Diffusion process with evolution and its parameter estimation
A discrete Markov process in an asymptotic diffusion environment with a uniformly ergodic embedded Markov chain can be approximated by an Ornstein–Uhlenbeck process with evolution. The drift parameter estimation is obtained using the stationarity of the Gaussian limit process. Показано, що дискретни...
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Інститут кібернетики ім. В.М. Глушкова НАН України
2020
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| Cite this: | Diffusion process with evolution and its parameter estimation / V.S. Koroliuk, D. Koroliouk, S.О. Dovgyi // Кибернетика и системный анализ. — 2020. — Т. 56, № 5. — С. 55–62. — Бібліогр.: 8 назв. — англ. |
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| author | Koroliuk, V.S. Koroliouk, D. Dovgyi S.О. |
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| citation_txt | Diffusion process with evolution and its parameter estimation / V.S. Koroliuk, D. Koroliouk, S.О. Dovgyi // Кибернетика и системный анализ. — 2020. — Т. 56, № 5. — С. 55–62. — Бібліогр.: 8 назв. — англ. |
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| description | A discrete Markov process in an asymptotic diffusion environment with a uniformly ergodic embedded Markov chain can be approximated by an Ornstein–Uhlenbeck process with evolution. The drift parameter estimation is obtained using the stationarity of the Gaussian limit process.
Показано, що дискретний марковський процес в асимптотичному дифузійному середовищі з рівномірним ергодичним вкладеним ланцюгом Маркова може бути наближений процесом Орнштейна-Уленбека з еволюцією. Оцінку параметра дрейфу отримано з використанням стаціонарності гаусівського граничного процесу.
Показано, что дискретный марковский процесс в асимптотической диффузионной среде с равномерной эргодической вложенной цепью Маркова может быть приближен процессом Орнштейна-Уленбека с эволюцией. Оценка параметра дрейфа получена с использованием стационарности гауссовского предельного процесса.
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UDC 519.24
V.S. KOROLIUK , D. KOROLIOUK, S.Î. DOVGYI
DIFFUSION PROCESS WITH EVOLUTION
AND ITS PARAMETER ESTIMATION
Abstract. A discrete Markov process in an asymptotic diffusion environment
with a uniformly ergodic embedded Markov chain can be approximated by
an Ornstein–Uhlenbeck process with evolution. The drift parameter estimation is
obtained using the stationarity of the Gaussian limit process.
Keywords: discrete Markov process, diffusion approximation, asymptotic
diffusion environment, Ornstein–Uhlenbeck process, phase merging, drift
parameter estimation.
We consider a random evolution �( )t , t � 0 , that depends on a random
environment Y t( ) , t � 0 , which in turn, is switched by an embedded Markov chain
X k , k � 0 . The connection between the continuous-time t � 0 and the discrete-time
k � 0 will be explained in the sequel.
The purpose of this work is to prove the convergence (in distribution) of the
process �( )t , t � 0 , to the Ornstein–Uhlenbeck process under some scaling of the
process and its time parameter.
The limit will be considered by a small series parameter �� 0 , � � 0 .
ASYMPTOTIC DIFFUSION ENVIRONMENT
Consider a discrete Markov process in a semi-Markov asymptotic diffusion
environment, determined by a solution of the following scaled difference stochastic
equation:
� � � �� �� � � � � � � �( ) ( ) ( ) ( ) ( )t V Y t Y t
n n n n n� �� � �
1
2
1
� , (1)
where t nn
� �:� 2 , hence t t
n n� � �
1
2� � � , n� 0 , �� 0 , for the process increments
�� � �� � � � � �( ): ( ) ( )t t t
n n n� �� �
1 1
, n � 0 .
The asymptotic diffusion environment Yn
� , n � 0 , is also a random evolution
process generated by a solution of the following scaled difference evolutionary
equation:
�Y t A Y X A Y X n
n n n n n
� � � � � �� �( ) ( ; ) ( ; ) ,� � � �
1 0
2 0 , (2)
with the embedded Markov chain X X tn n
� �: ( )� , n � 0 .
