Does all $L^2(\Omega ,\mathcal F_T,\mathbb P)$ processes is a martingale?

Question

In the Book of Schilling (Brownain motion), there is the following theorem

I'm quite surprised by this theorem. It looks to mean that all $L^2(\Omega ,\mathcal F_T,\mathbb P)$ is a Martingale (or local martingale). Is this really true ?

Answer 1

Given such a $Y$, consider the martingale $M_t:=E[Y|\mathscr F_t]$, $0\le t\le T$. Because of the assumption on the filtration, $(M_t)_{0\le t\le T}$ can be taken to be right continuous. And because of the martingale representation theorem for Brownian motion, you then have $M_t=M_0+\int_0^t X_s\,dB_s$, $\forall t\in[0,T]$, a.s., for a square-integrable process $(X_s)$ as you have indicated. Because $Y$ is $\mathscr F_T$ measurable, $$ y+\int_0^T X_s\,dB_s = M_T=E[Y|\mathscr F_T]=Y, $$ where $y:=M_0=E[Y]$.