Rough fractional diffusions as scaling limits of nearly unstable heavy\n tailed Hawkes processes

2015/04/13 by Thibault Jaisson, Jaisson, Thibault, Mathieu Rosenbaum +1 · 4 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Economics and business #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Statistical Finance (q-fin.ST) #Stochastic processes and statistical mechanics #Trading and Market Microstructure (q-fin.TR)

paper · pdf · doi:10.48550/arxiv.1504.03100

openalex publication_date 2015/04/13 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We investigate the asymptotic behavior as time goes to infinity of Hawkes\nprocesses whose regression kernel has L¹ norm close to one and power law\ntail of the form x^-(1+\α), with \α\∈(0,1). We in particular\nprove that when \α\∈(1/2,1), after suitable rescaling, their law\nconverges to that of a kind of integrated fractional Cox-Ingersoll-Ross\nprocess, with associated Hurst parameter H=\α-1/2. This result is in\ncontrast to the case of a regression kernel with light tail, where a classical\nBrownian CIR process is obtained at the limit. Interestingly, it shows that\npersistence properties in the point process can lead to an irregular behavior\nof the limiting process. This theoretical result enables us to give an\nagent-based foundation to some recent findings about the rough nature of\nvolatility in financial markets.\n

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