We analyse the strong approximation of the Cox-Ingersoll-Ross (CIR) process in the regime where the process does not hit zero by a positivity preserving drift-implicit Eulertype method. As an error criterion, we use the pth mean of the maximum distance between the CIR process and its approximation on a finite time interval. We show that under mild assumptions on the parameters of the CIR process, the proposed method attains, up to a logarithmic term, the convergence of order 1/2. This agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations with Lipschitz coefficients, despite the fact that the CIR process has a non-Lipschitz diffusion coefficient.

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.L p -error criterion. This strategy was also suggested by Alfonsi in [1]. Here, we extend that work in several ways:• Considering the maximum error in the discretization points, we prove that the drift-implicit Euler-Maruyama scheme for the CIR process strongly converges with rate one under slightly more restrictive conditions on the parameters of the process than in [6].• We provide a general framework for the strong order one convergence of the BEM scheme for SDEs with constant diffusion and one-sided Lipschitz drift coefficients.• Using this framework we present a detailed convergence analysis for several SDEs with sub-and super-linear coefficients.• We also show that BEM for the transformed SDE is closely related to a drift-implicit Milstein scheme for the original SDE, which has been introduced in [13]. In the case of the CIR process we provide a sharp error analysis for this scheme.

In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Lévy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Lévy area terms.

The authors of this paper study approximation methods for stochastic differential equations, and point out a simple relation between the order of convergence in the pth mean and the order of convergence in the pathwise sense: Convergence in the pth mean of order α for all p ≥ 1 implies pathwise convergence of order α – ε for arbitrary ε > 0. The authors then apply this result to several one-step and multi-step approximation schemes for stochastic differential equations and stochastic delay differential equations. In addition, they give some numerical examples.

In this paper, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H. We consider two cases. If H > 1/2, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which in particular depends on the Malliavin derivative of the solution. This result extends those contained in [17] and [18]. When 1/6 < H < 1/2, the exact rate of convergence of the Crank-Nicholson scheme is determined for a particular equation. Here we show convergence in law of the error to a random variable, which depends on the solution of the equation and an independent Gaussian random variable.