Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications *

TL;DR

This paper develops a method for optimal control of linear conditional McKean-Vlasov equations with random coefficients, providing explicit solutions and applications in finance such as portfolio optimization and tracking problems.

Contribution

It introduces semi closed-loop strategies and characterizes time-consistent optimal controls via backward stochastic Riccati equations for the first time in this context.

Findings

Explicit solutions for financial applications are derived.

The approach handles random coefficients and common noise.

Applications include mean-variance portfolio and tracking problems.

Abstract

We consider the optimal control problem for a linear conditional McKean-Vlasov equation with quadratic cost functional. The coefficients of the system and the weigh-ting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration. Semi closed-loop strategies are introduced, and following the dynamic programming approach in [32], we solve the problem and characterize time-consistent optimal control by means of a system of decoupled backward stochastic Riccati differential equations. We present several financial applications with explicit solutions, and revisit in particular optimal tracking problems with price impact, and the conditional mean-variance portfolio selection in incomplete market model.