[Paper Review] Solving stochastic differential equations with Cartan's exterior differential system

Paul Lescot, Hélène Quintard+1 more|arXiv (Cornell University)|Jan 1, 2015

Stochastic processes and financial applications14 references3 citations

TL;DR

This paper proposes a novel method for solving one-dimensional Itô stochastic differential equations by leveraging symmetries of the backward heat equation with potential, using Cartan's exterior differential system and the method of isovectors. The approach enables exact solutions for specific models, including a singular drift at the origin and a one-factor affine model in stochastic finance, demonstrating the power of algebro-geometric techniques in stochastic analysis.

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.

Motivation & Objective

• To apply Cartan's exterior differential system to solve stochastic differential equations arising in mathematical finance and quantum mechanics.
• To exploit symmetries of the backward heat equation with potential to derive exact solutions for specific SDEs.
• To demonstrate the effectiveness of the method of isovectors in solving one-dimensional Itô SDEs with non-trivial drift structures.
• To provide geometric and algebraic solutions for a singular drift model and a one-factor affine interest rate model.

Proposed method

• The method uses Cartan's exterior differential system to analyze the symmetry structure of the backward heat equation with potential.
• It applies the method of isovectors—infinitesimal symmetries of the equation—to reduce the complexity of the stochastic differential equation.
• The approach relies on identifying a Lie algebra of symmetries associated with the drift term, which is constrained by quantum mechanical considerations.
• The symmetry reduction leads to a system of ordinary differential equations that can be solved explicitly.
• The solution is then mapped back to the original stochastic differential equation to yield explicit expressions for the transition densities.
• The method is applied to two illustrative cases: a V-shaped drift singular at the origin and a one-factor affine model in finance.

Experimental results

Research questions

• RQ1 How can symmetries of the backward heat equation with potential be used to solve one-dimensional Itô stochastic differential equations?
• RQ2 What is the role of Cartan's exterior differential system in constructing exact solutions for SDEs with non-linear drift?
• RQ3 Can the method of isovectors provide closed-form solutions for SDEs with singular or affine drift structures?
• RQ4 How does the geometric approach via exterior calculus compare to standard analytical methods in stochastic calculus?
• RQ5 What are the implications of this method for modeling in stochastic finance and quantum mechanics?

Key findings

• The method successfully yields exact solutions for a stochastic differential equation with a drift singular at the origin, leveraging the underlying symmetry algebra.
• The one-factor affine model in stochastic finance is solved exactly using the same symmetry-based approach, confirming its compatibility with the framework.
• The use of Cartan's exterior differential system enables a systematic reduction of the SDE to solvable ordinary differential equations through symmetry analysis.
• The method of isovectors provides a powerful geometric alternative to traditional PDE-solving techniques in stochastic processes.
• The results demonstrate that algebro-geometric methods can yield explicit solutions where standard methods may fail or require strong assumptions.