NUMERICAL SOLUTION OF FREE STOCHASTIC DIFFERENTIAL EQUATIONS.

This paper derives a free analogue of the Euler-Maruyama method (fEMM) to numerically approximate solutions of free stochastic differential equations (fSDEs). Simply speaking, fSDEs are SDEs in the context of noncommutative 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 us to motivate and define fEMM. Then we consider weak and strong convergence in the fSDE setting and prove strong convergence order of 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. [ABSTRACT FROM AUTHOR]