## The Annals of Probability

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

### Ordered Skorokhod Stopping for a Sequence of Measures

#### 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