Limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case
The asymptotic behavior of the general type third order non-autonomous oscillating system under the action of small non-linear random periodic perturbations of "white" and "Poisson" types in resonance case is investigated.
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| Цитувати: | Limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case / O.D. Borysenko, O.V. Borysenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 17-26. — Бібліогр.: 8 назв.— англ. |
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irk-123456789-45652009-12-08T12:00:24Z Limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case Borysenko, O.D. Borysenko, O.V. The asymptotic behavior of the general type third order non-autonomous oscillating system under the action of small non-linear random periodic perturbations of "white" and "Poisson" types in resonance case is investigated. 2008 Article Limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case / O.D. Borysenko, O.V. Borysenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 17-26. — Бібліогр.: 8 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4565 en Інститут математики НАН України |
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The asymptotic behavior of the general type third order non-autonomous oscillating system under the action of small non-linear random periodic perturbations of "white" and "Poisson" types in resonance case is investigated. |
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Article |
| author |
Borysenko, O.D. Borysenko, O.V. |
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Borysenko, O.D. Borysenko, O.V. Limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case |
| author_facet |
Borysenko, O.D. Borysenko, O.V. |
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Borysenko, O.D. |
| title |
Limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case |
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Limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case |
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Limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case |
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Limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case |
| title_full_unstemmed |
Limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case |
| title_sort |
limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case |
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Інститут математики НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/4565 |
| citation_txt |
Limit behavior of non-autonomous random oscillating system of third order under random periodic external disturbances in resonance case / O.D. Borysenko, O.V. Borysenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 17-26. — Бібліогр.: 8 назв.— англ. |
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AT borysenkood limitbehaviorofnonautonomousrandomoscillatingsystemofthirdorderunderrandomperiodicexternaldisturbancesinresonancecase AT borysenkoov limitbehaviorofnonautonomousrandomoscillatingsystemofthirdorderunderrandomperiodicexternaldisturbancesinresonancecase |
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2025-07-02T07:46:40Z |
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2025-07-02T07:46:40Z |
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1836520466777374720 |
| fulltext |
Theory of Stochastic Processes
Vol.14 (30), no.3-4, 2008, pp.17-26
OLEKSANDR D. BORYSENKO AND OLGA V. BORYSENKO
LIMIT BEHAVIOR OF NON-AUTONOMOUS
RANDOM OSCILLATING SYSTEM OF THIRD
ORDER UNDER RANDOM PERIODIC
EXTERNAL DISTURBANCES
IN RESONANCE CASE
The asymptotic behavior of the general type third order non-autono-
mous oscillating system under the action of small non-linear random
periodic perturbations of “white”and “Poisson” types in resonance
case is investigated.
1. Introduction
The asymptotic behavior of the general type third order non-autonomous
oscillating system under the action of small non-linear random periodic per-
turbations of ”white” and ”Poisson” types in the non-resonance case is inves-
tigated in O.D.Borysenko, O.V.Borysenko [1]. The overview of papers de-
voted to the averaging method, proposed by N.M.Krylov, N.N.Bogolyubov
[2], and its applications to random oscillatory systems of different types is
presented in O.D.Borysenko, O.V.Borysenko [3] with corresponding refer-
ences.
