Stochastic impulsive processes on superposition of two renewal processes
Stochastic impulsive processes given by a sum of random variables on superposition of two renewal processes are considered on increasing time intervals. Algorithms of average, diffusion approximation and large deviation generators are realized in the series scheme with a small series parameter under...
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| Cite this: | Stochastic impulsive processes on superposition of two renewal processes / V.S. Koroliuk, R. Manca, G. D'Amico // Український математичний вісник. — 2014. — Т. 11, № 3. — С. 366-379. — Бібліогр.: 7 назв. — англ. |
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Koroliuk, V.S. Manca, R. D'Amico, G. 2017-09-26T16:20:04Z 2017-09-26T16:20:04Z 2014 Stochastic impulsive processes on superposition of two renewal processes / V.S. Koroliuk, R. Manca, G. D'Amico // Український математичний вісник. — 2014. — Т. 11, № 3. — С. 366-379. — Бібліогр.: 7 назв. — англ. 1810-3200 2010 MSC. 60J45, 60K05. https://nasplib.isofts.kiev.ua/handle/123456789/124466 Stochastic impulsive processes given by a sum of random variables on superposition of two renewal processes are considered on increasing time intervals. Algorithms of average, diffusion approximation and large deviation generators are realized in the series scheme with a small series parameter under suitable scalings. en Інститут прикладної математики і механіки НАН України Український математичний вісник Stochastic impulsive processes on superposition of two renewal processes Article published earlier |
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Stochastic impulsive processes given by a sum of random variables on superposition of two renewal processes are considered on increasing time intervals. Algorithms of average, diffusion approximation and large deviation generators are realized in the series scheme with a small series parameter under suitable scalings.
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Stochastic impulsive processes on superposition of two renewal processes / V.S. Koroliuk, R. Manca, G. D'Amico // Український математичний вісник. — 2014. — Т. 11, № 3. — С. 366-379. — Бібліогр.: 7 назв. — англ. |
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Український математичний вiсник
Том 11 (2014), № 3, 366 – 379
Stochastic impulsive processes on superposition
of two renewal processes
Vladimir S. Koroliuk, Raimondo Manca,
and Guglielmo D’Amico
Abstract. Stochastic impulsive processes given by a sum of random
variables on superposition of two renewal processes are considered on
increasing time intervals.
Algorithms of average, diffusion approximation and large deviation
generators are realized in the series scheme with a small series parameter
under suitable scalings.
2010 MSC. 60J45, 60K05.
Key words and phrases. Impulsive process, average, diffusion ap-
proximation, large deviation problem.
1. Introduction
The Stochastic Impulsive Process (SIP) given by a sum of random
variables on the Markov chains is described by the superposition on two
Renewal Processes (RP)
Sn = u+
n∑
k=1
αk(xk), n ≥ 0, S0 = u ∈ Rd (1.1)
The Markov chain xk, k ≥ 0 is defined by the Markov Renewal Process
(MRP) [1, ch. 1] on the space E = {±x; x > 0} with the sojourn times
θ±n (x) := θ±n ∧ x, x ∈ R+ = (0,+∞). Each of the two RP is defined by a
sum of positive i.i.d. random variables [2, S. 8.3]:
τ±n =
n∑
k=1
θ±k , n ≥ 0, τ±0 = 0, P±(t) = P{θ±k ≤ t}, t ≥ 0. (1.2)
Received 3.03.2014
ISSN 1810 – 3200. c⃝ Iнститут математики НАН України
V. S. Koroliuk, R. Manca, and G. D’Amico 367
The random variables (impulsive) α±
k (x), x ∈ R+ are given by the
distribution functions
G±(A) = P{α±
k (x) ∈ A}, A ∈ Rd.
The SIP is a particular case of the Random Evolution Process (REP)
[2].
In our previous work [3] the SIP was considered on the MRP with the
merging phase space Ê = {+,−}.
