Consider a stochastic differential equation
$$_ = _ + _(t), $$
where {W(t)} is a Wiener process and $, > 0$.
You are given also a process
$$_ = ^t(S_t)^b, a>0$$
Find a, if we know that $ = 0.3, = 3\%, = 0.73.\\$
My reasoning:
We have $V_t = f(t,S_t)$, so after using Ito's lemma we get:
$$dV_t=\underbrace{(lna+rb+\frac{\sigma^2b(b-1)}{2})}_{A}V_tdt+\sigma bV_tdW_t$$
Furthermore we know:
$$S_t=S_0e^{(r-\frac{\sigma^2}{2})t+\sigma W_t}$$
$$V_t=V_0e^{(A-\frac{(\sigma b)^2}{2})t+\sigma b W_t}=a^0(S_0)^be^{(A-\frac{(\sigma b)^2}{2})t+\sigma b W_t}$$
$$V_t=^t (S_t)^b=^t(S_0e^{(r-\frac{\sigma^2}{2})t+\sigma W_t})^b$$
and unfortunately it goes nowhere. We get $0=0$.
Maybe you guys have some ideas. I would appreciate.