A New Class of Processes for Formalizing and Generalizing Individual-Based Models: The Semi-Semi-Markov Processes

Abstract

Individual-based models are a \bottom-up" approach for calculating empirical distributions at the level of the population from simulated individual trajectories. We build a new class of stochastic processes for mathematically formalizing and generalizing these simulation models according to a \top-down" approach, when the individual state changes occur at countable random times. We allow individual population-dependent semi-Markovian transitions in a non closed population such as a branching population. These new processes are called Semi-Semi-Markov Processes (SSMP) and are generalizations of Semi-Markov processes. We calculate their kernel and their probability law, and we build a simulation algorithm from the kernel.