The increments of the SIP (1.1)
∆Sn = Sn − Sn−1 = α±
n , n ≥ 0, x ∈ R+
may be interpreted as a success α+
n , or as a failure α−
n . This is the natural
interpretation of the SIP in the risk theory [5].
The asymptotical behaviour of the SIP in the series scheme (average
and diffusion approximation [2]) and the scheme of asymptotically small
diffusion [4, 6] is considered.
The peculiarity of the MRP on the phase space E = {±x; x > 0}
is that the stationary distribution of the Markov chain xn, n ≥ 0 is
given in the explicit form (see Section 2). So, the algorithms of averaging
(Proposition 4.1) and diffusion approximation (Proposition 5.1) may be
realized effectively. Hence, the simplified models of the SIP may be used
in applications to the risk problems [5, S.6.5].
2. Superposition of two renewal processes
The renewal processes are defined by sum of positive valued random
variables, independent in common and identically distributed [1] (see also
[2, S. 8.3]):
τ±n =
n∑
k=1
θ±k , n ≥ 0, τ±0 = 0, (2.1)
P±(t) = P{θ±k ≤ t}, t ≥ 0, P±(0) = 0.
The renewal processes can be given by the counting processes:
ν±(t) := max{n > 0 : τ±n ≤ t}, t ≥ 0.
The superposition of two renewal processes (2.1) is defined by the count-
ing process
ν(t) = ν+(t) + ν−(t), t ≥ 0. (2.2)
368 Stochastic impulsive processes...
The superposition of two renewal processes (2.2) can be characterized
by the Markov Renewal Process (MRP) [1, ch. 1]
xn, θn, n ≥ 0,
given on the phase space
E = {±x; x > 0}, (2.3)
with the sojourn times
θ±n (x) = θ±n ∧ x, x ∈ R+ = (0,+∞).
The symbols + or − in (2.3) are fixed the renewal moment of one or
another renewal processes (2.1). The continuous component x is fixed
the remainder time up to the renewal moment other renewal process in
(2.2).
The embedded Markov chain xn, n ≥ 0, is given by the matrix of the
transition probabilities
P (x, dy) =
[
P+(x− dy) P+(x+ dy)
P−(x+ dy) P−(x− dy)
]
(2.4)
The specific property of the embedded Markov chain with the transi-
tion probabilities (2.4) is existence of the stationary distribution with the
densities
ρ±(x) = ρP∓(x), P∓(x) := 1− P∓(x),
ρ = (p+ + p−)
−1, p± :=
∞∫
0
P±(x) dx.
(2.5)
The stationary distribution on merged phase space Ê = {+,−}, is given
by
ρ± = ρp∓ = λ±/λ, λ± = 1/p±, λ = λ+ + λ−.
3. Storage Impulsive Process
The SIP on superposition of two renewal processes (2.2) is defined by
the sum of random variables take values in Euclidean space Rd, d ≥ 1
Sn = u+
n∑
k=1
αk(xk), n ≥ 0, S0 = u ∈ Rd. (3.1)
The random variables α±
k (x), k ≥ 1, x ∈ R+, are given by the distribution
functions on (Rd,Rd)
G±(A) = P{α±
k (x) ∈ A}, A ∈ Rd.
V. S. Koroliuk, R. Manca, and G. D’Amico 369
Example 3.1. The risk process (3.1) constructed by the (positive) input
random variables α+
k > 0, in the renewal moments of the renewal process
ν+(t), t ≥ 0, and by (negative) output random variables −α−
k > 0, in the
renewal moments of the renewal process ν−(t), t ≥ 0 that is
S(t) = u+
ν+(t)∑
k=1
α+
k −
ν−(t)∑
k′=1
α−
k′ .