In this paper we will investigate the behaviour, as ε→ 0, of the general
type third order non-autonomous oscillating system driven by stochastic
differential equation
x′′′(t) + ax′′(t) + b2x′(t) + ab2x(t) =
= εk0f0(μ0t, x(t), x
′(t), x′′(t)) + fε(t, x(t), x
′(t), x′′(t))
(1)
2000 Mathematics Subject Classifications. 60H10
Key words and phrases. Asymptotic behavior, third order autonomous oscillating
system, small non-linear random perturbations, resonance case
17
18 O.D.BORYSENKO AND O.V.BORYSENKO
with non-random initial conditions x(0) = x0, x
′(0) = x′0, x
′′(0) = x′′0, where
ε > 0 is a small parameter, fε(t, x, x
′, x′′) is a random function such that
t∫
0
fε(s, x(s), x
′(s), x′′(s)) ds =
=
m∑
j=1
εkj
t∫
0
fj(μjs, x(s), x
′(s), x′′(s)) dwj(s)+
+εkm+1
t∫
0
∫
R
fm+1(μm+1s, x(s), x
′(s), x′′(s), z) ν̃(ds, dz),
kj > 0, j = 0, m+ 1; a > 0, b > 0; fj, j = 0, m+ 1 are non-random
functions, periodic on μjt, j = 0, m+ 1 with period 2π; {wj(t), j = 1, m}
are independent one-dimensional Wiener processes; ν̃(dt, dy) = ν(dt, dy) −
Π(dy)dt, Eν(dt, dy) = Π(dy)dt, ν(dt, dy) is the Poisson measure indepen-
dent on wj(t), j = 1, m; Π(A) is a finite measure on Borel sets in R.
We will consider the equation (1) as a system of stochastic differential
equations
dx(t) = x′(t)dt
dx′(t) = x′′(t)dt
dx′′(t) = [−ax′′(t) − b2x′(t) − ab2x(t)+
+ εk0f0(μ0t, x(t), x
′(t), x′′(t))]dt+
+
m∑
j=1
εkjfj(μjt, x(t), x
′(t), x′′(t))dwj(t)+
+εkm+1
∫
R
fm+1(μm+1t, x(t), x
′(t), x′′(t), z)ν̃(dt, dz),
x(0) = x0, x
′(0) = x′0, x
′′(0) = x′′0.
(2)
In what follows we will use the constant K > 0 for the notation of different
constants, which are not depend on ε.
From Borysenko O. and Malyshev I. [4], using the obvious modifications
we obtain following results
Lemma. Let for each x ∈ Rd there exists
lim
T→∞
1
T
∫ T+A
A
f(t, x) dt = f̄(x)
uniformly with respect to A, the function f̄(x) is bounded, continuous, func-
tion f(t, x) is bounded and continuous in x uniformly with respect to (t, x) in
any region t ∈ [0,∞), |x| ≤ K, and stochastic processes ξ(t) ∈ Rd, η(t) ∈ R
are continuous, then
lim
ε→0
∫ t
0
f
(s
ε
+ η(s), ξ(s)
)
ds =
∫ t
0
f̄(ξ(s)) ds
LIMIT BEHAVIOR 19
almost surely for all arbitrary t ∈ [0, T ].
Remark. Let f(t, x, z) is bounded and uniformly continuous in x with
respect to t ∈ [0,∞) and z ∈ R in every compact set |x| ≤ K, x ∈ Rd. Let
Π(·) be a finite measure on the σ-algebra of Borel sets in R and let
lim
T→∞
1
T
∫ T+A
A
f(t, x, z) dt = f̄(x, z),
uniformly with respect to A for each x ∈ Rd, z ∈ R, where f̄(x, z) is
bounded, uniformly continuous in x with respect to z ∈ R in every compact
set |x| ≤ K. Then for any continuous processes ξ(t) ∈ Rd and η(t) ∈ R we
have
lim
ε→0
∫ t
0
∫
R
f
(s
ε
+ η(s), ξ(s), z
)
Π(dz)ds =
∫ t
0
∫
R
f̄(ξ(s), z) Π(dz)ds.
2. Main result
We will study the resonance case: μj =
pj
qj
· b for some j = 0, m+ 1,
where pj and qj are relatively prime integers. Let us consider the following
representation of processes x(t), x′(t), x′′(t):
x(t) = C(t) exp{−at} + A1(t) cos(bt) + A2(t) sin(bt),
x′(t) = −aC(t) exp{−at} − bA1(t) sin(bt) + bA2(t) cos(bt),
x′′(t) = a2C(t) exp{−at} − b2A1(t) cos(bt) − b2A2(t) sin(bt),
N(t) = C(t) exp{−at}.
