Let $M$ be a continuous martingale with $M_0=0$ and $S_n:=\inf \{t:|M_t|>n\}$. Show using Ito's Isometry that $\langle M\rangle_{S_n}$ has finite expectation for each $n\in\mathbb{N}$.
I know that $S_n$ is a stopping time. But abit stuck on how Ito's Isometry can be used to proceed.
In general I think it's quite difficult to gain intuition on this topic so that one could go about approaching the exercises. Can some one give pointers to good notes and textbooks?
Many thanks !
Hints: By definition, $(M_{t \wedge S_n}^2- \langle M \rangle_{t \wedge S_n})_{t \geq 0}$ is a martingale and therefore $$\mathbb{E}(\langle M \rangle_{t \wedge S_n}) = \mathbb{E}(M_{t \wedge S_n}^2).$$ Since $M$ has continuous sample paths, this implies
$$\mathbb{E}(\langle M \rangle_{t \wedge S_n}) \leq n^2.$$
By monotone convergence,
$$\mathbb{E}(\langle M \rangle_{S_n}) \leq n^2.$$