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

Julia Ackermann, Thomas Kruse, Petr Petrov, Alexandre Popier

TL;DR

This work addresses a multidimensional stochastic LQ control problem with random coefficients and a random linear terminal constraint, formalizing a Riccati BSDE with a singular terminal condition and constructing a minimal supersolution to characterize the value function via $v(t,x)=\langle x,Y_t x\rangle$. The authors implement a penalization scheme, analyze the penalized problems to obtain existence, bounds, and an explicit link between the Riccati solution and optimal controls, and then pass to the singular case to obtain a minimal supersolution and optimal strategy under broad conditions. Special cases are deeply explored: when the terminal constraint is $C=\{0\}$, explicit forms appear under uncorrelated multiplicative increments or Itô dynamics for $\eta$, including a closed-form $Y_t$ and a deterministic control in the former, and an asymptotic expansion in the latter. Overall, the paper extends the matrix Riccati BSDE framework to multidimensional settings with singular terminals and random terminal constraints, providing rigorous results and insights for applications such as multi-asset trade execution under stochastic liquidity and cross-asset effects.

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.

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

TL;DR

This work addresses a multidimensional stochastic LQ control problem with random coefficients and a random linear terminal constraint, formalizing a Riccati BSDE with a singular terminal condition and constructing a minimal supersolution to characterize the value function via . The authors implement a penalization scheme, analyze the penalized problems to obtain existence, bounds, and an explicit link between the Riccati solution and optimal controls, and then pass to the singular case to obtain a minimal supersolution and optimal strategy under broad conditions. Special cases are deeply explored: when the terminal constraint is , explicit forms appear under uncorrelated multiplicative increments or Itô dynamics for , including a closed-form and a deterministic control in the former, and an asymptotic expansion in the latter. Overall, the paper extends the matrix Riccati BSDE framework to multidimensional settings with singular terminals and random terminal constraints, providing rigorous results and insights for applications such as multi-asset trade execution under stochastic liquidity and cross-asset effects.

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.
Paper Structure (11 sections, 22 theorems, 161 equations)

This paper contains 11 sections, 22 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Solution approach
  3. Related literature and applications
  4. Outline
  5. Notation
  6. The penalized case
  7. The singular case
  8. The case $C=\{0\}$
  9. Processes with uncorrelated multiplicative increments
  10. The case when $\eta$ is an Itô process
  11. Two elementary lemmas

Key Result

Theorem 1.3

Assume that $A$ is a bounded process, that there exists $\delta \in (0, \infty)$ such that it holds $P$-a.s. for all $s \in [0, T]$ that and that Then there exists a minimal solution $(Y,Z)$ to the BSDE eq:BSDE_sing in the sense of def:supersol_bsde. Moreover, it holds for all $t\in [0,T)$, $x\in \mathbb{R}^d$$P$-a.s. that $v(t,x)=\langle x,Y_t x \rangle$. Finally, for every $x\in \mathbb{R}^d$,

Theorems & Definitions (54)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 44 more