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

Authors: Igor V. Malyk
Journal: Journal of Applied Mathematics
 (2015)

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## 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.


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## Body

## 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 η
t, t
≥
0, on the probability space Ω
,
F
,
P [11, 16, 17] in Euclidian space R
d, d
≥
1, which is generated by the Markov renewal process (MRP)(1)
η
n
,
τ
n
,
n
≥
0
;that is(2)
η
t
=
η
ν
t
,where ν
t
≔
max
⁡
n
≥
0
:
τ
n
<
t is the counting process.Denote the sojourn time in statesθ
n
≔
τ
n
-
τ
n
-
1, n
≥
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:(3)
Q
u
,
d
v
,
t
≔
P
Δ
η
n
+
1
∈
d
v
,
θ
n
+
1
≤
t
∣
η
n
=
u
=
Γ
u
,
d
v
F
u
t
,where u
∈
R
d, d
v
∈
β
R
d, t
≥
0, Δ
η
n
+
1
=
η
n
+
1
-
η
n, β
R
d is the Borel σ-algebra on R
d, and(4)
Γ
u
,
d
v
≔
P
Δ
η
n
+
1
∈
d
v
∣
η
n
=
u
,
F
u
t
≔
P
θ
n
+
1
≤
t
∣
η
n
=
u
.Note that in this section the important fact will be one when the kernelQ
u
,
d
v
,
t has decomposition:(5)
Q
u
,
d
v
,
t
=
Γ
u
,
d
v
F
u
tbecause in general this assumption is not valid.In this paper suppose that balance condition did not hold; it means that(6)
a
u
≔
∫
R
d
v
Γ
u
,
d
v
≠
0
.In this case it is impossible to consider the process in the scheme of diffusive approximation, which is defined by relation(7)
ζ
ε
t
≔
ε
η
ε
-
2
tbecause it is impossible to write the asymptotic representation of compensating operator (CO) of the process. That is why we consider semi-Markov process η
ε
t, t
≥
0, in averaging scheme, which is defined by equation(8)
η
ε
t
≔
ε
η
ε
-
1
t
.We can prove weak convergence of the processη
ε
⇒
ρ, where ρ is the solution of differential equation(9)
d
ρ
t
d
t
=
C
ρ
t
,where(10)
C
u
=
a
u
b
u
,
b
u
≔
λ
u
-
1
,
λ
u
≔
∫
0
∞
t
d
F
u
t
.Consider stochastic process:(11)
ξ
ε
t
≔
ε
-
1
ε
2
η
ε
-
2
t
-
ρ
t
.According to (11), we get(12)
τ
n
ε
=
ε
2
τ
n
,
θ
n
ε
=
ε
2
θ
n
.Let us define(13)
ξ
n
ε
≔
ε
η
τ
n
-
ε
-
1
ρ
ε
2
τ
n
,
ρ
n
ε
=
ρ
ε
2
τ
n
.Consider compensating operator for some process.Definition 1.
The compensating operatorΓ
ε for SMP ξ
ε
t, t
≥
0, is defined by the relation(14)
Γ
ε
φ
u
,
t
≔
E
φ
ξ
n
+
1
ε
,
τ
n
+
1
ε
-
φ
ξ
n
ε
,
τ
n
ε
∣
ξ
n
ε
=
u
,
τ
n
ε
=
t
E
θ
n
+
1
ε
∣
ξ
n
ε
=
uon the test-functions φ
u
,
t.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):(15)
l
i
m
T
→
∞
sup
u
∈
R
d
⁡
∫
T
∞
F
¯
t
d
t
=
0
.
(D2)
∃
C, C
1
>
0, that inequality(16)
E
e
-
ε
θ
u
≤
1
-
C
ε
,
D
θ
u
<
C
1
is uniform byu
∈
R
d and ε
>
0.
(D3)
Boundary of the second moment of jump’s value:(17)
s
u
p
u
∈
R
d
∫
R
d
v
v
T
Γ
ε
u
,
d
v
<
∞
.
