Applications to Biology and Medicine

Abstract

In the chapter on stochastic processes the Poisson process was introduced as an example of an RCLL nonexplosive counting process. Furthermore, we reviewed a general theory of counting processes as point processes on the real line within the framework of martingale theory and dynamics. Indeed, for these processes, under the usual regularity assumptions, we can invoke the Doob-Meyer decomposition theorem (see (2.79) onwards) and claim that any nonexplosive RCLL process $\left( {X_t } \right)t \in \mathbb{R}_ +$ satisfies a generalized stochastic differential equation of the form dX_t = dA_t + dM_t (6.1) subject to a suitable initial condition. Here A is the compensator of the process representing the model of “evolution” and M is a martingale representing the “noise.”