The terms A y x0 ( ; ) and A y x( ; ) are Lipschitz functions, together with the first
derivative A y xy0 ( ; ) .
Here the predictable evolutional component in (1) is determined by the following
conditional expectation [1]:
V Y t E t Y t tn n n n n n( ) ( ) : [ ( ) | , ( )] ( )� � � � � � � � � �� � � �� ��
1
,
where it is assumed that the drift regression function V z( ) is positive: V z( ) � 0
z.
ISSN 1019-5262. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2020, òîì 56, ¹ 5 55
© V.S. Koroliuk , D. Koroliouk, S.Î. Dovgyi, 2020
The martingale difference ��� �( )t
n�1
, n � 1, generated by the process �� � �( )t
n�1
,
n � 1, is determined by the following conditional second moment:
� � ��� � � � �� � � � � � � � �( ) : [( ( ) ( ) ( )) | ]Y E t V Y t Yn n n n n�
1
2 2 .
The embedded Markov chain X X tn n
� �: ( )� , t nn
� �:� 2 , n � 0 , is supposed to be
homogeneous ergodic Markov chain with transition probabilities P x B( , ) , x E� ,
B �� , having a stationary distribution �( )B , B ��, which satisfies the condition
� �( ) ( ) ( , )B dx P x B
E
� � ; �( )E �1.
The stochastic difference equations (1), (2) generate a discrete stochastic basis
[2, Ch. 1] with filtration Fm n nY t Y t n m( , ) { ( ), ( ), }� � �� � � � � ��
, m � 0 .
Now we consider three components ( ( ), ( ), )� � � � � �t Y t Xn n n , n � 0 , as piecewise
constant functions with continuous time:
� �
�
� � �
� � �
� �
( ) ( )
( ) ( ) (
t t
Y t Y t
X X
n t n
n
n
t n
�
�
�
�
�
�
�
�
�
�for 2 1 2)� .
In what follows, a solution of equations (1), (2) is given by martingale characterization
[3, Section 4.4] of three-component Markov process ( ( ), ( ), )� � �t Y t X t , t � 0 :
M t t Y t X Y Xt
� � � � �� � � �( ) ( ( ), ( ), ) ( ( ), ( ), )� � �0 0 0
�� L s Y s X ds
t
s
�� � � � �� �
0
2 2[ / ]
( ( ), ( ), ) ,
and the generator of three-component Markov process ( ( ), ( ), )� � �t Y t X t , t � 0 , is
represented as follows [4, Ch. 5]:
L c y x E c t Y t X
n n n
� � � � � �� � � �( , , ) : [ ( ( , ( ), )) |� � � � �
2
1 1 1
�
� � � � � �( ) , ( ) , ].t c Y t y X xn n n� � �
APPROXIMATION OF A DISCRETE MARKOV PROCESS
IN A SYMPTOTIC DIFFUSION ENVIRONMENT
Let the singular term A y x0 ( ; ) satisfies the balance condition
�
E
dx A y x� �( ) ( ; )0 0 . (3)
The approximation of a discrete Markov process in asymptotic diffusion
environment gives the following theorem.
Theorem 1. Let the Markov chain X n , n � 0 , be uniformly ergodic with the
stationary distribution �( )B , B ��.
The finite-dimensional distributions of the discrete Markov process (1), together
with asymptotical diffusion Y t� ( ) , t � 0 , converge, as � � 0 , to a diffusion
Ornstein–Uhlenbeck process with evolution:
( ( ), ( )) ( ( ), ( )) , ,� � �� �t Y t t Y t t T
D� �� �
0 0 0 0 .
