Backward stochastic differential equations with constraints on the gains-process

We consider backward stochastic differential equations with convex constrai nts on the gains (or intensity-of-noise) process. Existence and uniqueness of a minimal solution are established in the case of a drift coefficient wh ich is Lipschitz continuous in the state and gains processes and convex in the gains process. It is also shown that the minimal solution can be charac terized as the unique solution of a functional stochastic control-type equa tion. This representation is related to the penalization method for constru cting solutions of stochastic differential equations, involves change of me asure techniques, and employs notions and results from convex analysis, suc h as the support function of the convex set of constraints and its various properties.