### Frederi Viens

*Written by: Andrea Rau, Ph.D. candidate in Statistics*

A self-similar process is a stochastic process for which any part of its trajectory is invariant under time scaling. In many applications, these processes exhibit dependencies over a long range of time scales, which is known as long-range or long-memory dependency. That is, increments over intervals of the same length exhibit specific slowly decaying correlation, even if far apart. For self-similar processes, the most important modeling task is to determine the self-similarity parameter, because it is responsible for the long memory of the process and its regularity properties.

Although parameter estimation for stochastic differential equations driven by standard Brownian motion has been an active area of research in the last few decades, estimation questions concerning stochastic differential equations driven by the family of Gaussian processes known as fractional Brownian motion (fBm) are, in contrast, relatively new. Professor Frederi Viens, in collaboration with Professor Ciprian Tudor of the Sorbonne in Paris, France and Purdue doctoral student Alexandra Chronopoulou, has been working to study parameter estimation properties of fBm, as well as generalizations of fBm to non-Gaussian process known as the Rosenblatt process and other higher-order Hermite processes.

The self-similarity parameter of an fBm process, H in the interval (0,1), is known as the Hurst index. If H = ½, the process is a regular Brownian motion, if H > ½ the increments of the process are positively correlated, and if H < ½ the increments are negatively correlated. Professor Viens and his colleagues used the k-variations statistic to estimate H: they constructed a strongly consistent estimator and proved its normal and non-normal convergence, with the transition from asymptotic normality to non-normality occurring at H = 3/4 in the case of fBm. In the case of the Rosenblatt process in [2], they found only non-normal convergence holds, although a subtle compensation method allowed them to construct a distinct asymptotically normal estimator when H < 3/4. Using the variations method allowed them to then study the entire class of Hermite processes, showing that the compensation method works only in the case of the Rosenblatt process, and allowing them to "find out how the above central and non-central limit theorems fit into a larger picture" [1].

To establish these results, the researchers applied Malliavin calculus, Wiener-Itô chaos expansions, and recent results on the convergence of multiple stochastic integrals. Malliavin calculus, or the stochastic calculus of variations, was originally introduced by Paul Malliavin, of the French Academy of Sciences, to study the existence and regularity of the density of solutions to stochastic differential equations. Professors Tudor and Viens originally started applying Malliavin calculus in the context of statistical estimation for slightly different problems involving stochastic differential equations driven by fBm [3], in which the motivation to use this calculus was closer to Paul Malliavin's original contribution. Their current work is a testimony to the extraordinary versatility of the Academician's work.

Self-similar processes can be particularly useful in modeling phenomena such as internet traffic, hydrology, and economics. It is often argued that climate change has long memory, but what is unclear is whether it exhibits statistical long memory, deterministic trends, or both. A study of the properties of a climatic process could help reveal to what extent each of these effects play a role in climate change. Similarly, it is known that long-range dependency and correlation decay are observed in financial data. However, current pricing theory uses martingales and the Markov property which are, on the surface, in conflict with long-term memory. According to Viens, "adapting long-term pricing models to continuous-time self-similar processes in a financially consistent way should become an important new area of research."

Professor Viens is actively involved in advising students in the Statistics and Mathematics departments. He has graduated over 60 M.S. students at Purdue, principally in Computational Finance, and at any one time he typically advises around 6 Ph.D. students and 20 M.S. students. "I'm very proactive, and I usually don't leave students to their own devices," he said of his philosophy for advising doctoral students. "I work closely with them to form them as stochastic analysts."

Professor Viens obtained his Ph.D. from the Department of Mathematics at the University of California, Irvine in 1996. He has a joint appointment in the Department of Statistics and the Department of Mathematics and serves as the Director for the Computational Finance Program at Purdue. His research interests include stochastic processes, probability theory, and mathematical finance. For more information about Professor Viens, please visit his homepage.

Additional reading:

[1] Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes. Submitted, 2008. 36 pages. With A. Chronopoulou, C. Tudor.

[2] Variations and estimators for the selfsimilarity order through Malliavin calculus. 2007, 37 pages. Submitted. With C. Tudor.

[3] Statistical aspects of the fractional stochastic calculus. Annals of Statistics, Vol. 35 (3) (2007), 1183-1212. With C.A. Tudor.

*July 2008*