Compensating Operator and Weak Convergence of Semi-Markov Process to the Diffusion Process without Balance Condition

Weak convergence of semi-Markov processes in the diffusive approximation scheme is studied in the paper.This problem is not new and it is studied in many papers, using convergence of random processes. Unlike other studies, we used in this paper concept of the compensating operator. It enables getting sufficient conditions of weak convergence under the conditions on the local characteristics of output semi-Markov process.


Introduction
Weak convergence conditions for semi-Markov stochastic processes in the diffusion approximation scheme without balance condition are studied in the paper.Theory of Markov and semi-Markov processes is used in security market (Black-Sholes equation, Vasicek model, and their modifications) [1], queuing systems [2,3], engineering [4], biology [5], climate models [6,7], and publicity models [8].But numerous papers were devoted to problems of Markov processes convergence.This approach produces errors of mathematical model because exponential distribution of sojourn time in states is supposed.The supposition enables staying in the state any time with nonzero probability.This is unacceptable in physics systems.That is why the subject of this research is semi-Markov processes.Beside this, other techniques of weak convergence research are used in many papers.In this case authors get different sufficient conditions of the convergence.For example, in [9] author states his results as solution of some martingale problem.It complicates testing these conditions.In [3] author focuses on the convergence of characteristic functions and claims the convergence of characteristic functions for prelimited processes.In [10] author claims the convergence of the generators of the prelimited processes to the generator of some diffusion process.
In contrast to abovelisted works, only moment's conditions on the semi-Markov process local characteristic are used in this paper.Using the term of compensating operator makes it possible to not use convergence of the generators of prelimited processes.

Main Result
Consider the conditions of weak convergence of semi-Markov random processes (SMP) in diffusion approximation scheme.Consideration of these problems can be found in [3,[9][10][11][12][13][14][15].Let us consider SMP (),  ≥ 0, on the probability space (Ω, , ) [11,16,17] in Euclidian space   ,  ≥ 1, which is generated by the Markov renewal process (MRP) that is where ]() fl max{ ≥ 0 :   < } is the counting process.Denote the sojourn time in states   fl   −  −1 ,  ≥ 0. MRP is determined by semi-Markov kernel, which sets conditional probabilities of jump's values, and by distribution of the sojourn time in states: where  ∈   , V ∈    ,  ≥ 0, Δ +1 =  +1 −   ,    is the Borel -algebra on   , and Note that in this section the important fact will be one when the kernel (, V, ) has decomposition: because in general this assumption is not valid.
In this paper suppose that balance condition did not hold; it means that In this case it is impossible to consider the process in the scheme of diffusive approximation, which is defined by relation because it is impossible to write the asymptotic representation of compensating operator (CO) of the process.That is why we consider semi-Markov process   (),  ≥ 0, in averaging scheme, which is defined by equation We can prove weak convergence of the process   ⇒ , where  is the solution of differential equation where Consider stochastic process: According to (11), we get Let us define Consider compensating operator for some process.
In this case there is a weak convergence SMP in the scheme of diffusive approximation without balance conditions.
(D3) Boundary of the second moment of jump's value: (D4) Kernel Γ  (, V) satisfies the following conditions: where (D6) Convergence of the initial conditions is as follows: Then weak convergence takes place in ([0, ]),  < ∞, as  ↓ 0: where  0 (),  ≥ 0, is the diffusion process with generator ( Remark 3. Boundary operator depends on the averaged evolution ; that is why it is advisable to consider weak convergence of two-component evolution (  ,   ).But, according to [11,[18][19][20], we will prove theorem only for the process, which consists of parameter of series , in other words   .
The proof of Theorem 2 consists of two steps.
Step 1.Let us solve the problem of the singular perturbation for CO of process   as  ↓ 0.
Consider an evolution equation that corresponds with and with semigroup By analogues, an evolution equation corresponds with operator with semigroup Lemma 4. CO of two-component process (  ,   ) on testfunctions  ∈ ( 2 ) is given by where Proof.By Definition 1, we got a relation for values of jumps and time of renewals: Then So, according to the condition    = V we get For embedded chain    ,  ≥ 0, we get Then calculate So, finally we get representation of the compensating operator for two-component evolution (  ,   ): as we wanted to show.Lemma 4 is proved.
Consider asymptotic behavior of CO, from Lemma 4 as  ↓ 0. Lemma 5. On test-functions (, V) ∈  ∞ ( 2 ) CO of the process (  ,   ) has asymptotic representation where Γ 0 is given by the following relation: According to Lemma 4 and algebraic identity (40) we get Let us use the equations for semigroups [9]: Then for the term  1 , by semigroups properties and condition (D5), we get the following relation: Similarly for  2 +  3 we get From factorization [  2     2  Δ  − ], according to (40), terms   2  −, (   2  −)+(Δ  −) were considered.Γ 0 is built by these terms.It is easy to check that the sum of the rest of the terms is (1) as  ↓ 0, if conditions (D3)-(D5) hold.
Using representation for semigroups   2  ,    2  , and Δ  , it is easy to show that negligible term is (1) as  ↓ 0.
Lemma 5 is proved.
Step 2. Let us show the relative compactness of the processes family   as  > 0. We will use Theorem 1.4.6 from [21].Let us formulate and prove the statement we need to use this theorem.
Remark 6. Conditions of relative compactness can also be found in [22,23].
To prove condition (46) the properties of the function  0 remain to be remembered; namely

Conclusions
Weak convergence of semi-Markov processes in the diffusive approximation scheme on conditions on the local characteristics of semi-Markov process is studied in this paper.