2([0,𝑇]×ℝ^𝑛).

Proof. We only consider one dimensional case. Applying Lemma 12, for each ℎ,ℎ∈ℝ⧵{0} and 𝑘,𝑘∈ℝ, 𝐸sup_{[]𝑢∈𝑡,𝑇}𝑌^𝛾𝑘𝑡(𝑢)−𝑌^𝛾𝑘𝑡(𝑢)^𝑝≤𝐶_𝑝1+𝛾_𝑡^𝑞𝑘−𝑘^𝑝,𝐸^𝑇_𝑡𝑍^𝛾𝑘𝑡(𝑢)−𝑍^{𝛾𝑘𝑡𝑝}𝑑𝑢^𝑝/2≤𝐶₂1+𝛾_𝑡^𝑞𝑘−𝑘^𝑝,𝐸sup_{[]𝑢∈𝑡,𝑇}Δ^𝑖_ℎ𝑌^𝛾𝑘𝑡(𝑢)−Δ^𝑖_ℎ𝑌^{𝛾𝑘𝑡𝑝}≤𝐶₂1+𝛾_𝑡^𝑞+𝛾_𝑡^𝑞+|ℎ|^𝑞+ℎ^𝑞×𝑘−𝑘^𝑝+ℎ−ℎ^𝑝,𝐸^𝑇_𝑡Δ^𝑖_ℎ𝑍^𝛾𝑘𝑡(𝑢)−Δ^𝑖_ℎ𝑍^𝛾𝑘𝑡(𝑢)²𝑑𝑢^𝑝/2≤𝐶₂1+𝛾_𝑡^𝑞+𝛾_𝑡^𝑞+|ℎ|^𝑞+ℎ^𝑞×𝑘−𝑘^𝑝+ℎ−ℎ^𝑝. (45) By Kolmogorov's criterion, there exists a continuous derivative of 𝑌^𝛾𝑧𝑡(𝑠) with respect to 𝑧. There also exists a mean-square derivative of 𝑍^𝛾𝑧𝑡(𝑠) with respect to 𝑧, which is mean square continuous in 𝑧. We denote them by 𝐷_𝑧𝑌^𝛾𝑡,𝐷_𝑧𝑍^𝛾𝑡. (46) By Theorem 13 and Definition 9, (𝐷_𝑧𝑌^𝛾𝑡,𝐷_𝑧𝑍^𝛾𝑡) is the solution of the following BSDE: 𝐷_𝑥𝑌^𝛾𝑡(𝑠)=𝑔^′_𝛾𝑡𝑋^𝛾𝑡_𝑇−^𝑇_𝑠𝐷_𝑥𝑍^𝛾𝑡(𝑟)(𝑟)−𝑑𝑊_𝑇𝑠ℎ^′_𝛾𝑡𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)+ℎ^′_𝑦𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)𝐷_𝑥𝑌^𝛾𝑡(𝑟)+ℎ^′_𝑧𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)𝐷_𝑥𝑍^𝛾𝑡(𝑟)𝑑𝑟. (47) It is easy to check that the above BSDE has a unique solution. Thus the existence of a continuous second order derivative of 𝑌^𝛾𝑧𝑡(𝑠) with respect to 𝑧 is proved in a similar way.

Define 𝑢𝛾_𝑡∶=𝑌^𝛾𝑡(𝑡),for𝛾_𝑡∈Λ. (48) We have the following results about 𝑢(𝛾_𝑡).

Lemma 15. For all 𝑡₁≤𝑡≤𝑇, one has 𝑢(𝑋^𝛾𝑡1_𝑡)=𝑌^𝛾𝑡1(𝑡).

