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arXiv · 2210.00348

Numerical Solution of Free Stochastic Differential Equations

Georg Schluechtermann·Michael Wibmer
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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.

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BibTeXRIS

Georg Schluechtermann, Michael Wibmer. 2022-10-01. Numerical Solution of Free Stochastic Differential Equations. https://doi.org/10.1137/22m1526393

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