I am trying to wrap my head around this idea and am trying to understand why this might be useful. For instance, suppose I have some stochastic process like the Brownian Motion. Why might I be interested in knowing "how quickly the Brownian Motion might change" at some point (i.e. the derivative) and the "area that the Brownian Motion might cover over two time intervals" (i.e. the integral)?
I understand this is more complicated than evaluating derivatives and integrals on deterministic functions. In a deterministic function, you only need to take one derivative and one integral to answer your question. On the other hand, each time I simulate a Stochastic Process on a computer, each realization of this Stochastic Process will look different. Thus, it is likely more challenging to take the integral and derivative of a function that can take many forms, compared to a function that can only take a single form.
But this point aside - why is it useful to know the derivative and the integral of a stochastic process? What do I gain from knowing this - how might I be able to apply this information to some real world application? For example, in financial models such as the Black-Scholes Model, are we using Stochastic Calculus to infer the amount of "risk" or "volatility" (i.e. the area under the stochastic process) that the stochastic process corresponds to over some period of time?