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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)?

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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 $$ \mathbb E(|W_t|^a)=\frac{(2t)^{a/2}}{\sqrt{\pi}}\Gamma\left(\frac{a+1}2\right). $$

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