Stochastic Differential Equations on Manifolds

In [1] and [2], we studied the problem of the existence and uniqueness of a solution to some general BSDE on manifolds. In these two articles, we assumed some Lipschitz conditions on the drift f(b, x, z). The purpose of this article is to extend the existence and uniqueness results under weaker assumptions, in particular a monotonicity condition in the variable x. This extends well-known results for Euclidean BSDE. 1 Reminder of the problem Unless otherwise stated, we shall work on a fixed finite time interval [0;T ]; moreover, (Wt)0≤t≤T will always denote a Brownian Motion (BM for short) in R dw , for a positive integer dw. Moreover, Einstein’s summation convention will be used for repeated indices in lower and upper position. Let (B t )0≤t≤T denote the R -valued diffusion which is the unique strong solution of the following SDE : { dB t = b(B y t )dt+ σ(B y t )dWt B 0 = y, (1.1) where σ : R → Rw and b : R → R are C bounded functions with bounded partial derivatives of order 1, 2 and 3. 1 Let us recall the problem studied in [1] and [2]. We consider a manifold M endowed with a connection Γ, which defines an exponential mapping. On M , we study the uniqueness and existence of a solution to the equation (under infinitesimal form) (M +D)0 { Xt+dt = expXt(ZtdWt + f(B y t , Xt, Zt)dt) XT = U where Zt ∈ L(Rdw , TXtM) and f(B t , Xt, Zt) ∈ TXtM . For details about links with PDEs, the reader is referred to the introductions of [1] and [2]. In local coordinates (x), the equation (M+D)0 becomes the following backward stochastic differential equation (BSDE in short) (M +D) { dXt = ZtdWt + ( − 2 Γjk(Xt)([Zt] |[Zt]) + f(B t , Xt, Zt) ) dt XT = U. We keep the same notations as in [1] : (·|·) is the usual inner product in an Euclidean space, the summation convention is used, and [A] denotes the i row of any matrix A; moreover, Γjk(x) = 