Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients

Abstract

We study the approximation of stochastic differential equations on domains. For this, we introduce modified Itô–Taylor schemes, which preserve approximately the boundary domain of the equation under consideration. Assuming the existence of a unique non-exploding solution, we show that the modified Itô–Taylor scheme of order γ has pathwise convergence order γε for arbitrary ε > 0 as long as the coefficients of the equation are sufficiently differentiable. In particular, no global Lipschitz conditions for the coefficients and their derivatives are required. This applies for example to the so called square root diffusions.

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Correspondence to A. Neuenkirch.

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Partially supported by the DFG-project “Pathwise numerics and dynamics of stochastic evolution equations”.

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Jentzen, A., Kloeden, P.E. & Neuenkirch, A. Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients. Numer. Math. 112, 41–64 (2009). https://doi.org/10.1007/s00211-008-0200-8

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Mathematics Subject Classification (2000)

  • 65C30
  • 60H35