Generalized Risk-Sensitive Optimal Control and Hamilton–Jacobi–Bellman Equation

In this article, we consider the generalized risk-sensitive optimal control problem, where the objective functional is defined by the controlled backward stochastic differential equation (BSDE) with quadratic growth coefficient. We extend the earlier results of the risk-sensitive optimal control problem to the case of the objective functional given by the controlled BSDE. Note that the risk-neutral stochastic optimal control problem corresponds to the BSDE objective functional with linear growth coefficient, which can be viewed as a special case of the article. We obtain the generalized risk-sensitive dynamic programming principle for the value function via the backward semigroup associated with the BSDE. Then we show that the corresponding value function is a viscosity solution to the Hamilton–Jacobi–Bellman equation. Under an additional parameter condition, the viscosity solution is unique, which implies that the solution characterizes the value function. We apply the theoretical results to the risk-sensitive European option pricing problem.