What is $1_{\{\tau_n>0\}}X^{\tau_n}$ process saying?

Question

As title says, what is $1_{\{\tau_n>0\}}X^{\tau_n}$ process? I do have understanding of what stochastic processes are, but not sure what is this specific process saying.

Answer 1

Let $\tau$ a stopping time, then the stochastic process $X^{\tau}$ is defined as $$X^{\tau}_t(\omega) := X_{\tau \wedge t}(\omega) := \begin{cases} X(t,\omega) & t \leq \tau(\omega) \\ X(\tau(\omega),\omega) & t>\tau(\omega) \end{cases}$$ i.e. it's the stopped process.

Example Let $(B_t)_{t \geq 0}$ a Brownian motion and define $$\tau(\omega) := \inf\{t \geq 0; B(t,\omega) \geq a\}$$ for $a>0$. This means that, for each path $t \mapsto B(t,\omega)$ you look at which time the path gets (the first time) larger than $a$ - and exactly in this moment you stop this particular path. Thus $$B^{\tau}_t 1_{\{\tau>0\}}(\omega) \stackrel{\tau>0}{=} B^{\tau}_t(\omega) = \begin{cases} B(t,\omega) & t \leq \tau(\omega) \\ a & t>\tau(\omega) \end{cases}$$

Furthermore, you could consider a sequence $(\tau_n)_n$ of stopping times. Then $(1_{\{\tau_n>0\}} X^{\tau_n})_{n}$ is a sequence of stochastic processes. This is used for example to define local martingales.

$X_{\tau}$ denotes the random variable $$X_{\tau}(\omega) := X(\tau(\omega),\omega)$$ In the given example, $B_{\tau}=a$.

Remark In some literature, the notation $X^{\tau}$ is used for the random variable $X_{\tau}$ (as defined above) and vica versa.