Random Walk and Martingal Hi Guys I am trying to solve this exercice but I don't seem how to do that. Can anyone help me please?
On a probability space $(Ω, \mathcal{A}, P)$ a sequence $(X_n)_{n∈\Bbb{N}}$ of i.i.d random variables with values in the set {$-1, 0, 1$} , equally distributed on this three elemental set. Let $S = (S_n)_{n∈\Bbb{N}_0}$ with $S_n = \sum_{k=1}^{n} X_k $ its corresponding Random Walk.
Further $c ∈ \Bbb{N}$ and $T_c :=$ inf{$n ∈ \Bbb{N}_0| S_n ∈ ${$-c, 2c$}} .
(a) Find b ∈ R so that ($S^2_n - bn)_{n∈\Bbb{N}_0}$ is a martingale relating to the filtration
$\mathcal{F} = (\mathcal{F}_n)_{n∈\Bbb{N}_0}$ with $\mathcal{F}_n = σ(X_k : k ∈ [n])$ . The proof is needed
(b) Show $E[T_c] < ∞.$
(c) Calculate $P[T_c < ∞, S_{T_c} = -c]$ and $P[T_c < ∞, S_{T_c} = 2c]$.
(d) Calculate $E[T_c]$.}
 A: I'll write an answer for the first part.
$\mathbb{E}[S_n^2-bn|\mathcal{F_{n-1}}]=\mathbb{E}[(S_{n-1}^2+X_n)^2-bn|\mathcal{F_{n-1}}]=\mathbb{E}[(S_{n-1}+X_n)^2|\mathcal{F_{n-1}}]-bn=$
$=\mathbb{E}[S_{n-1}^2|\mathcal{F_{n-1}}]+\mathbb{E}[2S_{n-1}X_n|\mathcal{F_{n-1}}]+\mathbb{E}[X_n^2|\mathcal{F_{n-1}}]-bn$
Now we just have to evaluate all conditional expectations. Note that $S_{n-1}^2$ is measurable with respect to $\mathcal{F_{n-1}}$ and hence $\mathbb{E}[S_{n-1}^2|\mathcal{F_{n-1}}]=S_{n-1}^2$. Also, $X_n^2$ is independent of $\mathcal{F_{n-1}}$ and so $\mathbb{E}[X_n^2|\mathcal{F_{n-1}}]=\mathbb{E}[X_n^2]=\frac{2}{3}$. Finally, note that $2S_{n-1}$ is a bounded random variable which is measurable with respect to $\mathcal{F_{n-1}}$, and so:
$\mathbb{E}[2S_{n-1}X_n|\mathcal{F_{n-1}}]=2S_{n-1}\mathbb{E}[X_n|\mathcal{F_{n-1}}]=2S_{n-1}\mathbb{E}[X_n]=0$
Combining all of these, we get:
$\mathbb{E}[S_n^2-bn|\mathcal{F_{n-1}}]=S_{n-1}^2+\frac{2}{3}-bn$
And this is equal to $S_{n-1}^2-b(n-1)$ when $b=\frac{2}{3}$.
