Levy Noise Affects Ornstein-Uhlenbeck Memory.

Abstract

This paper investigates the memory of the Ornstein-Uhlenbeck process (OUP) via three ratios of the OUP increments: signal-to-noise, noise-to-noise, and tail-to-tail. Intuition suggests the following points: (1) changing the noise that drives the OUP from Gauss to Levy will not affect the memory, as both noises share the common 'independent increments' property; (2) changing the auto-correlation of the OUP from exponential to slowly decaying will affect the memory, as the change yields a process with long-range correlations; and (3) with regard to Levy driving noise, the greater the noise fluctuations, the noisier the prediction of the OUP increments. This paper shows that intuition is plain wrong. Indeed, a detailed analysis establishes that for each of the three above-mentioned points, the very converse holds. Hence, Levy noise has a significant and counter-intuitive effect on Ornstein-Uhlenbeck memory.