Field master equation theory of the self-excited Hawkes process

Kanazawa，Kiyoshi1; Sornette，Didier2,3,4

2020-09-21

10.1103/PhysRevResearch.2.033442

Physical Review Research   影响因子和分区

2643-1564

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 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.

EI

英语

其他

20204809556138

Exponential functions ; Earthquakes ; Stochastic systems ; Lagrange multipliers

Seismology:484 ; Control Systems:731.1 ; Mathematics:921 ; Probability Theory:922.1 ; Systems Science:961

2-s2.0-85096546039

Scopus

1.Faculty of Engineering,Information and Systems,University of Tsukuba,Tennodai, Tsukuba,Ibaraki,305-8573,Japan
2.ETH Zurich,Department of Management,Technology,and Economics,Zurich,8092,Switzerland
3.Tokyo Tech World Research Hub Initiative,Institute of Innovative Research,Tokyo Institute of Technology,Tokyo,152-8550,Japan
4.Institute of Risk Analysis Prediction,and Management,Academy for Advanced Interdisciplinary Studies,Southern University of Science and Technology,Shenzhen,518055,China

Kanazawa，Kiyoshi,Sornette，Didier. Field master equation theory of the self-excited Hawkes process[J]. Physical Review Research,2020,2(3).

Kanazawa，Kiyoshi,&Sornette，Didier.(2020).Field master equation theory of the self-excited Hawkes process.Physical Review Research,2(3).

Kanazawa，Kiyoshi,et al."Field master equation theory of the self-excited Hawkes process".Physical Review Research 2.3(2020).