backward stochastic differential equations ∗

In this paper, after recalling the definition of generalized anticipated backward stochastic differential equations (generalized anticipated BSDEs for short) and the existence and uniqueness theorem for their solutions, we show there is a duality between them and stochastic differential delay equations. We then provide a continuous dependence property for their solutions with respect to the parameters and finally establish a comparison result for the solutions of these equations.

(ii) there exists a constant L ≥ 0 such that for any t ∈ [0, T ] and nonnegative and integrable g(·), T t g(s + δ(s))ds ≤ L T +K t g(s)ds; T t g(s + ζ(s))ds ≤ L T +K t g(s)ds.
Further, for all s ∈ [0, T ], f (s, ω, y, z, ξ, η) : Ω×R m ×R m×d ×L 2 (F r ; R m )×L 2 (F r ; R m×d ) −→ L 2 (F s , R m ), where r, r ∈ [s, T + K], and f satisfies the following conditions: * Supported by the NSERC and the ARC. † Shandong University, China and University of Calgary, Canada. E-mail: yangzhezhe@gmail.com ‡ University of Calgary, Canada and University of Adelaide, Australia. E-mail: relliott@ucalgary.ca Some properties of generalized anticipated BSDEs (H1) there exists a constant C > 0, such that for all s ∈ [0, T ], y, y ∈ R m , z, z ∈ R m×d , ξ . , ξ . ∈ L 2 For these equations, [4] gives an existence and uniqueness result, and also proves some comparison theorems. Then Xu [5] gave a more general comparison theorem for anticipated BSDEs where the generators have less restrictions. In 2011 Xu [6] provided necessary and sufficient condition for the comparison theorem for multidimensional anticipated BSDEs. In 2013 Yang and Elliott [8] established a converse comparison theorem for anticipated BSDEs. In 2006 Yang [7] generalized anticipated BSDEs as follows: Here K > 0 is a given constant and for all t . It is obvious that the equations studied in [4] are a special type of this equation. In the same paper Yang deduces the existence and uniqueness theorem for solutions of the above equations. However, the notation in the above equations is not clear. In this paper we rewrite above equation as: where K > 0 is a given constant and for all t ∈ [0, T ] and f is a function defined on L 2 F (t, T + K; R m ) × L 2 F (t, T + K; R m×d ) with values in L 2 (F t , R m ). We call the above type of equation a generalized anticipated BSDE. In this paper, we discuss generalized anticipated BSDEs and derive a comparison theorem. This paper is organized as follows. After a brief presentation of some known results that we shall use in Section 2, Section 3 provides some properties of generalized anticipated BSDEs, including a duality between them and stochastic differential delay equations (SDDEs), a continuous dependence property with respect to parameters for their solutions, and an important result for generalized anticipated BSDEs, the comparison theorem.

Preliminaries
Let (Ω, F , P, F t , t ≥ 0) be a complete stochastic basis such that F 0 contains all Pnull elements of F and suppose that the filtration is generated by a d-dimensional standard Brownian motion W. = (W t ) t≥0 . Suppose T > 0 is given. For all n ∈ N, denote the Euclidean norm in R n by | · |. Denote: . Then L 2 and S 2 are separable Hilbert spaces.

Some properties of generalized anticipated BSDEs
The following four lemmas are quoted from Peng [3]. Lemma 2.1 below is Lemma 3.1 of Peng [3]. Lemma 2.2 is Theorem 3.2 of Peng [3], and is a basic result for BSDEs: an existence and uniqueness theorem. Lemma 2.3, which is a comparison result for solutions of BSDEs, is Theorem 3.3 of Peng [3], and can also be found in El Karoui, Peng and Quenez [1].
satisfying the following BSDE: We have the following basic estimate: (2.1) In particular, where β > 0 is an arbitrary constant. We also have where the constant k depends only on T .
Consider the following conditions for g = g(ω, t, y, z) : Ω×[0, T ]×R m ×R m×d −→ R m : (a) g(·, y, z) is an R m -valued and F t -adapted process satisfying Lipschitz condition in (y, z), i.e., there exists ρ > 0 such that for each y, y ∈ R m and z, z ∈ R m×d , Lemma 2.2. Assume that g satisfies (a) and (b). Then for any given terminal condition has a unique solution, i.e., there exists a unique pair of F t -adapted processes (Y . , Z . ) . , Z (1) . ) and (Y (2) . , Z (2) . ) be, respectively, the solutions of BSDEs as follows:

Recalling generalized anticipated BSDEs
We first recall the basic conditions for the existence and uniqueness of solutions to generalized anticipated BSDEs in [7]. Let K > 0 be a given constant. Consider the following generalized anticipated BSDE: (3.1) We wish to find a pair of F t -adapted processes (Y., Z.) ∈ S 2 Assume that for all t ∈ [0, T ], f (t, y . , z . ) : R m ), and f satisfies the conditions as follows where β ≥ 0 is an arbitrary constant.

Example 2. For any
That is, (H4 ) holds. Hence (H4) holds. On the other hand, for any t ∈ (0, T ], x ∈ [t, T + K], y ∈ [0, t), by Itô's formula we know Therefore, f does not satisfy (H1). The following is the main result of this section: the existence and uniqueness theorem for solutions of generalized anticipated BSDEs.

Theorem 3.2.
Suppose that f satisfies (H3), (H4) and (H5). Then for arbitrary pair of given terminal conditions ξ . ∈ S 2 F (T, T + K; R m ), η . ∈ L 2 F (T, T + K; R m×d ), the generalized anticipated BSDE (3.1) has a unique solution, that is, there exists a unique pair The proof of theorem 3.2 is similar to the proof of theorem 4.2 in [4] or can be found in [7].

A duality between SDDEs and generalized anticipated BSDEs
El Karoui, Peng and Quenez [1] showed that there is a duality between stochastic differential equations (SDEs) and BSDEs, that is, the solutions of linear BSDEs can expressed in terms of the solutions of SDEs. In [4] Peng and Yang proved a duality between SDDEs and anticipated BSDEs. We shall investigate whether there exists a duality between SDDEs and generalized anticipated BSDEs. For the generalized anticipated BSDEs considered below, the answer is positive.
They are obviously two linear functions, that is, for all s ∈ [t, , and for all α is a constant: Consider the following generalized anticipated BSDE: where for all s ∈ [0, T ], θ. ∈ L 2 F (s, T + K), θ . ∈ L 2 F (s, T + K; R 1×d ). Then A * s (θ) and B * s (θ ) are defined as follows:

Some properties of generalized anticipated BSDEs
Proof. There exists a unique solution to the above SDDE (see Theorem (2.1) of [2]).
Apply Itô's formula to X s Y s for s ∈ [t, T ], and take the conditional expectation under F t .

A continuous dependence property with respect to parameters for the solutions of generalized anticipated BSDEs
The basic estimates (2.1) and (2.2) can also be applied to study the continuous dependence property of the generalized anticipated BSDEs with respect to parameters.

Comparison theorem of 1-dimensional generalized anticipated BSDEs
Comparison theorems are fundamental results in the theory of BSDEs. It is natural to ask whether there is a comparison result for 1-dimensional generalized anticipated BSDEs. The answer is positive.
The following result is a comparison theorem for 1-dimensional generalized anticipated BSDEs. In this section, m = 1.