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On an Optimal Stopping Problem of an Insider

6 Pages Posted: 23 Apr 2014  

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

Zhou Zhou

University of Minnesota - Twin Cities

Date Written: April 21, 2014

Abstract

We consider the optimal problem $\sup_{\tau\in\mathcal{T}_{\eps,T}}\mathbb{E}\left[\sum_{i=1}^n \phi_{(\tau-\eps^i)^ }^i\right]$, where $T>0$ is a fixed time horizon, $(\phi_t^i)_{0\leq t\leq T}$ is progressively measurable with respect to the Brownian filtration, $\eps^i\in[0,T]$ is a constant, $i=1,\dotso,n$, and $\mathcal{T}_{\eps,T}$ is the set of stopping times that lie between a constant $\eps\in[0,T]$ and $T$. We solve this problem by conditioning and then using the theory of reflected backward stochastic differential equations (RBSDEs). As a corollary, we provide a solution to the optimal stopping problem $\sup_{\tau\in\mathcal{T}_{0,T}}\mathbb{E}B_{(\tau-\eps)^ }$ recently posed by Shiryaev at the International Conference on Advanced Stochastic Optimization Problems organized by the Steklov Institute of Mathematics in September 2012. We also provide its asymptotic order as $\eps\searrow 0$.

Keywords: optimal stopping problem of an insider, Reflected Backward Stochastic Differential Equations (RBSDEs), Levy's modulus for Brownian motion

Suggested Citation

Bayraktar, Erhan and Zhou, Zhou, On an Optimal Stopping Problem of an Insider (April 21, 2014). Available at SSRN: https://ssrn.com/abstract=2427270 or http://dx.doi.org/10.2139/ssrn.2427270

Erhan Bayraktar (Contact Author)

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

Zhou Zhou

University of Minnesota - Twin Cities ( email )

420 Delaware St. SE
Minneapolis, MN 55455
United States

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