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August 2018 Strong convergence of the symmetrized Milstein scheme for some CEV-like SDEs
Mireille Bossy, Héctor Olivero
Bernoulli 24(3): 1995-2042 (August 2018). DOI: 10.3150/16-BEJ918


In this paper, we study the rate of convergence of a symmetrized version of the Milstein scheme applied to the solution of the one dimensional SDE \[X_{t}=x_{0}+\int_{0}^{t}{b(X_{s})\,ds}+\int_{0}^{t}{\sigma\vert X_{s}\vert^{\alpha}\,dW_{s}},\qquad x_{0}>0,\sigma>0,\alpha\in[\frac{1}{2},1).\] Assuming $b(0)/\sigma^{2}$ big enough, and $b$ smooth, we prove a strong rate of convergence of order one, recovering the classical result of Milstein for SDEs with smooth diffusion coefficient. In contrast with other recent results, our proof does not relies on Lamperti transformation, and it can be applied to a wide class of drift functions. On the downside, our hypothesis on the critical parameter value $b(0)/\sigma^{2}$ is more restrictive than others available in the literature. Some numerical experiments and comparison with various other schemes complement our theoretical analysis that also applies for the simple projected Milstein scheme with same convergence rate.


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Mireille Bossy. Héctor Olivero. "Strong convergence of the symmetrized Milstein scheme for some CEV-like SDEs." Bernoulli 24 (3) 1995 - 2042, August 2018.


Received: 1 August 2015; Revised: 1 August 2016; Published: August 2018
First available in Project Euclid: 2 February 2018

zbMATH: 06839258
MathSciNet: MR3757521
Digital Object Identifier: 10.3150/16-BEJ918

Keywords: CEV models , CIR model , Milstein scheme , Multilevel Monte Carlo , non-Lipschitz diffusion coefficient , Stochastic differential equations , strong error analysis

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 3 • August 2018
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