The controller-and-stopper game for a linear diffusion.
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.
scientific article; zbMATH DE number 1906013
scientific article