56 ISSN 1019-5262. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2020, òîì 56, ¹ 5
The limit two-component diffusion process with evolution ( ( ), ( ))� 0 0t Y t , t � 0 , is
set by the generator
L0 21
2
( ) ( , ) ( ) ( , ) ( ) ( , )y c y V y c c y y c yc c� � � �� � � �
(4)
� � �� ( ) ( , ) � ( ) ( , ) , ( , ),A y c y B y c y C Ry y� � �
1
2
2
0
2 B
where by definition
� ( ): ( )[ ( ; ) ( ; )]A y dx A y x A y x
E
� �� � 1 , (5)
A y x A y x PR A y xy1 0 0 0( ; ): ( ; ) ( ; )� , (6)
� ( ) ( ) ( ; ),B y dx B y x
E
2 � � �
B y x A y x P R A y x( ; ) ( ; ) [ ] ( ; )� �0 0 0
1
2
� .
(7)
Here � is the standard identity matrix, P is the transition operator of the Markov
chain X t , t � 0 , and the potential kernel R 0 is defined as in [3, Section 5.2]:
R P x dx x
E
0
1� � � � � ��
�( ) , : , ( ): ( ) ( )� �Ï � �� � � � . (8)
Remark 1. The limit two-component diffusion process ( ( )� 0 t , Y t0 ( )) , t � 0 , set
by the generator (4)–(8), has a stochastic representation by the stochastic differential
equation
d t V Y t t dt Y t dW t� � �0 0 0 0( ) ( ( )) ( ) ( ( )) ( )� � � ,
dY t A Y t dt B Y t dW t0 0 0
0( ) � ( ( )) � ( ( )) ( ).� �
Consequently, the parameters of the limit diffusion � 0 ( )t , t � 0 , depend on the
diffusion process Y t0 ( ) , t � 0 .
Proof of Theorem 1. The basic idea is that any Markov process is determined by
its generator on the class of real-valued test functions, defined on the set of values of
Markov process [5].
First of all, the extended three-component Markov chain is used
( ( ), ( ), ( ) ) , ,� � �� � �t Y t X t X t nn n n n n� � �2 0 , (9)
with operator characterization in the following form.
Lemma 1. The extended Markov chain is determined by the generator
� � �
� � �� � �( ) ( , , ) [ ( ) ( ) ] ( , , )x c y x y A x c y x� ��2 � , (10)
where the transition operators are defined as follows:
� �� � � � �� � � �( ) ( ) : [ ( ( ; ))| ( ) , ( ) , ( )y c E c t y t c Y t y X t x� � � � � ],
�
� � � �� �( ) ( ) : [ ( ( ; ))| ( ; ) , ( ) ],x y E y Y t x Y t x y X t x� � � �� (11)
� � �( ) : ( , ) ( ) .x P x dz z
E
� �
ISSN 1019-5262. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2020, òîì 56, ¹ 5 57
The assertion of Lemma 1 follows from the next argumentation. The extended
three-component Markov chain (11), under the additional condition Y t y� ( ) � ,
X t x� ( ) � , has independent components. So its transition probabilities are given by the
product of transition probabilities of each component.
An essential step in the proof of Theorem 1 is realized in the next lemma.
Lemma 2. The generator (10), (11) of three-component Markov chain (9) on the
class of real-valued test functions �( , , )c y x , having bound derivatives up to the third
order inclusively, admits an asymptotic representation
�
� �( ) ( , , )x c y x � (12)
� � � � �� �[ ( ) ( ) ( ) ( ; )] ( , , );� � ��
2 1
0
0
� � � � � � �x x y P y x c y x
� �0 0( ) ( ) ( ; ) ( ) , ( ) ( ) ( ; ) ( );x y A y x y x y A y x y� � � �� � (13)
�
0 21
2
( ) ( ) ( ) ( ) ( ) ( )y c V y c c y c� � � �� � � .
The residual term is expressed as:
�� � � �( ; ) ( , , ) , , ( )y x c y x C R� � �0 0 3 2 .
Here one intends the uniform convergence for all the arguments.
