## Abstract

We consider the optimal stopping problem v^{(ε)}:= sup_{τ}_{∈T0,T} E B_{(τ−ε)+} posed by Shiryaev at the International Conference on Advanced Stochastic Optimization Problems organized by the Steklov Institute of Mathematics in September 2012. Here T > 0 is a fixed time horizon, (B_{t})_{0}_{≦t}_{≦T} is the Brownian motion, ε ∈ [0,T] is a constant, and T_{ε,T} is the set of stopping times taking values in [ε, T ]. The solution of this problem is characterized by a path dependent reflected backward stochastic differential equation, from which the continuity of ε → v^{(ε)} follows. For large enough ε, we obtain an explicit expression for v^{(ε)}, and for small ε we have lower and upper bounds. The main result of the paper is the asymptotics of v^{(ε)} as ε ↘ 0. As a by-product, we also obtain Lévy’s modulus of continuity result in the L^{1} sense.

Original language | English (US) |
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Pages (from-to) | 129-133 |

Number of pages | 5 |

Journal | Theory of Probability and its Applications |

Volume | 61 |

Issue number | 1 |

DOIs | |

State | Published - 2017 |

### Bibliographical note

Funding Information:This work was supported by the National Science Foundation under grant DMS 0955463.

Publisher Copyright:

© 2017 Society for Industrial and Applied Mathematics.

Copyright:

Copyright 2017 Elsevier B.V., All rights reserved.

## Keywords

- Lévy’s modulus of continuity for Brownian motion
- Optimal stopping problem of an insider