The SIP (3.1) can be characterized by the generator of the two compo-
nents Markov chain
Sn, xn, n ≥ 0. (3.2)
Lemma 3.1. The two component Markov chain (3.2) is characterized
by the generator given on the vector test function φ(u, x) = (φ+(u, x),
φ−(u, x)):
Lφ(u, x) = PGφ(u, x)− φ(u, x), (3.3)
where the operator P is given by the matrix (2.4) and
Gφ(u) =
[
G+ 0
0 G−
](
φ+(u)
φ−(u)
)
= (G+φ+(u),G−φ−(u)),
G±φ±(u) :=
∫
Rd
G±(dv)φ±(u+ v)
(3.4)
Remark 3.1. The generator (3.3) can be represented in scalar form:
L±φ(u, x) =
∫
R
G±(dv)
x∫
0
P±(dt)φ±(u+ v, x− t)
+
∫
R
G∓(dv)
∞∫
x
P±(dt)φ∓(u+ v, t− x)− φ±(u, x). (3.5)
Proof of Lemma 3.1. The conditional expectation
Lφ(u, x) = E[φ(Sn+1, xn+1)− φ(u, x)|Sn = u, xn = ±x]
is calculated directly:
L±φ(u, x) = E[φ(u+ αn+1, xn+1)− φ(u, x)]
= G±P±φ±(u, x) +G∓P±φ∓(u, x),
370 Stochastic impulsive processes...
where by definition
P±φ(x) :=
x∫
0
P±(dt)φ(x− t), P±φ(x) :=
∞∫
x
P±(dt)φ(t− x).
Remark 3.2. The generator (3.3) can be transformed as follows:
Lφ(u, x) = Qφ(·, x) +P[G− I]φ(u, x),
where by definition
Q := P− I,
is the generator of the embedded Markov chain xn, n ≥ 0, given by the
transition probabilities (2.4).
The two component Markov chain, given by the generator (3.3), is
characterized by the martingale with respect to the standard σ-algebras
Fn := σ{(Sk, xk), 0 ≤ k ≤ n}
µn+1 = φ(Sn+1, xn+1)− φ(u, x)−
n∑
k=1
Lφ(Sk, xk). (3.6)
The martingale characterization (3.6) of the SIP (3.1) will be used in
asymptotical analysis on increasing time intervals in the series scheme
with the small parameter series ε→ 0 (ε > 0).
4. SIP in the average scheme
The SIP in the series scheme with the small parameter series ε →
0 (ε > 0), is considered in the following scaling:
Sε(t) = u+ ε
[t/ε]∑
k=1
αk(xk), t ≥ 0, ε > 0, u ∈ Rd. (4.1)
The averaging behavior of the SIP is analyzed by using a martingale
characterization (3.6).
Lemma 4.1. The normalized SIP (4.1) can be characterized by the mar-
tingale
µε(t) = φ(Sε(t), xε(t))− φ(Sε(0), xε(0))−
ε[t/ε]∫
0
Lεφ(Sε(h), xε(h)) dh.
V. S. Koroliuk, R. Manca, and G. D’Amico 371
The compensating generator is such that
Lεφ(u, x) = ε−1Qφ(·, x) +P(x)Gεφ(u, x), (4.2)
where
P(x) =
[
P+(x) P+(x)
P−(x) P−(x)
]
, P±(x) = 1− P±(x)
Gε =
[
Gε
+ 0
0 Gε
−
]
, Gε
±φ(u) = ε−1
∫
Rd
G±(dv)[φ(u+ εv)− φ(u)].
(4.3)
Lemma 4.2. The generator (4.2)–(4.3) admits the following asymptotic
representation
Lεφ(u, x) = ε−1Qφ(·, x) +P(x)Gφ(u, x) + δεl (x)φ(u, x) (4.4)
with the negligible term
|δεl (x)φ(u)| → 0, ε→ 0, φ(u) ∈ C2(Rd).
