I am working on Karatzas&Shreve Brownian Motion and Stochastic Calculus P.36, Problem 5.19.
• (i) A local martingale of class DL is a martingale.
• (ii) A nonnegative local martingale is a supermartingale.
• (iii) If $M\in\mathscr{M}^{c,loc}$ and $S$ is a stopping time of $\left\{ \mathscr{F}_{t}\right\}$ then $E\left(M_{S}^{2}\right)\le E\left\langle M\right\rangle _{S}$ where $M_{\infty}^{2}={\displaystyle \liminf_{t\rightarrow\infty}M_{t}^{2}}$. $\mathscr{M}^{c,loc}$ donate the space of continuous local martingale.
I have solved (i) and (ii). But I am trapped in (iii). By definition, we know $M^{2}-\left\langle M\right\rangle\in\mathscr{M}^{c,loc}$. But I don't know what to next. Can anyone give me a hint?
Consider a localizing sequence $\{T_n\}$ for $X=M^2-\langle M\rangle$. Since $X^{S\wedge T_n}$ is a u.i. martingale, $\mathsf{E}X_{S\wedge T_n}=0$ for each $n\ge 1$. Using Fatou's lemma, $$ \mathsf{E}M_S^2\le \liminf_{n\to\infty}\mathsf{E}M_{S\wedge T_n}^2=\liminf_{n\to\infty}\mathsf{E}\langle M\rangle_{S\wedge T_n}=\mathsf{E}\langle M\rangle_S. $$