An inverse optimal stopping problem for diffusion processes

2014/06/01 by Kruse, Thomas, Strack, Philipp
#FOS: Economics and business #FOS: Mathematics #General Economics (econ.GN) #Optimization and Control (math.OC) #Probability (math.PR)

paper · doi:10.48550/arxiv.1406.0209

Abstract

Let X be a one-dimensional diffusion and let g\colon[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 sup_τ𝔼[g(τ,X_τ)+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 | X_t≥ 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 sup_τ𝔼[g(τ,X_τ)].

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