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

### 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)
``` |

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