Stochastic linear-quadratic control with a jump and regime switching on a random horizon

TL;DR

This paper develops explicit solutions for a stochastic linear-quadratic control problem with regime switching and jumps on a random horizon, using Riccati equations and filtration techniques, with applications to mean-variance hedging.

Contribution

It introduces a novel approach to solving SREs with jumps on a random horizon by decomposing the problem and solving related Riccati equations, including a case not previously addressed in literature.

Findings

Explicit optimal control and cost for the problem.

Solution of Riccati equations with jumps expressed via Brownian filtration.

Closed-form solutions for portfolio optimization in a stochastic environment.

Abstract

In this paper, we study a stochastic linear-quadratic control problem with random coefficients and regime switching on a horizon $[0,T\wedge\tau]$, where $\tau$ is a given random jump time for the underlying state process and $T$ is a constant. We obtain an explicit optimal state feedback control and explicit optimal cost value by solving a system of stochastic Riccati equations (SREs) with jumps on $[0,T\wedge\tau]$. By the decomposition approach stemming from filtration enlargement theory, we express the solution of the system of SREs with jumps in terms of another system of SREs involving only Brownian filtration on the deterministic horizon $[0,T]$. Solving the latter system is the key theoretical contribution of this paper and we establish this for three different cases, one of which seems to be new in the literature. These results are then applied to study a mean-variance hedging problem with random parameters that depend on both Brownian motion and Markov chain. The optimal portfolio and optimal value are presented in closed forms with the aid of a system of linear backward stochastic differential equations with jumps and unbounded coefficients in addition to the SREs with jumps.