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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:

enter image description here

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?

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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.

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  • $\begingroup$ 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. $\endgroup$ – An old man in the sea. Jun 10 at 20:25

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