(D4)
KernelΓ
ε
u
,
d
v satisfies the following conditions:(18)
a
ε
z
,
u
=
∫
R
d
v
Γ
ε
z
+
ε
u
,
d
v
=
a
z
+
ε
a
1
z
,
u
+
ε
δ
1
ε
,
B
ε
z
,
u
=
∫
R
d
v
v
T
Γ
ε
z
+
ε
u
,
d
v
=
B
z
+
ε
δ
2
ε
,
whereδ
1
ε
,
δ
2
ε
→
0, ε
→
0.
(D5)
Functionb
u satisfies the condition(19)
b
ρ
+
ε
u
=
b
ρ
+
ε
δ
3
ε
,
δ
3
ε
⟶
0
,
ε
⟶
0
.
(D6)
Convergence of the initial conditions is as follows:(20)
ξ
ε
0
⟶
ξ
0
,
sup
ε
>
0
⁡
E
ξ
ε
0
<
∞
.
Then weak convergence takes place inD
0
,
T, T
<
∞, as ε
↓
0:(21)
ξ
ε
⟹
ξ
0
,where ξ
0
t, t
≥
0, is the diffusion process with generator(22)
Γ
t
0
φ
u
=
C
1
u
,
ρ
t
φ
′
u
+
1
2
B
~
ρ
t
φ
′
′
u
,
C
1
u
,
ρ
t
≔
b
ρ
t
a
1
ρ
t
,
u
,
B
~
ρ
t
≔
b
ρ
t
σ
2
ρ
t
,
σ
2
u
≔
B
u
-
a
2
u
.Remark 3.
Boundary operator depends on the averaged evolutionρ; that is why it is advisable to consider weak convergence of two-component evolution ξ
ε
,
ρ
ε. But, according to [11, 18–20], we will prove theorem only for the process, which consists of parameter of series ε, in other words ξ
ε.The proof of Theorem2 consists of two steps.Step 1. Let us solve the problem of the singular perturbation for CO of process ξ
ε as ε
↓
0.Consider an evolution equation(23)
d
ρ
t
d
t
=
C
ρ
tthat corresponds with(24)
C
φ
v
=
C
v
φ
′
vand with semigroup(25)
C
t
φ
v
=
φ
v
+
∫
0
t
C
ρ
s
d
s
.By analogues, an evolution equation(26)
d
ρ
ε
t
d
t
=
-
ε
-
1
C
ρ
ε
tcorresponds with operator(27)
C
ε
φ
v
=
-
ε
C
v
φ
′
vwith semigroup(28)
C
t
ε
φ
v
=
φ
v
-
ε
-
1
∫
0
t
C
ρ
s
d
s
.Lemma 4.
CO of two-component processξ
ε
,
ρ
ε on test-functions φ
∈
C
R
2
d is given by(29)
Γ
ε
φ
u
,
v
=
ε
-
2
b
v
+
ε
u
∫
0
∞
F
v
+
ε
u
d
s
∫
R
d
C
ε
2
s
C
ε
2
s
ε
Δ
ε
z
φ
u
,
v
-
φ
u
,
v
Γ
ε
v
+
ε
u
,
d
z
,where(30)
Δ
ε
z
φ
u
,
v
≔
φ
u
+
ε
z
,
v
.Proof.
By Definition1, we got a relation for values of jumps and time of renewals:(31)
Δ
ξ
n
+
1
ε
=
ε
Δ
η
n
+
1
ε
-
ε
-
1
Δ
ρ
n
+
1
ε
,
τ
n
ε
=
ε
2
τ
n
.
Then(32)
Δ
ρ
n
+
1
ε
=
∫
0
ε
2
θ
n
+
1
C
ρ
ε
2
τ
n
+
h
d
h
.So, according to the condition ρ
n
ε
=
v we get(33)
φ
u
,
ρ
n
+
1
ε
=
φ
u
,
ρ
n
ε
+
Δ
ρ
n
+
1
ε
=
C
ε
2
θ
n
+
1
φ
u
,
v
.
