Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes
| Autor: | Mathieu Rosenbaum, Thibault Jaisson |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
nearly unstable processes
Statistics and Probability volatility 01 natural sciences Power law Point process FOS: Economics and business 010104 statistics & probability fractional stochastic equation 60F05 0502 economics and business FOS: Mathematics 60G22 Statistical physics Limit (mathematics) 0101 mathematics Brownian motion Mathematics Hurst exponent Statistical Finance (q-fin.ST) 050208 finance Quantitative Finance - Trading and Market Microstructure heavy tail Probability (math.PR) 05 social sciences limit theorems long memory Quantitative Finance - Statistical Finance fractional Cox–Ingersoll–Ross process Trading and Market Microstructure (q-fin.TR) Cox–Ingersoll–Ross model Heavy-tailed distribution Kernel (statistics) Statistics Probability and Uncertainty Hawkes processes Mathematics - Probability |
| Zdroj: | Ann. Appl. Probab. 26, no. 5 (2016), 2860-2882 |
| Popis: | 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 $\alpha\in(0,1)$. We in particular prove that when $\alpha\in(1/2,1)$, after suitable rescaling, their law converges to that of 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. Comment: 21 pages |
| Databáze: | OpenAIRE |
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