Diffusion approximation algorithms in merging phase space
Diffusion approximation algorithms for stochastic systems in split and merging phase space are represented in servey form. The main mathematical tools of such algorithms are described in our book ”Stochastic Systems in Merging Phase Space” (World Scientific Publishing, 2005).
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irk-123456789-44802009-11-20T12:00:32Z Diffusion approximation algorithms in merging phase space Koroliuk, V. Limnios, N. Diffusion approximation algorithms for stochastic systems in split and merging phase space are represented in servey form. The main mathematical tools of such algorithms are described in our book ”Stochastic Systems in Merging Phase Space” (World Scientific Publishing, 2005). 2007 Article Diffusion approximation algorithms in merging phase space / V. Koroliuk, N. Limnios // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 98-102. — Бібліогр.: 1 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4480 en Інститут математики НАН України |
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Diffusion approximation algorithms for stochastic systems in split and merging phase space are represented in servey form. The main mathematical tools of such algorithms are described in our book ”Stochastic Systems in Merging Phase Space” (World Scientific Publishing, 2005). |
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Article |
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Koroliuk, V. Limnios, N. |
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Koroliuk, V. Limnios, N. Diffusion approximation algorithms in merging phase space |
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Koroliuk, V. Limnios, N. |
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Koroliuk, V. |
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Diffusion approximation algorithms in merging phase space |
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Diffusion approximation algorithms in merging phase space |
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Diffusion approximation algorithms in merging phase space |
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Diffusion approximation algorithms in merging phase space |
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Diffusion approximation algorithms in merging phase space |
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diffusion approximation algorithms in merging phase space |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4480 |
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Diffusion approximation algorithms in merging phase space / V. Koroliuk, N. Limnios // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 98-102. — Бібліогр.: 1 назв.— англ. |
| work_keys_str_mv |
AT koroliukv diffusionapproximationalgorithmsinmergingphasespace AT limniosn diffusionapproximationalgorithmsinmergingphasespace |
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2025-07-02T07:42:54Z |
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2025-07-02T07:42:54Z |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.98-102
VLADIMIR KOROLIUK AND NIKOLAOS LIMNIOS
DIFFUSION APPROXIMATION ALGORITHMS IN
MERGING PHASE SPACE
Diffusion approximation algorithms for stochastic systems in split
and merging phase space are represented in servey form.
The main mathematical tools of such algorithms are described in our
book ”Stochastic Systems in Merging Phase Space” (World Scientific
Publishing, 2005).
1. Introduction
The general scheme is illustrated on the semi-Markov random evolution
given by a solution of the evolutionary equation
dUε(t)
dt
= Cε(Uε(t); Xε(t/ε2))dt. (1)
The velocity function
Cε(u; x) = C(u; x) + ε−1C0(u; x), u ∈ R
d, x ∈ E (2)
satisfies the conditions of existence of global solutions for associated deter-
ministic evolutionary equations
dUε
x(t)
dt
= Cε(Uε
x(t); x), x ∈ E. (3)
2. The semi-Markov switching process
The semi-Markov switching process Xε(t), t ≥ 0, is considered in the
series scheme with small parameter series ε → 0 (ε > 0) given on the
double split phase space (E, ξ) :
E = ∪N
k=1Ek, Ek = ∪Nk
r=1E
r
k, Er
k ∩ Er′
k = ∅, r �= r′, (4)
Invited lecture.
2000 Mathematics Subject Classifications. 60K15
Key words and phrases. Stochastic processes, normal distribution,...
98
DIFFUSION APPROXIMATION ALGORITHMS 99
by the semi-Markov kernel
Qε(x, B, t) = P ε(x, B)Fx(t), x ∈ E, B ∈ ξ, t ≥ 0. (5)
The stochastic kernels
P ε(x, B) = P (x, B) + εP1(x, B) + ε2P2(x, B) (6)
in (6) coordinated with split (4) as follows:
P (x, Er
k) = 1r
k(x) :=
{
1, x ∈ Er
k,
0, x �∈ Er
k,
P1(x, Ek) = 0, P2(x, E) = 0, x ∈ E.
