I'm currently studying Gaussian processes (in a physical manner - so maths for dummies) and have a few questions, because i lack the overview of how and what happens. I'll order my questions, i hope they can be answered!
1) I read about Ito Integrals and Ito's lemma. The definition of the integral seems very 'ad hoc' to me. Is it the general integral one uses for stochastic? For me, it does seem strange to define the integral that way.
2) I want to calculate $Y_t=\exp\left(\int_0^tX_t\mathrm{d}t\right)$ where $X_t$ is a gaussian process. Now my question: should i use ito's lemma on it, getting a differential equation for $Y_t$, should i calculate the moments explicitely (without knowing how many i need), should i assume that the resulting variable is a gaussian process again, needing only the expectation value and the covariance at different times? how do i, in general, approach such a problem? I also heard i could use the cummulants, but are they even properly defined for a gaussian process and different times?
Thanks already for your answers and thoughts!
Martin