1
$\begingroup$

I am to prove Doob's (d) in the red box below:


enter image description here


enter image description here


What I tried:

Since $T < \infty$ a.s., we have

$$E[X_T] = E[\lim X_{T \wedge n}].$$

By Fatou's Lemma, we have

$$E\left[\lim X_{T \wedge n}\right] = E\left[\liminf X_{T \wedge n}\right] \stackrel{\mbox{Fatou's}}{\le} \liminf E\left[X_{T \wedge n}\right] \le \liminf E[X_0] = E[X_0].$$

Is that right?

$\endgroup$
1
$\begingroup$

I think you just need to change your second-to-last $=$ in a $\leq$ as you have a super martingale and not a martingale. Other than that, I think you are right.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.