Optimal stopping times for a class of Ito diffusion bridges

TL;DR

This paper analyzes optimal stopping problems for a class of Ito diffusion bridges, providing explicit solutions and exploring alternative processes with similar optimal barriers, relevant for investment and liquidation strategies.

Contribution

It introduces explicit solutions for optimal stopping problems related to Ito diffusion bridges and explores alternative processes with similar barriers, enhancing modeling flexibility.

Findings

Explicit solutions for optimal stopping times are derived.

Alternative processes with similar barriers are identified.

A model for optimal liquidation timing is proposed.

Abstract

The scope of this paper is to study the optimal stopping problems associated to a stochastic process, which may represent the gain of an investment, for which information on the final value is available a priori. This information may proceed, for example, from insider trading or from pinning at expiration of stock options. We solve and provide explicit solutions to these optimization problems. As special case, we discuss different processes whose optimal barrier has the same shape as the optimal barrier of the Brownian bridge. So doing we provide a catalogue of alternatives to the Brownian bridge which in practice could be better adapted to the data. Moreover, we investigate if, for any given (decreasing) curve, there exists a process with this curve as optimal barrier. This provides a model for the optimal liquidation time, i.e. the optimal time at which the investor should liquidate a position in order to maximize the gain.