An Inverse Optimal Stopping Problem for Diffusion Processes

Let X be a one-dimensional diffusion and let g be a real-valued function depending on time and the value of X. This article analyzes the inverse optimal stopping problem of finding a time-dependent real-valued function π depending only on time such that a given stopping time τ^⋆ is a solution of the stopping problem ${sup}_{\tau }𝔼[g(\tau ,X_{\tau })+\pi (\tau )]$.

Under regularity and monotonicity conditions, there exists such a transfer π if and only if τ^⋆ is the first time when X exceeds a time-dependent barrier b. We prove uniqueness of the solution π and derive a closed form representation. The representation is based on an auxiliary process that 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 ${sup}_{\tau }𝔼[g(\tau ,X_{\tau })]$.