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A Controller And A Stopper Game With Degenerate
Variance Control
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Ananda P. Weerasinghe, Iowa State University
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Abstract
We consider a zero sum
stochastic differential game which involves two players, emph{the
controller} and emph{the stopper}. The stopper selects the
stopping rule which halts the game. The controller chooses the
diffusion coefficient of the corresponding state process which is
allowed to degenerate. At the end of the game, the controller pays
the stopper, the amount $ Eint_{0}^{tau} e^{-alpha t} C(Z_x(t))dt
$, where $Z_x(cdot)$ represents the state process with initial
position $x$ and $alpha $ is a positive constant. Here $C(cdot)$
is a reward function where the set $ {x: C(x)>0}$ is an open
interval which contains the origin. Under some assumptions on the
reward function $C(cdot)$ and the drift coefficient of the state
process, we show that this game has a value. Furthermore, this value
function is Lipschitz
continuous, but it fails to be a $C^1$ function.
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Full text: PDF
Pages: 89-99
Published on: July 4, 2006
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Electronic Communications in Probability. ISSN: 1083-589X |
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