Backward Stochastic Differential Equations in a Lie Group

Abstract

We study backward stochastic differential equations where the solution process lives in a finite dimensional Lie group. The group stucture makes this problem easier to deal with than in a general manifold, but the geometry still imposes interesting conditions. The main tools are the stochastic exponential and logarithm of Lie groups, used to change group-valued martingales into $\mathbb{R}^d$-valued martingales. We are first interested in getting a group-valued martingale with prescribed terminal value: existence and uniqueness are proved for nilpotent Lie groups by a constructive method; also a recursive construction of the solution is given and uniqueness is obtained for groups where a convex barycenter can be defined. We then study more general backward stochastic equations with a drift term.