An inverse optimal stopping problem for diffusion processes

Abstract

Let $X$ be a one-dimensional diffusion and let $g:[0,T]\times \mathbb{R}\to \mathbb{R}$ 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 $\ddot{I}€:[0,T]\to \mathbb{R}$ such that a given stopping time $\ddot{I}{„}^{\star }$ is a solution of the stopping problem ${\sup }_{\ddot{I}}„\mathbb{E}\left[g(\ddot{I}„,X_{\ddot{I}}„)+\ddot{I}€(\ddot{I}„)\right].$ Under regularity and monotonicity conditions, there exists a solution $\ddot{I}€$ if and only if $\ddot{I}{„}^{\star }$ is the first time when $X$ exceeds a time-dependent barrier $b$, i.e. $\ddot{I}{„}^{\star }=\inf \left\{t\ge 0|X_t\ge b(t)\right\}.$ We prove uniqueness of the solution $\ddot{I}€$ 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 ${\sup }_{\ddot{I}}„\mathbb{E}\left[g(\ddot{I}„,X_{\ddot{I}}„)\right]$.