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SYFT: Fat-Tail Distributions - Applications from Physics to Finance

SYFT 1: Fat-Tail Distributions - Applications from Physics to Finance

SYFT 1.2: Hauptvortrag

Donnerstag, 11. März 2004, 10:00–10:30, H1

Use and Abuse of Ito vs. non-Ito Stochastic Calculus — •Peter Hänggi — Universität Augsburg, Institut für Physik

Brownian motion dynamics, discovered first in 1789 by Jan Ingen-Housz on charcoal particles under his microscope and by Robert Brown in 1828 in his observations of pollen, inspired many physicists to develop guiding contributions to the theory of fluctuation phenomena and statistical mechanics of irreversible processes per se. The list includes distinguished names such as Langevin, Einstein, Smoluchowski, Ornstein, Uhlenbeck, to name but a few. The use of stochastic differential equations plays a key role in most phenomena involving noisy perturbations. The paper by K. Ito (Proc. Imp. Acad. Tokyo, 20: 519 (1944)) gave the corresponding stochastic integral a mathematical precise meaning while Stratonovich in his works introduced a different notion that is appealing for many from a physics point of view. In this talk I will survey the use – and abuse – of actually infinite many differing stochastic integral definitions for Gaussian white noise and, as well as, for white Poisson shot noise [1]. With this talk I elaborate on various pitfalls and comment on the use of colored noise (or correlated) driven flows. Moreover, I will clarify the connection with a path integral representation of the stochastic dynamics and clear up some confusion of Ito vs. Stratonovitch discretization rules. Finally, I present results of multiplicative Langevin equations to describe long time tail phenomena and anomalous diffusion.

[1] P. Hänggi and H. Thomas, Phys. Rep. 88: 207 (1982); sec. 2.4; P. Hänggi, Helv. Phys. Acta 51: 183 (1978).