[Paper Review] Singular control and optimal stopping of memory mean-field processes

Nacira Agram, Achref Bachouch+2 more|arXiv (Cornell University)|Feb 15, 2018

Stochastic processes and financial applications7 references3 citations

TL;DR

This paper establishes a connection between optimal singular control of memory mean-field SDEs, reflected advanced mean-field backward SDEs, and optimal stopping problems. It proves existence and uniqueness of solutions to reflected AMBSDEs, derives necessary and sufficient conditions for optimal singular control under partial information, and shows that the optimal control problem is equivalent to an optimal stopping problem via a dual representation involving the adjoint process and a reflected BSDE with memory and mean-field dependence.

ABSTRACT

The purpose of this paper is to study the following topics and the relation between them: (i) Optimal singular control of mean-field stochastic differential equations with memory, (ii) reflected advanced mean-field backward stochastic differential equations, and (iii) optimal stopping of mean-field stochastic differential equations. More specifically, we do the following: - We prove the existence and uniqueness of the solutions of some reflected advanced memory backward stochastic differential equations (AMBSDEs), - we give sufficient and necessary conditions for an optimal singular control of a memory mean-field stochastic differential equation (MMSDE) with partial information, and - we deduce a relation between the optimal singular control of a MMSDE, and the optimal stopping of such processes.

Motivation & Objective

• To establish the existence and uniqueness of solutions to reflected advanced mean-field backward SDEs with memory and mean-field dependence.
• To derive sufficient and necessary conditions for optimal singular control of memory mean-field SDEs under partial information.
• To demonstrate a duality between optimal singular control and optimal stopping problems in the context of memory mean-field SDEs.
• To unify the three core problems—singular control, reflected BSDEs, and optimal stopping—through a common mathematical framework involving adjoint processes and memory structures.

Proposed method

• Formulates a forward-backward system coupling a memory mean-field SDE with a reflected advanced mean-field BSDE.
• Applies stochastic maximum principles to derive necessary and sufficient conditions for optimal singular control under partial information.
• Uses the adjoint process and duality to link the singular control problem to an optimal stopping problem via a reflected BSDE with memory and mean-field terms.
• Establishes that the value process of the optimal stopping problem corresponds to the adjoint process in the singular control setting.
• Derives a representation of the optimal stopping time as the first hitting time of a boundary defined by the control cost and adjoint process.
• Employs advanced tools from stochastic analysis, including anticipative SDEs, mean-field expectations, and path-dependent stochastic calculus.

Experimental results

Research questions

• RQ1 Under what conditions does a reflected advanced mean-field backward SDE with memory and mean-field dependence admit a unique solution?
• RQ2 What are the necessary and sufficient conditions for an optimal singular control in a memory mean-field SDE with partial information?
• RQ3 How is the optimal singular control problem related to an optimal stopping problem in the context of memory mean-field SDEs?
• RQ4 Can the value process of the optimal stopping problem be represented via the adjoint process of the singular control problem?
• RQ5 What is the explicit form of the optimal stopping time in terms of the control and adjoint processes?

Key findings

• The paper proves the existence and uniqueness of solutions to a class of reflected advanced mean-field backward SDEs with memory and mean-field dependence.
• It establishes that the optimal singular control is characterized by a system of forward-backward SDEs involving an adjoint process and a reflected BSDE with memory and mean-field terms.
• The optimal control problem is shown to be equivalent to an optimal stopping problem, where the value process corresponds to the adjoint process and the stopping time is the first time the adjoint process hits a boundary defined by the control cost.
• The optimal stopping time is explicitly given as the first time the cumulative control process exceeds a threshold, or equivalently, the first time the reflected process hits its boundary.
• The duality between singular control and optimal stopping is formalized through a dual representation involving the adjoint process and the driver of the reflected BSDE.
• In the case of constant control cost and full information, the optimal stopping time is given by $ \hat{\tau}_0 = \inf\{s \in [0,T], Y(s) \leq S(s)\} \wedge T $, where $ S(s) = \frac{1}{\lambda_0} \hat{h}(s) $.