The operator is defined as follows:
G =
[
G+ 0
0 G−
]
, G±φ(u) = g±φ
′(u), g± :=
∫
Rd
vG±(dv). (4.5)
Proof of Lemma 4.2 follows from the Taylor expansion applied to the
formula (4.3).
Proposition 4.1. The finite-dimensional distributions of the SIP (4.1)
in the average scheme converges to
Sε(t) ⇒ S0(t) = u+ ĝt, t ≥ 0, ε→ 0, (4.6)
where the velocity
ĝ = ρ+g+ + ρ−g− = (λ+g+ + λ−g−)/λ.
Proof. The generator of the normalized SIP (4.4)–(4.5) has the singular
perturbation form. The singular perturbed operator Q = P − I is re-
ducible invertible [2, ch. 5] with the projector on the null-space given by
the stationary distribution (2.5):
Πφ(x) = ρ
[ ∞∫
0
P−(x)φ+(x) dx+
∞∫
0
P+(x)φ−(x) dx
]
.
372 Stochastic impulsive processes...
To get the limit operator the algorithm of solving of perturbation problem
[2, ch. 5] can be used.
The generator (4.4)–(4.5) is considered on the perturbed test function
φε(u, x) = φ(u) + εφ1(u, x), φ(u) := (φ(u), φ(u)).
Let’s calculate:
Lεφε(u, x) = ε−1Qφ(u) + [Qφ1 +P(x)Gφ(u)] + δεl (x)φ(u), (4.7)
with the negligible term
|δεl (x)φ(u)| → 0, ε→ 0, φ(u) ∈ C2(Rd).
The first term in (4.7) equals zero, because φ(u) is a constant for the
generator Q. The next term in (4.7) gives us a problem of singular per-
turbation
Qφ1(u, x) +P(x)Gφ(u) = L0φ(u). (4.8)
The limit operator L0 is determined by using solvability condition for
the equation (4.8) (see [2, ch. 5])
L0Π = ΠP(x)GΠ. (4.9)
Let’s calculate (4.9) taking in mind (4.4)–(4.5):
L0 = ρ
[
G+
∞∫
0
P−(x)P+(x) dx+G−
∞∫
0
P−(x)P+(x) dx
+G−
∞∫
0
P+(x)P−(x) dx+G+
∞∫
0
P+(x)P−(x) dx
]
= ρ[Gg+p− +G−p+] = ρ+G+ + ρ−G−
= (λ+G+ + λ−G−)/λ = ρ+G+ + ρ−G− = Ĝ.
That is the limit generator
L0φ(u) = ĝφ′(u),
defines the deterministic drift in (4.6)
S0(t) = u+ ĝt, t ≥ 0.
V. S. Koroliuk, R. Manca, and G. D’Amico 373
5. SIP in the diffusion approximation scheme
The SIP in the series scheme with the small parameter series ε → 0
(ε > 0), is considered in the following scaling:
Sε(t) = u+ ε
[t/ε2]∑
k=1
αk(xk), t ≥ 0, u ∈ Rd, (5.1)
under the additional Balance Condition (BC):
ĝ = ρ+g+ + ρ−g− = 0. (5.2)
Proposition 5.1. The SIP (5.1) under BC (5.2) converges weakly
Sε(t) ⇒ wσ(t), ε→ 0.
The limit Brownian motion wσ(t), t ≥ 0, is defined by the variance
σ2 = σ2b + σ2g , (5.3)
σ2b = ρ+B+ + ρ−B−, B± =
∫
Rd
v2G±(dv), (5.4)
σ2g = 2ΠP(x)GR0P(x)GΠ. (5.5)
The potential operator R0 is defined by a solution of the equation
QR0 = R0Q = Π− I.
Remark 5.1. The component of variance (5.5) can be calculated by us-
ing the reducible inverse operator R0 to the generator Q of the embedded
Markov chain. That is an open problem (see [3]).
Proof of Proposition 5.1. As in previous section 3 the martingale charac-
terization of the SIP (5.1) is a starting point in asymptotic analysis.