Then
N(t) =
b2x(t) + x′′(t)
a2 + b2
,
A1(t) = cosα cos(bt+ α)x(t) − sin bt
b
x′(t) − sinα sin(bt+ α)
b2
x′′(t),
A2(t) = cosα sin(bt+ α)x(t) +
cos bt
b
x′(t) +
sinα cos(bt+ α)
b2
x′′(t),
where α = arctg (b/a). We can apply Ito formula [5] to stochastic process
ξ(t) = (N(t), A1(t), A2(t)) and obtain for the process ξ(t) the system of
stochastic differential equations
dN(t) = −aN(t) dt+
1
a2 + b2
dM(t),
dA1(t) = −sinα sin(bt+ α)
b2
dM(t), dA2(t) =
sinα cos(bt+ α)
b2
dM(t),
20 O.D.BORYSENKO AND O.V.BORYSENKO
dM(t) = εk0 f̃0(μ0t, N(t), A1(t), A2(t), t)dt+ (3)
+
m∑
j=1
εkj f̃j(μjt, N(t), A1(t), A2(t), t)dwj(t)+
+εkm+1
∫
R
f̃m+1(μm+1t, N(t), A1(t), A2(t), t, z)ν̃(dt, dz)],
N(0) =
b2x0 + x′′0
a2 + b2
, A1(0) =
a2x0 − x′′0
a2 + b2
, A2(0) =
ax′′0 + (a2 + b2)x′0 + ab2x0
b(a2 + b2)
,
where f̃j(μjt, N,A1, A2, t) =
fj(μjt, N + A1 cos bt + A2 sin bt,−aN − bA1 sin bt + bA2 cos bt, a2N −
b2A1 cos bt − b2A2 sin bt), j = 0, m, f̃m+1(μm+1t, N,A1, A2, t, z) =
fm+1(μm+1t, N + A1 cos bt + A2 sin bt,−aN − bA1 sin bt + bA2 cos bt, a2N −
b2A1 cos bt− b2A2 sin bt, z).
Theorem. Let Π(R) < ∞, t ∈ [0, t0], k = min(k0, 2kj, j = 1, m+ 1). Let
us suppose, that functions fj , j = 0, m+ 1 bounded and satisfy Lipschitz
condition on x, x′, x′′. If given below matrix σ̄2(A1, A2) is positive definite,
then:
1. Let μj =
pj
qj
· b, for j = 0, m+ 1, where pj and qj some relatively
prime integers. If k0 = 2kj, j = 1, m+ 1, then the stochastic process
ξε(t) = ξ(t/εk) weakly converges, as ε → 0, to the stochastic process ξ̄(t) =
(0, Ā1(t), Ā2(t)), where Ā(t) = (Ā1(t), Ā2(t)) is the solution of the system of
stochastic differential equations
dĀ(t) = ᾱ(Ā(t))dt+ σ̄(Ā(t))dw̄(t), (4)
Ā(0) = (A1(0), A2(0)),
where ᾱ(Ā) = (ᾱ(1)(A1, A2), ᾱ
(2)(A1, A2)),
ᾱ(1)(A1, A2) = − 1
4π2b(a2 + b2)
×
∑
p0n+q0l=0
2π∫
0
2π∫
0
f̂0(ψ,A1, A2, t)(a sinψ + b cosψ)e−i(nψ+lt) dt dψ,
ᾱ(2)(A1, A2) =
1
4π2b(a2 + b2)
×
∑
p0n+q0l=0
2π∫
0
2π∫
0
f̂0(ψ,A1, A2, t)(a cosψ − b sinψ)e−i(nψ+lt) dt dψ,
LIMIT BEHAVIOR 21
σ̄(A1, A2) =
{
B̄(A1, A2)
} 1
2 =
⎧⎨⎩ 1
4π2b2(a2 + b2)2
×⎡⎣ m∑
j=1
∑
pjn+qj l=0
2π∫
0
2π∫
0
f̂ 2
j (ψ,A1, A2, t)B(ψ)e−i(nψ+lt) dt dψ+
∑
pm+1n+qm+1l=0
