Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion

By Andreas Neuenkirch, Ivan Nourdin. 2006

113 citations recorded by the source corpus.

Abstract In this article, 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 J. Complex. 22(4), 459–474, 2006 and C.R. Acad. Sci. Paris, Ser. I 340(8), 611–614, 2005. 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.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/8683efe798d622676e434e22735933c2bd32ef90

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Citations in the corpus

A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion (2010, background): “A detailed analysis of the one-dimensional case is given in [28], where the exact convergence rate n and the asymptotic error distribution are derived.”
Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case H=1/4 (2008, background): “More precisely: f ′′ ( B H s ) ds ; (1.5) for W a standard Brownian motion independent of B H ; • if H = for W a standard Brownian motion independent of B 3 / 4 ; • if H > then for Z the Rosenblatt process deﬁned by where I X 2 denotes the double stochastic integral with respect to the Wiener…”
Existence and Uniqueness of the Solution of Stochastic Differential Equation Involving Wiener Process and Fractional Brownian Motion with Hurst Index H > 1/2 (2011, unknown): “Numerical methods for pathwise stochastic differential equations with fBm were considered in [11, 12, 2, 4].”
Controlled differential equations as Young integrals: A simple approach (2010, background): “The properties of such equations are studied for example in [1, 5, 6, 8, 14, 15, 25–30, 32, 35].”
Stein's method meets Malliavin calculus: a short survey with new estimates (2009, background): “- Reference [2], by Bercu, Nourdin and Taqqu, contains some applications of the results of [18] to almost sure CLTs. -”
The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion (2007, background): “[9] it is proved that for an equation with homogeneous regular coefficients the error jXt 2 Yt jd122H almost surely converges to some stochastic process jt, which means that the estimate of the rate of convergence in Theorem 3.”
Central Limit Theorems for Multiple Skorokhod Integrals (2010, result): “Recent examples of application of these results are, among others, the study of p-variations of fractional stochastic integrals (Corcuera et al. [4]), quadratic functionals of bivariate Gaussian processes (Deheuvels et al. [6]), self-intersection local times of fractional Brownian motion (Hu and Nua”
Stein’s method on Wiener chaos (2007, background): “Remark 3.4 Theorem 3.3 and its generalizations have been applied to a variety of frameworks, such as: p-variations of stochastic integrals with respect to Gaussian processes [2, 12], quadratic functionals of bivariate Gaussian processes [14], self-intersection local times of fractional Brownian moti”
Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions (2014, methodology): “This has been proved in the one-dimensional case by Nourdin and Neuenkirch in [18] using the Doss representation of the solution; see also Theorem 10.1 below.”
Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion (2007, background): “1) is motivated, for instance, by the study of the exact rates of convergence of some approximation schemes of scalar stochastic differential equations driven by B (see [5, 10] and [11]), besides, of course, the traditional applications of quadratic variations to parameter estimation problems.”