Abstract
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.
1. 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.
2. 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, 9–15]. Let us consider SMP , , on the probability space [11, 16, 17] in Euclidian space , , which is generated by the Markov renewal process (MRP)that iswhere is the counting process.
Denote the sojourn time in states , . 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 , , , , is the Borel -algebra on , and
Note that in this section the important fact will be one when the kernel 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 relationbecause it is impossible to write the asymptotic representation of compensating operator (CO) of the process. That is why we consider semi-Markov process , , in averaging scheme, which is defined by equation
We can prove weak convergence of the process , where is the solution of differential equationwhere
Consider stochastic process:
According to (11), we get
Let us define
Consider compensating operator for some process.
Definition 1. The compensating operator for SMP , , is defined by the relationon the test-functions .
In this case there is a weak convergence SMP in the scheme of diffusive approximation without balance conditions.
Theorem 2. Let the following conditions be satisfied:(D1)Uniform integrability (bounded time in states):(D2), , that inequality is uniform by and .(D3)Boundary of the second moment of jump’s value: (D4)Kernel satisfies the following conditions: where , .(D5)Function satisfies the condition(D6)Convergence of the initial conditions is as follows:Then weak convergence takes place in , , as :where , , 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, 18–20], 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 .
Consider an evolution equationthat corresponds withand with semigroup
By analogues, an evolution equationcorresponds with operatorwith semigroup
Lemma 4. CO of two-component process on test-functions is given bywhere
Proof. By Definition 1, we got a relation for values of jumps and time of renewals:ThenSo, according to the condition we getFor embedded chain , , we getThen calculateSo, 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 .
Lemma 5. On test-functions CO of the process has asymptotic representationwhere is given by the following relation:and for the negligible term,is true.
Proof. Let us use an algebraic identityAccording to Lemma 4 and algebraic identity (40) we getLet us use the equations for semigroups [9]:Then for the term , by semigroups properties and condition (D5), we get the following relation:Similarly for we getFrom factorization , according to (40), terms , were considered. is built by these terms. It is easy to check that the sum of the rest of the terms is as , if conditions (D3)–(D5) hold.
Using representation for semigroups , , and , it is easy to show that negligible term is as .
Lemma 5 is proved.
Step 2. Let us show the relative compactness of the processes family as . 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].
Lemma 7. There is an inequalityfor test-functions , where constant depends only on function .
For function there is a boundwhere constant depends only on the function , independent from .
Proof. Let us use the result of Lemma 5:According to the definition of test-function we have . To prove the lemma, condition (D3) for a boundary of the first and the second moments, condition (D4), and condition (D2) remain to be used, from which it follows that .
Thenwhere constant depends only on , .
To prove condition (46) the properties of the function remain to be remembered; namelyLemma 7 is proved.
Lemma 8. , , is relatively compact family.
Proof. To determine the relative compactness of the family , , according to Theorem 1.4.6 [21] submartingality of the stochastic process for nonnegative infinitely differentiable and for some constant and inequality (45) and (46) remains to be shown, where .
Let us prove that stochastic process is nonnegative submartingale relatively to the stream of -algebras , :Two last terms tend to 0 as . By Lemma 7Measurability of the process relatively to the stream is obvious. So, is nonnegative submartingale.
Lemma 8 is proved.
According to Lemma 8 , , is a relatively compact family. To complete the proof of the theorem the family that converges to martingale remains to be shown. Consider stochastic processes:
Then
According to Lemma 7 the third term satisfies the relation
By the same way we can prove that the first and the fourth terms tend to 0, because is continuous.
The last term tends to 0 by Lemma 5, becauseon test-functions , which have uniform bounded derivatives of any order.
The second term is equal toand has a martingale condition according to Lemma 6.1 [11].
Let us use its martingale condition:
Finally we havewhere as .
Now from theorem’s condition (D6) we getin another words, is martingale.
So, we have checked all conditions of the weak convergence, namely, the compactness of the processes family and the martingality of the limited process. Beside this according to Lemma 5 CO converges to the generator of diffusion process.
Theorem 2 is proved.
3. Numerical Example
Consider semi-Markov process , , in . For this process has uniform distribution on and has Bernoulli distribution with parameters . Thenand average evolution , , has representationwhere are initial condition. It is easy to verify the conditions of the theorem. Prelimited processes are shown in Figures 1 and 2 for .
4. 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.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.