I understand that result should be 0, but I don't understand how to get to it
Thanks a lot in advance
(ADDinfo)
I am actually trying to prove that the following process can be written as stochastic integrals with respect to B:
The solutions then give the answer:
And so however I try, I cannot understand how the equation for Yt with no dt arises
If we write in shorthand $$ dX_t =- a^2/2\cos^2(at)dt -a \cos(at)dB_t $$ and we define $$ Y_t = f(X_t) = \exp(X_t), $$ then we can use Ito's Lemma to get \begin{align*} dY_t &= \left(- a^2/2\cos^2(at) -a^2/2 \cos^2(at)\right)Y_t d_t -a \cos(at)Y_tdB_t\\ &= - a^2\cos^2(at) Y_t d_t -a \cos(at)Y_tdB_t \end{align*} because
$\frac{\partial f}{\partial t} = 0$
$\frac{\partial f}{\partial x} = \exp(X_t) = Y_t$
$\frac{\partial^2 f}{\partial x^2} = Y_t$.
Hopefully this is correct. I'm not totally sure what people do when they divide through by $dt$. It must be that $dB_t/dt$ has some meaning I don't know about.