ROUGH FRACTIONAL DIFFUSIONS AS SCALING LIMITS OF NEARLY UNSTABLE HEAVY TAILED HAWKES PROCESSES

• 成果类型: Article
• 署名作者: Jaisson, Thibault; Rosenbaum, Mathieu
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique; Sorbonne Universite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/15-AAP1164
• 发表日期: 2016
• 页码: 2860-2882
• 关键词: point-processes; THEOREMS; SPECTRA

摘要

We investigate the asymptotic behavior as time goes to infinity of Hawkes processes whose regression kernel has L-1 norm close to one and power law tail of the form x(-(1+alpha)), with alpha is an element of (0, 1). We in particular prove that when alpha is an element of (1/2, 1), after suitable rescaling, their law asymptotically behaves as a kind of integrated fractional Cox Ingersoll Ross process, with associated Hurst parameter H = alpha - 1/2. This result is in contrast to the case of a regression kernel with light tail, where a classical Brownian CIR process is obtained at the limit. Interestingly, it shows that persistence properties in the point process can lead to an irregular behavior of the limiting process. This theoretical result enables us to give an agent-based foundation to some recent findings about the rough nature of volatility in financial markets.