Self-exciting negative binomial distribution process and critical properties of intensity distribution
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We study the continuous time limit of a self-exciting negative binomial process and discuss the critical properties of its intensity distribution. The discrete time process is equivalent to a self-exciting Poisson process with conditionally gamma distributed intensities. The process becomes a Hawkes process with conditionally gamma distributed intensities or a marked Hawkes process in a continuous time limit. It has a parameter ω that controls the variance of the gamma distributed intensities; the process is reduced to a pure Hawkes process in the limit ω→ 0. We study the Lagrange–Charpit equations for the master equations in the Laplace representation close to the critical point of the marked Hawkes process, and we extend the previous results of power-law scaling of the probability density function of the intensities in the intermediate asymptotic regime. We develop an efficient sampling method for the continuous self-exciting negative binomial process based on the time-rescaling theorem. The numerical results verify the power-law exponent for small ω. For ω>>1, we observe large discrepancies.
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