# Conditional Expectations (Stochastic Calculus)

1. Jul 14, 2007

### irony of truth

1. The problem statement, all variables and given/known data

Let (X_n; for all counting number n) be a sequence of independent random
variables. We focus on the random walk S_n := X_1 + . . . + X_n and set
F_n = 'sigma-algebra' of (S_1, . . . , S_n).

1. Compute E[S_(n+1) \ F_n]
2. For any z belonging to the complex plane C, show that:

E[ z^S_(n+1) \ F_n] = z^S_n E(z^X_(n+1))

2. Relevant equations

3. The attempt at a solution

Here's what I've done:

1. E[S_(n+1) \ F_n] = E[S_n + X_(n+1) \ F_n]

= E[S_n \ F_n] + E[X_(n+1) \ F_n]

= S_n + E[X_(n+1) \ F_n] <- S_n is F_n - measurable

The last line is my final answer. Should I leave it as is? Is there
something I can do with the second term?

2. Note that X_(n+1) is independent of X_1, X_2, ..., X_n. Also,
S_(n+1) = Sn + X_(n+1). Thus,

z^S_(n+1) = z^{Sn + X_(n+1)}

= (z^Sn)(z^X_(n+1))

Hence, E[ z^S_(n+1) \ F_n] = E[(z^Sn)(z^X_(n+1)) \ F_n]
Because S_n is F_n - measurable,

E[ z^S_(n+1) \ F_n] = (z^S_n)E[z^X_(n+1)\ F_n]

= z^S_n E(z^X_(n+1))

...because X_(n+1) is independent of X_1, X_2, ..., X_n.

Please inform me if there are some mistakes in my solution. Hoping for your
consideration.