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October, 1990 Ordered Skorokhod Stopping for a Sequence of Measures
C. T. Shih
Ann. Probab. 18(4): 1623-1634 (October, 1990). DOI: 10.1214/aop/1176990637


Let $X$ be a transient right (Markov) process on a compact metric space including a death point. Let $\mu$ and $\nu_n$ be finite measures whose potentials satisfy $\mu U \geq \cdots \geq \nu_nU \geq \cdots \geq \nu_1 U$. We prove that there exists a right-continuous stochastic process $Y = (\tilde\Omega, \mathscr{M}, \tilde\mathscr{M}_t, Y_t Q)$ that is a version of $X$ with initial measure $\nu_\infty(\cdot) = Q(Y_0 \in \cdot)$ and in which there are $(\tilde\mathscr{M}_t)$-stopping times $\tilde{\tau}_n \downarrow 0$ with $Q(Y(\tilde{\tau}_n) \in \cdot, \tilde{\tau}_n < \infty) = \nu_n(\cdot)$. Furthermore, a canonical representation of $Y$ and $(\tilde{\tau}_n)$ is given in which one has a better understanding of the tail behavior of the sequence $\tilde{\tau}_n$. Based on this representation an open question is posed whose answer in the positive would permit defining in $X$, assuming it admits a continuous real random variable independent of the path, decreasing stopping times $T_n$ such that $P^\mu(X(T_n) \in \cdot, T_n < \infty) = \nu_n(\cdot)$. These $T_n$ would satisfy the Markov property $T_n = T_{n+1} + S_n \circ \theta(T_{n+1}), S_n$ a stopping time linking $\nu_n$ and $\nu_{n+1}$. Fitzsimmons has now proved the existence of a desired decreasing sequence $T_n$ in $X$ for any given $\mu$ and $\nu_n$ as above, using a very different approach. His $T_n$, however, do not satisfy the Markov property.


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C. T. Shih. "Ordered Skorokhod Stopping for a Sequence of Measures." Ann. Probab. 18 (4) 1623 - 1634, October, 1990.


Published: October, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0725.60081
MathSciNet: MR1071814
Digital Object Identifier: 10.1214/aop/1176990637

Primary: 60J40
Secondary: 60G40 , 60J45

Keywords: randomization of stopping times , Right processes , Skorokhod stopping

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 4 • October, 1990
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