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WROCŁAW UNIVERSITY
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TECHNOLOGY

Contents of PMS, Vol. 4, Fasc. 2,
pages 185 - 196
 

STOPPING GAMES FOR SYMMETRIC MARKOV PROCESSES

J. Zabczyk

Abstract: Let E be a Dirichlet form corresponding to a symmetric Markov process M  = (Q,M, x ,Px)
            t acting on a state space X. Let g and h,g < h , be quasi-continuous elements of the corresponding Dirichlet space F , and n a quasi-continuous solution of the variational inequality

E (n,u -n) > 0  for all u  (-  F, g < u < h,
 a
where a > 0 and Ea(u,n) = E(u,n)+ a(u,n) for all u,n  (-  F . It is shown in the paper that if Jx(t,s) is defined for all x  (-  X and all stopping times t and s by

Jx(t,s) = Ex(e-at /\ s(It<ah(xt)+ It>sg(xs))),
then for quasi-every x  (-  X we have

n(x) = inf sup Jx(t,s) = sup infJx(t,s).
       t s            s  t
Moreover, for quasi-every x  (-  X the pair (^t,^s) such that

^t = inf(t > 0;h(xt) = n(xt)),s^= inf(t > 0;g(xt) = n(xt))
is the saddle point of the game

Jx(^t,s) < Jx(^t,^s) < Jx(t,^s)
for all stopping times t, s and quasi-every x  (-  X .

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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