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Numerical Solution of Free Stochastic Differential Equations

Georg Schluechtermann, Michael Wibmer

TL;DR

This work addresses numerically solving free stochastic differential equations (fSDEs) in non-commutative probability spaces by deriving a free Euler-Maruyama method (fEMM) through a free Itô formula built via multiple operator integrals. It proves strong convergence with order $p=\tfrac{1}{2}$ and weak convergence with order $p=1$, and shows that large-$N$ matrix implementations converge in distribution to the operator-valued solution, with the limits $N\to\infty$ and $\Delta t\to 0$ commuting. The authors provide a detailed numerical framework and validate the method on examples including the free Ornstein–Uhlenbeck process, a geometric-like free Brownian motion, and the free CIR process, reproducing known spectral properties and demonstrating applicability where no analytic solution is available. Overall, the paper delivers a first numerical methodology for fSDEs with rigorous convergence results and practical relevance for simulating spectral dynamics in large random matrices.

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 Itô formula from Taylor expansion of operator valued functions. Iterating the free Itô 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.

Numerical Solution of Free Stochastic Differential Equations

TL;DR

This work addresses numerically solving free stochastic differential equations (fSDEs) in non-commutative probability spaces by deriving a free Euler-Maruyama method (fEMM) through a free Itô formula built via multiple operator integrals. It proves strong convergence with order and weak convergence with order , and shows that large- matrix implementations converge in distribution to the operator-valued solution, with the limits and commuting. The authors provide a detailed numerical framework and validate the method on examples including the free Ornstein–Uhlenbeck process, a geometric-like free Brownian motion, and the free CIR process, reproducing known spectral properties and demonstrating applicability where no analytic solution is available. Overall, the paper delivers a first numerical methodology for fSDEs with rigorous convergence results and practical relevance for simulating spectral dynamics in large random matrices.

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 Itô formula from Taylor expansion of operator valued functions. Iterating the free Itô formula allows to motivate and define fEMM. Then we consider weak and strong convergence in the fSDE setting and prove strong convergence order of and weak convergence order of . Numerical examples support the theoretical results and show solutions for equations where no analytical solution is known.
Paper Structure (18 sections, 6 theorems, 67 equations, 8 figures)

This paper contains 18 sections, 6 theorems, 67 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Free Stochastic Calculus
  3. Free Probability Theory
  4. Free Brownian Motion
  5. Stochastic Integration with Respect to Free Brownian Motion
  6. Free Itô Formula and - Process
  7. Free Stochastic Differential Equation (fSDE)
  8. Free Itô-Formula in Functional Form
  9. Free analog of Euler-Maruyama Method (fEMM)
  10. Convergence Results
  11. Strong convergence of fEMM
  12. Weak convergence of fEMM
  13. Examples
  14. Free Ornstein-Uhlenbeck Equation
  15. Geometric Brownian Motion I
  16. ...and 3 more sections

Key Result

Theorem 4.1

Suppose $a, b, c$ are continuous functions $\mathcal{A}\rightarrow\mathcal{A}$ in the operator norm such that $a(\mathcal{A}^{sa})\subset\mathcal{A}^{sa}, b(\mathcal{A}^{sa})\subset\mathcal{A}^{sa}, c(\mathcal{A}^{sa})\subset\mathcal{A}^{sa}$. Furthermore $b,c$ are so that the product $b(X_t)dW_tc(X where the operators $L^0,L^1:\mathcal{A}^{sa}\rightarrow\mathcal{A}^{sa}$ are introduced as an abbr

Figures (8)

  • Figure 1: Diagram of the approximation scheme of fEMM. Implementation of \ref{['freeEM-Definition']} is realized in $\mathcal{M}_N^{sa}(\mathbb{R})$ (bottom right), which gives an approximation to the solution $X_t$ of \ref{['intro-freeSDE-diffform']} (top left). The limits $N\rightarrow\infty$ according to the size of random matrices and the step size limit $\Delta t$ do commute and give convergence of $\overline{X}_k^N$ in distribution to $X_t\in\mathcal{A}^{sa}$.
  • Figure 2: Distribution of the eigenvalues of the solution $\overline{X}_L^N$ at $T=1$ of \ref{['OU-equations-numerics']} for $\theta=\sigma=1$, $L=1024$ and $N=500$. The exact solution $X_T$ at $T=1$ is semicircle with $R\approx 3.575$.
  • Figure 3: Strong and weak convergence properties of fEMM applied to the free Ornstein-Uhlenbeck \ref{['OU-equations-numerics']} at $T=1$.
  • Figure 4: Spectral Distribution of $\overline{X}_t^N$ of \ref{['eq:geoI-numerics']} approximated by fEMM at different time points. The red line is the spectral distribution of the exact solution $X_t$ recovered from it's Cauchy transform.
  • Figure 5: Comparison of boundaries of support interval $[R1=I^1(t),R2=I^2(t)]$ of the density of spectral distribution of \ref{['eq:geoI-numerics']} for $\theta=1$, $N=100$.
  • ...and 3 more figures

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Remark 1
  • Definition 2.3
  • Remark 2
  • Definition 3.1
  • Remark 3
  • Remark 4
  • Theorem 4.1: Free Itô Formula in Integral Form
  • Remark 5
  • ...and 15 more