Hybrid Markov Processes

Our focus so far has been on continuous-time, discrete-state-space Markov jump processes. These stochastic processes are conceptually suitable to biochemical reaction networks because the reaction events occur randomly on a continuous time scale and bring about discrete changes in the species abundances. We have also seen how a jump process can be approximated by a diffusion process, which is a continuous process, using approaches including the chemical Langevin equation (and the associated Fokker–Planck equation) and the system size expansion. Apart from the two extremes is an intermediate possibility, a hybrid process that is essentially a diffusion process but with occasional jumps. More appropriate terms for such hybrid processes in the context of systems biology are “switch plus diffusion” and “diffusion process with Markovian switching” [72, 134]. This happens in two-time-scale systems wherein events occurring on the slower time scale contribute a jump process (Markovian switching) superimposed on a diffusion process arising from events occurring on the faster time scale. A sample path of such a hybrid process is a continuously varying time course with occasional jumps, as illustrated in Figure 8.1.