If $\tau$ is a stopping time, then $E(X_{\tau})=?$

Question

Let $\{X_n \in \mathbb{N}: n \in \mathbb{N}\}$ be a sequence of r.v. and $\tau_k=\min\{n\in \mathbb{N}:X_n=k\}$

Does $E(X_{\tau_k})=E(k)=k$?

Any help would be appreciated.

Answer 1

Use inf not min in the case that we have $\tau_k = \inf \{\emptyset\}$

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Hint:

1. Show that $\tau_k < \infty$ a.s. (Thus we can use min instead of inf)

2. Note that

$$E[X_{\tau_k}] = \sum_{t=0}^{\infty} P(\tau_k = t)E[X_{\tau_k} | \tau_k = t]$$

where

$$E[X_{\tau_k} | \tau_k = t] = \frac{E[X_{\tau_k} 1_{\tau_k = t}]}{P(\tau_k = t)} = \frac{E[X_{t} 1_{\tau_k = t}]}{P(\tau_k = t)}$$

3. Convince yourself that

$$E[X_{t} 1_{\tau_k = t}] = E[k 1_{\tau_k = t}]$$

4. Finally note that

$$\sum_{t=0}^{\infty} P(\tau_k = t) = 1$$

Note that 2, 3 and 4 make use of 1.