Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations

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Abstract

Influenced by Higham, Mao and Stuart [9], several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations (SDEs) under the local Lipschitz condition. These numerical methods include the tamed Euler–Maruyama (EM) method, the tamed Milstein method, the stopped EM, the backward EM, the backward forward EM, etc. Recently, we developed a new explicit method in [23], called the truncated EM method, for the nonlinear SDE dx(t) = f (x(t))dt + g(x(t))dB(t) and established the strong convergence theory under the local Lip- schitz condition plus the Khasminskii-type condition xT f (x) + p−1 |g(x)|2 ≤ K(1 + |x|2). However, due to the page limit there, we did not study the convergence rates for the method, which is the aim of this paper. We will, under some additional conditions, discuss the rates of Lq -convergence of the truncated EM method for 2 ≤ q < p and show that the order of Lq -convergence can be arbitrarily close to q/2.
Original languageEnglish
Pages (from-to)362-375
Number of pages14
JournalJournal of Computational and Applied Mathematics
Volume296
Early online date13 Oct 2015
Publication statusPublished - Apr 2016

Keywords

  • stochastic differential equation
  • local Lipschitz condition
  • Khasminskii-type condition
  • truncated Euler-Maruyama method
  • convergence rate

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