Can someone point me into the right direction to calculate this one: $E(B^4_t)=3t^2$
I had tried using the following property with no luck:
$E(B^4_t)=E(B^2_tB^2_t)=E(\int B^2 dt )E(\int B^2 dt )=[E(\int B^2 dt )]^2=[\int E(B^2) dt]^2=[\int t dt]^2$
Any other suggestion will be appreciated. Thanks!
Answer
Apply Itô's Lemma to $W_t^4$: $$ \text{d}(W_t^4)=4W_t^3\text{d}W_t+6W_t^2\text{d}t$$
Integrate: $$ W_t^4=4\int_0^tW_s^3\text{d}W_s+6\int_0^tW_s^2\text{d}s$$
The first term is an Itô integral, which is by construction a martingale, with expectation $0$ hence: $$E[W_t^4]=6\int_0^tE[W_s^2]\text{d}s=6\int_0^ts\text{d}s=3t^2$$
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