Average and diffusion approximation of stochastic evolutionary systems in an asymptotic split state space (Q1431563)

This is a new advance in the authors' study of average and diffusion approximation of stochastic evolutionary systems. A first process \(x^{\varepsilon}\) (\(\varepsilon\) is a small parameter) takes values in \(\{0\}\cup E\) where \(\{0\}\) is an absorbing state and \(E\) is a compact space splitted into a finite number of subspaces. It is a nonergodic jump Markov process and the switched stochastic evolutionary system process has \(R^d\)-valued locally independent increments that depend on the switching process \(x^{\varepsilon}\). The three level stochastic systems considered, that is the switching process, the switched process and additive functionals such as the reward of operating time, all depend upon the parameter \(\varepsilon\). The average and diffusion approximation limit theorems are established in both simple and double merging. Particular cases and examples are presented, including stochastic integral functionals, dynamical systems, storage jump processes and compound Poisson processes.