You are welcome to attend the following Statistics and Stochastic colloquium (part of the Colloquium Series of the Department of Mathematics and Statistics) at La Trobe University.
Title: From Fractional Brownian Motion to Multifractional Processes
Speaker: Prof. Antoine Ayache, Université de Lille (France)
Time & Date: 12:00 noon Thursday 8 August 2019
Venue: Room 310, Physical Sciences 2, La Trobe University, Melbourne Campus.
Fractional Brownian Motion (FBM), which was first introduced by Kolmogorov in the 1940s and later made popular by Mandelbrot and Van Ness in the 1970s, is the unique self-similar and stationary increments Gaussian process. Basically it only depends on one constant parameter, belonging to the interval (0,1), denoted by H and called the Hurst parameter, with reference to the hydrologist Hurst. FBM reduces to the well known Brownian Motion (BM) when H = 1/2, generally speaking, it can be viewed as a quite natural extension of the latter process; however there is a major difference between both processes: in contrast with BM, the increments of FBM are not independent and they even display long range dependence when H > 1/2.
FBM has turned out to be quite useful in many applied areas including signal and image processing as well as telecommunications. Yet, this model does not seem to be realistic enough. Indeed, an important limitation of FBM comes from the fact that the roughness of its path remains everywhere the same; more precisely the critical local Hölder regularity of the path [usually measured through the pointwise Hölder exponent] does not change from one place to another, since it is equal to the constant Hurst parameter H. In order to overcome this drawback, Multifractional Processes were introduced starting from the mid-1990s. The main goal of our talk is to present the most classical one of them.