I have informally heard that a Stochastic Random Variable (e.g. Brownian Motion) is by definition not differentiable in the same way that a deterministic function is differentiable . As a result, a whole branch of mathematics called "Stochastic Calculus" was created to solve math problems involving derivatives/integrals of such Stochastic Random Variables.
I am interested in the underlying thought behind this idea : Even though this might sound obvious - why is a Stochastic Random Variable not differentiable using the standard notion of a derivative?
If I were to take a guess, I would guess that a Stochastic Random Variable is not-differentiable because it is not "sufficiently smooth" and contains too many "discontinuities". In a way, I am guessing that perhaps a Stochastic Random Variable inherits many of the same obstacles that the Weierstrass Function that prevents it from being differentiable in the standard way. Apart from , the other idea that I had as to why a Stochastic Random Variable may not be differentiable in the standard way is due to its stochastic nature - unlike a deterministic function that has a "static form", a Stochastic Random Variable could have "infinite forms", thus making the differentiation of a function with Stochastic Random Variables ambiguous .
- Can someone please provide some thoughts on this - mathematically, can we show that a Stochastic Random Variable has certain mathematical properties that renders it non-differentiable using standard calculus?
Thank you!
The question of "defining derivatives" of stochastic processes is indeed never abandoned in the theory of stochastic processes. However, the scope and the abstractness of this problem are subtler than your question.
For your question, you can look at the following property of Brownian motions:
Proposition: For any $\gamma>1/2$, (almost surely) a path of Brownian motion is nowhere $\gamma$-Holder continuous.