An Inverse Optimal Stopping Problem for Diffusion Processes

28 Pages Posted: 28 Apr 2017

See all articles by Thomas Kruse

Thomas Kruse

Justus Liebig University Giessen

Philipp Strack

Yale, Department of Economics

Date Written: April 27, 2017

Abstract

Let X be a one-dimensional diffusion and let g : [0, T ] × ℝ → ℝ be 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 π : [0, T ] → ℝ such that a given stopping time τ* is a solution of the stopping problem
supτ 𝔼 [ g (τ, Xτ ) + π(τ)].

Under regularity and monotonicity conditions, there exists a solution π if and only if τ* is the first time when X exceeds a time-dependent barrier b, i.e. τ* = inf {t ≥ 0 | Xt ≥ b(t)}. We prove uniqueness of the solution π and derive a closed form representation. The representation is based on an auxiliary process which is a version of the original diffusion X reflected at b towards the continuation region. The results lead to a new integral equation characterizing the stopping boundary b of the stopping problem supτ 𝔼 [ g (τ, Xτ )].

Keywords: Optimal Stopping, Reflected Stochastic Processes, Dynamic Mechanism Design, Dynamic Implementability

Suggested Citation

Kruse, Thomas and Strack, Philipp, An Inverse Optimal Stopping Problem for Diffusion Processes (April 27, 2017). Available at SSRN: https://ssrn.com/abstract=2959702 or http://dx.doi.org/10.2139/ssrn.2959702

Thomas Kruse

Justus Liebig University Giessen ( email )

Licher Str. 64
Giessen, 35394
Germany

Philipp Strack (Contact Author)

Yale, Department of Economics ( email )

28 Hillhouse Ave
New Haven, CT 06520-8268
United States

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