Problem about Random walk and Stopping time.

Question

Here is an example in "Probability with Martingales"

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My questions are:

(1)Does equation (a) hold for $T=\infty$?

(2)The equation:$$\mathbb{E}M_T^\theta=1=\mathbb{E}[(sech \theta)^Te ^\theta]$$

The author said when $T=\infty$ ,$\mathbb{E}[(sech \theta)^Te ^\theta]=0$

So the equation doesn't hold??

(3)Why if $T=\infty$ ,$(sech\theta)^T \uparrow 0$?

In my opinion,if $T=\infty$ ,$(sech\theta)^T \equiv 0$.

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Thanks for regards.

Answer 1

The author did not say that the expectation is $0$: he rather meant that $M_{T(\omega)}^\theta=0$ if $T(\omega)=\infty$. This is justified because $0\lt \mathrm{sech}(x)\lt 1 $ for each $x$.

We thus have $$\mathbb E[M_{T}^\theta]=\mathbb E[M_T^\theta\chi\{T<\infty\}].$$