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Mean field optimal stopping with uncontrolled state

Andrea Cosso, Laura Perelli

TL;DR

The paper studies finite-horizon mean field optimal stopping with an uncontrolled McKean-Vlasov state, addressing time-inconsistency by embedding the problem into an enlarged state space with an extended value function $\tilde{V}$. It proves that the original problem’s Snell envelope and optimal stopping rule can be represented via $\tilde{V}$, and that $\tilde{V}$ satisfies a dynamic programming principle on the enlarged space. A Hamilton-Jacobi-Bellman equation on the Wasserstein space, expressed as a second-order variational inequality with Lions derivatives, characterizes $\tilde{V}$ in the viscosity sense (with a disintegration link to the original value). These results provide a rigorous framework for time-consistent analysis and pave the way for numerical and analytical treatment of mean-field stopping problems with McKean-Vlasov dynamics.

Abstract

We study a specific class of finite-horizon mean field optimal stopping problems by means of the dynamic programming approach. In particular, we consider problems where the state process is not affected by the stopping time. Such problems arise, for instance, in the pricing of American options when the underlying asset follows a McKean-Vlasov dynamics. Due to the time inconsistency of these problems, we provide a suitable reformulation of the original problem for which a dynamic programming principle can be established. To accomplish this, we first enlarge the state space and then introduce the so-called extended value function. We prove that the Snell envelope of the original problem can be written in terms of the extended value function, from which we can derive a characterization of the smallest optimal stopping time. On the enlarged space, we restore time-consistency and in particular establish a dynamic programming principle for the extended value function. Finally, by employing the notion of Lions measure derivative, we derive the associated Hamilton-Jacobi-Bellman equation, which turns out to be a second-order variational inequality on the product space $[0, T ] \times \mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^d)$; under suitable assumptions, we prove that the extended value function is a viscosity solution to this equation.

Mean field optimal stopping with uncontrolled state

TL;DR

The paper studies finite-horizon mean field optimal stopping with an uncontrolled McKean-Vlasov state, addressing time-inconsistency by embedding the problem into an enlarged state space with an extended value function . It proves that the original problem’s Snell envelope and optimal stopping rule can be represented via , and that satisfies a dynamic programming principle on the enlarged space. A Hamilton-Jacobi-Bellman equation on the Wasserstein space, expressed as a second-order variational inequality with Lions derivatives, characterizes in the viscosity sense (with a disintegration link to the original value). These results provide a rigorous framework for time-consistent analysis and pave the way for numerical and analytical treatment of mean-field stopping problems with McKean-Vlasov dynamics.

Abstract

We study a specific class of finite-horizon mean field optimal stopping problems by means of the dynamic programming approach. In particular, we consider problems where the state process is not affected by the stopping time. Such problems arise, for instance, in the pricing of American options when the underlying asset follows a McKean-Vlasov dynamics. Due to the time inconsistency of these problems, we provide a suitable reformulation of the original problem for which a dynamic programming principle can be established. To accomplish this, we first enlarge the state space and then introduce the so-called extended value function. We prove that the Snell envelope of the original problem can be written in terms of the extended value function, from which we can derive a characterization of the smallest optimal stopping time. On the enlarged space, we restore time-consistency and in particular establish a dynamic programming principle for the extended value function. Finally, by employing the notion of Lions measure derivative, we derive the associated Hamilton-Jacobi-Bellman equation, which turns out to be a second-order variational inequality on the product space ; under suitable assumptions, we prove that the extended value function is a viscosity solution to this equation.

Paper Structure

This paper contains 6 sections, 13 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. The mean field optimal stopping problem
  3. Extended mean field optimal stopping problem
  4. Snell envelope and optimal stopping time
  5. Dynamic programming principle
  6. Hamilton-Jacobi-Bellman equation

Key Result

Proposition 2.1

For every $t\in[0,T]$ and $\xi\in L^2({\cal F}_t;\mathbb{R}^d)$, there exists a unique (up to $\mathbb{P}$-indistinguishability) process $X^{t,\xi}\in\mathbb{S}^2_{t,T}$ solution to MKV SDE that satisfies the following estimate: there exists a constant $C>0$ such that

Theorems & Definitions (21)

  • Remark 2.1
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.3
  • Definition 3.1
  • Proposition 3.1
  • Remark 3.1
  • Proposition 3.2
  • ...and 11 more