Mean of Hawkes process jumps.

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Description

The function returns the theoretical mean of the number of jumps of a Hawkes process on a time interval of length tau.

Usage

1
jumpMean(lambda0, alpha, beta, tau)

Arguments

lambda0

Vector of initial intensity, a scalar in the monovariate case.

alpha

Matrix of excitation, a scalar in the monovariate case. Excitation values are all positive.

beta

Vector of betas, a scalar in the monovariate case.

tau

Time interval length.

Details

Notice that in the scalar case, one must have beta>alpha for the process to be stable, and in the multivariate case, the matrix (diag(beta)-alpha) must have eigen values with strictly positive real parts for the process to be stable.

Value

Returns a vector containing the mean number of jumps of every process component.

References

Jose Da Fonseca and Riadh Zaatour Hawkes Process : Fast Calibration, Application to Trade Clustering and Diffusive Limit. Journal of Futures Markets, Volume 34, Issue 6, pages 497-606, June 2014.

Jose Da Fonseca and Riadh Zaatour Clustering and Mean Reversion in Hawkes Microstructure Models.

Examples

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#One dimensional Hawkes process
lambda0<-0.02
alpha<-0.05
beta<-0.06
tau<-60#one minute
h<-jumpMean(lambda0,alpha,beta,tau)

#Multivariate Hawkes process
lambda0<-c(0.02,0.02)
alpha<-matrix(c(0.05,0,0,0.05),byrow=TRUE,nrow=2)
beta<-c(0.06,0.06)
tau<-60#one minute
h<-jumpMean(lambda0,alpha,beta,tau)