Stochastic target problems, dynamic programming, and viscosity solutions

In this paper, we define and study a new class of optimal stochastic control problems which is closely related to the theory of backward SDEs and forward-backward SDEs. The controlled process (X^ν, Y^ν) takes values in ℝ^d × ℝ and a given initial data for X^ν(0). Then the control problem is to find the minimal initial data for Y^ν so that it reaches a stochastic target at a specified terminal time T. The main application is from financial mathematics, in which the process X^ν is related to stock price, Y^ν is the wealth process, and ν 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 first 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^ν.