Stochastic Calculus via Stopping Derivatives

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 operations on processes, culminating in a version of the multi-dimensional It\^o 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.