As homework for this week I've got the task to use Itô's formula to write stochastic processes as integrals with respect to W and t and I'm a bit overwhelmed at what to do exactly. ($W = W_t$ is a Brownian motion.)
E.g. I have got the following process:
$X_t = ln(sin(t-5W_t)+10)$
My idea would be to compute the (first (and second)) partial derivatives regarding t and W, here:
$ \frac{dX}{dW}= \frac{-5 cos(t-5W_t)}{sin(t-5W_t)+10} :=I$
$ \frac{d^2X}{dW^2}=\frac{-25 (sin(t-5W_t)+10)(sin(t-5W_t))+25(cos^2(t-5W_t))}{(sin(t-5W_t)+10)^2} := T$
$ \frac{dX}{dt}=\frac{cos(t-5W_t)}{sin(t-5W_t)+10}:=O$
and then by using the Itô formula
( $F(t,W_t) = F(0,W_0) + \int_0^t \frac{\partial}{\partial x} F(s,W_s) dW_s +\int_0^t \frac{\partial^2}{\partial x^2} F(s,W_s) ds + \int_0^t \frac{\partial}{\partial t} F(s,W_s) ds $ )
I would arrive at $X_t = ln(sin(0-5*0) +10) (=X(t=0,W_0=0)) + \int_0^t I dW_s +\int_0^t T ds + \int_0^t O ds $ ? Is this supposedly all I've to do? (please don't check the partial derivatives, this was just a proxy example.)
(The not so important part of my question would then be, what exactly am I doing here? I started with a stochastic process (X) and now I arrived at somethign that looks (to me) vastly different than what I've started with. If I would solve the integrals (which I am allowed to do? They are Riemann integrals, aren't they?) now, my equation would look very different than my initial equation).
To the first part: Yes, you are doing it correctly and it is also the only thing to do. To my modest opinion, it is an easy exercise to learn about applying ito's formula, which can be a bit confusing sometimes and you have to watch out to plug in $W_s$ and $s$ in your partial derivatives, so that it would be better to write
$X_t=X_0+\int_0^t I(s,W_s) dW_s + \int_0^t T(s,W_s) ds + \int_0^t O(s,W_s) ds,$
which leads to the second part of your question: You have just used some famous identity to write stochastic processes as integrals, which can be interpreted as somme type of stochastic Taylor-approximation (see the proof). The latter two integrals are still random processes, but also Riemann-Integrals, the first Integral is not a Riemann-Integral, but an Ito-Integral and, therefore, also a local Martingale.