Intuition underlying stopped martingales

Question

Let $X$ be a martingale and $T$ a stopping time. Define the stopped martingale $X_{\min\{T,n\}}$. What is the intuition underlying this process? It is quite confusing here. $X$ is random and $T$ is also random and depends on $X$. Then what does one get putting two together?

Answer 1

The Gambler's Ruin example is probably helpful for intuition. Consider a Markov chain $X_n$ whose state space is the integers which jumps 1 unit to the left or to the right with equal probability. Now let $\tau = \inf \{ n : X_n = 0 \}$. Then the stopped process is the same as the original process until the gambler runs out of money, after which he presumably leaves the casino. By contrast, in the original process, the gambler would be allowed to play into debt rather than stopping when he went bankrupt.