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 infinite-dimensional 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 PDF of the intensities in an intermediate asymptotics 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 generalisation of the Hawkes process, such as nonlinear Hawkes processes previously proposed to describe the multifractal properties of earthquake seismicity and of financial volatility.
Highlights
The self-excited conditional Poisson process introduced by Hawkes [1,2,3] has progressively been adopted as a useful firstorder model of intermittent processes with time clustering, such as those occurring in seismicity and financial markets
We have presented an analytical framework of the Hawkes process for an arbitrary memory kernel, based on the master equation governing the behavior of auxiliary field variables
While the Hawkes point process is non-Markovian by construction, the introduction of auxiliary field variables provides a formulation in terms of linear stochastic partial differential equations that are Markovian
Summary
The self-excited conditional Poisson process introduced by Hawkes [1,2,3] has progressively been adopted as a useful firstorder model of intermittent processes with time (and space) clustering, such as those occurring in seismicity and financial markets. We propose a natural formulation of the Hawkes process in the form of a field theory of probability density functionals taking the form of a field master equation This formulation is found to be ideally suited to investigate the hitherto ignored properties of the distribution of Hawkes intensities, which we analyze in depth in a series of increasingly sophisticated forms of the memory kernel characterizing how past events influence the triggering of future events. These sections are complemented by seven Appendixes, in which the detailed analytical derivations are provided
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