A Differentiation Theory for It\^o's Calculus

TL;DR

This paper introduces a pathwise stochastic derivative for semimartingales with respect to Brownian motion, establishing a differentiation theory that parallels elementary calculus and enhances stochastic calculus with new fundamental theorems.

Contribution

It defines a novel stochastic derivative based on quadratic covariation, leading to a fundamental theorem, chain rule, and mean value theorem in stochastic calculus.

Findings

Derives a stochastic chain rule including convex functions

Establishes stochastic mean value and Rolle's theorems

Provides a natural calculus for semimartingales

Abstract

A peculiar feature of It\^o's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to It\^o's integral calculus? From It\^o's definition of his integral, such a derivative must be based on the quadratic covariation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle's theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications some of which we address in an upcoming article.