Prove $M_{S(k) \wedge n}$ is bounded in $\mathscr L^2$

Question

Probability with Martingales:

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To prove $$\sup E[M_{S(k) \wedge n}^2] < \infty,$$ how can we use 12.12c? There aren't any stopping times there.

Answer 1

$$\sup E[M_{S(k) \wedge n}^2]$$

$$ = \sup (E[N_{S(k) \wedge n}] + E[A_{S(k) \wedge n}])$$

$$ \le \sup (E[N_{S(k) \wedge n}]) + \sup(E[A_{S(k) \wedge n}])$$

Since $S(k)$ is a stopping time and $N$ is a martingale, we have

$$E[N_{S(k) \wedge n}] = E[N_{n}] (= 0)$$

We have $E[A_{S(k) \wedge n}] \le E[k] = k$

Thus,

$$\sup E[M_{S(k) \wedge n}^2] \le \sup (E[N_{S(k) \wedge n}]) + \sup(E[A_{S(k) \wedge n}]) = \sup(0) + \sup(k) = 0 + k = k < \infty$$