Representation theorem for generators of BSDEs driven by G-Brownian motion and its applications

We obtain a representation theorem for the generators of BSDEs driven by G-Brownian motions, and then we use the representation theorem to get a converse comparison theorem for G-BSDEs and some equivalent results for nonlinear expectations generated by G-BSDEs.


Introduction
In 1997, Peng ([1]) introduced g−expectations basing on Backward Stochastic Differential Equations (BSDEs) ( [2]). One of the important properties of g−expectations is comparison theorem or monotonicity. Chen ([3]) first consider a converse result of BSDEs under equal case. After that Briand et. al. ([4]) obtained a converse comparison theorem for BSDEs under general case. They also derived a representation theorem for the generator g. Following this paper, Jiang ([5]) discussed a more general representation theorem, then in his another paper ( [6]) showed a more general converse comparison theorem. Here the representation theorem is an important method in solving the converse comparison problem and other problems (see Jiang [7]).
Recently, Hu et. al. ([8]) proved an existence and uniqueness result on BSDEs driven by G−Brownian motions (G-BSDEs), further (in [9]) they gave a comparison theorem for G-BSDEs. In this paper we consider the representation theorem for generators of G-BSDEs, and then consider the converse comparison theorem of G-BSDEs and some equivalent results for nonlinear expectations generated by G-BSDEs. In the following, In Section 2, we review some basic concepts and results about G−expectations. We give the representation theorem of G-BSDEs in Section 3; In Section 4, we consider the applications of representation theorem of G-BSDEs, which contain the converse comparison theorem and some equivalent results for nonlinear expectations generated by G-BSDEs.

Preliminaries
We review some basic notions and results of G-expectation, the related spaces of random variables and the backward stochastic differential equations driven by a G-Browninan motion. The readers may refer to [8], [10], [11], [12], [13], [14] for more details.
whereX is an independent copy of X, i.e.,X d = X andX⊥X. Here the letter G denotes the function where S d denotes the collection of d × d symmetric matrices.
Peng [13] showed that X = (X 1 , · · ·, X d ) is G-normally distributed if and only if for each is the solution of the following G-heat equation: where S + d denotes the collection of nonnegative elements in S d . In this paper, we only consider non-degenerate G-normal distribution, i.e., there exists some Let G : S d → R be a given monotonic and sublinear function. G-expectation is a sublinear expectation defined byÊ For each fixed T > 0, we set . Let π N t = {t N 0 , · · · , t N N }, N = 1, 2, · · · , be a sequence of partitions of [0, t] such that µ(π N t ) = max{|t N i+1 − t N i | : i = 0, · · · , N − 1} → 0, the quadratic variation process of B a is defined by For each fixed a,ā ∈ R d , the mutual variation process of B a and Bā is defined by Definition 2.6. For fixed T > 0, let M 0 G (0, T ) be the collection of processes in the following form: for a given partition We consider the following type of G-BSDEs (in this paper we always use Einstein convention): where satisfy the following properties: (H1): There exists some β > 1 such that for any y, z, f (·, ·, y, z), g ij (·, ·, y, z) ∈ M β G (0, T ). (H2): There exists some L > 0 such that For simplicity, we denote by S α Definition 2.7. Let ξ ∈ L β G (Ω T ) and f satisfy (H1) and (H2) for some β > 1. A triplet of processes (Y, Z, K) is called a solution of equation (1) if for some 1 < α ≤ β the following properties hold: Theorem 2.1. ( [8]) Assume that ξ ∈ L β G (Ω T ) and f , g ij satisfy (H1) and (H2) for some β > 1. Then equation (1) has a unique solution (Y, Z, K). Moreover, for any . We have the following estimates. (1). Then there exists a constant C α > 0 depending on α, T , G, L such that Proposition 2.2. ( [15,8]) Let α ≥ 1 and δ > 0 be fixed. Then there exists a constant C depending on α and δ such that Theorem 2.2. ( [9]) Let (Y l , Z l , K l ), l = 1, 2, be the solutions of the following G-BSDEs: where ξ ∈ L β G (Ω T ), f and g ij satisfy (H1) and (H2) for some In this paper, we also need the following assumptions for G-BSDE (1).

Representation theorem of generators for G-BSDEs
We consider the following type of G-FBSDEs: where h ij = h ji and g ij = g ji , 1 ≤ i, j ≤ d.
We now give the main result in this section.

Some applications
4.1. Converse comparison theorem for G-BSDEs. We consider the following G-BSDEs: where g l ij = g l ji . We first generalized the comparison theorem in [9]. If Proof. From the above G-BSDEs, we have By the assumption, it is easy to check that (V t ) t≤T is a decreasing process. Thus, using Remark 4.1. Suppose d = 1, and let f 1 = 10|z|, f 2 = |z|, g 1 = |z| and g 2 = 2|z|. It is easy to check that . Now we give the converse comparison theorem.
Proof. For simplicity, we take the notationẼ l .
Proof. Taking η ε as in Theorem 4.1, since f l and g l ij are deterministic, we could getẼ l t [η ε ] = E l [η ε ], for l = 1, 2. And the proof in Theorem 4.1 still holds true. 4.2. Some equivalent relations. We consider the following G-BSDE: where g ij = g ji . We use the notationẼ t [ξ] = Y t .