Lemma 5.1. The normalized SIP (5.1) can be characterized by the mar-
tingale
µε(t) = φ(Sε(t), xε(t))− φ(u, x)−
ε2[t/ε2]∫
0
Lεφ(Sε(h), xε(h)) dh. (5.6)
The generator L of the two component Markov chain
Sεn = Sε(τ εn), xεn = xε(τ εn), n ≥ 0,
374 Stochastic impulsive processes...
is represented by
Lεφ(u, x) = [ε−2Q+P(x)Gε]φ(u, x), (5.7)
Gε =
[
Gε
+ 0
0 Gε
−
]
, Gε
±φ(u) =
∫
Rd
G±(dv)[φ(u+ εv)− φ(u)]. (5.8)
Lemma 5.2. The generator (5.7)–(5.8) admit the asymptotic represen-
tation on the smooth enough test function φ(u) ∈ C3(Rd) :
Lεφ(u, x) = ε−2Qφ(·, x) + ε−1P(x)Gφ(u, x)
+P(x)Bφ(u, x) + δεe(x)φ(u, x), (5.9)
with the negligible term
|δεe(x)φ(u, ·)| → 0, ε→ 0, φ(u, ·) ∈ C3(Rd).
Here
B =
[
B+ 0
0 B−
]
, B±φ(u) :=
1
2
B±φ
′′(u).
Now a solution of singular perturbation problem [2, ch. 5, Proposi-
tion 5.2] is used on the perturbed test function
φε(u, x) = φ(u) + εφ1(u, x) + ε2φ2(u, x). (5.10)
Lemma 5.3. The generator (5.9) on the perturbed test function (5.10)
admit the asymptotic representation
Lεφε(u, x) = L0φ(u) + δεe(x)φ(u),
with the negligible term
|δεe(x)φ(u)| → 0, ε→ 0, φ(u) ∈ C3(Rd).
The limit generator
L0φ(u) =
1
2
σ2φ′′(u),
is the generator of the Brownian motion wσ(t), t ≥ 0, with the variance
(5.3)–(5.5).
Proof. Let’s calculate
Lεφε(u, x) = ε−2Qφ(u) + ε−1[Qφ1(u, x) +P(x)Gφ(u)]
+
[
Qφ2 +P(x)Gφ1(u, x) +
1
2
B(x)φ(u)
]
+ δεe(x)φ(u),
V. S. Koroliuk, R. Manca, and G. D’Amico 375
with the negligible term
|δεe(x)φ(u)| → 0, ε→ 0, φ(u) ∈ C3(Rd).
The BC (5.2) can be represented as follows:
ΠP(x)GΠ = 0.
Hence there exists solution of the equation
Qφ1(u, x) +P(x)Gφ(u) = 0,
that is
φ1(u, x) = R0G(x)φ(u).
Now the next equation
Qφ2 +
[
P(x)GR0P(x)G+
1
2
B(x)
]
φ(u) = L0φ(u),
gives the limit generator
L0φ(u) =
1
2
σ2φ′′(u),
by using solvability condition:
L0φ(u) =
[
ΠP(x)GR0P(x)GΠ+
1
2
ΠB(x)Π
]
φ(u) =
1
2
σ2φ′′(u).
6. Large deviation problem
The SIP in the scheme of asymptotically small diffusion is considered
in the following scaling:
Sε(t) = u+ ε2
[t/ε3]∑
k=1
αk(xk), t ≥ 0, u ∈ Rd (6.1)
under additional BC (5.2).
Proposition 6.1. The large deviation problem for SIP (6.1) under the
BC (5.2) is realized by the exponential generator of asymptotically small
diffusion
Hφ(u) =
1
2
σ2[φ′(u)]2. (6.2)
The variance is defined in (5.3)–(5.5).
Proof. The SIP (6.1) can be characterized by the exponential martingale
[4, Part I].
