  Hawkes point processes are first-order non-Markovian stochastic models of
intermittent bursty dynamics with applications to physical, seismic, epidemic,
biological, financial, and social systems. While accounting for positive
feedback loops that may lead to critical phenomena in complex systems, the
standard linear Hawkes process only describes excitative phenomena. To describe
the co-existence of excitatory and inhibitory effects (or negative feedbacks),
extensions involving nonlinear dependences of the intensity as a function of
past activity are needed. However, such nonlinear Hawkes processes have been
found hitherto to be analytically intractable due to the interplay between
their non-Markovian and nonlinear characteristics, with no analytical solutions
available. Here, we present various exact and robust asymptotic solutions to
nonlinear Hawkes processes using the field master equation approach. We report
explicit power law formulas for the steady state intensity distributions
$P_{\mathrm{ss}}(\lambda)\propto \lambda^{-1-a}$, where the tail exponent $a$
is expressed analytically as a function of parameters of the nonlinear Hawkes
models. We present three robust interesting characteristics of the nonlinear
Hawkes process: (i) for one-sided positive marks (i.e., in the absence of
inhibitory effects), the nonlinear Hawkes process can exhibit any power law
relation either as intermediate asymptotics ($a\leq 0$) or as true asymptotics
($a>0$) by appropriate model selection; (ii) for distribution of marks with
zero mean (i.e., for balanced excitatory and inhibitory effects), the Zipf law
($a\approx 1$) is universally observed for a wide class of nonlinear Hawkes
processes with fast-accelerating intensity map; (iii) for marks with a negative
mean, the asymptotic power law tail becomes lighter with an exponent increasing
above 1 ($a>1$) as the mean mark becomes more negative.

Statistical Mechanics;Trading and Market Microstructure