Stochastic target problems, dynamic programming, and viscosity solutions

Abstract

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-nu, Y-nu) takes values in R-d x R and a given initial data for X-nu (0). Then the control problem is to find the minimal initial data for Y-nu 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-nu is related to stock price, Y-nu is the wealth process, and nu 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-nu.