Stochastic Calculus :Conditional Expectation

[WMD]

Suppose [imath]X_1,X_2 \dots[/imath]are independent random variables with [imath]\mathbb{P}[X_j= 1] =1- \mathbb{P}[X_j=-1]=\frac13[/imath]

Let [imath]S_n = X_1 + \dots + X_n[/imath]


1. Find [imath] \mathbb{E}[S_n], \mathbb{E}[S^2_n] , \mathbb{E}[S^3_n][/imath]. Answers to 1 are given in #2.
2.Find If m < n [imath] E[S_n| \mathcal{F}_m], E[S^2_n| \mathcal{F}_m], E[S^3_n| \mathcal{F}_m][/imath]

3. Find If m < n [imath] E[ X_m | \mathcal{F}_n] [/imath]

My answers to 2.

[math]E[(S_n -S_m)^2|\mathcal{F}_m] = \mathbb{E}[(S_n -S_m)^2] = E[X^2_j](n-m) =1(n-m)[/math]and hence,
[math]E[S^2_n|\mathcal{F}_m] = S^2_m -\frac23 {S_m} + (n-m)[/math]

 

[WMD]

These are the correct answers.

[math]\mathbb{E}[S_n]=-\frac{n}{3}, \mathbb{E}[S^2_n]=n + \frac{n(n-1)}{9}, \mathbb{E}[S^3_n]= -\frac{n}{3} -n(n-1) - \frac{n(n-1)(n-2)}{27}[/math]