Optimal Controls of Stochastic Differential Equations with Jumps and Random Coefficients: Stochastic Hamilton–Jacobi–Bellman Equations with Jumps

We study the stochastic Hamilton–Jacobi–Bellman (HJB) equation with jump, which arises from a non-Markovian optimal control problem recursive utility cost functional. The solution to is predictable triplet of random fields. show that value function problem, under some regularity assumptions, HJB equation; and classical this characterizes control. With additional assumptions on coefficients, an existence uniqueness result in sense Sobolev space shown by recasting as backward evolution Hilbert spaces Brownian motion Poisson jump.