Proof. For given 𝛾_𝑡1, 𝑡₁<𝑡, set 𝑋_𝑡1=𝛾_𝑡1. Consider the solutions of FBSDE (5) and (6) on [𝑡,𝑇]: 𝑋^𝛾𝑡1(𝑠)=𝑋^𝛾𝑡1(𝑡)+^𝑠_𝑡𝑏𝑋^𝛾𝑡1_𝑟𝑑𝑟+^𝑠_𝑡𝜎𝑋^𝛾𝑡1_𝑟(𝑟),𝑌𝑑𝑊_𝛾𝑡1(𝑠)=𝑔𝑋^𝛾𝑡1_𝑇−^𝑇_𝑠ℎ𝑋^𝛾𝑡1_𝑟,𝑌^𝛾𝑡1(𝑟),𝑍^𝛾𝑡1(𝑟)𝑑𝑟−^𝑇_𝑠𝑍^𝛾𝑡1(𝑟)𝑑𝑊(𝑟),𝑠∈[𝑡,𝑇]. (49) We need to prove 𝑢(𝑋^𝛾𝑡1_𝑡)=𝑌^𝛾𝑡1(𝑡).
For each fixed 𝑡∈[0,𝑇] and positive integer 𝑛, we introduce a mapping 𝛾^𝑛(𝛾_𝑠)∶Λ_𝑠↦Λ_𝑠𝛾^𝑛𝛾_𝑠(𝑟)=𝛾_𝑠(𝑟)𝐼_[0,𝑡1)+^𝑛−1_𝑘=0𝛾_𝑠𝑡^𝑛_𝑘+1∧𝑠𝐼_[𝑡^𝑛_{𝑘∧𝑠,𝑡}^𝑛_{𝑘+1∧𝑠)}(𝑟)+𝛾_𝑠(𝑠)𝐼_{𝑠}(𝑟),𝑠∈[0,𝑡], (50) where 𝑡^𝑛_𝑘=𝑡₁+(𝑘(𝑡−𝑡₁)/𝑛), 𝑘=0,1,…,𝑛,
Denote 𝛾^𝑛𝑋^𝛾𝑡1_𝑡(𝑟)=𝑋^{𝑛,𝛾𝑡1}(𝑟),𝑡₁≤𝑟≤𝑡. (51) Set 𝑋^{𝑛,𝑁;𝛾𝑡1}_𝑡∶=^𝑁_𝑖=1𝐼_𝐴𝑖𝑥^𝑖_𝑡, (52) where {𝐴_𝑖}^𝑁_𝑖=1 is a division of ℱ_𝑡, 𝑥^𝑖_𝑡∈Λ_𝑡, 𝑖=1,2,…,𝑁. For any 𝑖, (𝑌^𝑥𝑖𝑡(𝑠),𝑍^𝑥𝑖𝑡(𝑠)) is the solution of the following BSDE: 𝑌^𝑥𝑖𝑡(𝑠)=𝑔𝑋^𝑥𝑖𝑡_𝑇−^𝑇_𝑠ℎ𝑋^𝑥𝑖𝑡_𝑟,𝑌^𝑥𝑖𝑡(𝑟),𝑍^𝑥𝑖𝑡(𝑟)−𝑑𝑟_𝑇𝑠𝑍^𝑥𝑖𝑡(𝑟)𝑑𝑊(𝑟),𝑠∈[𝑡,𝑇]. (53) Multiplying by 𝐼_𝐴𝑖 and adding the corresponding terms, we obtain ^𝑁_𝑖=1𝐼_𝐴𝑖𝑌^𝑥𝑖𝑡(𝑠)=𝑔^𝑁_𝑖=1𝐼_𝐴𝑖𝑋^𝑥𝑖𝑡_𝑇−^𝑇_𝑠ℎ^𝑁_𝑖=1𝐼_𝐴𝑖𝑋^𝑥𝑖𝑡_𝑟,^𝑁_𝑖=1𝐼_𝐴𝑖𝑌^𝑥𝑖𝑡(𝑟),^𝑁_𝑖=1𝐼_𝐴𝑖𝑍^𝑥𝑖𝑡(𝑟)−𝑑𝑟_{𝑇𝑠𝑁𝑖=1}𝐼_𝐴𝑖𝑍^𝑥𝑖𝑡(𝑟)𝑑𝑊(𝑟),𝑠∈[𝑡,𝑇]. (54) By the uniqueness and existence theorem of BSDE, we have 𝑌^{𝑋𝑡1𝑡𝑛,𝑁;𝛾}(𝑠)=^𝑁_𝑖=1𝐼_𝐴𝑖𝑌^𝑥𝑖𝑡(𝑠),𝑍^{𝑋𝑡1𝑡𝑛,𝑁;𝛾}(𝑠)=^𝑁_𝑖=1𝐼_𝐴𝑖𝑍^𝑥𝑖𝑡(𝑠),𝑃-a.s. (55)Then, by the definition of 𝑢, we get 𝑌^{𝑋𝑡1𝑡𝑛,𝑁;𝛾}(𝑡)=^𝑁_𝑖=1𝐼_𝐴𝑖𝑌^𝑥𝑖𝑡(𝑡)=^𝑁_𝑖=1𝐼_𝐴𝑖𝑢𝑥^𝑖_𝑡=𝑢𝑋^{𝑛,𝑁;𝛾𝑡1}_𝑡. (56) Note that lim_{𝑛,𝑁→∞}𝑋^{𝑛,𝑁;𝛾𝑡1}_𝑡=𝑋^𝛾𝑡1_𝑡,𝑃-a.s. (57) This completes the proof.

By Theorem 13 and 14 and the definition of vertical derivative, we have the following corollary.

Corollary 16. 𝑢(𝛾_𝑡) is Λ-continuous and 𝐷_𝑥𝑢(𝛾_𝑡),𝐷_𝑥𝑥𝑢(𝛾_𝑡) exist; moreover they are both Λ-continuous.

