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Expectations of Brownian motion (simple, I hope)

  1. Sep 24, 2011 #1
    Let [itex]B_t[/itex] be Brownian motion in [itex]\mathbb R[/itex] beginning at zero. I am trying to find expressions for things like [itex]E[(B^n_s - B^n_t)^m][/itex] for [itex]m,n\in \mathbb N[/itex]. So, for example, I'd like to know [itex]E[(B^2_s - B^2_t)^2][/itex] and [itex]E[(B_s - B_t)^4][/itex]. Here are the only things I know:
    1. [tex]E[B_t^{2k}] = \frac{(2k)!}{2^k \cdot k!}[/tex]
    2. [tex]E[B_s B_t] = \min(s,t)[/tex]
    3. [tex]E[(B_s - B_t)^2] = |s-t|[/tex]
    4. Brownian motion has independent increments.
    But I'm having a hard time getting expressions for the expectations I listed in the start of the question using these facts. Can someone help?
  2. jcsd
  3. Sep 24, 2011 #2
    For example, in trying to compute [itex]E[(B_s - B_t)^4][/itex], one comes up against things like [itex]4E[B_s^3B_t][/itex]. How in the world am I supposed to deal with that?
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