First order strong approximations of scalar SDEs with values in a domain
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 analysing classical examples of SDEs with sublinear coefficients (CIR, CEV models and Wright-Fisher diffusion) and also with superlinear coefficients (3/2-volatility, Ait-Sahalia model). Our goal is to justify an efficient Multi-Level Monte Carlo (MLMC) method for a rich family of SDEs, which relies on good strong convergence properties.
References listed on IDEAS
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- Roger Lord & Remmert Koekkoek & Dick van Dijk, 2006.
"A Comparison of Biased Simulation Schemes for Stochastic Volatility Models,"
Tinbergen Institute Discussion Papers
06-046/4, Tinbergen Institute, revised 07 Jun 2007.
- Roger Lord & Remmert Koekkoek & Dick Van Dijk, 2010. "A comparison of biased simulation schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 177-194.
- Kristian Stegenborg Larsen & Michael Sørensen, 2007. "Diffusion Models For Exchange Rates In A Target Zone," Mathematical Finance, Wiley Blackwell, vol. 17(2), pages 285-306.
- Alfonsi Aurélien, 2005. "On the discretization schemes for the CIR (and Bessel squared) processes," Monte Carlo Methods and Applications, De Gruyter, vol. 11(4), pages 355-384, December.
- Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 513-536.
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