Numerical Solution of Free Stochastic Differential Equations

TL;DR

This paper introduces a numerical method called free Euler-Maruyama for solving free stochastic differential equations, extending classical stochastic calculus to non-commutative variables like large random matrices.

Contribution

It develops a free analog of the Euler-Maruyama method using free Itô calculus, providing convergence proofs and numerical validation for non-commutative stochastic equations.

Findings

Established strong convergence order of 1/2

Established weak convergence order of 1

Numerical solutions for equations without known analytical solutions

Abstract

This paper derives a free analog of the Euler-Maruyama method (fEMM) to numerically approximate solutions of free stochastic differential equations (fSDEs). Simply speaking fSDEs are stochastic differential equations in the context of non-commutative random variables (e.g. large random matrices). By applying the theory of multiple operator integrals we derive a free It\^{o} formula from Taylor expansion of operator valued functions. Iterating the free It\^{o} formula allows to motivate and define fEMM. Then we consider weak and strong convergence in the fSDE setting and prove strong convergence order of $\frac{1}{2}$ and weak convergence order of ${1}$. Numerical examples support the theoretical results and show solutions for equations where no analytical solution is known.