Stochastic Calculus via Stopping Derivatives
Alex Simpson
TL;DR
This work develops stochastic calculus from stopping derivatives, defining drift and variance rate as right derivatives of conditional expectation and conditional variance with respect to stopping times. It establishes a complete derivative-based calculus, including a multi-dimensional Itô formula, and reveals a deep link to classical stochastic calculus by showing that continuous zero-drift processes are random translations of continuous local martingales. A Fundamental Theorem of Calculus for stopping derivatives connects the drift and diffusion parameters to stochastic-integral representations, enabling construction and uniqueness results for Itô-type processes via stochastic integration. The framework localises to individual stopping times, bypasses integrability barriers, and recovers standard objects such as quadratic variation and Brownian motion within a unified, derivative-driven view with broad implications for diffusion theory and potential extensions to generalized generators and diffusion-characteristics.
Abstract
We show that a substantial portion of stochastic calculus can be developed along similar lines to ordinary calculus, with derivative-based concepts driving the development. We define a notion of stopping derivative, which is a form of right derivative with respect to stopping times. Using this, we define the drift and variance rate of a process as stopping derivatives for (generalised) conditional expectation and conditional variance respectively. Applying elementary, derivative-based methods, we derive a calculus of rules describing how drift and variance rate transform under constructions on processes, culminating in a version of the multi-dimensional Itô formula. Our approach connects with the standard machinery of stochastic calculus via a theorem establishing that continuous processes with zero drift coincide with random translations of continuous local martingales. This equivalence allows us to derive a Fundamental Theorem of Calculus for stopping derivatives, which relates the quantities of drift and variance rate, defined as stopping derivatives, to parameters used in the description of a process as a stochastic integral.