论文标题
使用Karhunen-Loève扩展和矩阵函数对随机微分方程进行采样
Sampling of Stochastic Differential Equations using the Karhunen-Loève Expansion and Matrix Functions
论文作者
论文摘要
我们考虑使用Karhunen-Loève扩展的随机微分方程的线性化。我们通过使用矩阵函数理论将扩展作为一系列矩阵矢量产物来获得线性化。此外，我们将解作为线性微分方程系统的解。我们在截断过程中获得了强大而弱的错误界限，并表明，在适当的条件下，均方根错误的顺序$ \ Mathcal {o}（\ frac {1} {M}）$，第二瞬间的收敛顺序较弱。我们还讨论了有效的数值线性代数技术，以近似矩阵函数和微分方程的线性化系统。与标准方法（例如Euler-Maruyama方案）相比，显示了我们算法的有效性的实验支持这些理论结果。
We consider linearizations of stochastic differential equations with additive noise using the Karhunen-Loève expansion. We obtain our linearizations by truncating the expansion and writing the solution as a series of matrix-vector products using the theory of matrix functions. Moreover, we restate the solution as the solution of a system of linear differential equations. We obtain strong and weak error bounds for the truncation procedure and show that, under suitable conditions, the mean square error has order of convergence $\mathcal{O}(\frac{1}{m})$ and the second moment has a weak order of convergence $\mathcal{O}(\frac{1}{m})$, where $m$ denotes the size of the expansion. We also discuss efficient numerical linear algebraic techniques to approximate the series of matrix functions and the linearized system of differential equations. These theoretical results are supported by experiments showing the effectiveness of our algorithms when compared to standard methods such as the Euler-Maruyama scheme.
