Inverse Optimal Stopping
Let $X$ be a one-dimensional diffusion and $g$ 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 $\pi:[0,T]\to\mathbb{R}$ such that a given stopping time $\tau^{\star}$ is a solution of the stopping problem $\sup_{\tau\in[0,T]}\mathbb{E}\left[g(\tau,X_{\tau})+\pi(\tau)\right]$. Under regularity and monotonicity conditions, there exists a solution $\pi$ if and only if $\tau^{\star}$ is the first time $X$ exceeds a time-dependent cut-off $b$, i.e. $\tau^{\star}=\inf\left\{ t\ge0\,|\, X_{t}\ge b(t)\right\}\wedge T \,.$ We prove uniqueness of the solution $\pi$ and derive a closed form representation. The representation is based on the process $\tilde{X}$ which is a version of the original diffusion $X$ reflected at $b$, \[ \pi(t)=\mathbb{E}\left[\int_{t}^{T}(\partial_{t}+\mathcal{L}g)(s,\tilde{X}_{s})\mathrm{d}s\,|\,\tilde{X}_{t}=b(t)\right]\,. \] The results lead to a new integral equation characterizing the stopping boundary $b$ of the stopping problem $\sup_{\tau\in\mathcal{T}}\mathbb{E}\left[g(\tau,X_{\tau})\right]$.
Year of publication: |
2014-06
|
---|---|
Authors: | Kruse, Thomas ; Strack, Philipp |
Institutions: | arXiv.org |
Saved in:
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