I am studying stochastic process and of course Brownian Motion takes a major role. From the beginning it is proved that its paths are almost everywhere non-differentiable and because of that theories of stochastic integrals, such as Itô integrals, are introduced in order to work with Stochastic differential Equations.
The issue of non-differentiability of the Brownian Motion is not a topological one, that is, if we consider some weaker notion of convergence between random variables, still the BM will not be differentiable (in the sense of the usual quotient limit). However, weakening the topology in order to study differentiability does not seem a bad idea to me at first, at least when we are not strictly interested in martingale processes.
Hence, my question is, does someone work with the notion of a derivative for a stochastic process in the "distribution sense"? That is, is the following definition
A stochastic process $X_t$ is differentiable at $t$ in the sense of distribution if there exists a random variable $Y$ such that $$ \frac{X_{t+h} - X_t}{h} \to Y, $$ where the convergence is understood in the distribution sense.
studied? If the answer is yes, can someone provide me a reference and an interpretation/applications of such notion? I am really glad!