Proof. By Theorem 14 we know that 𝐷_𝑥𝑢(𝛾_𝑡) and 𝐷_𝑥𝑥𝑢(𝛾_𝑡) exist. In the following, we only prove 𝑢(𝛾_𝑡) is Λ-continuous. The proof for the continuous property of 𝐷_𝑥𝑢(𝛾_𝑡) and 𝐷_𝑥𝑥𝑢(𝛾_𝑡) is similar. Taking expectation on both sides of (6), 𝑢𝛾_𝑡=𝐸𝑔𝑋^𝛾𝑡_𝑇−𝐸^𝑇_𝑡ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)𝑑𝑟. (58) For 𝛾_𝑡,𝛾_𝑡∈Λ,𝑡≥𝑡, we have 𝑢𝛾_𝑡−𝑢𝛾_𝑡≤𝐸𝑔𝑋^𝛾𝑡_𝑇−𝑔𝑋^𝛾𝑡_𝑇+𝐸^𝑡_𝑡ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)𝑑𝑟+𝐸^𝑇_𝑡ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)−ℎ𝑋^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)𝐶𝑑𝑟≤𝐸₁1+𝑋^𝛾𝑡_𝑇^𝑘+𝑋^𝛾𝑡_𝑇^𝑘𝛾_𝑡−𝛾_𝑡+3𝑡−𝑡^1/2𝑡_𝑡ℎ𝑋^𝛾𝑡(𝑟),0,0²+𝐶𝑌^𝛾𝑡(𝑟)²+𝐶𝑌^𝛾𝑡(𝑟)²𝑑𝑟^1/2+𝐶^𝑇_𝑡𝑌^𝛾𝑡(𝑟)−𝑌^𝛾𝑡(𝑟)+𝑍^𝛾𝑡(𝑟)−𝑍^𝛾𝑡(𝑟)𝑑𝑟. (59) By Theorem 13, for some constant 𝐶₁ depending on 𝐶,𝑘, and 𝑇, 𝑢𝛾_𝑡−𝑢𝛾_𝑡≤𝐶₁1+𝛾_𝑡^𝑘+𝛾_𝑡^𝑘𝛾_𝑡−𝛾_𝑡+𝑡−𝑡^1/2. (60) This completes the proof.

3.2. Path Regularity of Process 𝑍

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: 𝑍^𝛾𝑡(𝑠)=∇_𝑥𝑢𝑠,𝛾_𝑡(𝑡)+𝑊(𝑠)−𝑊(𝑡),𝑃-a.s. (61) Under the conditions (𝐇₁) and (𝐇₂) in [8], Peng and Wang extend this result to the path-dependent case. The corresponding BSDE is 𝑌^𝛾𝑡(𝑠)=𝑔𝑊^𝛾𝑡_𝑇−^𝑇_𝑠ℎ𝑊^𝛾𝑡_𝑟,𝑌^𝛾𝑡(𝑟),𝑍^𝛾𝑡(𝑟)−𝑑𝑟_𝑇𝑠𝑍^𝛾𝑡(𝑟)𝑑𝑊(𝑟),𝑠∈[𝑡,𝑇], (62) where 𝑊^𝛾𝑡_𝑇=𝐼_𝑠≤𝑡𝛾_𝑡(𝑠)+𝐼_{𝑡<𝑠≤𝑇}𝛾_𝑡(𝑡)+𝑊(𝑠)−𝑊(𝑡). (63) Then under some assumptions, they obtained 𝑍^𝛾𝑡(𝑠)=𝐷_𝑥𝑢𝑊^𝛾𝑡_𝑠,𝑃-a.s. (64) In our context, we have the following theorem.

Theorem 17. Under Assumptions 1, 10, and 11, for each 𝛾_𝑡∈Λ, the process (𝑍^𝛾𝑡(𝑠))_{𝑠∈[𝑡,𝑇]} has a continuous version with the form 𝑍^𝛾𝑡(𝑠)=𝜎𝑋^𝛾𝑡_𝑠𝐷_𝑥𝑢𝑋^𝛾𝑡_𝑠,𝑓𝑜𝑟𝑠∈[𝑡,𝑇]𝑃-a.s. (65)

To prove the above theorem, we need the following lemma essentially from Pardoux and Peng [2].

