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

In this talk, we investigate optimal relaxed control problem which cost functional is a general risk measure, instead an expectation. We develop a stochastic maximum principle for this kind of optimal control problems using a variational method. Also, a sufficient condition without assumption of the concavity is established. Then, under the expectation optimization objective,  a dynamic programing 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 a optimal investment problem.