I am trying to understand the concept of strong Markov property quoted from Wikipedia:
Suppose that $X=(X_t:t\geq 0)$ is a stochastic process on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with natural filtration $\{\mathcal{F}\}_{t\geq 0}$. Then $X$ is said to have the strong Markov property if, for each stopping time $\tau$, conditioned on the event $\{\tau < \infty\}$, the process $X_{\tau + \cdot}$ (which maybe needs to be defined) is independent from $\mathcal{F}_{\tau}:=\{A \in \mathcal{F}: \tau \cap A \in \mathcal{F}_t ,\, \ t \geq 0\}$ and $X_{\tau + t} − X_{\tau}$ has the same distribution as $X_t$ for each $t \geq 0$.
Here are some questions that make me stuck:
- In $\mathcal{F}_{\tau}:=\{A \in \mathcal{F}: \tau \cap A \in \mathcal{F}_t ,\, \ t \geq 0\} $, what does $\tau \cap A $ mean? $\tau$ is a stopping time and therefore a random variable and $A$ is a $\mathcal{F}$-measurable subset, but what does $\tau \cap A$ mean?
- How is the process $X_{\tau + \cdot}$ defined from the process $X_{\cdot}$ ? Is it the translated version of the latter by $\tau$?
How is the conditional independence between a process, such as $X_{\tau + \cdot}$, and the sigma algebra, such as $\mathcal{F}_{\tau}$, given an event, such as $\{\tau < \infty\}$, defined?
Related question, is independence between a random variable and a sigma algebra defined as independence between the sigma algebra of the random variable and the sigma algebra?
- Is "$X_{\tau+ t} − X_{\tau}$ has the same distribution as $X_t$ for each $t \geq 0$" also conditional on the event $\{\tau < \infty\}$?
Thanks and regards!