A probability space of continuous-time discrete value stochastic process with Markov property

Miloslawa Sokol

doi:10.24297/jam.v12i3.519

Abstract

Getting acquainted with the theory of stochastic processes we can read the following statement: "In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it". The classical results for limited stochastic and intensity matrices goes back to Kolmogorov at least late 40-s. But for some infinity matrices the sum of probabilities of all trajectories is less than 1. Some years ago I constructed physical models of simulation of any stochastic processes having a stochastic or an intensity matrices and I programmed it. But for computers I had to do some limitations - set of states at present time had to be limited, at next time - not necessarily. If during simulation a realisation accepted a state out of the set of limited states - the simulation was interrupted. I saw that I used non-quadratic, half-infinity stochastic and intensity matrices and that the set of trajectories was bigger than for quadratic ones. My programs worked good also for stochastic processes described in literature as without probability space. I asked myself: did the probability space for these experiments not exist or were only set of events incompleted? This paper shows that the second hipothesis is true.

Highlights

For each continuous-time discrete value stochastic process we can define instantaneous probability rates
For a continuous-time discrete value stochastic Markov process it can be defined as a right differential coefficient of the conditional probability of change from one state to another
Half-infinite stochastic and intensity matrices give the impression of unnecessary mathematical entities, this paper shows that all continuous-time discrete value stochastic processes with Markov property defined by half-infinite intensity matrices have a correct defined probability spaces as Kolmogorov conception

Summary

Introduction

For each continuous-time discrete value stochastic process we can define instantaneous probability rates. For every time t we can consider a matrix (finite or infinite) of instantaneous probability rates: [Pk,n ' (t)]k,n N It is a wellknown intensity matrix [11], [1] or state-transition matrix [7], [12] used by mathematicians to create Kolmogorov equations. The aim of this paper is to prove that for all matrices of integrability functions [ fk,n (t)]k,n I limited on each time period [0,T ] (T < ) , non-negative for k n and non-positive for k = n , which satisfy the condition k fk,n '(t) = 0 for all t and n we can construct a probability space for a continuous-time discrete value stochastic Markov process whose instantaneous probability rates Pk,n ' (t) are equal to the fk,n (t)

Half-infinite stochastic and intensity matrices

General theorem

Base function P on B

Measure P is the probability

Conclusion