A Stochastic Maximum Principle for Relaxed Control with General Risk Measure and Its Application in Finance

Abstract

This paper deals with optimal relaxed control problem where the cost functional is a general risk measure instead of an expectation. We develop a stochastic maximum principle for this kind of optimal control problems using a variational method. Then, under the expectation optimization objective, a dynamic programming principle is studied and its connections to the adjoint process is shown. At last, the result is applied to two examples. One is a linear quadratic problem and the other is an optimal investment problem.