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We propose a new Kalikow decomposition for continuous time multivariatecounting processes, on potentially infinite networks. We prove the existence ofsuch a decomposition in various cases. This decomposition allows us to derivesimulation algorithms that hold either for stationary processes withpotentially infinite network but bounded intensities, or for processes withunbounded intensities in a finite network and with empty past before 0. TheKalikow decomposition is not unique and we discuss the choice of thedecomposition in terms of algorithmic efficiency in certain cases. We applythese methods on several examples: linear Hawkes process, age dependent Hawkesprocess, exponential Hawkes process, Galves-L\"ocherbach process.

Kalikow decomposition for counting processes with stochastic intensity and application to simulation algorithms