# On an Optimal Stopping Problem of an Insider

6 Pages Posted: 23 Apr 2014

See all articles by Erhan Bayraktar

## Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

## Zhou Zhou

The University of Sydney

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