Indefinite stochastic linear-quadratic optimal control problems with random jumps and related stochastic Riccati equations

Abstract

We discuss the stochastic linear-quadratic (LQ) optimal controlproblem with Poisson processes under the indefinite case. Based on thewellposedness of the LQ problem, the main idea is expressed by thedefinition of relax compensator that extends the stochasticHamiltonian system and stochastic Riccati equation with Poissonprocesses (SREP) from the positive definite case to the indefinite case. Wemainly study the existence and uniqueness of the solution for thestochastic Hamiltonian system and obtain the optimal control withopen-loop form. Then, we further investigate the existence anduniqueness of the solution for SREP in some special case and obtainthe optimal control in close-loop form.