The intrinsic "sense" of stochastic differential equations

TL;DR

The paper investigates the intrinsic interpretation of stochastic differential equations, demonstrating how different integration senses affect solutions and establishing the Stratonovich sense as a natural choice through variable transformations.

Contribution

It reveals the paradoxical dependence of solutions on the integration sense and identifies the Stratonovich sense as intrinsic using specific variable changes.

Findings

Changing the integration sense alters the number of solutions.

A specific transformation neutralizes the sense, leading to a constant coupling with noise.

The Stratonovich sense is singled out as intrinsic via the Ito formula.

Abstract

A free choice of the integration sense would lead to the paradox that the number of possible equations (thus of solutions for a given model) can vary under a mere change of the variables. This is shown by a specific change which neutralizes the sense (by establishing a constant coupling with the noise). Its inverse singles out the Stratonovich sense, by means of the Ito formula.