Supposed the following: $(X_n)_{n\geq1}$ independent random variable adapted to the $\sigma X_n$ $\forall n$. Consider that: $$P(X_n=1)=P(X_n=-1)=\frac{1}{2} $$ and that $X_0=a$ with $1\leq a\leq K-1$. Now define $S_n=\sum^n_{i=0} X_i$ and $$\tau=\min\left[n\in N: S_n=0\,or\, S_n=K \right]$$ How can I show that $\tau$ is integrable?
Integrability of the Stopping Time in a Random Walk
0
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probability-theory
stochastic-processes
martingales
stopping-times
1 Answers
2
Hints:
- Show that $$M_n := S_n^2-\frac{n}{2}$$ is a martingale.
- Apply the optional stopping theorem to show that $(M_{n \wedge \tau})_{n \in \mathbb{N}}$ is a martingale; in particular, $$\mathbb{E}(S_{n \wedge \tau}^2) = \frac{1}{2} \mathbb{E}(n \wedge \tau).$$
- Show that $|S_{n \wedge \tau}| \leq K$. Use the dominated convergence theorem and the monotone convergence theorem to show that $$\mathbb{E}(S_{\tau}^2) = \frac{1}{2} \mathbb{E}(\tau).$$
- Since $\mathbb{E}(S_{\tau}^2) \leq K^2$, this proves that $\tau$ is integrable.