Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How can we easily compute $\mathbb{E} [ \left|W_t\right| ^\alpha]$, where $\alpha \in \mathbb R^*_+ $ and $W = (W_t)_{t \geq 0}$ is the one dimensional standard Brownian motion (or wiener process)?

share|cite|improve this question
up vote 2 down vote accepted

By scaling, $\mathbb E(|W_t|^a)=t^{a/2}\mathbb E(|W_1|^a)$ for each $a\gt-1$, where $$ \mathbb E(|W_1|^a)=\frac2{\sqrt{2\pi}}\int_0^{+\infty}x^a\mathrm e^{-x^2/2}\mathrm dx\stackrel{x^2=2u}{=}\frac1{\sqrt{\pi}}2^{a/2}\int_0^{+\infty}u^{(a-1)/2}\mathrm e^{-u}\mathrm du, $$ hence $$ \mathbb E(|W_t|^a)=\frac{(2t)^{a/2}}{\sqrt{\pi}}\Gamma\left(\frac{a+1}2\right). $$

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.