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.