Abstract
The matrices of non-homogeneous Markov processes consist of time-dependent functions whose values at time form typical intensity matrices. For solvingsome problems they must be changed into stochastic matrices. A stochas-tic matrix for non-homogeneous Markov process consists of time-dependent functions, whose values are probabilities and it depend on assumed time pe- riod. In this paper formulas for these functions are derived. Although the formula is not simple, it allows proving some theorems for Markov stochastic processes, well known for homogeneous processes, but for non-homogeneous ones the proofs of them turned out shorter.
Highlights
Non-homogeneous Markov stochastic processes often appear in scientific literature as solutions of many real problems due to twenty-four hour or seasonal fluctuations of probability of many real events [2], [4], [5], [6], [10], [11], [12], [14], [17]
They can be defined by a probability space (, ( ), P) for a Markov stochastic process ( X t )t [0,T ) by the formulas: fk,n (t) = lim 0
The first aim of this study is to construct a stochastic matrix for a continuous-time discrete value stochastic process with Markov property, whose intensity matrix (e.g. matrix of the function fk,n (t) wchich satisfy the properties 1, 2 and 3) and time period is known
Summary
Non-homogeneous (sometimes called inhomogeneous) Markov stochastic processes often appear in scientific literature as solutions of many real problems due to twenty-four hour or seasonal fluctuations of probability of many real events [2], [4], [5], [6], [10], [11], [12], [14], [17]. The theorems in this paper apply to all continuous-time, discrete value Markov stochastic processes. The intensity matrix specifies all features of continuous-time discrete value stochastic processes with Markov property. In particular it should clearly define a stochastic matrix of this process for any fixed time-period. The first aim of this study is to construct a stochastic matrix for a continuous-time discrete value stochastic process with Markov property, whose intensity matrix (e.g. matrix of the function fk,n (t) wchich satisfy the properties 1, 2 and 3) and time period is known. The third aim is showing that for homogeneous processes similar formulas are too complicated for applications It shows that Markov stochastic processes theory should be taught starting with non-homogeneous processes
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