Proof of Lemma 2. We use transformation of generator (12) by the formula
�
� �( ) ( , , )x c y x �
� � � � � ��� �� �
�
2 [ ( ( ) ) ( ( ) ) ( ; )] ( , , )� � � � � �A x y y x c y x� . (14)
The residual term has the following form:
� � � ��
� �� �( ; ) ( , , ) ( ( ) )( ( ) ) ( , , )y x c y x y A x c y x� � �� .
Then we calculate
� � � � � � �� � �� �� � � � �2 2[ ( ) ] ( ) { [ ( ( ; ) ) | ( ; ) ] (� �y c E c t y t y c� c)} �
� �[ ( ) ( ; )] ( )� �
0 y y c c� � ;
The next term in (17) has the next representation:
� � � � �� � �� �� � � � �2 2[ ( ) ] ( ) { [ ( ( ; )) | ( ; ) ] (A x y E y Y t y Y t y y� � y)} �
� � ��� � � � �� �
�
2 2 21
2
E Y t y y E Y t y y x[ ( ; )] ( ) [ ( ; )] ( ) ( )� � � ( )y
�
��
�
��
�
� � � ��[ ( ; ) ( ; )] ( ) ( ; ) ( ) ( )� � � ��
1
0 0 0
21
2
� � � �y x y x y y x y x ( )y ,
gives the asymptotic expansion in Lemma 2:
� � � � � �
� � � �( ) ( , , ) [ ( ) ( )x c y x x x� � � �� �2 1
0
� � �
1
2
0
2 0( ( ) ) ( ) ] ( , , ) ( ; ) ( , , )� � � �x y c y x y x c y x� �� .
Next, we use the solution of singular perturbation problem for the truncated
operator [3].
58 ISSN 1019-5262. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2020, òîì 56, ¹ 5
Lemma 3. The solution of the singular perturbation problem for the truncated
operator is realized on perturbed test functions:
� � � � �
0
2 1
0
� �� � �( ) ( , , ) [ ( ) ( )x c y x x A x� � � �� �
� � � �
1
2
0
2 0
1
2
2( ( ) ) ( )][ ( , ) ( , , ) ( , , )]� � �x y c y c y x c y x� �� � � �
� �L0 ( ) ( , ) ( ) ( , ).y c y x c y� ��� (15)
The averaging parameters are determined by the formulas (4)–(8).
The limit operator is calculated by the formula
L0 0 21
2
( ) ( , ) ( ) ( , ) � ( ) ( , ) � ( )y c y y c y A y c y B yy y� � � �� � � � ( , )c y . (16)
Proof of Lemma 3. To solve the singular perturbation problem for the truncated
operator, consider the asymptotic representation by the powers of � :
� � �
0
2 1
1
� �� � � � �( ) ( , , ) ( , ) [ ( , , )x c y x c y c y x� � �� �
� � � �� � � �0 2 0 1( ) ( , )] [ ( , , ) ( ) ( , , )x c y c y x x c y x� � �
� � � �[ ( ) ( ( ) ) ( )] ( , )] ( ) ( , ).� � � � �x x y c y x c y
1
2
0
2 0 � ��
Obviously that � �( , )c y � 0 .
The balance condition (3) is then used. The solution of the equation
� �� �1 0 0( , , ) ( ) ( , )c y x x c y� �
is given by the formula [4, Section 5.4]:
� �1 0 0( , , ) ( ) ( , )c y x x c y� � � .
Lemma 3 implies the following equation
� � � �� � �2
0 0( , , ) [ ( ) ( ) ( )] ( , ) ( ) ( , ).c y x x x y c y y c y� � � � L (17)
Here, by definition
� � � � � �( ) : ( ) ( ) ( ( ) )x x R x x� �0 0 0 0
21
2
.