376 Stochastic impulsive processes...
Lemma 6.1. The two component Markov process Sε(t), xε(t) := x[t/ε3],
t ≥ 0, is characterized by the exponential martingale
exp{φ(Sε(t), xε(t))/ε− φ(Sε(0), xε(0))/ε
− ε−1
ε3[t/ε3]∫
0
Hε(Sε(h), xε(h)) dh} = µε(t)
is Ft(S, x)-martingale.
The exponential generator
Hεφ(u, x) = ε−2 ln[e−φ(u,x)/εLεeφ(u,x)/ε + 1]. (6.3)
The compensative generator
Lεφ(u, x) = [Q+P(x)Gε]φ(u, x),
P(x) =
[
P+(x) P+(x)
P−(x) P−(x)
]
, Gε =
[
Gε
+(x) 0
0 Gε
−(x)
]
, (6.4)
Gε
±φ(u) =
∫
Rd
G±(dv)[φ(u+ ε2v)− φ(u)].
Note that the generators Gε
± admit the asymptotic representation
Gε
±φ(u) = ε2g±φ
′(u) + ε2δεgφ(u), (6.5)
with the negligible form |δεgφ| → 0, ε→ 0, φ ∈ C2(Rd).
Note the exponential generator (6.3)–(6.5) is considered on the per-
turbing test function
φε(u, x) = φ(u) + ε ln[1 + εφ1(u, x) + φ2(u, x)]. (6.6)
Lemma 6.2. The compensating operator (6.4) on the perturbing test
function (6.6) admits the asymptotic representation
Hε
Lφ
ε(u, x) := e−φ
ε/εLεeφ
ε/ε = ε[Qφ1 + P (x)Gφ]
+ ε2[Qφ2 − φ1Qφ1 +Hb(x)φ] + δεl φ, (6.7)
with the negligible term |δεl φ| → 0, ε→ 0, φ ∈ C3(Rd).
V. S. Koroliuk, R. Manca, and G. D’Amico 377
Proof. The asymptotic representation (6.7) is the consequence of the
following asymptotic relations:
e−φ
ε/εQeφ
ε/ε = e−φ/ε[1 + εφ1 + ε2φ2]
−1Q[1 + εφ1 + ε2φ2]e
φ/ε
= εQφ1 + ε2[Qφ2 − φ1Qφ1] + ε2δεqφ,
e−φ
ε/εP(x)Geφ
ε/ε = e−φ/ε[1+εφ1+ε
2φ2]
−1P(x)G[1+εφ1+ε
2φ2]e
φ/ε
= εP(x)Gφ+ ε2[P(x)Gφ1 + P (x)Hb(x)φ] + δεgφ,
with the negligible terms δεqφ and δεgφ.
Now the singular perturbation problems are used for the equations
Qφ1 +P(x)Gφ = 0, Qφ2 − φ1Qφ1 +Hb(x)φ = Hφ. (6.8)
The first equation (6.8) under the BC (5.2) has the solution
φ1 = R0P(x)Gφ, Qφ1 = −P(x)Gφ.
Hence
−φ1Qφ1 = Hg(x)φ(u) :=
1
2
σ2g(x)[φ
′(u)]2. (6.9)
The second equation (6.8) with (6.9) is transformed to
Qφ2 + [Hg(x) +Hb(x)]φ(u) = Hφ(u). (6.10)
The right part side in (6.10) is determined by the solvability condition:
H = Π[Hg(x) +Hb(x)]Π. (6.11)
Corollary 6.1. The exponential generator (6.3) admits the asymptotic
representation
Hεφε(u, x) = Hφ(u) + δεhφ(u), (6.12)
with the negligible term |δεhφ| → 0, ε→ 0, φ ∈ C3(Rd).
