Stochastic theta methods for free stochastic differential equations

TL;DR

This work develops free stochastic theta methods (free STMs) for numerically solving free SDEs within the framework of free probability. Under operator Lipschitz assumptions on the drift and diffusion coefficients, the free STMs achieve strong convergence of order $1/2$ and exhibit exponential mean-square stability depending on the theta parameter, with θ in $[1/2,1]$ guaranteeing unconditional stability. The authors prove uniform boundedness of the numerical solution, provide a rigorous convergence analysis, and demonstrate improved stability relative to the free Euler–Maruyama method, including a backward-Euler-like variant (θ=1). Numerical experiments on free OU, free GBM I/II, free CIR, and a nonlinear drift example corroborate the theory and illustrate practical performance and spectral behavior. Overall, the results establish a robust, implicit-like numerical framework for free SDEs and open avenues for further structure-preserving and Milstein-type methods in the non-commutative setting.

Abstract

We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations in this work. Assume that the drift coefficient of the free stochastic differential equations is operator Lipschitz and the diffusion coefficients are locally operator Lipschitz, we prove the strong convergence of the numerical methods. Moreover, we investigate the exponential stability in mean square of the equations and the numerical methods. In particular, the free stochastic theta methods with $θ\in [1/2, 1]$ can inherit the exponential stability of original equations for any given step size. Our methods offer better stability than the free Euler-Maruyama method. Numerical results are reported to confirm these theoretical findings and show the efficiency of our methods compared with the free Euler-Maruyama method.

Figures (16)

• Figure 1: The spectrum distribution of the numerical solution $\bar{U}_P^{(N)}$ of \ref{['ou']} generated by free BEM with $\mu=-2$ and $\sigma=1$.
• Figure 2: Strong convergence order simulations for the free STMs applied to \ref{['ou']} with $\mu=-2$ and $\sigma=1$.
• Figure 3: The exponential stability in mean square of free STMs applied to \ref{['ou']}.
• Figure 4: The spectrum distribution of the numerical solution $\bar{U}_P^{(N)}$ of \ref{['geo1']} generated by free BEM with $\mu=-1$.
• Figure 5: Comparison of boundaries of support interval $[I1=R_1(t),I2=R_2(t)]$ of the density of spectrum distribution of \ref{['geo1']} with $\mu=-1, h =2^{-10},$ and $N=500$.

Theorems & Definitions (35)

• Definition 1: Free stochastic integral bianeStochasticCalculusRespect1998karginFreeStochasticDifferential2011
• Definition 2
• Remark 1
• Theorem 1: Theorem 3.1, karginFreeStochasticDifferential2011
• Definition 3
• Remark 2
• Definition 4: AP2016schluechtermannNumericalSolutionFree2022a
• Lemma 2
• proof
• Proposition 1