# Exponential: Reproduction kernels for the Hawkes processes In hawkesbow: Estimation of Hawkes Processes from Binned Observations

## Description

These classes are derived from the class Model, each implementing a different reproduction kernel for the Hawkes process. They inherit all fields from Model.

## Details

• The kernel Exponential has density function

h^\ast(t) = β \exp(-β t) 1_{\{t ≥ 0\}}.

Its vector of parameters must be of the form (η, μ, β). Both loglik, its derivatives, and whittle can be used with this reproduction kernel.

• The kernel SymmetricExponential has density function

h^\ast(t) = 0.5 β \exp(-β |t|).

Its vector of parameters must be of the form (η, μ, β). Only whittle can be used with this reproduction kernel.

• The kernel Gaussian has density function

h^\ast(t) = \frac{1}{σ √{2π}}\exp≤ft(-\frac{(t-ν)^2}{2σ^2}\right).

Its vector of parameters must be of the form (η, μ, ν, σ^2). Only whittle is available with this reproduction kernel.

• The kernel PowerLaw has density function

h^\ast(t) = θ a^θ (t+a)^{-θ-1} 1_{\{θ > 0 \}}.

Its vector of parameters must be of the form (η, μ, θ, a). Both loglik, its derivatives, and whittle can be used with this reproduction kernel.

• The kernels Pareto3, Pareto2 and Pareto1 have density function

h_θ^\ast(t) = θ a^θ t^{-θ - 1} 1_{\{t > a\}},

with θ = 3, 2 and 1 respectively. Their vectors of parameters must be of the form (η, μ, a). Only whittle is available with this reproduction kernel.