[Paper Review] Inverse Optimal Stopping

Thomas Kruse, Philipp Strack|arXiv (Cornell University)|Jun 1, 2014

Stochastic processes and financial applications15 references3 citations

TL;DR

This paper solves the inverse optimal stopping problem for one-dimensional diffusions by characterizing the time-dependent perturbation function π that makes a given stopping time τ⋆ optimal. It proves that τ⋆ must be the first hitting time of a time-dependent boundary b(t), and derives a closed-form expression for π in terms of the reflected diffusion Ŷ, leading to a new integral equation for b(t).

ABSTRACT

Let $X$ be a one-dimensional diffusion and $g$ a payoff function depending on time and the value of $X$. The paper analyzes the inverse optimal stopping problem of finding a time-dependent function $\pi:[0,T] o\mathbb{R}$ such that a given stopping time $ au^{\star}$ is a solution of the stopping problem $\sup_{ au\in[0,T]}\mathbb{E}\left[g( au,X_{ au})+\pi( au) ight]$. Under regularity and monotonicity conditions, there exists a solution $\pi$ if and only if $ au^{\star}$ is the first time $X$ exceeds a time-dependent cut-off $b$, i.e. $ au^{\star}=\inf\left\{ t\ge0\,|\, X_{t}\ge b(t) ight\}\wedge T \,.$ We prove uniqueness of the solution $\pi$ and derive a closed form representation. The representation is based on the process $ ilde{X}$ which is a version of the original diffusion $X$ reflected at $b$, \[ \pi(t)=\mathbb{E}\left[\int_{t}^{T}(\partial_{t}+\mathcal{L}g)(s, ilde{X}_{s})\mathrm{d}s\,|\, ilde{X}_{t}=b(t) ight]\,. \] The results lead to a new integral equation characterizing the stopping boundary $b$ of the stopping problem $\sup_{ au\in\mathcal{T}}\mathbb{E}\left[g( au,X_{ au}) ight]$.

Motivation & Objective

• To determine the conditions under which a given stopping time τ⋆ is optimal for a time-dependent payoff g(t, X_t).
• To solve the inverse problem: find a time-dependent function π(t) such that τ⋆ remains optimal for the perturbed payoff g(τ, X_τ) + π(τ).
• To characterize the structure of the optimal stopping boundary b(t) in terms of the diffusion X and payoff g.
• To derive a closed-form representation for the perturbation π(t) using the reflected diffusion Ŷ.

Proposed method

• Assumes τ⋆ is the first hitting time of a time-dependent boundary b(t), i.e. τ⋆ = inf{t ≥ 0 | X_t ≥ b(t)} ∧ T.
• Introduces the reflected diffusion Ŷ, a version of X reflected at b(t), to analyze boundary behavior.
• Derives the representation π(t) = ℰ[∫_t^T (∂_t + ℒ)g(s, Ŷ_s) ds | Ŷ_t = b(t)], where ℒ is the generator of X.
• Uses regularity and monotonicity conditions on g and b to ensure existence and uniqueness of π.
• Establishes that π exists and is unique if and only if τ⋆ is a hitting time of b(t).
• Derives a new integral equation for the boundary b(t) by combining the representation of π with the optimality condition.

Experimental results

Research questions

• RQ1 Under what conditions does a given stopping time τ⋆ admit a time-dependent perturbation π(t) that preserves its optimality for the payoff g(τ, X_τ)?
• RQ2 What is the explicit form of the perturbation π(t) that makes τ⋆ optimal in the inverse problem?
• RQ3 How is the optimal stopping boundary b(t) characterized in terms of the diffusion X and the payoff g?
• RQ4 Can the inverse problem be reduced to a representation involving the reflected diffusion Ŷ?
• RQ5 What integral equation governs the boundary b(t) in the original optimal stopping problem?

Key findings

• A stopping time τ⋆ is optimal for the perturbed problem if and only if it is the first hitting time of a time-dependent boundary b(t).
• The perturbation function π(t) has a closed-form representation involving the conditional expectation of the time-space generator applied to g along the reflected diffusion Ŷ.
• The representation π(t) = ℰ[∫_t^T (∂_t + ℒ)g(s, Ŷ_s) ds | Ŷ_t = b(t)] is both necessary and sufficient for optimality under regularity and monotonicity conditions.
• The solution π is unique under the stated conditions.
• The results yield a new integral equation that characterizes the optimal stopping boundary b(t) for the original problem without perturbations.
• The reflected diffusion Ŷ plays a central role in expressing π and linking the inverse and direct optimal stopping problems.