The controller-and-stopper game for a linear diffusion.
type: Item
instance of: scholarly article
title: The controller-and-stopper game for a linear diffusion.
zbMATH Open document ID: 1039.60043
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: The authors consider a zero-sum stochastic game on linear diffusion processes with two players, one called controller and the other called stopper. Let \(X(\cdot)\) with the state space \(I= (l, r)\), which is a nonempty bounded open interval, be the diffusion process governed by the equation \[ dX(t)= \beta(t)dt + \sigma(t)dW(t), \] \(X(0)= x\in I\) on a filtered probability space \((\Omega,{\mathcal F},P,({\mathcal F}_t))\), where \(W(\cdot)\) is a standard Brownian motion. The first player chooses a pair \((\beta(t), \sigma(t))\) from some given class and the second player selects a stopping time \(\tau\), in which the first (second) player minimizes (resp. maximizes) an expectation \(E[e^{-\alpha\tau(X)} u(X(\tau(X)))]\), where \(\alpha\geq 0\) is a discount factor and \(u\) is a real-valued continuous function on \([l, r]\). They then give a saddle-point and show the game has a value, that is, \[ \inf_{X(\cdot)}\sup_\tau\, E[e^{-\alpha\tau(X)} u(X(\tau(X)))]= \sup_\tau\inf_{X(\cdot)} E[e^{-\alpha\tau(X)} u(X(\tau(X)))] \] for an undiscounted case \((\alpha= 0)\) and a discounted case \((\alpha> 0)\) under some assumptions.
reviewed by: Yoshio Ohtsubo
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 62L15
zbMATH DE Number: 1906013
zbMATH Keywords: stochastic game
zbMATH Keywords: optimal stopping
zbMATH Keywords: diffusion process
zbMATH Keywords: excessive function
author: Ioannis Karatzas
author: William D. Sudderth
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1015345598
OpenAlex ID: W2117732143
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DOI: 10.1214/AOP/1015345598
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description: scientific article; zbMATH DE number 1906013