Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes

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• 作者：Wolf-Jürgen Beyn ; Elena Isaak ; Raphael Kruse
• 刊名：Journal of Scientific Computing
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：67
• 期：3
• 页码：955-987
• 全文大小：807 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691
• 卷排序：67

文摘

This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion coefficient functions such as the stochastic Ginzburg–Landau equation and the 3/2-volatility model from mathematical finance. Our analysis of the mean-square error of convergence is based on a suitable generalization of the notions of C-stability and B-consistency known from deterministic numerical analysis for stiff ordinary differential equations. An important feature of our stability concept is that it does not rely on the availability of higher moment bounds of the numerical one-step scheme. While the convergence theorem is derived in a somewhat more abstract framework, this paper also contains two more concrete examples of stochastically C-stable numerical one-step schemes: the split-step backward Euler method from Higham et al. (SIAM J Numer Anal 40(3):1041–1063, 2002) and a newly proposed explicit variant of the Euler–Maruyama scheme, the so called projected Euler–Maruyama method. For both methods the optimal rate of strong convergence is proven theoretically and verified in a series of numerical experiments.KeywordsStochastic differential equationsGlobal monotonicity conditionSplit-step backward EulerProjected Euler–MaruyamaStrong convergence ratesC-stabilityB-consistency