Divergence of the backward Euler method for ordinary stochastic differential equations

Abstract

This paper is based on the analysis of the backward Euler method for stochastic differential equations. It is motivated by the paper (Hutzenthaler et al. Proc. R. Soc. A 467, 1563–1576, 2011), where authors studied the equations with superlinearly growing coefficients. The main goal of this paper is to reveal sufficient conditions of the strong and weak Lp-divergence of the backward Euler method at finite time, for all \(p\in (0,\infty )\). Theoretical results are supported by examples.

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Acknowledgements

The author is very thankful to the reviewers for their valuable suggestions which improved the paper.

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Correspondence to Marija Milošević.

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The author’s research is supported by Grant No. 174007 of the Ministry of Science, Republic of Serbia.

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Milošević, M. Divergence of the backward Euler method for ordinary stochastic differential equations. Numer Algor 82, 1395–1407 (2019). https://doi.org/10.1007/s11075-019-00661-6

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Keywords

  • Ordinary stochastic differential equations
  • Backward Euler method
  • Strong L p-divergence
  • Super-linear growth conditions
  • One-sided Lipschitz condition

Mathematics Subject Classification (2010)

  • 60H10