The limit operator is calculated using the balance condition
L0 0( ) ( , ) [ ( ) ( )] ( ) ( , ).y c y x x y c y� �� � �� �� � � (18)
Recall the projector’s operation:
� ��( ) ( ) ( , )x dx B y x
E
� � � , B y x A y x A y x A y x( , ) ( , ) ( , ) ( ( , ))� �0 0 0 0
21
2
� � .
Taking into account the definition of evolutionary operators (13), the limit
generator is determined by formula (16).
The limit operator (18) provides a solution of equation (17), which is a function of
�2 ( , , )c y x . The existence of perturbing functions � i ( )� , i �1 2, , ensures the
asymptotical representation (15). That completes the proof of Theorem 1.
The volatility is generated by introduction of a random environment Y t0 ( ) , t � 0 ,
into the diffusion parameter of the Ornstein–Uhlenbeck diffusion process � 0 ( )t , t � 0 .
ISSN 1019-5262. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2020, òîì 56, ¹ 5 59
PARAMETER ESTIMATION OF THE LIMIT PROCESS
The limit Ornstein–Uhlenbeck diffusion process parameters estimation is
substantiated in this section without the assumption of volatility, which greatly
changes the kind of estimates. The stationarity of the Gaussian statistical
experiment is essentially used [6].
It is known [7], that a diffusion type processes are given by stochastic differential
d dt dWt t t t
� �( ) . (19)
The predictable component satisfies the conditions
P dt Tt
T
t
2
0
1� � ��
!
"
#
$ � � �( ) , , (20)
P dtt t
2
0
1
�
� � ��
!
"
#
$ �( ) ,
which ensures the absolute continuity of the measure � � ( ) : { : }B P B� � and the
measure � �W B P W B( ) : { : }� � for all B T� = � ( t : 0
t T ) .
The Radon–Nicodemus derivative specifies the density of the measure
�
�
�
T
W
d
d
T( ) : ( , )� , (21)
which for processes of diffusion type (19) has the following representation.
Theorem 2 [7]. The measure density (21) for processes of diffusion type (19)
with additional conditions (20) is given by exponential martingale
d
d
T d dt
W
t t
T
t t t
T
t
�
�
( , ) exp ( ) ( )� �
�
��
�
��� �0
2
0
1
2
. (22)
In particular, the exponential martingale (21) is determined by a solution of the
stochastic Dol�ans–Dade equation [2]
d dT T T T� �
� ( ) ( ) ( ) , ( )� �0 1 , (23)
or in equivalent form:
� �
T T T Td( ) ( ) ( )� �1 .
The relationship of the density (22) with the stochastic Dol�ans–Dade equation
(23) can be explained, using the It�o formula for exponential function � ( ) �
� � % &exp[ ( ) ( ) ]� � T T
1
2
, with �
T t
T
dt( ): ( )� �0 , % & � ��
( ) : ( )T t
T
dt2
0
, namely
(see [2]):
d d d dT T T T T T� � � � � � ( ) ( )[ ( ) ( ) ] ( ) ( )� � % & � % &
1
2
1
2
.
Taking into account equality � � � ( ) ( ) ( )T T T� � for exponential function
� ( ) , we have a stochastic Dol�ans–Dade differential equation for exponential
martingale (22). According to the results of the previous section, the limit diffusion
process for normalized discrete Markov processes is the Ornstein–Uhlenbeck process
with a linear predictable component
d V dt dW t Tt t t
�� � �
0 0, ,
Without limiting of generality, let us put � �1.
60 ISSN 1019-5262. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2020, òîì 56, ¹ 5
The maximum likelihood method for estimating the parameter V0 of a diffusion
process with a stochastic differential (19) (� �1) is realized for the logarithm of the
measure density (22):
L V T V dt
V
dtT t
T
t
T
( , ) : ln ( )� � � �� ��
0
2
2
02
.
Therefore the equation for estimating the maximum likelihood method is
max ( , ) /
0 2 0
2
0
0
' ' � � � �� �
V
t
T
T t
T
L V T V dt V dt
,
and the estimate of the maximum likelihood method has the following form:
V d dtT t
T
t t
T
� �� �
0
2
0
/ .