2π∫
0
2π∫
0
∫
R
f̂ 2
m+1(ψ,A1, A2, t, z)B(ψ)e−i(nψ+lt) Π(dz) dt dψ
⎤⎦⎫⎬⎭
1
2
,
B(ψ) = (Bij(ψ), i, j = 1, 2), B11(ψ) = (a sinψ + b cosψ)2,
B12(ψ) = B21(ψ) = −(a sinψ + b cosψ)(a cosψ − b sinψ),
B22(ψ) = (a cosψ − b sinψ)2,
f̂j(ψ,A1, A2, t) = f̃j(ψ, 0, A1, A2, t), j = 0, m
f̂m+1(ψ,A1, A2, t, z) = f̃m+1(ψ, 0, A1, A2, t, z),
w̄(t) = (w̄j(t), j = 1, 2), w̄j(t), j = 1, 2 – independent one-dimensional
Wiener processes.
2. If k < k0 then in the averaging equation (4) we must put f̂0 ≡ 0; if
k < 2kj for some 1 ≤ j ≤ m + 1, then in the averaging equation (4) we
must put f̂j ≡ 0 for all such j.
3. If μj
= pj
qj
· b for some j = 0, m+ 1 and arbitrary relatively prime in-
tegers pj and qj, then in averaging coefficients in (4) we must put l = n = 0
in corresponding sums containing f̂j.
Proof. Let us make a change of variable t→ t/εk in equation (3) and obtain
for the process ξε(t) = (Nε(t), A
ε
1(t), A
ε
2(t)) = (N(t/εk), A1(t/ε
k), A2(t/ε
k))
the system of stochastic differential equations
dNε(t) =
[
− a
εk
Nε(t) +
εk0−k
a2 + b2
f̃0(μ0t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t), t/ε
k)
]
dt+
+
m∑
j=1
εkj−k/2
a2 + b2
f̃j(μjt/ε
k, Nε(t), A
ε
1(t), A
ε
2(t), t/ε
k)dwεj(t)+
+
εkm+1
a2 + b2
∫
R
f̃m+1(μm+1t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t), t/ε
k, z)ν̃ε(dt, dz),
dAε1(t) = −sinα sin(bt/εk + α)
b2
[εk0−kf̃0(μ0t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t))dt+ (5)
+
m∑
j=1
εkj−k/2f̃j(μjt/εk, Nε(t), A
ε
1(t), A
ε
2(t))dw
ε
j(t)+
22 O.D.BORYSENKO AND O.V.BORYSENKO
+εkm+1
∫
R
f̃m+1(μm+1t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t), z)ν̃ε(dt, dz)],
dAε2(t) =
sinα cos(bt/εk + α)
b2
[εk0−kf̃0(μ0t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t), t/ε
k)dt+
+
m∑
j=1
εkj−k/2f̃j(μjt/εk, Nε(t), A
ε
1(t), A
ε
2(t), t/ε
k)dwεj(t)+
+εkm+1
∫
R
f̃m+1(μm+1t/ε
k, Nε(t), A
ε
1(t), A
ε
2(t), t/ε
k, z)ν̃ε(dt, dz)],
where wεj(t) = εk/2wj(t/ε
k), ν̃ε(t, A) = ν(t/εk, A)−Π(A)t/εk, here A is Borel
set in R. For any ε > 0 the processes wεj(t), j = 1, m are the independent
Wiener processes and ν̃ε(t, A) is the centered Poisson measure independent
on wεj(t), j = 1, m.
