Almost sure stability of the Euler-Maruyama method with random variable stepsize for stochastic differential equations

Wei Liu, Xuerong Mao

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In this paper, the Euler–Maruyama (EM) method with random variable stepsize is studied to reproduce the almost sure stability of the true solutions of stochastic differential equations. Since the choice of the time step is based on the current state of the solution, the time variable is proved to be a stopping time. Then the semimartingale convergence theory is employed to obtain the almost sure stability of the random variable stepsize EM solution. To our best knowledge, this is the first paper to apply the random variable stepsize (with clear proof of the stopping time) to the analysis of the almost sure stability of the EM method.
Original languageEnglish
Pages (from-to)1-20
Number of pages20
JournalNumerical Algorithms
Early online date9 Jun 2016
Publication statusE-pub ahead of print - 9 Jun 2016


  • stopping time
  • almost sure stability
  • Euler-Maruyama
  • variable stepsize
  • semi-martingale convergence theory

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