Malliavin derivative and sensitivity for optimal liquidation

TL;DR

The paper addresses the problem of Malliavin differentiability for the minimal supersolution $Y$ of a BSDE with terminal singularity arising in a stochastic optimal liquidation model. It develops a detailed approximation framework using truncated BSDEs $Y^n$ and proves that $Y$ admits a Malliavin derivative with an explicit representation $D_\theta Y_t = \dfrac{D_\theta \eta_t}{(T-t)^{p-1}} + \dfrac{D_\theta H_t}{(T-t)^p}$, where $H$ solves a singular BSDE with generator $F$; near the terminal time the derivative exhibits a singularity driven by $D_\theta \eta_T$. The work then proves convergence of $D_\theta Y^n$ to $D_\theta Y$ and of $D_\theta H^n$ to $D_\theta H$, under suitable integrability assumptions, and derives implications for the gradient of the related PDE and for sensitivity analysis in the liquidation problem. The results provide new theoretical insights into Malliavin differentiability in the presence of terminal and generator singularities and furnish practical tools for computing PDE gradients and Greeks in optimal liquidation settings.

Abstract

We prove that the solution of the backward stochastic differential equation with terminal singularity has a Malliavin derivative, which is the limit of the derivative of the approximating sequence. We also provide the asymptotic behavior of this derivative close to the terminal time. We apply this result to the regularity of the related partial differential equation and to the sensitivity of the liquidation problem.

Theorems & Definitions (43)

• Remark 1
• Lemma 1
• proof
• Proposition 1
• Proposition 2
• proof
• Theorem 1
• Lemma 2
• proof
• Lemma 3