(7)
The associated Markov process X0(t), t ≥ 0 given by the generator
Qϕ(x) = q(x)
∫
E
P (x, dy)[ϕ(y)− ϕ(x)],
q(x) := 1/g(x), g(x) :=
∫ ∞
0
F x(t)dt, F x(t) := 1 − Fx(t),
(8)
is considered uniformly ergodic in every class Er
k, 1 ≤ r ≤ Nk, 1 ≤ k ≤ N,
with stationary distributions
πr
k(dx), 1 ≤ r ≤ Nk, 1 ≤ k ≤ N. (9)
Introduce the merging functions
V̂ (x) = V r
k , x ∈ Er
k,
̂̂
V (x) = k, x ∈ Ek (10)
and operators
Qlϕ(x) = q(x)
∫
E
Pl(x, dy)ϕ(y), l = 1, 2. (11)
2. Phase merging principle
Theorem 4.3 (double merging principle) [1, Section 4.2.4]. Under the
merging conditions MD1–MD3 the following weak convergence take place
V̂ (Xε(t/ε)) ⇒ X̂(t), ε → 0,̂̂
V (Xε(t/ε2)) ⇒ ̂̂
X(t), ε → 0.
(12)
The limit Markov processes X̂(t) and
̂̂
X(t), on the merged phase spaces
Ê =
⋃N
k=1 Êk, Êk = {V r
k , 1 ≤ r ≤ Nk, 1 ≤ k ≤ N} and Ê = {1, 2, . . . , N}
are given by the generating matrices Q̂1 and
̂̂
Q2 correspondedly determined
as follows:
Q̂1Π = ΠQ1Π,
̂̂
Q2Π̂ = Π̂ΠQ2ΠΠ̂. (13)
100 VLADIMIR KOROLIUK AND NIKOLAOS LIMNIOS
The projectors Π and Π̂ are defined as follows:
Πϕ(x) =
∑N
k=1
∑Nk
r=1 ϕ̂r
k1
r
k(x), ϕ̂r
k =
∫
Er
k
πr
k(dx)ϕ(x)
Π̂ϕ̂k =
∑Nk
r=1 π̂r
kϕ̂
r
k.
(14)
Here π̂k = (π̂r
k, 1 ≤ r ≤ Nk), 1 ≤ k ≤ N, are the stationary distributions of
the merged Markov process X̂(t), t ≥ 0.
The uniformly ergodicity of the double merged Markov process
̂̂
X(t),
t ≥ 0, is assumed also with stationary distribution ̂̂π = (̂̂πk, 1 ≤ k ≤ N)
defines the corresponding projector
̂̂
Πϕ̂ =
N∑
k=1
̂̂πkϕ̂k.
4. Split and merging scheme
In what follows the following generators of semigroups are used:
C0(x)ϕ(u) = C0(u; x)ϕ′(u)
C(x)ϕ(u) = C(u; x)ϕ′(u)
Here for simplicity the following equality is used:
C(u; x)ϕ′(u) :=
d∑
i=1
Ci(u; x)∂ϕ(u)/∂Ui
4.1. Split & Merging
Theorem 4.7 [1, Section 4.4.1]. Under the conditions of Section 4.4.1 [1]
the weak convergence takes place:
Uε(t) ⇒ ζ(t), ε → 0.
The limit diffusion process ζ(t), t ≥ 0, switched by the Markov process X̂(t),
t ≥ 0, is defined by the generator
L̂ϕ(u, k) = L̂0ϕ(u, · ) + Q̂1ϕ( · , k),
L̂0ϕ(u) = b̂(u; k)ϕ′(u) +
1
2
B̂(u; k)ϕ′′(u).
The vector of drift b̂(u; k) and the covariance matrix B̂(u; k) are defined by
a solution of singular perturbation problem for the operator
L
ε = ε−2Q + ε−1
C0(x)P + C(x)P + Q1,
given by the following formulae:
L̂ = Ĉ0 + Ĉ + Q̂1,
Ĉ0 = ΠC0(x)PR0C0(x)Π, Ĉ = ΠC(x)Π.
