Backward stochastic differential equations on manifolds

Backward Stochastic Differential Equations on Manifolds

The problem of finding a martingale on a manifold with a fixed random terminal value can be solved by considering BSDEs with a generator with quadratic growth. We study here a generalization of these equations and we give uniqueness and existence results in two different frameworks, using differential geometry tools. Applications to PDEs are given, including a certain class of Dirichlet problem...

Backward Stochastic Differential Equations on Manifolds II

In [1], we have studied a generalization of the problem of finding a martingale on a manifold whose terminal value is known. This article completes the results obtained in the first article by providing uniqueness and existence theorems in a general framework (in particular if positive curvatures are allowed), still using differential geometry tools.

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 result...

On Solving Backward Doubly Stochastic Differential Equations

Anticipated Backward Stochastic Differential Equations

In this paper, we discuss a new type of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and ...