# Hawkes process simulation Function

### Description

The function simulates a Hawkes process for the given parameter, and until a time horizon.

### Usage

1 | ```
simulateHawkes(lambda0, alpha, beta, horizon)
``` |

### 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. |

`horizon` |
Time horizon until which the simulation is to be conducted. |

### 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 of jump times in the monovariate case, and a list of such vectors for every component in the multivariate case.

### References

Y. Ogata. (1981)
On Lewis simulation method for point processes.
*IEEE Transactions on Information Theory*, **31**

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
#One dimensional Hawkes process
lambda0<-0.2
alpha<-0.5
beta<-0.7
horizon<-3600#one hour
h<-simulateHawkes(lambda0,alpha,beta,horizon)
#Multivariate Hawkes process
lambda0<-c(0.2,0.2)
alpha<-matrix(c(0.5,0,0,0.5),byrow=TRUE,nrow=2)
beta<-c(0.7,0.7)
horizon<-3600#one hour
h<-simulateHawkes(lambda0,alpha,beta,horizon)
``` |

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