Ecole d’été régionale franco-roumaine en mathématiques appliquées, Sinaia, 2 - 11 Juillet 2017
Self-similar processs: stochastic and statistical analysis
Ciprian A. Tudor - Université Lille 1
Self-similar processes are stochastic processes that are invariant in
distribution under suitable scal- ing of time and space. This property
is crucial in applications such as network traffic analysis,
mathematical finance, astrophysics, hydrology or image processing. For
these reasons, their analysis constitutes an important research
direction in probability theory since a while.
Our purpose is to discuss the basic properties of several classes of
(Gaussian or non-Gaussian) self- similar stochastic processes. The
main example is the fractional Brownian motion which the most known
self- similar process with stationary increments. It includes the
standard Brownian motion as a particular case. The applications of
this process are now widely recognized. We survey the basic properties
of the process and several other processes related with it that
emerged recently in the scientific research.
On the other hand, the self-similar stochastic processes are well
suited to model various phenomena where scaling and long memory are
important factors (internet traffic, hydrology, econometrics, among
other). The most important modeling task is then to determine or
estimate the self- similarity parameter, because it is also typically
responsible for the process?s long memory and its regularity
properties. Studying them is thus an important research direction in
theory and practice. Our purpose is to present recent results in this
direction. The approach we use is based on the so-called Malliavin
calculus and multiple Wiener-Ito integrals.
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