An inverse optimal stopping problem for diffusion processes

Let X be a one-dimensional diffusion and let gcolon[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 τstar is a solution of the stopping problem supτmathbbEleft[g(τ,Xτ)+π(τ)right],. Under regularity and monotonicity conditions, there exists a solution π if and only if τstar is the first time when X exceeds a time-dependent barrier b, i.e. τstar=infleft t≥0,|,Xt≥ b(t)right ,. 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 supτmathbbEleft[g(τ,Xτ)right].