| jumpVariance {hawkes} | R Documentation |
Variance of Hawkes process jumps.
Description
The function returns the theoretical variance matrix of the number of jumps of a Hawkes process on a time interval of length tau.
Usage
jumpVariance(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 matrix containing the variance of the 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
#One dimensional Hawkes process
lambda0<-0.02
alpha<-0.05
beta<-0.06
tau<-60#one minute
h<-jumpVariance(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<-jumpVariance(lambda0,alpha,beta,tau)