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

In this paper, we study a stochastic linear-quadratic control problem with random coefficients and regime switching on a horizon $[0,T\wedgeτ]$, where $τ$ 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 $[0,T\wedgeτ]$. 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.