Estimating a parametric trend component in a continuous-time jump-type process

We consider stochastic processes with continuous time parameter and discrete state space processing an intensity process. We assume that the intensity process depends on a parameter [beta], the maximum likelihood (m.l.) estimator of which enjoys the usual asymptotic properties. Now a trend is defined by a factor multiplied to the intensity which may depend on a parameter [alpha]. We present two different types of trend functions (polynominal and reciprocal functions) under which the asymptotic properties of are inherited by the m.l. estimator () of ([alpha],[beta]). These trend functions, in particular, can be consistently estimated. Examples where the theory presented applies are Markov processes of jump-type, Markov branching processes with immigration and linear OM- (or learning-) processes.