The least square method estimation of parameter V0 of diffusion process with
stochastic differential (19) (� �1) is implemented using equality
t
T
t t
T
t
T
td V dt dW
0
0
2
0 0� � �� � � .
So we have a relationship
V V dW dtT t
T
t t
T
0
0
2
0
� � � �
/ .
The estimation of the least squares method has a representation
V d dt
T t
T
t t
T0
0
2
0
� � �
/ .
Corollary 1. The estimates of maximum likelihood and least squares coincide:
V VT T
� 0 .
Corollary 2. Estimation by the least squares method, and hence estimation by the
method of maximum likelihood of the parameter V0 are strongly consistent:
P V V
T T
1 0
0lim
��
� . (24)
Remark 2. In the presence of volatility (see [8]), the maximum likelihood
estimate and the least squares estimate are different, but the property of strong
consistency (24) is retained.
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Íàä³éøëà äî ðåäàêö³¿ 22.03.2020
Â.Ñ. Êîðîëþê , Ä. Êîðîëþê, Ñ.Î. Äîâãèé
ÄÈÔÓDzÉÍÈÉ ÏÐÎÖÅÑ Ç ÅÂÎËÞÖ²ªÞ ÒÀ ÎÖ²ÍÞÂÀÍÍß ÉÎÃÎ ÏÀÐÀÌÅÒÐÀ
Àíîòàö³ÿ. Ïîêàçàíî, ùî äèñêðåòíèé ìàðêîâñüêèé ïðîöåñ â àñèìïòîòè÷íîìó
äèôóç³éíîìó ñåðåäîâèù³ ç ð³âíîì³ðíèì åðãîäè÷íèì âêëàäåíèì ëàíöþãîì
Ìàðêîâà ìîæå áóòè íàáëèæåíèé ïðîöåñîì Îðíøòåéíà–Óëåíáåêà ç åâî-
ëþö³ºþ. Îö³íêó ïàðàìåòðà äðåéôó îòðèìàíî ç âèêîðèñòàííÿì ñòàö³îíàð-
íîñò³ ãàóñ³âñüêîãî ãðàíè÷íîãî ïðîöåñó.
Êëþ÷îâ³ ñëîâà: äèñêðåòíèé ìàðêîâñüêèé ïðîöåñ, äèôóç³éíà àïðîêñèìàö³ÿ,
àñèìïòîòè÷íå äèôóç³éíå ñåðåäîâèùå, ïðîöåñ Îðíøòåéíà–Óëåíáåêà, ôàçîâå
óêðóïíåííÿ, îö³íêà ïàðàìåòðà çñóâó.
Â.Ñ. Êîðîëþê , Ä. Êîðîëþê, Ñ.À. Äîâãèé
ÄÈÔÔÓÇÈÎÍÍÛÉ ÏÐÎÖÅÑÑ Ñ ÝÂÎËÞÖÈÅÉ È ÎÖÅÍÊÀ ÅÃÎ ÏÀÐÀÌÅÒÐÀ
Àííîòàöèÿ. Ïîêàçàíî, ÷òî äèñêðåòíûé ìàðêîâñêèé ïðîöåññ â àñèìïòîòè÷åñ-
êîé äèôôóçèîííîé ñðåäå ñ ðàâíîìåðíîé ýðãîäè÷åñêîé âëîæåííîé öåïüþ
Ìàðêîâà ìîæåò áûòü ïðèáëèæåí ïðîöåññîì Îðíøòåéíà–Óëåíáåêà ñ ýâîëþ-
öèåé. Îöåíêà ïàðàìåòðà äðåéôà ïîëó÷åíà ñ èñïîëüçîâàíèåì ñòàöèîíàðíîñòè
ãàóññîâñêîãî ïðåäåëüíîãî ïðîöåññà.