Lemma 18. Let 𝛾_𝑡 and some 𝑡∈[𝑡,𝑇] be given. Suppose that 𝑔𝛾=𝜑𝛾𝑡,𝛾(𝑇)−𝛾𝑡, (66) where 𝜑 is in 𝐶³_𝑝(ℝ^2𝑑×ℝ^𝑚;ℝ^𝑚). For 𝑏,𝜎,ℎ, suppose that 𝑏𝛾_𝑠=𝑏₁𝑠,𝛾_𝑠(𝑠)𝐼_[0,𝑡)(𝑠)+𝑏₂𝑠,𝛾_𝑠(𝑠)−𝛾_𝑠𝑡𝐼_[𝑡,𝑇](𝑠),𝜎𝛾_𝑠=𝜎₁𝑠,𝛾_𝑠(𝑠)𝐼_[0,𝑡)(𝑠)+𝜎₂𝑠,𝛾_𝑠(𝑠)−𝛾_𝑠𝑡𝐼_[𝑡,𝑇](𝑠),ℎ𝛾_𝑠,𝑦,𝑧=ℎ₁𝑠,𝛾_𝑠(𝑠),𝑦,𝑧𝐼_[0,𝑡)(𝑠)+ℎ₂𝑠,𝛾_𝑠(𝑠)−𝛾_𝑠𝑡,𝑦,𝑧𝐼_[𝑡,𝑇](𝑠), (67) where 𝑏_𝑖,𝜎_𝑖,ℎ_𝑖∈𝐶^0,3,𝑖=1,2. Then for each 𝑠∈[𝑡,𝑇], 𝑍^𝛾𝑡(𝑠)=𝜎𝑋^𝛾𝑡_𝑠𝐷_𝑥𝑢𝑋^𝛾𝑡_𝑠,𝑓𝑜𝑟𝑠∈[𝑡,𝑇]𝑃-a.s. (68)

