• DOI:10.1137/S0036142901389530
• Corpus ID: 10050445

Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations

@article{Higham2002StrongCO,
  title={Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations},
  author={Desmond J. Higham and Xuerong Mao and Andrew M. Stuart},
  journal={SIAM J. Numer. Anal.},
  year={2002},
  volume={40},
  pages={1041-1063}
}

• D. Higham, X. Mao, A. Stuart
• Published 1 March 2002
• Mathematics
• SIAM J. Numer. Anal.

Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler--Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p >2. As an application of this general theory we show that an implicit variant of Euler--Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler--Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.