Backward Stochastic Diierential Equations with Constraints on the Gains-process

We consider Backward Stochastic Diierential Equations with convex constraints on the gains (or intensity-of-noise) process. Existence and uniqueness of a minimal solution are established in the case of a drift coeecient which 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 characterized as the unique solution of a functional stochastic control-type equation. This representation is related to the penalization method for constructing solutions of stochastic diierential equations, involves change of measure techniques, and employs notions and results from convex analysis, such as the support function of the convex set of constraints and its various properties.