Optimal switching between a pair of Brownian motions
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.
scientific article; zbMATH DE number 4172066