Linear-quadratic optimal control under non-Markovian switching

TL;DR

This paper develops a solution for linear-quadratic optimal control problems involving non-Markovian switching mechanisms, using a Riccati equation driven by both Brownian motion and a marked point process.

Contribution

It introduces a novel approach to solve LQ control problems with non-Markovian switching via a backward stochastic Riccati equation.

Findings

The control problem is solvable using a backward stochastic Riccati equation.

The model includes general switching mechanisms not limited to Markovian assumptions.

The approach extends classical LQ control to more complex, stochastic switching systems.

Abstract

We study a finite-dimensional continuous-time optimal control problem on finite horizon for a controlled diffusion driven by Brownian motion, in the linear-quadratic case. We admit stochastic coefficients, possibly depending on an underlying independent marked point process, so that our model is general enough to include controlled switching systems where the switching mechanism is not required to be Markovian. The problem is solved by means of a Riccati equation, which a backward stochastic differential equation driven by the Bronwian motion and by the random measure associated to the marked point process.