For embedded chainξ
n
ε, n
≥
0, we get(34)
E
φ
ξ
n
+
1
ε
,
v
∣
ξ
n
ε
=
u
=
E
φ
ξ
n
ε
+
Δ
ξ
n
+
1
ε
,
v
∣
ξ
n
ε
=
u
=
E
φ
ξ
n
+
1
ε
,
v
∣
ξ
n
ε
=
u
=
E
φ
ξ
n
ε
+
Δ
η
n
+
1
ε
-
ε
-
1
∫
0
ε
2
θ
n
+
1
C
ρ
ε
2
τ
n
+
h
d
h
,
v
∣
ξ
n
ε
=
u
=
C
ε
2
s
ε
Δ
ε
θ
n
+
1
φ
u
,
v
.
Then calculate(35)
Γ
ε
φ
u
,
v
=
E
φ
ξ
n
+
1
ε
,
ρ
n
+
1
ε
-
φ
u
,
v
∣
ξ
n
ε
=
u
,
ρ
n
ε
=
v
E
θ
n
+
1
ε
∣
ξ
n
ε
=
u
=
ε
-
2
b
v
+
ε
u
E
φ
ξ
n
ε
+
Δ
ξ
n
+
1
ε
,
ρ
n
ε
+
Δ
ρ
n
+
1
ε
-
φ
u
,
v
∣
η
n
=
u
+
ε
v
.
So, finally we get representation of the compensating operator for two-component evolutionξ
ε
,
ρ
ε:(36)
Γ
ε
φ
u
,
v
=
ε
-
2
b
v
+
ε
u
∫
0
∞
F
v
+
ε
u
d
s
∫
R
d
C
ε
2
s
C
ε
2
s
ε
Δ
ε
z
φ
u
,
v
-
φ
u
,
v
Γ
ε
v
+
ε
u
,
d
z
,as we wanted to show.
Lemma4 is proved.Consider asymptotic behavior of CO, from Lemma4 as ε
↓
0.Lemma 5.
On test-functionsφ
u
,
v
∈
C
∞
R
2
d CO of the process ξ
ε
,
ρ
ε has asymptotic representation(37)
Γ
ε
φ
u
,
v
=
Γ
0
φ
u
,
v
+
R
ε
φ
u
,
v
,where Γ
0 is given by the following relation:(38)
Γ
0
φ
u
,
v
=
C
1
v
,
u
φ
u
′
u
,
v
+
1
2
B
v
φ
u
′
′
u
,
v
+
C
v
φ
v
′
u
,
vand for the negligible term,(39)
l
i
m
ε
↓
0
sup
u
,
v
∈
R
d
⁡
R
ε
φ
u
,
v
=
0is true.Proof.
Let us use an algebraic identity(40)
a
b
c
-
1
=
a
-
1
+
b
-
1
+
c
-
1
+
a
-
1
b
-
1
+
a
-
1
c
-
1
+
b
-
1
c
-
1
+
a
-
1
b
-
1
c
-
1
.
According to Lemma4 and algebraic identity (40) we get(41)
Γ
ε
φ
u
,
v
=
ε
-
2
b
v
+
ε
u
∫
0
∞
F
v
+
ε
u
d
s
∫
R
d
C
ε
2
s
-
I
+
C
ε
2
s
ε
-
I
+
Δ
ε
z
-
I
+
o
ε
φ
u
,
v
Γ
ε
v
+
ε
u
,
d
z
=
I
1
+
I
2
+
I
3
+
I
4
.
Let us use the equations for semigroups [9]:(42)
C
ε
2
s
-
I
=
C
∫
0
ε
2
s
C
ε
2
h
d
h
,
C
ε
2
s
ε
-
I
=
C
ε
∫
0
ε
2
s
C
ε
2
h
ε
d
h
.
Then for the termI
1, by semigroups properties and condition (D5), we get the following relation:(43)
I
1
=
ε
-
2
b
v
+
ε
u
∫
0
∞
F
v
+
ε
u
d
s
∫
R
d
C
∫
0
ε
2
s
C
ε
2
h
d
h
φ
u
,
v
Γ
ε
v
+
ε
u
,
d
z
=
b
v
+
ε
u
∫
0
∞
F
v
+
ε
u
d
s
∫
R
d
C
s
φ
u
,
v
Γ
ε
v
+
ε
u
,
d
z
=
C
v
φ
v
′
u
,
v
+
o
ε
.