Êëþ÷åâûå ñëîâà: äèñêðåòíûé ìàðêîâñêèé ïðîöåññ, äèôôóçèîíàÿ àïïðîêñè-
ìàöèÿ, àñèìïòîòè÷åñêàÿ äèôôóçèîííàÿ ñðåäà, ïðîöåññ Îðíøòåéíà–Óëåíáå-
êà, ôàçîâîå óêðóïíåíèå, îöåíêà ïàðàìåòðà ñäâèãà.
Koroliuk Volodymyr Semenovich,
Dr. of Sciences, Academiciam of NAS Ukraine, Emer. Prof., Advisor to the Directorate, Institute of Mathe-
matics of the National Academy of Sciences of Ukraine, Kyiv.
Koroliouk Dmytro,
Dr. of Sciences, Lead Researcher, Institute of Telecommunications and Global Information Space of
the National Academy of Sciences of Ukraine, Kyiv, e-mail: dimitri.koroliouk@ukr.net.
Dovgyi Stanislav Îleksiyovich
Dr. of Sciences, Academiciam of NAS of Ukraine, Director of Dpt. of the Institute of Telecommunications
and Global Information Space of the National Academy of Sciences of Ukraine, Kyiv,
e-mail: pryjmalnya@gmail.com.
62 ISSN 1019-5262. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2020, òîì 56, ¹ 5
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| id | nasplib_isofts_kiev_ua-123456789-190452 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1019-5262 |
| language | English |
| last_indexed | 2025-12-07T18:25:10Z |
| publishDate | 2020 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Koroliuk, V.S. Koroliouk, D. Dovgyi S.О. 2023-06-08T15:19:07Z 2023-06-08T15:19:07Z 2020 Diffusion process with evolution and its parameter estimation / V.S. Koroliuk, D. Koroliouk, S.О. Dovgyi // Кибернетика и системный анализ. — 2020. — Т. 56, № 5. — С. 55–62. — Бібліогр.: 8 назв. — англ. 1019-5262 https://nasplib.isofts.kiev.ua/handle/123456789/190452 519.24 A discrete Markov process in an asymptotic diffusion environment with a uniformly ergodic embedded Markov chain can be approximated by an Ornstein–Uhlenbeck process with evolution. The drift parameter estimation is obtained using the stationarity of the Gaussian limit process. Показано, що дискретний марковський процес в асимптотичному дифузійному середовищі з рівномірним ергодичним вкладеним ланцюгом Маркова може бути наближений процесом Орнштейна-Уленбека з еволюцією. Оцінку параметра дрейфу отримано з використанням стаціонарності гаусівського граничного процесу. Показано, что дискретный марковский процесс в асимптотической диффузионной среде с равномерной эргодической вложенной цепью Маркова может быть приближен процессом Орнштейна-Уленбека с эволюцией. Оценка параметра дрейфа получена с использованием стационарности гауссовского предельного процесса. en Інститут кібернетики ім. В.М. Глушкова НАН України Кибернетика и системный анализ Системний аналіз Diffusion process with evolution and its parameter estimation Дифузійний процес з еволюцією та оцінювання його параметра Диффузионный процесс с эволюцией и оценка его параметра Article published earlier |
| spellingShingle | Diffusion process with evolution and its parameter estimation Koroliuk, V.S. Koroliouk, D. Dovgyi S.О. Системний аналіз |
| title | Diffusion process with evolution and its parameter estimation |
| title_alt | Дифузійний процес з еволюцією та оцінювання його параметра Диффузионный процесс с эволюцией и оценка его параметра |
| title_full | Diffusion process with evolution and its parameter estimation |
| title_fullStr | Diffusion process with evolution and its parameter estimation |
| title_full_unstemmed | Diffusion process with evolution and its parameter estimation |
| title_short | Diffusion process with evolution and its parameter estimation |
| title_sort | diffusion process with evolution and its parameter estimation |
| topic | Системний аналіз |
| topic_facet | Системний аналіз |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/190452 |
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