Solving stochastic differential equations with Cartan's exterior differential systems

TL;DR

This paper introduces an algebraic-geometric method based on Cartan's exterior differential systems to solve certain stochastic differential equations by exploiting their symmetries, with applications in quantum mechanics and finance.

Contribution

It systematically applies Cartan's method of isovectors to solve stochastic differential equations, connecting geometric symmetry methods with stochastic analysis.

Findings

Successfully solves specific SDEs using symmetry methods

Demonstrates the approach on quantum-inspired and financial models

Provides a new geometric perspective on stochastic equations

Abstract

The aim of this work is to use systematically the symmetries of the (one dimensional) bacward heat equation with potentiel in order to solve certain one dimensional It\^o's stochastic differential equations. The special form of the drift (suggested by quantum mechanical considerations) gives, indeed, access to an algebrico-geometric method due, in essence, to E.Cartan, and called the Method of Isovectors. A V singular at the origin, as well as a one-factor affine model relevant to stochastic finance, are considered as illustrations of the method.