Stochastic Control with Random Coefficients under Recursive-Type Objective Functionals

We consider the stochastic optimal control problem with random coefficients under recursive-type objective functionals captured by backward stochastic differential equations (with random coefficients). The associated Hamilton- Jacobi-Bellman (HJB) equation obtained from the dynamic programming principle is a second-order nonlinear stochastic HJB (SHJB) equation (or stochastic PDE). The solvability of the SHJB equation, together with Itô-Kunita's formula, leads to the verification theorem that is the sufficient condition for optimality. We also show the existence and uniqueness of the (weak) solution to the SHJB equation via the Sobolev space technique.