Consider a classical symmetric random walk that starts at origin $x=0$ and lasts for $N$ steps. If $x=-1$ is reached, a walk is terminated. What is the expected value of this process?
I divided a walk into 2 parts.
- A path hits $x=-1$ at some step $n \le N$. Then, I can compute a part of a total expected value summing the probabilities with weight $-1$ as it was presented in https://math.stackexchange.com/a/182376
- However, I encounter a problem with a part giving rise from walking only in the nonnegative part of the axis. How should I proceed?
Best Answer
Let $\{\xi_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variables with $\mathbb{P}(\xi_i=1)=\mathbb{P}(\xi_i=-1)=\frac{1}{2}$, and let $X_0=0$, $X_n=\xi_1+\dots+\xi_n$ for $n\geq 1$. Then $\{X_n\}$ is a martingale with respect to the natural filtration $\mathcal{F}_n=\sigma(\xi_1,\dots,\xi_n)$.
Define a stopping time $T_{-1}=\min\{n\in\mathbb{N}:X_n=-1\}$, and let $T=\min\{T_{-1},N\}$. Then $T$ is a bounded stopping time, hence by the Optional Stopping Theorem, $$ \mathbb{E}(X_T)=\mathbb{E}(X_0)=0 $$