Brownian motion. Solve stoc. integral by using Ito's lemma

Question

I want to show that following statement is true by using Ito's lemma to solve stochastic integrals:

I define the functions in Ito's model: a()=0, b()= (2wt-2)^2. f(t)=Integrate[(2wt-2)^2]

Then df=(b^2/2)(d^2/dwt^2)+(bdf/dst). But it doesn’t add up. How do I show it by using Ito's lemma?

Answer 1

Try Ito's formula for $(2W_t+1)^3$, and then integrate. More specifically, note that \begin{align*} d\left( (2W_t+1)^3 \right) &= 6(2W_t+1)^2 dW_t + 12 (2W_t+1) dt, \end{align*} then \begin{align*} (2W_T+1)^3 - 1 = \int_0^T 6(2W_t+1)^2 dW_t + 12 \int_0^T (2W_t+1) dt. \end{align*} The remaining is obvious.