Autocorrelation of Hawkes process jumps on nonoverlapping time intervals with lag.

Share:

Description

The function returns the theoretical autocorrelation of the number of jumps of a Hawkes process on nonoverlapping time intervals with lag

Usage

1
jumpAutocorrelation(lambda0, alpha, beta, tau,lag)

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.

lag

Time lag.

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 matrix containing the autocorrelation of the number of jumps of process components.

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

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
#One dimensional Hawkes process
lambda0<-0.02
alpha<-0.05
beta<-0.06
tau<-60#one minute
lag<-0#adjacent non overlappingintervals
h<-jumpAutocorrelation(lambda0,alpha,beta,tau,lag)

#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
lag<-0#adjacent non overlappingintervals
h<-jumpAutocorrelation(lambda0,alpha,beta,tau,lag)