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

$dX\left(t\right)=\beta \left(t\right)dt+\sigma \left(t\right)dW\left(t\right),$

$X\left(0\right)=x\in I$ on a filtered probability space $\left({\Omega },ℱ,P,\left({ℱ}_{t}\right)\right)$, where $W\left(·\right)$ is a standard Brownian motion. The first player chooses a pair $\left(\beta \left(t\right),\sigma \left(t\right)\right)$ from some given class and the second player selects a stopping time $\tau$, in which the first (second) player minimizes (resp. maximizes) an expectation $E\left[{e}^{-\alpha \tau \left(X\right)}u\left(X\left(\tau \left(X\right)\right)\right)\right]$, where $\alpha \ge 0$ is a discount factor and $u$ is a real-valued continuous function on $\left[l,r\right]$. They then give a saddle-point and show the game has a value, that is,

$\underset{X\left(·\right)}{inf}\underset{\tau }{sup}\phantom{\rule{0.166667em}{0ex}}E\left[{e}^{-\alpha \tau \left(X\right)}u\left(X\left(\tau \left(X\right)\right)\right)\right]=\underset{\tau }{sup}\underset{X\left(·\right)}{inf}E\left[{e}^{-\alpha \tau \left(X\right)}u\left(X\left(\tau \left(X\right)\right)\right)\right]$

for an undiscounted case $\left(\alpha =0\right)$ and a discounted case $\left(\alpha >0\right)$ under some assumptions.

##### MSC:
 60G40 Stopping times; optimal stopping problems; gambling theory 93E20 Optimal stochastic control (systems) 62L15 Optimal stopping (statistics)