Proof. We only consider the one dimensional case.
For 𝑠∈[𝑡,𝑇], the FBSDE (5), (6) can be rewritten as 𝑋^𝛾𝑠(𝑢)=𝛾_𝑠(𝑠)+^𝑢_𝑠𝑏₂𝑟,𝑋^𝛾𝑠(𝑠)−𝛾_𝑠𝑡+𝑑𝑟_𝑢𝑠𝜎₂𝑟,𝑋^𝛾𝑠(𝑠)−𝛾_𝑠𝑡(𝑟),𝑌𝑑𝑊_𝛾𝑠(𝑢)=𝜑𝛾_𝑠𝑡,𝑋^𝛾𝑠(𝑇)−𝛾_𝑠𝑡−^𝑇_𝑢𝑍^𝛾𝑠(𝑟)(𝑟)−𝑑𝑊_𝑇𝑢ℎ₂𝑟,𝛾_𝑠𝑡,𝑋^𝛾𝑠(𝑟)−𝛾_𝑠𝑡,𝑌^𝛾𝑠(𝑟),𝑍^𝛾𝑠(𝑟)[].𝑑𝑟,𝑢∈𝑠,𝑇 (69) For 𝑠∈[𝑡,𝑡], 𝑋^𝛾𝑠(𝑢)=𝛾_𝑠(𝑠)+^𝑢_𝑠𝑏₁𝑟,𝑋^𝛾𝑠(𝑠)+𝑑𝑟_𝑢𝑠𝜎₁𝑟,𝑋^𝛾𝑠(𝑠)𝑑𝑊(𝑟),𝑢∈𝑠,𝑡,𝑋^𝛾𝑠(𝑢)=𝑋^𝛾𝑠𝑡+^𝑢_𝑡𝑏₂𝑟,𝑋^𝛾𝑠(𝑠)−𝑋^𝛾𝑠𝑡+𝑑𝑟_𝑢𝑠𝜎₂𝑟,𝑋^𝛾𝑠(𝑠)−𝑋^𝛾𝑠𝑡𝑑𝑊(𝑟),𝑢∈,𝑌𝑡,𝑇_𝛾𝑠(𝑢)=𝜑𝑋^𝛾𝑠𝑡,𝑋^𝛾𝑠(𝑇)−𝑋^𝛾𝑠𝑡−^𝑇_𝑢𝑍^𝛾𝑠(𝑟)(𝑟)−𝑑𝑊_𝑇𝑢ℎ₂𝑟,𝑋^𝛾𝑠𝑡,𝑋^𝛾𝑠(𝑟)−𝑋^𝛾𝑠𝑡,𝑌^𝛾𝑠(𝑟),𝑍^𝛾𝑠(𝑟)𝑑𝑟,𝑢∈,𝑌𝑡,𝑇_𝛾𝑠(𝑢)=𝑌^𝛾𝑠𝑡−^𝑡_𝑢ℎ₁𝑟,𝑋^𝛾𝑠𝑡,𝑌^𝛾𝑠(𝑟),𝑍^𝛾𝑠(𝑟)𝑑𝑟−^𝑡_𝑢𝑍^𝛾𝑠(𝑟)𝑑𝑊(𝑟),𝑢∈𝑠,𝑡. (70)
Now consider the following system of quasilinear parabolic differential equations, which is defined on [𝑡,𝑇]×ℝ² and parameterized by 𝑥∈ℝ: 𝜕_𝑠𝑢₂𝑠,𝑥,𝑦+ℒ𝑢₂𝑠,𝑥,𝑦=ℎ₂𝑠,𝑥,𝑦,𝑢₂𝑠,𝑥,𝑦,𝜕_𝑦𝑢₂𝑠,𝑥,𝑦𝜎𝑟_𝑠,𝑢₂𝑇,𝑥,𝑦=𝜑𝑥,𝑦, (71) where ℒ=(1/2)𝜎²(𝜕²/𝜕_𝑥𝑥)+𝑏(𝜕/𝜕_𝑥). The other one is defined on [𝑡,𝑡]×ℝ: 𝜕_𝑠𝑢₁(𝑠,𝑥)+ℒ𝑢₁(𝑠,𝑥)=ℎ₁𝑟,𝑥,𝑢₁(𝑠,𝑥),𝜕_𝑦𝑢₁()𝜎(𝑟),𝑢𝑠,𝑥₁𝑡,𝑥=𝑢₂𝑡,𝑥,0, (72) 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∈𝐶^1,2([𝑡,𝑇]×ℝ²;ℝ), 𝑢₁∈𝐶^1,2([𝑡,𝑡]×ℝ;ℝ), and 𝑢𝛾_𝑠=𝑢₁𝑠,𝛾_𝑠(𝑠)𝐼_[𝑡,𝑡)(𝑠)+𝑢₂𝑠,𝛾_𝑠𝑡,𝛾_𝑠(𝑠)−𝛾_𝑠𝑡𝐼_[𝑡,𝑇](𝑠). (73) Then we obtain 𝑌^𝛾𝑡(𝑠)=𝑢₁𝑠,𝑋^𝛾𝑡(𝑠),𝑡≤𝑠<𝑌𝑡,_𝛾𝑡(𝑠)=𝑢₂𝑠,𝑋^𝛾𝑡𝑡,𝑋^𝛾𝑡(𝑠)−𝑋^𝛾𝑡𝑡,𝑍𝑡≤𝑠≤𝑇,_𝛾𝑡(𝑠)=𝜕_𝑥𝑢₁𝑠,𝑋^𝛾𝑡(𝑠)𝜎₁𝑋^𝛾𝑡_𝑠,𝑡≤𝑠<𝑍𝑡,_𝛾𝑡(𝑠)=𝜕_𝑥𝑢₂𝑠,𝑋^𝛾𝑡𝑡,𝑋^𝛾𝑡(𝑠)−𝑋^𝛾𝑡𝑡𝜎₂𝑋^𝛾𝑡_𝑠,𝑡≤𝑠≤𝑇. (74) Finally, for each 𝑠∈[𝑡,𝑇], 𝑍^𝛾𝑡(𝑠)=𝜎𝑋^𝛾𝑡_𝑠𝐷_𝑥𝑢𝑋^𝛾𝑡_𝑠,𝑃-a.s. (75) In particular, 𝑍^𝛾𝑡(𝑡)=𝜎𝛾_𝑡𝐷_𝑥𝑢𝛾_𝑡,𝛾_𝑡∈Λ. (76) This completes the proof.

Now we give the proof of Theorem 17.

