The Wellposedness of FBSDEs (II)

This paper is a continuation of \cite{zhang}, in which we established the wellposedness result and a comparison theorem for a class of one dimensional Forward-Backward SDEs. In this paper we extend the wellposedness result to high dimensional FBSDEs, and weaken the key condition in \cite{zhang} significantly. Compared to the existing methods in the literature, our result has the following features: (i) arbitrary time duration; (ii) random coefficients; (iii) (possibly) degenerate forward diffusion; and (iv) no monotonicity condition.


Introduction and Main Result
Assume (Ω, F , P ) is a complete probability space, F 0 ⊂ F , and W is a d-dimensional standard Brownian motion independent of F 0 . Let F △ = {F t } 0≤t≤T be the filtration generated by W and F 0 , augmented by the null sets as usual. We study the following FBSDE:  where Θ △ = (X, Y, Z) and * denotes the transpose. We assume that X 0 ∈ F 0 , b, σ, f, g are progressively measurable, and for any θ △ = (x, y, z), b, σ, f are F-adapted and g(·, x) ∈ F T . For simplicity we will always omit the variable ω in b, σ, f, g.
Theorem 1.1 Assume that all processes are one dimensional; that b, σ, f, g are differentiable with respect to x, y, z with uniformly bounded derivatives; and that If I 2 0 < ∞, then FBSDE (1.1) has a unique solution Θ such that After [10] has been accepted for publication, we find that Theorem 1.1 can be improved significantly. In the sequel we assume Here W, Y, et al are considered as column vectors. Let ∂ denote partial derivatives with appropriate dimensions; and | · | denote the Euclidian norm. For example, Our main result is the following theorem. Theorem 1.2 Assume that b, σ, f, g are uniformly Lipschitz continuous in x, y, z; and that there exists a constant c > 0 such that for any y ∈ IR d such that |y| = 1, where (1.7) If I 2 0 < ∞, then FBSDE (1.1) has a unique solution Θ such that Θ 2 ≤ CI 2 0 , where C depends on c and the Lipschitz constant of the coefficients.
We note that we only assume those partial derivatives involved in (1.7) exist.
Moreover, when one part of a product vanishes, we do not need to assume the other part to be differentiable. For example, if ∂ z b = 0, then we do not need ∂ x σ. In fact, we can even weaken (1.6) further by using approximating coefficients (see (2.13) at below). Remark 1.3 Following are three sufficient conditions for (1.6): (1.8) where Id n ∈ IR n×n is the n × n identity matrix.
Remark 1.4 (i) A necessary condition to ensure (1.8) is n ≤ d; (ii) There are two typical cases for (1.9). One is that ∂ y σ = 0, then (1.1) becomes the standard decoupled FBSDE. The other one is that ∂ z f = 0, then (1.1) becomes (1.11) We note that in this case it is allowed to have n > d. However, we should point out that our method does not work when X is high dimensional, mainly due to the non-commuting property of matrices multiplication.
We would leave this case for future research.

Small Time Duration
In this section we establish some important results for FBSDEs with small time duration T . First we recall a wellposedness result due to Antonelli [1].
Lemma 2.1 Assume b, σ, f have a uniform Lipschitz constant K, and g has a uniform Lipschitz constant K 0 . There exist constants δ 0 and C 0 , depending only on K and K 0 , such that for T ≤ δ 0 , if I 2 0 < ∞, then (1.1) has a unique solution Θ and it holds that Θ ≤ C 0 I 0 .
The following lemma, which estimates the C 0 at above in terms of (K, K 0 ), is the key step for the proof of Theorem 1.2.
Lemma 2.2 Consider the following linear FBSDE: for any y ∈ IR such that |y| = 1, Let δ 0 be as in Lemma 2.1. There exists a constant C K , depending only on K but independent of K 0 , such that for any T ≤ δ 0 , the solution to FBSDE (2.1) satisfies In the sequel we use C K to denote a generic constant which depends only on K and may vary from line to line. Recalling (1.7) one can easily check that, for linear Recalling that |Z| 2 △ = tr (ZZ * ). Then DenoteȲ t △ =Ỹ t |Ỹ t | −1 when |Ỹ t | = 0, and arbitrary unit vector otherwise. Then |Ȳ t | = 1 and Step 2. The arguments in this step are similar to those for Lemma 3.2 in [10], so we will only sketch the main idea. Denote Then τ n ↑ τ and X t > 0 for t ∈ [0, τ ). Recall (2.5) for t ∈ [0, τ ). By Lemma 2.1 one can easily prove that |Y t | ≤ C 0 |X t |, and thus Then by (2.6) one gets In light of (2.5) we define for the C K in (2.9). By (2.7) M is a martingale. Moreover, thanks to the obvious fact that L t > 0, M t > 0.
We note that estimate (2.8) is essential for the wellposedness of FBSDEs.
Example 1 Consider the following one dimensional FBSDE (2.11) Then Actually one can prove in this example thatỸ t > 0 for any t, then We would also like to mention that (2.8) is consistent with the four step scheme (see [5] and [3]) in the following sense. Assume an FBSDE in the four step scheme framework has two solutions Θ 1 , and u is uniformly Lipschitz continuous in x, where u is the solution to the corresponding PDE. ThenỸ t is uniformly bounded and thus (2.8) holds true.

Corollary 2.3
Assume that all the conditions in Lemma 2.1 as well as (1.6) hold true with c = 1 K . Let T ≤ δ 0 as in Lemma 2.1, and Θ i , i = 0, 1, be the solution to FBSDEs: Then Proof. We first assume that all the coefficients are differentiable. For 0 ≤ λ ≤ 1, let be the solutions to FBSDEs: tr (∂ z f j * (s, Θ λ s )∇Z λ * s ) ds; (2.12) respectively. One can easily prove that In particular, Next we consider the following FBSDE over [T m−2 , T m−1 ]: Similarly we may define g m−2 (x) such that Repeat the arguments for i = m, · · · , 1, we may define g i such that Now for any X 0 ∈ L 2 (F 0 ), we may construct the solution to FBSDE (1.1) piece by piece over subintervals [T i−1 , T i ] with terminal condition g i , i = 1, · · · , n. Since on each subinterval the solution is unique, we obtain the uniqueness of the solution to FBSDE (1.1). Finally, the estimate Θ ≤ CI 0 can also be obtained by piece by piece estimates, as done in [10].
Finally we state the stability result whose proof is exactly the same as in [10] and thus is omitted.