An exactly solvable multiple stochastic optimal stopping problem

Hidekazu Yoshioka

doi:10.1186/s13662-018-1626-7

Abstract

A new kind of multiple stochastic optimal stopping problem is formulated and its associated recursive variational inequalities are derived. We show that these variational inequalities can be solved exactly in a cascading manner. The relevance of the present problem in analyzing animal migration, which is an ecologically important problem, is also briefly discussed.

Highlights

Stochastic optimal stopping models are useful mathematical tools for analyzing decisionmaking processes in the fields of financial [1, 2], environment [3, 4], and ecology [5,6,7].Multiple optimal stopping problems based on stochastic differential equations (SDEs) are among the ones that have been analyzed most in detail because of their rich mathematical structures [8, 9]
We are interested in solvability of a multiple optimal stopping problem that has not been focused on so far, which is related to animal migration: an important ecological problem
We show that an application of the dynamic programming principle reduces the present optimal stopping problem to a series of variational inequalities (VIs)

Summary

Introduction

Stochastic optimal stopping models are useful mathematical tools for analyzing decisionmaking processes in the fields of financial [1, 2], environment [3, 4], and ecology [5,6,7].Multiple optimal stopping problems based on stochastic differential equations (SDEs) are among the ones that have been analyzed most in detail because of their rich mathematical structures [8, 9]. I–1 j=1 δj (τj where E is the expectation operator, δi > 0, qi > 0, and 0 < α < βM < βM–1 < · · · < β1 < 1 are given constants and z ≥ 0 is the initial condition of Zt. We show that an application of the dynamic programming principle reduces the present optimal stopping problem to a series of variational inequalities (VIs). The VIs, and the present multiple optimal stopping problem, turn out to be exactly solvable.

Results

Conclusion