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Homework Help: Conditional Expectations (Stochastic Calculus)

  1. Jul 14, 2007 #1
    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

    -N.A.- (Hoping for your consideration)

    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
  2. jcsd
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