Proof. For each fixed 𝑡∈[0,𝑇] and positive integer 𝑛, we introduce a mapping 𝛾^𝑛(𝛾_𝑠)∶Λ_𝑠↦Λ_𝑠𝛾^𝑛𝛾_𝑠(𝑟)=𝛾_𝑠(𝑟)𝐼_[0,𝑡)+^𝑛−1_𝑘=0𝛾_𝑠𝑡^𝑛_𝑘+1∧𝑠𝐼_[𝑡^𝑛_{𝑘∧𝑠,𝑡}^𝑛_{𝑘+1∧𝑠)}(𝑟)+𝛾_𝑠(𝑠)𝐼_{𝑠}(𝑟),𝑠∈[0,𝑇], (77) where 𝑡^𝑛_𝑘=𝑡+(𝑘(𝑇−𝑡)/𝑛), 𝑘=0,1,…,𝑛𝑔^𝑛𝛾∶=𝑔𝛾^𝑛𝛾,ℎ^𝑛𝛾_𝑠,𝑦,𝑧∶=ℎ𝛾^𝑛𝛾_𝑠,𝑦,𝑧. (78) For each 𝑛, there exist some functions 𝜑_𝑛 defined on Λ_𝑡×ℝ^𝑛×𝑑 and 𝜓_𝑛,𝜙_𝑛,𝜙_𝑛 defined on [𝑡,𝑇]×Λ_𝑡×ℝ^𝑛×𝑑×ℝ^𝑚×ℝ^𝑚×𝑑 such that 𝑔^𝑛𝛾∶=𝜑_𝑛𝛾_𝑡,𝛾𝑡^𝑛₁−𝛾(𝑡),…,𝛾𝑡^𝑛_𝑛−𝛾𝑡^𝑛_𝑛−1,𝑏^𝑛𝛾_𝑠∶=𝜙_𝑛𝑠,𝛾_𝑡,𝛾_𝑠𝑡^𝑛₁∧𝑠−𝛾_𝑠(𝑡),…,𝛾_𝑠𝑡^𝑛_𝑛−∧𝑠𝛾_𝑠𝑡^𝑛_𝑛−1,𝜎∧𝑠_𝑛𝛾_𝑠∶=𝜙_𝑛𝑠,𝛾_𝑡,𝛾_𝑠𝑡^𝑛₁∧𝑠−𝛾_𝑠(𝑡),…,𝛾_𝑠𝑡^𝑛_𝑛−∧𝑠𝛾_𝑠𝑡^𝑛_𝑛−1,ℎ∧𝑠_𝑛𝛾_𝑠,𝑦,𝑧∶=𝜓_𝑛𝑠,𝛾_𝑡,𝛾_𝑠𝑡^𝑛₁∧𝑠−𝛾_𝑠(𝑡),…,𝛾_𝑠𝑡^𝑛_𝑛−∧𝑠𝛾_𝑠𝑡^𝑛_𝑛−1∧𝑠,𝑦,𝑧. (79) Indeed, if we set 𝜑_𝑛𝛾_𝑡,𝑥₁,…,𝑥_𝑛∶=𝑔𝛾_𝑡(𝑠)𝐼_[0,𝑡)(𝑠)+^𝑛_𝑘=1𝑥_𝑘𝐼_[𝑡^𝑛_{𝑘−1,𝑡}^𝑛_𝑘)(𝑠)+𝑥_𝑛𝐼_{𝑇}(𝑠)_{0≤𝑠≤𝑇},𝜑_𝑛𝛾_𝑡,𝑥₁,…,𝑥_𝑛∶=𝜑_𝑛𝛾_𝑡,𝛾_𝑡+𝑥₁,𝛾_𝑡(𝑡)+𝑥₁+𝑥₂,…,𝛾_𝑡(𝑡)+^𝑛_𝑖=1𝑥_𝑖, (80)
then by Assumptions 1, 10, and 11, we obtain that, for each fixed 𝛾_𝑡, 𝜑_𝑛(𝛾_𝑡,𝑥₁,…,𝑥_𝑛) is a 𝐶³_𝑝-function of 𝑥₁,…,𝑥_𝑛. In particular, for each 𝛾∈Λ, 𝜕_𝑥𝑖𝜑_𝑛𝛾_𝑡,𝛾𝑡^𝑛₁−𝛾(𝑡),…,𝛾𝑡^𝑛_𝑛−𝛾𝑡^𝑛_𝑛−1=𝑔^′_{𝛾𝑡𝑛𝑖−1}𝛾^𝑛𝛾. (81)
For any 𝑡≥𝑡,𝛾_𝑡∈Λ_𝑡, we consider the following FBSDE: 𝑋^{𝑛,𝛾𝑡}(𝑠)=𝛾_𝑡𝑡+^𝑠_𝑡𝑏^𝑛𝑋^{𝑛,𝛾𝑡}_𝑟𝑑𝑟+^𝑠_𝑡𝜎^𝑛𝑋^{𝑛,𝛾𝑡}_𝑟(𝑟),𝑌𝑑𝑊_{𝑛,𝛾𝑡}(𝑠)=𝑔^𝑛𝑋^{𝑛,𝛾𝑡}_𝑇−^𝑇_𝑠ℎ^𝑛𝑋^{𝑛,𝛾𝑡}_𝑟,𝑌^{𝑛,𝛾𝑡}(𝑟),𝑍^{𝑛,𝛾𝑡}(𝑟)−𝑑𝑟_𝑇𝑠𝑍^{𝑛,𝛾𝑡}(𝑟)𝑑𝑊(𝑟),𝑡≤𝑠. (82) We denote 𝑢^𝑛𝛾_𝑡∶=𝑌^{𝑛,𝛾𝑡}(𝑡),𝛾_𝑡∈Λ. (83) Following the argument as in Lemma 18, for each 𝑠∈[𝑡,𝑇], we have 𝑍^{𝑛,𝛾𝑡}(𝑠)=𝜎^𝑛𝑋^{𝑛,𝛾𝑡}_𝑠𝐷_𝑥𝑢^𝑛𝛾_𝑡,𝑃-a.s. (84) By the definition of 𝑔^𝑛, 𝑏^𝑛, 𝜎^𝑛, ℎ^𝑛 and (82) we have lim_𝑛𝑋^{𝑛,𝛾𝑡}_𝑇=𝑋^𝛾𝑡_𝑇,𝑃-a.s,lim_𝑛𝑌^{𝑛,𝛾𝑡}(𝑠),𝑍^{𝑛,𝛾𝑡}(𝑠)=𝑌^𝛾𝑡(𝑠),𝑍^𝛾𝑡(𝑠),a.e.𝑠∈[𝑡,𝑇],𝑃-a.s,lim_𝑛𝑢^𝑛𝛾_𝑡=𝑢𝛾_𝑡,lim_𝑛𝐷_𝑥𝑢^𝑛𝛾_𝑡=𝐷_𝑥𝑢𝛾_𝑡,lim_𝑛𝐷_𝑥𝑥𝑢^𝑛𝛾_𝑡=𝐷_𝑥𝑥𝑢𝛾_𝑡,lim_𝑛𝑢^𝑛𝑋^{𝑛,𝛾𝑡}_𝑠,𝐷_𝑥𝑢^𝑛𝑋^{𝑛,𝛾𝑡}_𝑠,𝐷_𝑥𝑥𝑢^𝑛𝑋^{𝑛,𝛾𝑡}_𝑠=𝑢𝑋^𝛾𝑡_𝑠,𝐷_𝑥𝑢𝑋^𝛾𝑡_𝑠,𝐷_𝑥𝑥𝑢𝑋^𝛾𝑡_𝑠, (85)(86)a.e.𝑠∈[𝑡,𝑇],𝑃-a.s.
Therefore 𝑍^𝛾𝑡(𝑠)=𝜎𝑋^𝛾𝑡_𝑠𝐷_𝑥𝑢𝑋^𝛾𝑡_𝑠,a.e.𝑠∈[𝑡,𝑇],𝑃-a.s. (87)
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: 𝐷_𝑡𝑢𝛾_𝑡+ℒ𝑢𝛾_𝑡−ℎ𝛾_𝑡,𝑢𝛾_𝑡,𝜎𝛾_𝑡𝐷_𝑥𝑢𝛾_𝑡𝑢𝛾=0,_𝑇=𝑔𝛾_𝑇,𝛾_𝑇∈Λ^𝑛, (88) where 1ℒ𝑢=2tr𝜎𝜎^𝑇𝐷_𝑥𝑥𝑢+𝑏,𝐷_𝑥𝑢. (89)

