Stochastic Target Problems, Dynamic Programming, and Viscosity Solutions
SIAM Journal of Control and Optimization
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
... 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 .