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

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 a set of events incompleted? This paper shows that the second hipothesis is true. space for stochastic processes


Introduction
For each continuous-time discrete value stochastic process we can define instantaneous probability rates. That idea is used by biologists [9], [6] and it corresponds to the intensity of probability used in the theory of stochastic processes. 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.   Instantaneous probability rates can have values greater than 1 (but always non-negative for n k  and non-positive for n k = ) and they are measured by 1 (5) instantaneous probability rates fit the condition: . It is a wellknown intensity matrix [11], [1] or state-transition matrix [7], [12] used by mathematicians to create Kolmogorov equations. In this article such a matrix will be treated differently. This matrix is a based object creating probability space for Markov stochastic processes.
The aim of this paper is to prove that for all matrices of integrability functions

Half-infinite stochastic and intensity matrices
A modelling of Markov stochastic processes is an important branching in theoretical biology [5], [3]. But  Although 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. These probability spaces allow to define a probability space for all (finite or infinite) intensity matrices and for infinite time period.

General theorem
For all matrices of integrability functions , where K can be finite or infinite, which satisfy the following condition: and their graphs on the coordinate system (Fig.1). We will consider functions constant on left closed Such a function will be called the realization of the forming of a stochastic process. All those realizations form a set  .     i . Therefore, we will use a notation Sometimes the same base set of realizations can be formed by different coverings and sequences of states. For instance, the covering: with the sequence of states )) ), with the sequence of states B is a subset of  2 (all subsets of  ). Basis B is required to define  -algebra on  as minimal  -algebra consisting B , but in this paper the construction of  -algebra for probability space is a little different. It is similar to construction of Lebesgue measure domain [8], [10], [2], [4].
The properties of:

Base function P on B
For any interval

Theorem 1 For any
According to this inequality the theorem 1 point 4 is truthful.

Definition 3 The base function
[0,1] :  B P is a function which for all base sets A formed by the covering Theorem 2 The definition of base function P is good.
Proof. We must prove that for the optimal covering The same base set can be formed by The value of P for the base set is the same as the value for its optimal covering. So the P is a well-defined function. Theorem 3 Base function P has the following properties:         Basis B and base function P satisfy all of the assumption of Caratheodory's theorem [8], [10], [2], [4]. It means that function ) [0, 2 : is an outer measure. A set of subsets satisfaying Caratheodory condition: is a  -algebra and restriction * P to B is a measure. Moreother for base set B  A : The set B is a  -algebra for the created probability space. Measure The countable sum of all sets from B formed by all coverings Then: Above inequolities are true if all realizations, also finishing before time T due to last state L E for K L > , are taken into account.
The finished inequality has the following form: Using the same notation for as in proof of theorem 5, we can note: In the end: . The T can be equal to infinity, but for each or such limits exist in the subset of values of  . Such function will be called realizations and will be noted ...
,..., , , The base set in probabilistic space of for any k and t . Then time t can be treated as construction of the probability space on this set is the same. It will be noted: . This probability space limited by time T only will be noted: For all sets A is a set of restricting realizations.

Stochastic process defined by the probability space
is a probability space of events, which changed in time. But if we put: At the end, the last theorem will be proved.

Theorem 7
For any k , n and t the instantaneous probability rate of stochastic process defined by matrix

Conclusion
The true value of the work is to describe the detailed construction of a probability space in which the events are realizations of the stochastic process. Equations (11) and (14) allow calculating the probability of occurrence of a given type of realization of a stochastic process during the simulation. It is a skill comparable to the possibility of calculating the volume of a cube, what allows calculation of the volume of other geometric solids.
Probability space exists for all discrete stochastic processes with Markov property. It exists also for processes