Matrix Riccati BSDEs with singular terminal condition and stochastic LQ control with linear terminal constraint

TL;DR

This paper studies multidimensional stochastic control problems with linear constraints, deriving Riccati BSDEs with singular terminal conditions, and establishes existence and characterization of solutions using penalization methods.

Contribution

It introduces a novel approach to handle singular terminal conditions in Riccati BSDEs for constrained stochastic control problems.

Findings

Existence of a minimal supersolution to the Riccati BSDE with singular terminal condition.

Characterization of the value function and optimal control via the supersolution.

Analysis of asymptotic behavior and special cases with closed-form solutions.

Abstract

We analyze a class of multidimensional linear-quadratic stochastic control problems with random coefficients, motivated by multi-asset optimal trade execution. The problems feature non-diffusive controlled state dynamics and a terminal constraint that restricts the terminal state to a prescribed random linear subspace. We derive the associated Riccati backward stochastic differential equation (BSDE) and identify a suitable formalization of its singular terminal condition. Via a penalization approach, we establish existence of a minimal supersolution of the Riccati BSDE and use it to characterize both the value function and the optimal control. We analyze the asymptotic behavior of the supersolution near terminal time and discuss special cases where closed-form solutions can be obtained.