Since we have relationship Nε(t) = exp{−at/εk}C(t/εk) and process
Cε(t) = C(t/εk) satisfies the stochastic equation
Cε(t) = C(0) + εk0−k
∫ t
0
eas/ε
k
a2 + b2
f̃0(μ0s/ε
k, Nε(s), A
ε
1(s), A
ε
2(s), s/ε
k) ds+
+
m∑
j=1
εkj−k/2
∫ t
0
eas/ε
k
a2 + b2
f̃j(μjs/ε
k, Nε(s), A
ε
1(s), A
ε
2(s), s/ε
k) dwεj(s)+
+εkm+1
∫ t
0
∫
R
eas/ε
k
a2 + b2
f̃m+1(μm+1s/ε
k, Nε(s), A
ε
1(s), A
ε
2(s), s/ε
k, z) ν̃ε(dt, dz),
where C(0) =
b2x0+x′′0
a2+b2
, we can obtain estimate
E|Nε(t)|2 ≤ K[e−2at/εk
+ εk(1 − e−2at/εk
)(tε2(k0−k) +
m+1∑
j=1
ε2kj−k)].
Therefore limε→0 E|Nε(t)|2 = 0 and it is sufficient to study the behaviour,
as ε→ 0, of solution to the system of stochastic differential equations
dAε1(t) = −sinα sin(bt/εk + α)
b2
[εk0−kf̂0(μ0t/ε
k, Aε1(t), A
ε
2(t))dt+
+
m∑
j=1
εkj−k/2f̂j(μjt/εk, Aε1(t), A
ε
2(t))dw
ε
j(t)+
+εkm+1
∫
R
f̂m+1(μm+1t/ε
k, Aε1(t), A
ε
2(t), z)ν̃ε(dt, dz)],
dAε2(t) =
sinα cos(bt/εk + α)
b2
[εk0−kf̂0(μ0t/ε
k, Aε1(t), A
ε
2(t), t/ε
k)dt+ (6)
LIMIT BEHAVIOR 23
+
m∑
j=1
εkj−k/2f̂j(μjt/εk, Aε1(t), A
ε
2(t), t/ε
k)dwεj(t)+
+εkm+1
∫
R
f̂m+1(μm+1t/ε
k, Aε1(t), A
ε
2(t), t/ε
k, z)ν̃ε(dt, dz)],
with initial conditions Aε1(0) = A1(0), Aε2(0) = A2(0).
Let us denote Aε(t) = (Aε1(t), A
ε
2(t)). Using conditions on coefficients of
equation (6) and properties of stochastic integrals we obtain estimates
E||Aε(t)||2 ≤ K
(
1 + t2ε2(k0−k) + t
m+1∑
j=1
ε2kj−k
)
,
E||Aε(t) − Aε(s)||2 ≤ K
(
|t− s|2ε2(k0−k) + |t− s|
m+1∑
j=1
ε2kj−k
)
.
Similarly for the process ζε(t) = (ζε1(t), ζ
ε
2(t)), where
ζε1(t) = −
m∑
j=1
εkj−k/2
∫ t
0
sinα sin( bs
εk + α)
b2
f̂j(
μjs
εk
, Aε1(s), A
ε
2(s),
s
εk
)dwεj(s)−
−εkm+1
∫ t
0
∫
R
sinα sin( bs
εk + α)
b2
f̂m+1(
μm+1s
εk
, Aε1(s), A
ε
2(s),
s
εk
, z)ν̃ε(ds, dz)],
ζε2(t) =
m∑
j=1
εkj−k/2
∫ t
0
sinα cos( bs
εk + α)
b2
f̂j(
μjs
εk
, Aε1(s), A
ε
2(s),
s
εk
)dwεj(s)+
+εkm+1
∫ t
0
∫
R
sinα cos( bs
εk + α)
b2
f̂m+1(
μm+1s
εk
, Aε1(s), A
ε
2(s),
s
εk
, z)ν̃ε(ds, dz)]
we derive estimates
E||ζε(t)||2 ≤ Kt
m+1∑
j=1
ε2kj−k, E||ζε(t) − ζε(s)||2 ≤ K|t− s|
m+1∑
j=1
ε2kj−k.
