Friday, January 27, 2017

stochastic calculus - Variance of Brownian Motion


Can someone point me into the right direction to calculate this one: E(B4t)=3t2


I had tried using the following property with no luck:


E(B4t)=E(B2tB2t)=E(B2dt)E(B2dt)=[E(B2dt)]2=[E(B2)dt]2=[tdt]2


Any other suggestion will be appreciated. Thanks!



Answer



Apply Itô's Lemma to W4t: d(W4t)=4W3tdWt+6W2tdt


Integrate: W4t=4t0W3sdWs+6t0W2sds


The first term is an Itô integral, which is by construction a martingale, with expectation 0 hence: E[W4t]=6t0E[W2s]ds=6t0sds=3t2


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