Nonparametric inference for fractional diffusion

A non-parametric diffusion model with an additive fractional Brownian motion noise is considered in this work. The drift is a non-parametric function that will be estimated by two methods. On one hand, we propose a locally linear estimator based on the local approximation of the drift by a linear function. On the other hand, a Nadaraya-Watson kernel type estimator is studied. In both cases, some non-asymptotic results are proposed by means of deviation probability bound. The consistency property of the estimators are obtained under a one sided dissipative Lipschitz condition on the drift that insures the ergodic property for the stochastic differential equation. Our estimators are first constructed under continuous observations. The drift function is then estimated with discrete time observations that is of the most importance for practical applications.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ509 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion

International audienceWe prove the Malliavin regularity of the solution of a stochastic differential equation driven by a fractional Brownian motion of Hurst parameter H > 0:5. The result is based on the Fréchet differentiability with respect to the input function for deterministic differential equations driven by Hölder continuous functions. It is also shown that the law of the solution has a density with respect to the Lebesgue measure, under a suitable nondegeneracy conditio

Time reversal for drifted fractional Brownian motion with Hurst index H>1/2

International audienceLet X be a drifted fractional Brownian motion with Hurst index H > 1/2. We prove that there exists a fractional backward representation of X, i.e. the time reversed process is a drifted fractional Brownian motion, which continuously extends the one obtained in the theory of time reversal of Brownian diffusions when H = 1/2. We then apply our result to stochastic differential equations driven by a fractional Brownian motion

Diagnostic checking in FARIMA models with uncorrelated but non-independent error terms

This work considers the problem of modified portmanteau tests for testing the adequacy of FARIMA models under the assumption that the errors are uncorrelated but not necessarily independent (i.e. weak FARIMA). We first study the joint distribution of the least squares estimator and the noise empirical autocovariances. We then derive the asymp-totic distribution of residual empirical autocovariances and autocorrelations. We deduce the asymptotic distribution of the Ljung-Box (or Box-Pierce) modified portmanteau statistics for weak FARIMA models. We also propose another method based on a self-normalization approach to test the adequacy of FARIMA models. Finally some simulation studies are presented to corroborate our theoretical work. An application to the Standard & Poor's 500 and Nikkei returns also illustrate the practical relevance of our theoretical results. AMS 2000 subject classifications: Primary 62M10, 62F03, 62F05; secondary 91B84, 62P05

Estimating FARIMA models with uncorrelated but non-independent error terms

In this paper we derive the asymptotic properties of the least squares estimator (LSE) of fractionally integrated autoregressive moving-average (FARIMA) models under the assumption that the errors are uncorrelated but not necessarily independent nor martingale differences. We relax considerably the independence and even the martingale difference assumptions on the innovation process to extend the range of application of the FARIMA models. We propose a consistent estimator of the asymptotic covariance matrix of the LSE which may be very different from that obtained in the standard framework. A self-normalized approach to confidence interval construction for weak FARIMA model parameters is also presented. All our results are done under a mixing assumption on the noise. Finally, some simulation studies and an application to the daily returns of stock market indices are presented to corroborate our theoretical work

Diagnostic checking in FARIMA models with uncorrelated but non-independent error terms

This work considers the problem of modified portmanteau tests for testing the adequacy of FARIMA models under the assumption that the errors are uncorrelated but not necessarily independent (i.e. weak FARIMA). We first study the joint distribution of the least squares estimator and the noise empirical autocovariances. We then derive the asymp-totic distribution of residual empirical autocovariances and autocorrelations. We deduce the asymptotic distribution of the Ljung-Box (or Box-Pierce) modified portmanteau statistics for weak FARIMA models. We also propose another method based on a self-normalization approach to test the adequacy of FARIMA models. Finally some simulation studies are presented to corroborate our theoretical work. An application to the Standard & Poor's 500 and Nikkei returns also illustrate the practical relevance of our theoretical results. AMS 2000 subject classifications: Primary 62M10, 62F03, 62F05; secondary 91B84, 62P05