Self-exciting negative binomial distribution process and critical properties of intensity distribution

Abstract

We study the continuous time limit of a self-exciting negative binomial process and discuss the critical properties of its intensity distribution. In this limit, the process transforms into a marked Hawkes process. The probability mass function of the marks has a parameter $\omega$, and the process reduces to a “pure” Hawkes process in the limit $\omega \rightarrow 0$. We investigate the Lagrange–Charpit equations for the master equations of the marked Hawkes process in the Laplace representation close to its critical point and extend the previous findings on the power-law scaling of the probability density function (PDF) of intensities in the intermediate asymptotic regime to the case where the memory kernel is the superposition of an arbitrary finite number of exponentials. We develop an efficient sampling method for the marked Hawkes process based on the time-rescaling theorem and verify the power-law exponents.

Appendices

A Method of Characteristics

The method of characteristics is a standard method to solve first-order partial differential equations (Gardiner 2009; Kanazawa and Sornette 2020b, a). The equation for the characteristic function

$$\begin{aligned} \phi (s)=\int ^{\infty }_{-\infty }\mathrm {e^{isx}}p(x,t|x_0,0)dx \end{aligned}$$

is

$$\begin{aligned} \partial _t\phi +ks\partial _s\phi =-\frac{1}{2}Ds^2\phi . \end{aligned}$$

(A.1)

We consider the corresponding Lagrange–Charpit equations

$$\begin{aligned} \frac{dt}{dl}=-1, \quad \frac{ds}{dl}=-ks, \quad \frac{d\phi }{dl}=\frac{1}{2}Ds^2\phi \end{aligned}$$

with the parameter l encoding the position along the characteristic curves. These equations are equivalent to an invariant form in terms of l

$$\begin{aligned} \frac{dt}{1}=\frac{ds}{ks}=-\frac{d\phi }{\frac{1}{2}Ds^2\phi }. \end{aligned}$$

The method of characteristics can be used to solve this equation. Namely, if

$$\begin{aligned} u(s,t,\phi )=a, \qquad \textrm{and} \qquad v(s,t,\phi )=b \end{aligned}$$

are two integrals of the subsidiary equation (with a and b arbitrary constants), then a general solution of (A.1) is given by

$$\begin{aligned} f(u(s,t,\phi ),v(s,t,\phi ))=0, \end{aligned}$$

with an arbitrary function f(a, b), which is determined by the initial or boundary condition of the PDE (A.1). This method can be readily generalized to systems with many variables.

B Addendum

In a recent publication by K.Kanazawa and D.Sornette (Kanazawa and Sornette 2023), the power-law exponent of the intensity distribution of general marked Hawkes process was given. We explain the correspondence for the reader’s convenience.

In (Kanazawa and Sornette 2023), the power-law exponent is given using the second moment of the mark’s PDF E$[m^2]$ as

$$\begin{aligned} P_{SS}(\nu )\propto \nu ^{-1-a},a=\frac{2\tau \nu _0}{\text{ E }[m^2]}. \end{aligned}$$

The normalization of $\rho (m)$ with E$[m]=1$ is adopted. In our model, a is given as

$$\begin{aligned} a=\frac{2\tau \nu _0}{\text{ E }[m^2]/\text{E }[m]}. \end{aligned}$$

We use $\rho (m)$ in Eq.(6) to estimate the power-law exponent. As E$[m]=\omega /\ln (\omega +1)$, E$[m^2]=\omega (\omega +1)/\ln (\omega +1)$, we obtain

$$\begin{aligned} P_{SS}(\nu )\propto \nu ^{-1-a},a=\frac{2\tau \nu _0}{\text{ E }[m^2]/\text{E }[m]}=2\tau \nu _0/(\omega +1). \end{aligned}$$

The result is consistent with ours in (Hisakado et al. 2022a).