Let me set the stage;
Consider a stochastic PDE, which has to following form
$$\partial_t h(x,t) = H(x,t) + \chi(x,t),$$ where $H$ is a deterministic function, and $\chi(x,t)$ is a random variable.
In my case, the approximate solution of to this sPDE is known (via experimental and numerical simulations):
$$h(x,t) \approx G(x,t) + \epsilon(x,t),$$ where $\epsilon$ is a stochastic variable.
Of course, the solution of $h$ is not differentiable in the usual sense, but if the underlying distribution of $\epsilon$ is a symmetric distribution, such as a Gaussian one, if you observe $\partial_t h$ for long enough times, the deviations of $h$ from $G$ will be cancel out, so that you can experimentally or numerically determine $G$ quite accurately.
However, this requires us to know the distribution of the changes in the values $\epsilon$.
In this sense, this is "taking the derivative of" $\epsilon$.
A short analysis revealed me that, if the underlying probability distribution of $\epsilon$ is $g$ (let just assume that $\epsilon$ is a function of only t for the sake of the argument), then
the probability that $z-w$ change to occur is $g(z)g(w)$, because if the value of $\epsilon$ at the time $t$ is $z$, then at $t + dt$, the probability that $\epsilon (t+dt) = w$ is $g(w)$; therefore, considering that the probability that $\epsilon(t) = z$ in the first place is $g(z)$, so the probability that (Attention: abuse of notation) $d \epsilon = z-w$ is $g(w)g(z)$ (of course, this need some proper normalisation, but it is irrelevant to what I want to ask in here).
For example, if my analysis is correct, the "derivative" of a Gaussian random variable is still a Gaussian one.
Question:
Considering how "elementary" this idea made me wonder, is there a theory that captures the calculus on random variable in this sort ? I would also like to integrate random variables (though I haven't thought what that would mean physically or intuitively). I'm looking for references/papers dealing with this kind of theory; not just "taking derivative" of a random variable in a random sense, I need exactly above way of thinking in the theory.
I mean, I'm aware of the existence of Ito calculus, and Malliavin calculus, but whenever I tried to learn what is all about, or what is the underlying idea (such as what does taking derivative physically means in those theories) people would start to throw at me terminology that I'm not familiar. Don't get me wrong, I'm a also mathematics student, but in mathematics, doing theory without giving any motivation or the basic idea is so common, and I hate in such a way that I even don't read math books anymore.