Therefore for stochastic process ηε(t) = (Aε(t), ζε(t)) conditions of weak
compactness [6] are fulfilled
lim
h↓0
lim
ε→0
sup
|t−s|<h
P{|ηε(t) − ηε(s)| > δ} = 0 for any δ > 0, t, s ∈ [0, T ],
lim
N→∞
lim
ε→0
sup
t∈[0,T ]
P{|ηε(t)| > N} = 0,
and for any sequence εn → 0, n = 1, 2, . . . there exists a subsequence
εm = εn(m) → 0, m = 1, 2, . . ., probability space, stochastic processes
24 O.D.BORYSENKO AND O.V.BORYSENKO
Āεm(t) = (Āεm
1 (t), Āεm
2 (t)), ζ̄εm(t), Ā(t) = (Ā1(t), Ā2(t)), ζ̄(t) defined on this
space, such that Āεm(t) → Ā(t), ζ̄εm(t) → ζ̄(t) in probability, as εm → 0,
and finite-dimensional distributions of Āεm(t), ζ̄εm(t) are coincide with finite-
dimensional distributions of Aεm(t), ζεm(t). Since we interesting in limit
behaviour of distributions, we can consider processes Aεm(t), and ζεm(t)
instead of Āεm(t), ζ̄εm(t). From (6) we obtain equation
Aεm(t) = A(0) +
t∫
0
αεm(s, Aεm(s)) ds+ ζεm(t), A0 = (A1(0), A2(0)), (7)
where αε(t, A) = (α
(1)
ε (t, A1, A2), α
(2)
ε (t, A1, A2)),
α(1)
ε (t, A1, A2) = −εk0−k sinα sin(bt/εk + α)
b2
f̂0(μ0t/ε
k, A1, A2, t/ε
k),
α(2)
ε (t, A1, A2) = εk0−k
sinα cos(bt/εk + α)
b2
f̂0(μ0t/ε
k, A1, A2, t/ε
k).
It should be noted that process ζε(t) is the vector-valued square integrable
martingale with matrix characteristic
〈ζ (l)
ε , ζ
(n)
ε 〉(t) =
m∑
j=1
t∫
0
σ(l,j)
ε (s, Aε1(s), A
ε
2(s))σ
(n,j)
ε (s, Aε1(s), A
ε
2(s)) ds+
+
1
εk
t∫
0
∫
R
γ(l)
ε (s, Aε1(s), A
ε
2(s), z)γ
(n)
ε (s, Aε1(s), A
ε
2(s), z) Π(dz)ds, l, n = 1, 2,
where
σ(1,j)
ε (s, A1, A2) = −εkj−k/2 sinα sin( bs
εk + α)
b2
f̂j(
μjs
εk
, A1, A2,
s
εk
),
σ(2,j)
ε (s, A1, A2) = εkj−k/2 sinα cos( bs
εk + α)
b2
f̂j(
μjs
εk
, A1, A2,
s
εk
),
γ(1)
ε (s, A1, A2, z) = −εkm+1
sinα sin( bs
εk + α)
b2
f̂m+1(
μm+1s
εk
, A1, A2,
s
εk
, z),
γ(2)
ε (s, A1, A2, z) = εkm+1
sinα cos( bs
εk + α)
b2
f̂m+1(
μm+1s
εk
, A1, A2,
s
εk
, z).
