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In this paper, we study backward doubly stochastic recursive optimal controlproblem where the cost function is described by the solution of a backwarddoubly stochastic differential equation. We give the dynamical programmingprinciple for this kind of optimal control problem and show that the valuefunction is the unique Sobolev weak solution for the corresponding stochasticHamilton-Jacobi-Bellman equation.

Dynamic Programming Principle for Backward Doubly Stochastic Recursive  Optimal Control Problem and Sobolev Weak Solution of The Stochastic  Hamilton-Bellman Equation