Inference on the Hurst parameter and variance of diffusions driven by fractional Brownian motion
Résumé
The use of diffusion models driven by fractional noise has become popular for more than two decades. The reasons that produced this situation have been varied in nature. We can mention, among others, those that come from mathematics and other from the applications. With respect to the first group, it should be noted that fractional Brownian motion (fBm) has interesting properties. First, it is self-similar. This property implies that such a process is, from the standpoint of its distribution, invariant with respect to scale transformations. Moreover the fractional noise, the process of increments of the fBm taking in a mesh of equally spaced points, satisfies a strong dependence condition that is a notion away from independence and mixing. Using this last property, it has been possible to model natural phenomena, which exhibit temporal correlations tending to zero so slowly that their sum tends to infinity. With regard to the applications, we should mention that fractional models have become popular for modeling real-life events such as the value assets in financial markets, models of chaos in quantum physics, rivers flow along the time, irregular images, weather events and contaminant transport problems, among others. The fBm is a mean zero Gaussian process with stationary increments and whose covariance function is uniquely determined by the Hurst's parameter, that we denote by $H$ and that is between zero and one. The value $H=\frac12$ is important because the associate process results is the Brownian motion (Bm). The parameter $H$ determines the smoothness of the fBm trajectories. More regular are the trajectories as closer to one is the parameter. The exact opposite happening if $H$ is near zero. In the forties and fifties of the twentieth century, in the study of Bm, the introduction of the stochastic integral by Kiyosi Itô and Paul Levy was the key to the definition of diffusion processes. This important event led to the development of a whole area of probability and mathematics. Similarly, the introduction of several definitions of stochastic integrals with respect to fBm, from the nineties, has led to the definition of pseudo-diffusion processes driven by this noise. As in the case of a Bm the introduction of these processes significantly enriched the theory and the horizon of their applications. In these notes we develop estimation techniques for the parameter $ H $ and the local variance (volatility) of the pseudo-diffusion. The estimation of the two parameters is made simultaneously. We will use the observation of the process in a discrete mesh of points. Then the study of the asymptotics will be done when the mesh's norm tends to zero. We start by defining the second order increments of the process. Using these increments, we build the order $ p $ variations. These variations allow the definition of an estimator of the parameter $H$ for all its range. The reason to work with second order increments, instead of the first order increments, is that variations built through them are asymptotically Gaussian in all the range 0 < H <1, instead of 0 < H < 3/4 that is the case for the variations constructed by using first order increments. From the asymptotic normality of the variations, we deduce that the estimators of H are asymptotically Gaussian, for all their possible values.
After estimating the Hurst parameter, we study the local variance estimation in four pseudo-diffusion models. For each of them, we construct a local variance estimator and study its asymptotic normality. If we do not know in advance the value of the Hurst parameter, this procedure will reduce the rate of convergence in the central limit theorem (CLT) for the estimator of the variance.
Then we assume H known and try to estimate local variance functionals for more general models. For instance in the case where the variance is not constant, for this estimation procedure we will recover the lost speed, noted in the previous paragraph.
Finally, one of the main purposes of these notes is to provide a set of tools for computational statistics: efficient simulation of the processes, assessment of the goodness of fit of the estimators and the selection of the best estimator in each of the presented situations.
We will develop this program once the asymptotic properties of estimators have been studied. We will discuss simulations and their computer implementation as well as some of the codes developed.