The Annals of Probability

Ordered Skorokhod Stopping for a Sequence of Measures

C. T. Shih

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Abstract

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.

Article information

Source
Ann. Probab., Volume 18, Number 4 (1990), 1623-1634.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990637

Digital Object Identifier
doi:10.1214/aop/1176990637

Mathematical Reviews number (MathSciNet)
MR1071814

Zentralblatt MATH identifier
0725.60081

JSTOR
links.jstor.org

Subjects
Primary: 60J40: Right processes
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Keywords
Skorokhod stopping right processes randomization of stopping times

Citation

Shih, C. T. Ordered Skorokhod Stopping for a Sequence of Measures. Ann. Probab. 18 (1990), no. 4, 1623--1634. doi:10.1214/aop/1176990637. https://projecteuclid.org/euclid.aop/1176990637


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