论文标题
游离随机微分方程的数值解
Numerical Solution of Free Stochastic Differential Equations
论文作者
论文摘要
本文推导了Euler-Maruyama方法（FEMM）的自由类似物，以在数值上近似于自随机微分方程（FSDES）的近似于解决方案。简单地说FSD是在非交换随机变量（例如大型随机矩阵）中的随机微分方程。通过应用多个运算符积分的理论，我们从泰勒的泰勒扩展中得出了一个自由的iTôula公式。迭代自由itô奶酪可以激励和定义女性。然后，我们考虑在FSDE设置中弱和强收敛，并证明$ \ frac {1} {2} $的强收敛顺序和$ {1} $的弱收敛顺序。数值示例支持理论结果，并显示了不知道分析解决方案的方程解决方案。
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.