Theorem 19. Suppose that Assumptions 1, 10, and 11 are fulfilled, and if 𝑢∈ℂ^1.2(Λ), and that 𝑢 is the solution of (88), 𝑢 is uniformly Lipschitz continuous, and bounded by 𝐶(1+‖𝛾_𝑡‖), then the solution is unique, and for any 𝛾_𝑡∈Λ, 𝑢(𝛾_𝑡) is determined by (5) and (6).

Proof. By the assumptions of this theorem, we know that 𝑏(𝛾_𝑡) and 𝜎(𝛾_𝑡) are uniformly Lipschtiz continuous and the following SDE has a uniqueness solution: 𝑑𝑋^𝛾𝑡(𝑠)=𝑏𝑋^𝛾𝑡_𝑠𝑑𝑠+𝜎𝑋^𝛾𝑡_𝑠(𝑠),𝑋𝑑𝑊_𝑡=𝛾_𝑡,𝑠∈[𝑡,𝑇]. (90) Set 𝑌(𝑠)=𝑢(𝑋^𝛾𝑡_𝑠),𝑡≤𝑠≤𝑇. Applying Itô’s formula to 𝑌(𝑠)=𝑢(𝑋^𝛾𝑡_𝑠), we have (𝑠)𝑋𝑑𝑌=−ℎ^𝛾𝑡_𝑠,𝑌(𝑠),𝑍(𝑠)𝑑𝑟−𝜎𝑋^𝛾𝑡_𝑠𝐷_𝑥𝑢𝑋^𝛾𝑡_𝑠(𝑠).𝑌(𝑇)𝑋𝑑𝑊=𝑔^𝛾𝑡_𝑇𝑠∈[𝑡,𝑇]. (91) 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 Assumptions 1, 10, and 11, the function 𝑢(𝛾_𝑡)=𝑌^𝛾𝑡(𝑡) is the unique ℂ^1,2(Λ)-solution of the path-dependent PDE (88).