The proof of Proposition 6.1 is finished as follows. We consider (see
(6.7))
Hεφε(u, x) = ε−2 ln[1 +Hε
Lφ
ε(u, x)]
= ε−2 ln[1 + ε2Hφ(u) + ε2δεl φ(u)] = Hφ(u) + δεhφ,
with the negligible term δεhφ, φ ∈ C3(Rd). The limit exponential gener-
ator H is calculated in (6.11) using the representation (see (5.3)–(5.5))
Hg(x)φ(u) =
1
2
σ2g(x)[φ
′(u)]2, σ2g = 2ΠGR0P(x)G1,
Hb(x)φ(u) =
1
2
σ2b (x)[φ
′(u)]2, σ2b = ΠP(x)B1.
378 Stochastic impulsive processes...
7. Conclusion
This paper contains the three simplified approximation schemes for
the SIP given by a sum (1.1) of random variables defined on the Markov
renewal process: average (Section 4), diffusion approximation (Section 5)
and the scheme of asymptotically small diffusion (Section 6).
The considered simplification schemes may be effectively used in ap-
plications, partially in financial mathematics [7]. The initial SIP (1.1) or
(3.1), defined by two distribution functions G±(A) of jumps and two dis-
tribution functions P±(t), given the renewal processes, may be simplified
in the average scheme by the deterministic drift process (4.6), given by
the four constants ρ± and g± which are the first moments of the renewal
times (ρ±) and the average jumps (g±).
The fluctuations of the SIP on increasing time intervals in diffusion
approximation scheme (Section 4) is described by the limit Brownian
motion (Proposition 5.1) given by the variance (5.3)–(5.5) defined by the
first two moments of jumps (g± and B±). The scheme of asymptotically
small diffusion (Section 6) represents the exponential generator of large
deviations used in the analysis of asymptotically small probabilities (see
[4]).
The initial definition of the SIP given in Section 2 may be easily
interpreted as the logistic problem. The positive jumps α+
k (x), k ≥ 0
are interpreted as a profits and the negative jumps α−
k (x), k ≥ 0 are the
losses.
The SIP on increasing time intervals may be approximated by the
well-known emery drift (Section 3) and in addition by the Brownian mo-
tion process of fluctuation.
References
[1] V. S. Korolyuk, V. V. Korolyuk, Stochastic Models of Systems, Kluwer Academic
Publisher, 1999.
[2] V. S. Koroliuk, and N. Limnios, Stochastic Systems in Merging Phase Space,
Singapore: VSP, 2005.
[3] V. S. Koroliuk, R. Manca, G. D’Amico, Storage impulsive processes in merging
phase space // Journal of Mathematical Sciences, 196 (2013), No. 5, 644–651.
[4] J. Feng and T. G. Kurtz, Large Deviation for Stochastic Processes, AMS, 131,
RI, 2006.
[5] W. Feller, Introduction to Probability Theory and its Applications, Vol. 2, NY:
Willey, 1966.
[6] V. S. Koroliuk, Random evolutions with locally independent increments on in-
creasing time intervals // Journal of Mathematical Sciences, 179 (2011), No. 2,
273–289.
[7] A. N. Shyryaev, Essentials of Stochastic Finance: Facts, Models, Theory, World
Scientific, 1999.
V. S. Koroliuk, R. Manca, and G. D’Amico 379
Contact information
Vladimir S.
Koroliuk
Institute of Mathematics of the NASU,
3, Tereshchenkivska Str.,
Kyiv, 01601,
Ukraine
E-Mail: vskorol@yahoo.com
Raimondo Manca Faculty of Economics,
Sapienza University of Rome
Via del Castro Laurenziano, 9
00161 - Rome,
Italy
E-Mail: raimondo.manca@uniroma1.it
Guglielmo D’Amico Department of Pharmacy,
University “G. d’Annunzio”
of Chieti-Pescara,
Via dei Vestini - 66013 Chieti,
Italy
E-Mail: g.damico@unich.it
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