Stochastic differentiation

A simple introduction with Python.

In my previous article https://oscarnieves100.medium.com/the-building-blocks-of-stochastic-calculus-part-ii-f7ab2704a8b1 I discussed Poisson processes and their functions, as well as some examples of Brownian motion with jumps. With those tools at our disposal, we can now delve into the topic of stochastic differentiation.

Let us begin by mentioning one crucial property of stochastic processes: they are nowhere differentiable. Let us look at a simple random walk X(t) = σW(t) or Brownian motion:

Figure 1: Multiple realizations of X(t) = σW(t)

Does X(t) look differentiable anywhere? Let us recall that for a function f(t) to be differentiable somewhere it has to be smooth AND continuous. The ‘jaggedness’ of X(t) in this case suggests it is nowhere smooth, therefore nowhere differentiable, right? Not quite so. In fact, any function of W(t) can be thought of as a fractal, because zooming into it arbitrarily reveals non-smoothness throughout, and the same sort of jaggedness repeats itself over and over again no matter how much we zoom into it.

Enter Kiyosi Itô, a Japanese mathematician who is considered the father of stochastic calculus. He developed a branch of calculus known as Itô calculus, which explains how to compute derivatives and integrals of stochastic processes like W(t). By contrast, there is an equivalent (yet different)…