DIFFUSION APPROXIMATION ALGORITHMS 101
4.2. Split & Double merging
Theorem 4.9 [1, Section 4.4.2]. Under the conditions of Section 4.4 in [1]
the weak convergence takes place
Uε(t/ε) ⇒ ζ̂(t), ε → 0.
The limit diffusion process ζ̂(t), t ≥ 0, is defined by the generator
L̂ϕ(u) = b̂(u)ϕ′(u) +
1
2
B̂(u)ϕ′′(u).
The drift-coefficient b̂(u) and the covariance matrix B̂(u) are defined by a
solution of singular perturbation problem for the operator
L
ε = ε−3Q + ε−2Q1 + ε−1
C0(x)P + C(x)P,
given by the following formulae:
̂̂
L =
̂̂
C0 +
̂̂
C,
̂̂
C0 = Π̂Ĉ0R̂0Ĉ0Π̂, Ĉ0 = ΠC0(x)Π,̂̂
C = Π̂ĈΠ̂, Ĉ = ΠC(x)Π.
4.3. Double split & Merging
Theorem 4.10 [1, Section 4.4.4]. Under the conditions of Section 4.4 in
[1] the weak convergence takes place
Uε(t/ε) ⇒ ̂̂
ζ(t), ε → 0.
The limit diffusion process
̂̂
ζ(t), t ≥ 0, is defined by the generator
̂̂
Lϕ(u, k) =
̂̂
b(u; k)ϕ′(u) +
1
2
̂̂
B(u; k)ϕ′′(u).
The drift-coefficient
̂̂
b(u) and the covariance matrix
̂̂
B(u) are defined by a
solution of singular perturbation problem for the operator
L
ε = ε−3Q + ε−2Q1 + ε−1
C0(x) + C(x) + Q2,
given by the following formulae:
̂̂
L =
̂̂
C0 +
̂̂
C +
̂̂
Q2,
̂̂
C0 = Π̂Ĉ0R̂0Ĉ0Π̂, Ĉ0 = ΠC0(x)Π,̂̂
C = Π̂ĈΠ̂, Ĉ = ΠC(x)Π.
102 VLADIMIR KOROLIUK AND NIKOLAOS LIMNIOS
4.4. Double split & Double merging
Theorem 4.11 [1, Section 4.4.5]. Under the conditions of Section 4.4 in
[1] the weak convergence takes place
Uε(t/ε2) ⇒ ̂̂̂
ζ(t), ε → 0.
The limit diffusion process
̂̂̂
ζ(t), t ≥ 0, is defined by the generator
̂̂̂
Lϕ(u) =
̂̂̂
b(u)ϕ′(u) +
1
2
̂̂̂
B(u)ϕ′′(u).
The drift-coefficient
̂̂̂
b(u) and the covariance matrix
̂̂̂
B(u) are defined by a
solution of singular perturbation problem for the operator
L
ε = ε−4Q + ε−3Q1 + ε−2Q2 + ε−1
C0(x) + C(x),
given by the formulae: ̂̂̂
L =
̂̂̂
C0 +
̂̂̂
C,̂̂̂
C0 =
̂̂
Π
̂̂
C0
̂̂
Π,
̂̂
C0 = Π̂Ĉ0Π̂,̂̂̂
C =
̂̂
Π
̂̂
C
̂̂
Π,
̂̂
C = Π̂ĈΠ̂.
5. Additional comments
Diffusion approximation algorithms in split and merging phase space are
constructed due to specific dependency of the generator of the associated
Markov process on the parameter series ε. According to conditions of Section
4.4 in [1] the asymptotic extension of the compensative operator of the
random evolution process is used to construct the truncated operators L
ε
in Theorems 4.7 – 4.11. A solution of singular perturbation problems given
in Chapter 5, [1], are used to construct the limit generators. Verification of
weak convergences is realized on the scheme represented in Chapter 6, [1].
Bibliography
1. Koroliuk, V.S., Limnios, N., Stochastic Systems in Merging Phase Space,
World Scientific Publishing, (2005).
Institute of Mathematics, Kyiv, Ukraine.
University of Technology of Compiegne, France
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