Optimal switching between a pair of Brownian motions
type: Item
instance of: scholarly article
title: Optimal switching between a pair of Brownian motions
zbMATH Open document ID: 0712.60046
author: Robert J. Vanderbei
author: Avishai Mandelbaum
author: Larry Shepp
published in: The Annals of Probability
publication date: 1990-01-01T00:00:00Z
review text: Let \((B^ i,{\mathcal F}^ i,P^ i)\), \(i=1,2\), be a pair of Brownian motions adapted to the filtration \({\mathcal F}^ i=\{{\mathcal F}^ i_{s_ i},s_ i\geq 0\}\) on the space of continuous functions. The process \(B^ i=\{B^ i_{s_ i},s_ i\geq 0\}\) evolves on the interval \([0,a_ i]\) and is absorbed at the endpoints. The time evolution of the processes can be controlled by the switching strategy [see \textit{J. B. Walsh}, Processus aléatoires à deux indices, Colloq. E.N.S.T.-C.N.E.T., Paris 1980, Lect. Notes Math. 863, 172-201 (1981; Zbl 0456.60051) and \textit{A. Mandelbaum} and \textit{R. J. Vanderbei}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 57, 253-264 (1981; Zbl 0445.60036)] (running process \(B^ 1\) and freezing \(B^ 2\) or running \(B^ 2\) and freezing \(B^ 1)\) and choosing stopping time \(\tau\). On the faces of \(D=[0,a_ 1]\times [0,a_ 2]\) only one of the Brownian motions can be move. When the controlled processes are stopped, we collect a payoff depending on the location of the pair \((B^ 1,B^ 2)\) within D at the time of stopping. The goal is to maximize the expected payoff. A special form of the problem is considered. It is assumed that the payoff for stopping in the interior \(D^ 0\) of D is zero and that on the boundary \(\partial D\) of D the reward is specified by a nonnegative continuous payoff function \(f(x_ 1,x_ 2)\). In this case the stopping rule part of the problem has a simple solution: stop at the first hitting time of \(\partial D\). It is shown that the optimal switching strategy in the interior of the rectangle is determined by a partition into three sets: a horizontal control set, a vertical control set and an indifference set. An explicit characterization of these sets in the case when the payoff function is either linear or strongly concave on each face is given.
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60G07
Mathematics Subject Classification ID: 34F05
zbMATH DE Number: 4172066
zbMATH Keywords: optimal stopping
zbMATH Keywords: two parameter processes
zbMATH Keywords: nonlinear Dirichlet problem
zbMATH Keywords: optional increasing path
zbMATH Keywords: Brownian motions
zbMATH Keywords: switching strategy
reviewed by: Krzysztof J. Szajowski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990734
OpenAlex ID: W2040503715
DOI: 10.1214/AOP/1176990734
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description: scientific article; zbMATH DE number 4172066