A Risk-Averse Analog of the Hamilton-Jacobi-Bellman Equation

In this paper, we study the risk-averse control problem for diffusion processes. We make use of a forward–backward system of stochastic differential equations to evaluate a fixed policy and to formulate the optimal control problem. Weak formulation is established to facilitate the derivation of the risk-averse dynamic programming equation. We prove that the value function of the risk-averse control problem is a viscosity solution of a risk-averse analog of the Hamilton– Jacobi–Bellman equation. On the other hand, a verification theorem is provedwhen the classical solution of the equation exists.