Proof. We only study the one dimensional case. 𝑢∈ℂ^0,2(Λ) follows from Corollary 16. Let 𝛿>0 satisfying 𝑡+𝛿≤𝑇. By Lemma 15 we can get 𝑢𝑋^𝛾𝑡_𝑡+𝛿=𝑌^𝛾𝑡(𝑡+𝛿). (92) Hence 𝑢𝛾_{𝑡,𝑡+𝛿}−𝑢𝛾_𝑡=𝑢𝛾_{𝑡,𝑡+𝛿}−𝑢𝑋^𝛾𝑡_𝑡+𝛿+𝑢𝑋^𝛾𝑡_𝑡+𝛿−𝑢𝛾_𝑡, (93) By the proof of Theorem 17, we obtain 𝑢𝛾_{𝑡,𝑡+𝛿}−𝑢𝛾_𝑡=lim_𝑛→∞𝑢^𝑛𝛾_{𝑡,𝑡+𝛿}−𝑢^𝑛𝑋^𝛾𝑡_𝑡+𝛿+_𝑡^𝑡+𝛿ℎ𝑋^𝛾𝑡_𝑠,𝑌^𝛾𝑡(𝑠),𝑍^𝛾𝑡(𝑠)+𝑑𝑠_{𝑡𝑡+𝛿}𝑍^𝛾𝑡(𝑠)𝑑𝑊(𝑠). (94)
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 𝑢^𝑛𝛾_{𝑡,𝑡+𝛿}−𝑢^𝑛𝑋^𝛾𝑡_𝑡+𝛿=_𝑡^𝑡+𝛿𝐷_𝑠𝑢^𝑛𝛾_𝑡,𝑠𝑑𝑠−_𝑡^𝑡+𝛿𝐷_𝑠𝑢^𝑛𝑋^𝛾𝑡_𝑠−𝑑𝑠_{𝑡𝑡+𝛿}𝐷_𝑥𝑢^𝑛𝑋^𝛾𝑡_𝑠𝑑𝑋^{𝑛,𝛾𝑡}(𝑠)−12_𝑡^𝑡+𝛿𝐷_𝑥𝑥𝑢^𝑛𝑋^𝛾𝑡_𝑠𝑑𝑋^{𝑛,𝛾𝑡}(𝑠). (95) Thus by (86) and the dominated convergence theorem, we have 𝑢𝛾_{𝑡,𝑡+𝛿}−𝑢𝛾_𝑡=−_𝑡^𝑡+𝛿𝐷_𝑥𝑢𝑋^𝛾𝑡_𝑠𝑑𝑋^𝛾𝑡(𝑠)−12_𝑡^𝑡+𝛿𝐷_𝑥𝑥𝑢𝑋^𝛾𝑡_𝑠𝑑𝑋^𝛾𝑡𝑠)+_{𝑡𝑡+𝛿}(ℎ𝑋^𝛾𝑡_𝑠,𝑌^𝛾𝑡(𝑠),𝑍^𝛾𝑡(𝑠+)𝑑𝑠_{𝑡𝑡+𝛿}𝑍^𝛾𝑡(𝑠)𝑑𝑊(𝑠)+lim_𝑛→∞𝐶^𝑛, (96) where 𝐶^𝑛=_𝑡^𝑡+𝛿𝐷_𝑠𝑢^𝑛𝛾_𝑡,𝑠𝑑𝑠−_𝑡^𝑡+𝛿𝐷_𝑠𝑢^𝑛𝑋^𝛾𝑡_𝑠𝑑𝑠. (97) Note that 𝑢^𝑛(𝛾_𝑡)∈ℂ^0,2_{𝑙,𝑙𝑖𝑝}(Λ). By Lemma 12, we have 𝐷_𝑠𝑢^𝑛𝛾_𝑡,𝑠−𝐷_𝑠𝑢^𝑛𝑋^𝛾𝑡_𝑠≤𝑐𝛾_𝑡−𝑋^𝛾𝑡_𝑠,a.e.𝑠∈[𝑡,𝑇],𝑃-a.s (98) for some constant 𝑐 depending on 𝐶,𝑇,𝛾_𝑡, and 𝑘. Hence 𝐶^𝑛≤𝑐𝛿sup_{[]𝑠∈𝑡,𝑡+𝛿}𝑋^𝛾𝑡(𝑠)−𝛾_𝑡(𝑡),𝑃-a.s. (99) Taking expectation on both sides of (96), we have lim_𝛿→0𝑢𝛾_{𝑡,𝑡+𝛿}−𝑢𝛾_𝑡𝛿=−ℒ𝑢𝛾_𝑡+ℎ𝛾_𝑡,𝑢𝛾_𝑡,𝐷_𝑥𝑢𝛾_𝑡𝜎𝛾_𝑡. (100) Thus 𝑢(𝛾_𝑡) belongs to ℂ^1,2(Λ) and satisfies (88).

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 11171187 and no. 11221061); by the Programme of Introducing Talents of Discipline to Universities of China (no. B12023); and by Program for New Century Excellent Talents in University of China.