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The controller-and-stopper game for a linear diffusion. (English) Zbl 1039.60043

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(·) with the state space I=(l,r), which is a nonempty bounded open interval, be the diffusion process governed by the equation

dX(t)=β(t)dt+σ(t)dW(t),

X(0)=xI on a filtered probability space (Ω,,P,( t )), where W(·) is a standard Brownian motion. The first player chooses a pair (β(t),σ(t)) from some given class and the second player selects a stopping time τ, in which the first (second) player minimizes (resp. maximizes) an expectation E[e -ατ(X) u(X(τ(X)))], where α0 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(·) sup τ E[e -ατ(X) u(X(τ(X)))]=sup τ inf X(·) E[e -ατ(X) u(X(τ(X)))]

for an undiscounted case (α=0) and a discounted case (α>0) under some assumptions.

MSC:
60G40Stopping times; optimal stopping problems; gambling theory
93E20Optimal stochastic control (systems)
62L15Optimal stopping (statistics)