Process Sampled at a Random Time and Stopped Process

Let $\Omega$ be a set, let $T>0$ be a real number, let $X=(X_t)_{t\in[0,T]}$ be a family of real-valued functions on $\Omega$, and let $\tau:\Omega\to[0,T]$ be a function (a random time).

The process $X$ sampled at $\tau$ is the real-valued function $X_\tau$ on $\Omega$ defined by

$$X_\tau(\omega)=X_{\tau(\omega)}(\omega)\qquad(\omega\in\Omega).$$

The process $X$ stopped at $\tau$ is the family $X^\tau=(X^\tau_t)_{t\in[0,T]}$ of real-valued functions on $\Omega$ defined by

$$X^\tau_t(\omega)=X_{\min(t,\tau(\omega))}(\omega)\qquad(t\in[0,T],\ \omega\in\Omega).$$