Constrained stochastic linear quadratic control under regime switching with controlled jump size

TL;DR

This paper addresses a complex stochastic control problem involving regime switching and jumps, deriving explicit optimal controls and values through advanced stochastic calculus and Riccati equations.

Contribution

It introduces a novel control framework with regime-dependent jump size control, providing explicit solutions and establishing existence and uniqueness of solutions for associated BSDEJs.

Findings

Explicit optimal control and value functions derived

Established existence and uniqueness of solutions for multi-dimensional BSDEJs

Developed comparison theorems for multidimensional BSDEJs

Abstract

In this paper, we examine a stochastic linear-quadratic control problem characterized by regime switching and Poisson jumps. All the coefficients in the problem are random processes adapted to the filtration generated by Brownian motion and the Poisson random measure for each given regime. The model incorporates two distinct types of controls: the first is a conventional control that appears in the continuous diffusion component, while the second is an unconventional control, dependent on the variable $z$, which influences the jump size in the jump diffusion component. Both controls are constrained within general closed cones. By employing the Meyer-It\^o formula in conjunction with a generalized squares completion technique, we rigorously and explicitly derive the optimal value and optimal feedback control. These depend on solutions to certain multi-dimensional fully coupled stochastic Riccati equations, which are essentially backward stochastic differential equations with jumps (BSDEJs). We establish the existence of a unique nonnegative solution to the BSDEJs. One of the major tools used in the proof is the newly established comparison theorems for multidimensional BSDEJs.