Stochastic monotonicity and duality for continuous time Markov chains with general q-Matrix

The key problems in discussing stochastic monotonicity and duality for continuous time Markov chains are to give the criteria for existence and uniqueness and to construct the associated monotone processes in terms of their infinitesimal q -matrices. In their recent paper, Chen and Zhang [6] discussed these problems under the condition that the given q-matrix Q is conservative. The aim of this paper is to generalize their results to a more general case, i.e., the given q-matrix Q is not necessarily conservative. New problems arise 'in removing the conservative assumption. The existence and uniqueness criteria for this general case are given in this paper. Another important problem, the construction of all stochastically monotone Q-processes, is also considered.

Uniqueness criteria for continuous-time Markov chains with general transition structures

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

Stochastic comparability and dual q-functions

In this paper we discuss the relationship and characterization of stochastic comparability, duality, and Feller–Reuter–Riley transition functions which are closely linked with each other for continuous time Markov chains. A necessary and sufficient condition for two Feller minimal transition functions to be stochastically comparable is given in terms of their density q-matrices only. Moreover, a necessary and sufficient condition under which a transition function is a dual for some stochastically monotone q-function is given in terms of, again, its density q-matrix. Finally, for a class of q-matrices, the necessary and sufficient condition for a transition function to be a Feller–Reuter–Riley transition function is also given.

Existence, uniqueness and constructions for stochastically monotone Q-processes

The key problems in discussing duality and monotonicity for continuous-time Markov chains are to find conditions for existence and uniqueness and then to construct corresponding processes in terms of infinitesimal characteristics, i.e., q-matrices. Such problems are solved in this paper under the assumption that the given q-matrix is conservative. Some general properties of stochastically monotone Q-process ( Q is not necessarily conservative) are also discussed.

A Markov modulated Poisson process model for rainfall increments

The problems encountered when using traditional rectangular pulse hierarchical point processmodels for fine temporal resolution and the growing number of available tip-time records suggest that rainfall increments from tipping-bucket gauges be modelled directly. Poisson processes are used with an arrival rate modulated by a Markov chain in Continuous time. The paper shows how, by using two or three states for this chain, much of the structure of the rainfall intensity distribution and the wet/dry sequences can be represented for time-scales as small as 5 minutes.

Some models for discretized series of events

A discretized series of events is a binary time series that indicates whether or not events of a point process in the line occur in successive intervals. Such data are common in environmental applications. We describe a class of models for them, based on an unobserved continuous-time discrete-state Markov process, which determines the rate of a doubly stochastic Poisson process, from which the binary time series is constructed by discretization. We discuss likelihood inference for these processes and their second-order properties and extend them to multiple series. An application involves modeling the times of exposures to air pollution at a number of receptors in Western Europe.