Stochastic optimal control with random coefficients and associated stochastic Hamilton\u2013Jacobi\u2013Bellman equations

Abstract

We consider the optimal control problem for stochastic differential equations (SDEs) with random coefficients under the recursive-type objective functional captured by the backward SDE (BSDE). Due to the random coefficients, the associated Hamilton–Jacobi–Bellman (HJB) equation is a class of second-order stochastic PDEs (SPDEs) driven by Brownian motion, which we call the stochastic HJB (SHJB) equation. In addition, as we adopt the recursive-type objective functional, the drift term of the SHJB equation depends on the second component of its solution. These two generalizations cause several technical intricacies, which do not appear in the existing literature. We prove the dynamic programming principle (DPP) for the value function, for which unlike the existing literature we have to use the backward semigroup associated with the recursive-type objective functional. By the DPP, we are able to show the continuity of the value function. Using the Itô–Kunita’s formula, we prove the verification theorem, which constitutes a sufficient condition for optimality and characterizes the value function, provided that the smooth (classical) solution of the SHJB equation exists. In general, the smooth solution of the SHJB equation may not exist. Hence, we study the existence and uniqueness of the solution to the SHJB equation under two different weak solution concepts. First, we show, under appropriate assumptions, the existence and uniqueness of the weak solution via the Sobolev space technique, which requires converting the SHJB equation to a class of backward stochastic evolution equations. The second result is obtained under the notion of viscosity solutions, which is an extension of the classical one to the case for SPDEs. Using the DPP and the estimates of BSDEs, we prove that the value function is the viscosity solution to the SHJB equation. For applications, we consider the linear-quadratic problem, the utility maximization problem, and the European option pricing problem. Specifically, different from the existing literature, each problem is formulated by the generalized recursive-type objective functional and is subject to random coefficients. By applying the theoretical results of this paper, we obtain the explicit optimal solution for each problem in terms of the solution of the corresponding SHJB equation.