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=4∫t0W3sdWs+6∫t0W2sds
The first term is an Itô integral, which is by construction a martingale, with expectation 0 hence: E[W4t]=6∫t0E[W2s]ds=6∫t0sds=3t2
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