On an Optimal Stopping Problem of an Insider

TL;DR

This paper analyzes an optimal stopping problem involving Brownian motion, providing explicit solutions for large delays, bounds for small delays, and asymptotic behavior as the delay approaches zero, with implications for continuity properties.

Contribution

It characterizes the problem using path-dependent reflected BSDEs, derives explicit solutions for large delays, bounds for small delays, and asymptotics as delay tends to zero.

Findings

Explicit expression for large ps

Bounds for small ps

Asymptotic behavior as ps rrows 0

Abstract

We consider the optimal stopping problem $v^{(\eps)}:=\sup_{\tau\in\mathcal{T}_{0,T}}\mathbb{E}B_{(\tau-\eps)^+}$ 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, $(B_t)_{0\leq t\leq T}$ is the Brownian motion, $\eps\in[0,T]$ is a constant, and $\mathcal{T}_{\eps,T}$ is the set of stopping times taking values in $[\eps,T]$. The solution of this problem is characterized by a path dependent reflected backward stochastic differential equations, from which the continuity of $\eps \to v^{(\eps)}$ follows. For large enough $\eps$, we obtain an explicit expression for $v^{(\eps)}$ and for small $\eps$ we have lower and upper bounds. The main result of the paper is the asymptotics of $v^{(\eps)}$ as $\eps\searrow 0$. As a byproduct, we also obtain L\'{e}vy's modulus of continuity result in the $L^1$ sense.