Field master equation theory of the self-excited Hawkes process

Abstract

A field theoretical framework is developed for the Hawkes self-excited point process with arbitrary memory kernels by embedding the original non-Markovian one-dimensional dynamics onto a Markovian infinitedimensional one. The corresponding Langevin dynamics of the field variables is given by stochastic partial differential equations that are Markovian. This is in contrast to the Hawkes process, which is non-Markovian (in general) by construction as a result of its (long) memory kernel. We derive the exact solutions of the Lagrange-Charpit equations for the hyperbolic master equations in the Laplace representation in the steady state, close to the critical point of the Hawkes process. The critical condition of the original Hawkes process is found to correspond to a transcritical bifurcation in the Lagrange-Charpit equations. We predict a power law scaling of the probability density function (PDF) of the intensities in an intermediate asymptotic regime, which crosses over to an asymptotic exponential function beyond a characteristic intensity that diverges as the critical condition is approached. We also discuss the formal relationship between quantum field theories and our formulation. Our field theoretical framework provides a way to tackle complex generalization of the Hawkes process, such as nonlinear Hawkes processes previously proposed to describe the multifractal properties of earthquake seismicity and of financial volatility.

Pólya urn with memory kernel and asymptotic behaviours of autocorrelation function

“…Our analysis is very similar to the one that studies the Hawkes process using a stochastic differential equation [14,15]. The Hawkes process is a non-Markov self-excited point process [16].…”

Exact asymptotic solutions to nonlinear Hawkes processes: a systematic classification of the steady-state solutions

“…In this article, we have studied various analytical solutions to NLHawkes processes by generalizing the field master equation approach recently developed in Refs. [16,17]. We have derived the field master equation for the general NLHawkes processes and have formulated its functional Kramers-Moyal expansion and the corresponding diffusive approximation.…”

“…(i) In the absence of inhibitory effects (i.e., when events all have positive marks), we find a non-universal power law relation for the intensity distribution at criticality, with exponent a that can take any value, i.e., corresponding to a genuine power law (a > 0) or to an intermediate power law asymptotics (a ≤ 0). This is in contrast to the LHawkes model, where only a negative exponent a < 0 exists [16,17]. (ii) In the presence of inhibitory effects (i.e., both positive and negative marks coexist), in the case where the mark distribution has zero mean corresponding to a balance between inhibitory and excitatory effects, a wide class of NLHawkes processes exhibit Zipf's law (a ≈ 1) for their intensity distributions.…”

“…The long-standing problem of combining inhibitory and excitatory effects in point processes requires considering nonlinear extensions of the Hawkes processes in order to fulfill the condition that the intensity (a probability per unit time) remains positive. Our recent Letter [27] has presented a step toward a general theory of nonlinear Hawkes processes by applying the framework of the field-master-equation [16,17]. In this letter, we discovered the existence of an asymptotic universal Zipf's law for NLHawkes processes in the case of mark distributions with zero mean (of which symmetric distributions are an important special example), corresponding to positive and negative feedback effects being statistically balanced.…”

“…Recently, however, a new theoretical scheme was developed to address such non-Markovian stochastic processes directly, in particular for the Hawkes process [16,17]. This scheme is based on a mapping from the non-Markovian Hawkes model to an equivalent stochastic partial differential equation (SPDE).…”