A System of BSDEs with Singular Terminal Values Arising in Optimal Liquidation with Regime Switching

TL;DR

This work analyzes a regime-switching stochastic control problem for optimal liquidation in markets with dark pools, leading to a novel multidimensional BSDE system with jumps and singular terminal values due to the terminal constraint $X_T=0$. The authors establish a penalization-based existence result by solving truncated BSDEs and using a multidimensional comparison theorem to control the limit, obtaining a positive, bounded solution $(Y^i,Z^i,\Psi^i)$. They then solve the constrained control problem, show the value function is $V_t(x,i)=Y^i_t x^2$, and derive explicit feedback controls for the traditional and dark-pool venues; a verification argument demonstrates that the obtained solution is minimal and, crucially, unique among admissible solutions. The results extend the literature on BSDEs with singular terminal values to a regime-switching, jump-diffusion setting and provide a rigorous foundation for optimal liquidation with regime-dependent liquidity and execution costs.

Abstract

We study a stochastic control problem with regime switching arising in an optimal liquidation problem with dark pools and multiple regimes. The new feature of this model is that it introduces a system of BSDEs with jumps and with singular terminal values, which appears in literature for the first time. The existence result for this system is obtained. As a result, we solve the stochastic control problem with regime switching. More importantly, the uniqueness result of this system is also obtained, in contrast to merely minimal solutions established in most related literature.

Theorems & Definitions (17)

• Lemma 2.1
• proof
• Corollary 2.2
• Remark 2.3
• Definition 3.1
• Theorem 3.2
• Proposition 3.3
• Remark 3.4
• Lemma 3.5
• proof