Partially observed time‐inconsistent stochastic linear‐quadratic control with random jumps

Abstract

SummaryThis paper studies a partially observed time‐inconsistent stochastic linear‐quadratic control system, in which the state follows a stochastic differential equation driven by a Brownian motion and an independent Poisson random measure. The cost functional contains a state‐dependent term and a quadratic term of the conditional expected state process, which will cause the time inconsistency in dynamic systems. By virtue of a classical spike variation approach, we define an equilibrium and derive a sufficient condition for the equilibrium in the fully observed system with stochastic coefficients. Then, we obtain the equilibrium with an explicit feedback form in deterministic coefficients case and discuss the existence and uniqueness of the solution of corresponding Riccati equations. Furthermore, we get filtering equations of the partially observed system and get the explicit equilibrium in some special case.