Let $X_t$ be a brownian motion
define: $Y(t) = t^2X_t - 2 \int_0^t sX_s \ ds$ Is $Y$ a martingale? I am trying to use Ito's lemma, and show that the drift is 0, however I am having troubles differentiating the integral.
I can do it if I use fundamental thoerem of calculus but I am told there is an easier way.
Any help please
I will assume that $X$ is a standard brownian motion (without drift and starting from $0$.) In differential notation we have $dY_t = 2t X_t dt + t^2 d X_t -2 t X_t dt = t^2 d X_t$ so that $d\langle Y,Y\rangle_t = t^2 d\langle X,X\rangle_t = t^2 dt$ (Lévy) as $X$ is a brownian motion. So $Y$ cannot be a brownian motion, as were it be we would have (Lévy) $d\langle Y,Y\rangle_t = dt$. But $Y$ is nevertheless a martingale, as it has no drift (the "something $\times dt$ part) as we just saw.