We are interested in strong approximations of one-dimensional SDEs which have non-Lipschitz coefficients and which take values in a domain. Under a set of general assumptions we derive an implicit scheme that preserves the domain of the SDEs and is strongly convergent with rate one. Moreover, we show that this general result can be applied to many SDEs we encounter in mathematical finance and bio-mathematics. We will demonstrate flexibility of our approach by analyzing classical examples of SDEs with sublinear coefficients (CIR, CEV models and Wright–Fisher diffusion) and also with superlinear coefficients (3/2-volatility, Aït-Sahalia model). Our goal is to justify an efficient Multilevel Monte Carlo method for a rich family of SDEs, which relies on good strong convergence properties.
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We would like to thank the referees for their valuable and insightful comments and remarks. Moreover, we would like to thank Martin Altmayer for helpful comments on an earlier version of the manuscript.
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Neuenkirch, A., Szpruch, L. First order strong approximations of scalar SDEs defined in a domain. Numer. Math. 128, 103–136 (2014). https://doi.org/10.1007/s00211-014-0606-4
Mathematics Subject Classification (2000)