1) Which equation does the following process satisfy: $$Y_t:=W_t^{4}$$ Where $W_t$ is Wiener process.
2) Prove that $$\mathbb{E}W_t^{4}=3t^2$$
Using Ito formula for $Y_t$ is a good point to start?
My attempt:
Obviously $$ dW_t=0dt+1dW_t$$ Let $$F(W_t,t)=W_t^4$$ Now using Ito formula $$dF(W_t,t)=(0+0+6W_t^2)dt+4W_t^3dW_t$$ Now we integrate each side $$Y_t=\int_0^t 6W_s^2ds+\int_0^t4W_s^3dW_s$$ And when it comes to 1) that is our equation?
2) $$\mathbb{E}Y_t=\mathbb{E}\int_0^t 6W_s^2ds+\mathbb{E}\int_0^t4W_s^3dW_s$$ I don't know what I can use next. Any hint? Can i go with expectation inside the integral?