Hunt process

Definition

Let $E$ be a locally compact Hausdorff space with a countable base, and let $E_\Delta = E \cup \set{\Delta}$ be its one-point compactification; the point $\Delta$ is the cemetery. Let $X = (X_t)_{t \ge 0}$ be a stochastic process with values in $E_\Delta$, adapted to a filtration $(\mathcal{F}_t)$, with a family of probability measures $(P^x)_{x \in E_\Delta}$. $X$ is a Hunt process if:

1. $X$ is a Markov process with respect to $(\mathcal{F}_t)$ under each $P^x$, with $P^x(X_0 = x) = 1$, and $\Delta$ is absorbing: $X_t = \Delta$ implies $X_s = \Delta$ for all $s \ge t$.
2. Each path $t \mapsto X_t$ is right-continuous on $[0, \infty)$ and has left limits in $E_\Delta$ on $(0, \infty)$.
3. $X$ has the strong Markov property at every stopping time of $(\mathcal{F}_t)$, where $(\mathcal{F}_t)$ is right-continuous and suitably completed.
4. $X$ is quasi-left-continuous: if $T_n$ are stopping times increasing to $T$, then $X_{T_n} \to X_T$ almost surely on $\set{T < \infty}$, under each $P^x$.