Is a continuous supermartingale with vanishing drift already a martingale? In my concrete problem, I have a continuous nonnegative local martingale $ (X_t) $ on $ \left[0, T\right] $ which is bounded from below by 0 (and hence a supermartingale) and which in addition satisfies $ sup_{t \in \left[0,T\right]} \mathbb{E} \left[ X_t^p \right] < + \infty$, for some $p > 1$. When I apply Ito, I find
$dX_t = 0 dt + H dW_t$,
where W is a Brownian motion and H is some predictable process. Now, I am wondering if $(X_t)$ is in fact a true martingale (the reason is that I would like to apply the Doob inequality for martingales in $L^p$).
Thanks a lot for your help! Simon
Take $dX_t= f(t)dt+ dW_t$ such that :
-$F(t)=\int_0^t f(s)ds$ is strictly decreasing (determisitic)
-$f(t)\to 0$ as $t\to \infty$
Then $X_t= F(t)+ W_t$ and as the sum of a martingale term (the Brownian motion) plus a deterministic supermartingale $F(t)$, $X$ it is a supermaringale with a drift term that goes to 0, and $X$ is not a martingale (unless F is null).
For example $f(t)=-\frac{1}{(1+t)^2}$ gives $F(t)=\frac{1}{1+t}-1$ seems to work fine.
Best regards