# Understanding a sufficient and necessary condition for a nonnegative supermartingale to be a potential.

By Potential, the author means a nonnegative supermartingale $$X_t$$, such that $$\lim E(X_t)=0$$.

I'm trying to understand the following remark: I'm trying to prove the sufficiency of the condition to conclude that $$X$$ is a potential.

If I apply Fatou's lemma (several times), then I get $$E(\lim \inf X_t\mid \mathcal{F}_n)\leq \lim \inf E(X_t\mid \mathcal{F}_n)\leq \lim \inf X_n=0$$

and now $$0=E(\lim \inf X_t) \leq E(\lim \inf E(X_t\mid \mathcal{F}_n))\leq \lim \inf E(X_t) \leq 0$$

So, now I have $$\lim \inf X_t=0$$ ... this is not exactly want we want.

Any hints?

## 1 Answer

You can have a non-negative martingale $$X$$ with $$X_0=1$$ (and so $$E(X_t)=1$$ for all $$t\ge0$$) such that $$\lim_{t\to\infty}X_t =0$$ a.s. (Example: $$X_t=\exp(B_t-t/2)$$ with $$B$$ a standard Brownian motion.) Remark 4.6.4 is incorrect. (Source?)

Your author seems to have gotten the implication of Fatou the wrong way around.

• The source is the book by Cohen and Elliot, Stochastic Calculus and Applications. Thanks for your answer. I still haven't studied(by myself) well Brownian motion, so, it may take some time until I accept your answer. – An old man in the sea. Jun 10 at 20:25