Similarly forI
2
+
I
3 we get(44)
I
2
+
I
3
=
ε
-
2
b
v
+
ε
u
∫
0
∞
F
v
+
ε
u
d
s
∫
R
d
C
ε
2
s
ε
-
I
+
Δ
ε
z
-
I
φ
u
,
v
Γ
ε
v
+
ε
u
,
d
z
=
C
1
v
,
u
φ
u
′
u
,
v
+
1
2
B
v
φ
u
′
′
u
,
v
+
o
ε
.
From factorizationC
ε
2
s
C
ε
2
s
ε
Δ
ε
z
-
I, according to (40), terms C
ε
2
s
-
I, C
ε
2
s
ε
-
I
+
Δ
ε
z
-
I were considered. Γ
0 is built by these terms. It is easy to check that the sum of the rest of the terms is o
1 as ε
↓
0, if conditions (D3)–(D5) hold.
Using representation for semigroupsC
ε
2
s, C
ε
2
s
ε, and Δ
ε
z, it is easy to show that negligible term is o
1 as ε
↓
0.
Lemma5 is proved.Step 2. Let us show the relative compactness of the processes family ξ
ε as ε
>
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].Lemma 7.
There is an inequality(45)
Γ
ε
φ
u
,
v
≤
C
φfor test-functions φ
∈
C
0
3,2
R
2
d, where constant C
φ depends only on function φ.
For functionφ
0
u
,
v
=
1
+
u
2
+
v
2 there is a bound(46)
Γ
ε
φ
0
u
,
v
≤
C
l
φ
0
u
,
v
,
u
≤
l
,where constant C
l depends only on the function φ
0, independent from ε.Proof.
Let us use the result of Lemma5:(47)
Γ
ε
φ
u
,
v
≤
Γ
0
φ
u
,
v
+
R
ε
φ
u
,
v
≤
s
u
p
u
,
v
∈
R
2
d
b
v
a
1
u
,
v
φ
u
′
+
C
v
φ
v
′
+
sup
u
,
v
∈
R
2
d
⁡
1
2
b
v
B
v
+
∫
0
∞
s
2
F
v
d
s
C
2
v
-
2
C
v
a
v
φ
u
u
′
′
+
sup
u
,
v
∈
R
2
d
⁡
R
ε
φ
u
,
v
.
According to the definition of test-functionφ we have sup
u
,
v
∈
R
2
d
⁡
max
⁡
φ
,
φ
u
′
,
φ
v
′
,
φ
u
u
′
′
≤
K
<
∞. 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 b
u
<
c
<
∞.
Then(48)
Γ
ε
φ
u
,
v
≤
c
K
1
1
+
ε
<
C
φ
,where constant K
1
=
K
1
φ depends only on φ, C
φ
=
2
c
K
1.
To prove condition (46) the properties of the function φ
0 remain to be remembered; namely(49)
φ
0
u
′
≤
1
≤
φ
0
,
φ
0
v
′
≤
1
≤
φ
0
,
φ
0
u
u
′
≤
1
≤
φ
0
,
φ
0
v
v
′
≤
1
≤
φ
0
.
Lemma7 is proved.Lemma 8.
ξ
ε,
ε
>
0, is relatively compact family.Proof.
To determine the relative compactness of the familyξ
ε, ε
>
0, according to Theorem 1.4.6 [21] submartingality of the stochastic process α
ε
t
=
φ
ζ
ε
t
+
C
φ
t for nonnegative infinitely differentiable φ and for some constant C
φ and inequality (45) and (46) remains to be shown, where ζ
ε
t
=
ξ
ε
t
,
ρ
t.
Let us prove that stochastic processζ
ε
t is nonnegative submartingale relatively to the stream of σ-algebras F
t
ε
≔
σ
τ
+
ε
s
,
s
≤
t, τ
+
ε
t
=
τ
+
t
+
1:(50)
E
α
ε
t
-
α
ε
s
∣
F
t
ε
=
E
φ
ζ
ε
t
-
φ
ζ
ε
s
∣
F
t
ε
+
C
φ
t
-
s
=
E
∫
s
t
Γ
ε
φ
ζ
ε
u
d
u
∣
F
t
ε
+
C
φ
t
-
s
=
E
∫
τ
+
ε
s
τ
+
ε
t
Γ
ε
φ
ζ
ε
u
+
C
d
u
∣
F
t
ε
+
E
∫
s
τ
+
ε
s
+
∫
τ
+
ε
t
t
​
Γ
ε
φ
ζ
ε
u
+
C
d
u
∣
F
t
ε
+
C
t
-
τ
+
ε
t
-
s
+
τ
+
ε
s
.
