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\text { Exercise 2. Let } X=\left(X_n\right)_{n \geq 0} \text { be a martingale. }

(1) Suppose that $T$ is a stopping time, show that $X^T$ is also a martingale. In particular, $\mathbb{E}\left[X_{T \wedge n}\right]=\mathbb{E}\left[X_0\right]$.
(2) Suppose that $S \leq T$ are bounded stopping times, show that $\mathbb{E}\left[X_T \mid \mathscr{F}S\right]=X_S$, a.s. In particular, $\mathbb{E}\left[X_T\right]=\mathbb{E}\left[X_S\right]$ (3) Suppose that there exists an integrable random variable $Y$ such that $\left|X_n\right| \leq Y$ for all $n$, and $T$ is a stopping time which is finite a.s., show that $\mathbb{E}\left[X_T\right]=\mathbb{E}\left[X_0\right]$. (4) Suppose that $X$ has bounded increments, i.e. $\exists M>0$ such that $\left|X{n+1}-X_n\right| \leq M$ for all $n$, and $T$ is a stopping time with $\mathbb{E}[T]<\infty$, show that $\mathbb{E}\left[X_T\right]=\mathbb{E}\left[X_0\right]$.

Exercise 3. Let $X=\left(X_n\right){n \geq 0}$ be Gambler’s ruin with state space $\Omega={0,1,2, \ldots, N}$ : $$X_0=k, \quad \mathbb{P}\left[X{n+1}=X_n+1 \mid X_n\right]=\mathbb{P}\left[X_{n+1}=X_n-1 \mid X_n\right]=1 / 2, \quad \tau=\min \left{n: X_n=0 \text { or } N\right} .$$

$$X_0=k, \quad \mathbb{P}\left[X_{n+1}=X_n+1 \mid X_n\right]=\mathbb{P}\left[X_{n+1}=X_n-1 \mid X_n\right]=1 / 2, \quad \tau=\min \left{n: X_n=0 \text { or } N\right}$$
(1) Show that $Y=\left(Y_n:=X_n^2-n\right)_{n \geq 0}$ is a martingale.
(2) Show that $Y$ has bounded increments.
(3) Show that $\mathbb{E}[\tau]<\infty$.
(4) Show that $\mathbb{E}[\tau]=k(N-k)$.

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