Source: Ann. Probab.
Volume 26, Number 4
We consider backward stochastic differential 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
coefficient 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 differential 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
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