For processes Aε(t) and ζε(t) following estimates hold
E||Aε(t) −Aε(s)||4 ≤ K
[
ε4(k0−k)|t− s|4 + E||ζε(t) − ζε(s)||4
]
, (8)
LIMIT BEHAVIOR 25
E||ζε(t) − ζε(s)||4 ≤ K
[
m+1∑
j=1
ε4kj−2k|t− s|2+
+ε4km+1−3k/2|t− s|3/2 + ε4km+1−k|t− s|
]
, (9)
E||Aε(t) −Aε(s)||8 ≤ K, E||ζε(t) − ζε(s)||8 ≤ K. (10)
Since Aεm(t) → Ā(t), ζεm(t) → ζ̄(t) in probability, as εm → 0, then, using
(10), from (8) and (9) we obtain estimates
E||Ā(t) − Ā(s)||4 ≤ K(|t− s|4 + |t− s|2), E||ζ̄(t) − ζ̄(s)||4 ≤ C|t− s|2.
Therefore processes Ā(t) and ζ̄(t) satisfy the Kolmogorov’s continuity con-
dition [7].
Let us consider the case k0 = 2kj, j = 1, m+ 1. Under these conditions
we have for l, n = 1, 2
lim
ε→0
1
t
t∫
0
α(l)
ε (s, A1, A2)ds = ᾱ(l)(A1, A2),
lim
ε→0
1
t
t∫
0
[
m∑
j=1
σ(l,j)
ε (s, A1, A2)σ
(n,j)
ε (s, A1, A2)+ (11)
+
1
εk
∫
R
γ(l)
ε (s, A1, A2, z)γ
(n)
ε (s, A1, A2, z)Π(dz)
⎤⎦ ds = B̄ln(A1, A2),
where functions ᾱ(l)(A1, A2) and B̄(A1, A2) = {B̄ln(A1, A2), l, n = 1, 2} are
defined in the condition of theorem. Since processes Ā(t), ζ̄(t) are continu-
ous, then from Lemma and relationships (7), (11) it follows
Ā(t) = A(0) +
t∫
0
ᾱ(Ā1(s), Ā2(s))ds+ ζ̄(t), A(0) = (A1(0), A2(0)), (12)
where ζ̄(t) is continuous vector-valued martingale with matrix characteristic
〈ζ̄ (l), ζ̄ (n)〉(t) =
t∫
0
B̄ln(Ā1(s), Ā2(s))ds, l, n = 1, 2.
Hence [8] there exists Wiener process w̄(t) = (w̄j(t), j = 1, 2), such that
ζ̄(t) =
t∫
0
σ̄(Ā1(s), Ā2(s)) dw̄(s), σ̄(A1, A2) =
{
B̄(A1, A2)
}1/2
. (13)
26 O.D.BORYSENKO AND O.V.BORYSENKO
Relationships (12), (13) mean, that process Ā(t) satisfies equation (4). Un-
der conditions of theorem the equation (4) has unique solution. There-
fore process Ā(t) does not depend on choosing of sub-sequence εm → 0,
and finite-dimensional distributions of process Aεm(t) converge to finite-
dimensional distributions of process Ā(t). Since processes Aεm(t) and Ā(t)
are Markov processes, then using the conditions for weak convergence of
Markov processes [7], we complete the proof of statement 1 of theorem.
Let us consider the case k < k0. Then coefficients α
(i)
ε (t, A1, A2), i = 1, 2
of equation (7) tend to zero, as ε → 0. Repeating with obvious modifications
the proof of statement 1) of theorem we obtain proof of the first statement
of 2).
In the case k < 2kj, j = 1, m in (11) we have
σ(l,j)
ε (t, A1, A2)σ
(n,j)
ε (t, A1, A2) = O(ε2kj−k), l, n = 1, 2.
Then we can complete the proof in this case as above. In the same way we
consider the case k < 2km+1. The statement 3) follows from result of [1].�
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Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail: odb@univ.kiev.ua
Department of Mathematical Physics, National Technical Univer-
sity ”KPI“, Kyiv, Ukraine
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