Two last terms tend to 0 asε
↓
0. By Lemma 7
(51)
E
∫
τ
+
ε
s
τ
+
ε
t
Γ
ε
φ
ζ
ε
u
+
C
d
u
∣
F
t
ε
≥
0
.
Measurability of the processα
ε relatively to the stream F
ε is obvious. So, α
ε is nonnegative submartingale.
Lemma8 is proved.According to Lemma8  ζ
ε, ε
>
0, is a relatively compact family. To complete the proof of the theorem the family ζ
ε that converges to martingale remains to be shown. Consider stochastic processes:(52)
ζ
+
ε
t
≔
ζ
ε
τ
+
ε
t
,
ζ
τ
ε
t
≔
ζ
ε
τ
ε
t
,
t
≥
0
,
μ
t
ε
≔
φ
ζ
ε
t
-
∫
0
t
Γ
0
φ
ζ
ε
s
d
s
.Then(53)
E
μ
t
ε
=
E
φ
ζ
ε
t
-
∫
0
t
Γ
0
φ
ζ
ε
s
d
s
=
E
φ
ζ
ε
t
-
φ
ζ
+
ε
t
+
E
φ
ζ
+
ε
t
-
∫
0
t
Γ
ε
φ
ζ
+
ε
s
d
s
+
E
∫
t
τ
+
ε
t
Γ
0
φ
ζ
ε
s
d
s
+
E
∫
t
τ
+
ε
t
Γ
ε
φ
ζ
+
ε
s
-
φ
ζ
ε
s
d
s
+
E
∫
t
τ
+
ε
t
Γ
ε
φ
ζ
ε
s
-
Γ
0
φ
ζ
ε
s
d
s
.According to Lemma7 the third term satisfies the relation(54)
E
∫
t
τ
+
ε
t
Γ
0
φ
ζ
ε
s
d
s
⟶
0
.By the same way we can prove that the first and the fourth terms tend to 0, becauseφ is continuous.The last term tends to 0 by Lemma5, because(55)
lim
ε
↓
0
⁡
Γ
ε
φ
u
=
Γ
0
φ
uon test-functions φ, which have uniform bounded derivatives of any order.The second term is equal to(56)
ϑ
t
ε
≔
φ
ζ
+
ε
t
-
∫
0
t
Γ
ε
φ
ζ
+
ε
s
d
sand has a martingale condition according to Lemma 6.1 [11].Let us use its martingale condition:(57)
E
ϑ
t
ε
≔
E
ζ
ε
0
=
E
φ
ξ
ε
0
.Finally we have(58)
E
μ
t
ε
=
E
φ
ζ
ε
0
+
r
ε
,where r
ε
→
0 as ε
↓
0.Now from theorem’s condition (D6) we get(59)
E
μ
t
=
E
φ
ζ
0
;in another words, ζ 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 Lemma5 CO converges to the generator of diffusion process.Theorem2 is proved.


## 3. Numerical Example

Consider semi-Markov processη
t, t
≥
0, in R
1. For this process Γ
u
,
d
v has uniform distribution on u
-
1
/
2
,
u
+
1
/
2 and θ
n
+
1
∣
η
n
=
u has Bernoulli distribution with parameters 4
,
1
/
2. Then(60)
a
u
=
u
≠
0and average evolution ρ
t, t
>
0, has representation(61)
ρ
t
=
ρ
0
e
t
/
2
,where ρ
0 are initial condition. It is easy to verify the conditions of the theorem. Prelimited processes are shown in Figures 1 and 2 for ε
=
1
0
-
1
,
1
0
-
2.Figure 1
Diffusion approximation with epsilon = 0.1.Figure 2
Diffusion approximation with epsilon = 0.01.


## 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.



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*Source: 563060-2015-12-06.xml*