A note on costs minimization with stochastic target constraints

Abstract

We study the minimization of the expected costs under stochastic constraint at the terminal time. Our first main result says that for a power type of costs, the value function is the minimal positive solution of a second order semi-linear ordinary differential equation (ODE). Our second main result establishes the optimal control. In addition we show that the case of exponential costs leads to a trivial optimal control. All our proofs are based on the martingale approach.