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:(adsbygoogle = window.adsbygoogle || []).push({});

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?

- [tex]E[B_t^{2k}] = \frac{(2k)!}{2^k \cdot k!}[/tex]
- [tex]E[B_s B_t] = \min(s,t)[/tex]
- [tex]E[(B_s - B_t)^2] = |s-t|[/tex]
- Brownian motion has independent increments.

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Expectations of Brownian motion (simple, I hope)

Loading...

Similar Threads - Expectations Brownian motion | Date |
---|---|

A Value at Risk, Conditional Value at Risk, expected shortfall | Mar 9, 2018 |

I Conditional Expectation Value of Poisson Arrival in Fixed T | Dec 21, 2017 |

I Poisson distribution regarding expected distance | Oct 22, 2017 |

B The expectation value of superimposed probability functions | Sep 9, 2017 |

Expectation of a product of Brownian Motions | May 29, 2012 |

**Physics Forums - The Fusion of Science and Community**