On the pathwise approximation of stochastic differential equations

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• 作者：Tony Shardlow ; Phillip Taylor
• 关键词：Stochastic differential equations ; Numerical methods ; Rough path theory
• 刊名：BIT Numerical Mathematics
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：56
• 期：3
• 页码：1101-1129
• 全文大小：1,243 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Computational Mathematics and Numerical Analysis
  Numeric Computing
  Mathematics
  
• 出版者：Springer Netherlands
• ISSN：1572-9125
• 卷排序：56

文摘

We consider one-step methods for integrating stochastic differential equations and prove pathwise convergence using ideas from rough path theory. In contrast to alternative theories of pathwise convergence, no knowledge is required of convergence in pth mean and the analysis starts from a pathwise bound on the sum of the truncation errors. We show how the theory is applied to the Euler–Maruyama method with fixed and adaptive time-stepping strategies. The assumption on the truncation errors suggests an error-control strategy and we implement this as an adaptive time-stepping Euler–Maruyama method using bounded diffusions. We prove the adaptive method converges and show some computational experiments.