Stochastic Target Problems, Dynamic Programming, and Viscosity Solutions

In this paper, we de ne and study a new class of optimal stochastic control problems which is closely related to the theory of Backward SDE's and forward-backward SDE's. The controlled process (X ; Y ) takes values in IRd IR and a given initial data for X (0). Then, the control problem is to nd the minimal initial data for Y so that it reaches a stochastic target at a speci ed terminal time T . The main application is from nancial mathematics in which the process X is related to stock price, Y is the wealth process, and is the portfolio. We introduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation. The boundary conditions are also shown to solve a rst order variational inequality in the discontinuous viscosity sense. This provides a unique characterization of the value function which is the minimal initial data for Y .