I am having the following problem:
Consider the following stratonovich equation:
$dY_t=-\sin(Y_t)\cdot dt+s\cos(Y_t)\circ dB_t$
where $B_t$ is a regular brownian motion.
Then define the process $Z_t=\sin(Y_t)$.
I would now like to find the Ito-equation and the Stratonovich equation, that $Z_t$ satisfies.
My approach has been straighforward:
first setting $dZ_t=\sin'(Y_t)\cdot dY_t=\cos(Y_t)\cdot dY_t$.
Then, I use various rules from analysis to get
$dZ_t=\cos(Y_t)\cdot dY_t$
$dZ_t=\cos(\arcsin(Z_t))\cdot (-\sin(Y_t)\cdot dt+s\cos(Y_t)\circ dB_t)$
which after calculations give:
$dZ_t=-Z_t\cdot \sqrt{(1-Z_t^2)}\cdot dt+s(1-Z_t^2)\circ dB_t$.
Is that then the stratonovich equation that satisfies $Z_t$, or should I do something else?
Also, how can I from this find the Ito equation, which satisfies $Z_t$?
Thanks in advance.