Simulation of random processes with known correlation function with the help of Karhunen-Loeve decomposition
A theorem is proved that allows to use approximations for construction of the Karhunen-Loeve model of stochastic process with known correlation function.
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irk-123456789-45192009-11-25T12:00:38Z Simulation of random processes with known correlation function with the help of Karhunen-Loeve decomposition Moklyachuk, O. A theorem is proved that allows to use approximations for construction of the Karhunen-Loeve model of stochastic process with known correlation function. 2007 Article Simulation of random processes with known correlation function with the help of Karhunen-Loeve decomposition / O. Moklyachuk // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 163–169. — Бібліогр.: 3 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4519 en Інститут математики НАН України |
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A theorem is proved that allows to use approximations for construction of the Karhunen-Loeve model of stochastic process with known correlation function. |
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Article |
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Moklyachuk, O. |
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Moklyachuk, O. Simulation of random processes with known correlation function with the help of Karhunen-Loeve decomposition |
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Moklyachuk, O. |
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Moklyachuk, O. |
| title |
Simulation of random processes with known correlation function with the help of Karhunen-Loeve decomposition |
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Simulation of random processes with known correlation function with the help of Karhunen-Loeve decomposition |
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Simulation of random processes with known correlation function with the help of Karhunen-Loeve decomposition |
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Simulation of random processes with known correlation function with the help of Karhunen-Loeve decomposition |
| title_full_unstemmed |
Simulation of random processes with known correlation function with the help of Karhunen-Loeve decomposition |
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simulation of random processes with known correlation function with the help of karhunen-loeve decomposition |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4519 |
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Simulation of random processes with known correlation function with the help of Karhunen-Loeve decomposition / O. Moklyachuk // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 163–169. — Бібліогр.: 3 назв.— англ. |
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2025-07-02T07:44:42Z |
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2025-07-02T07:44:42Z |
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1836520343763681280 |
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Theory of Stochastic Processes
Vol.13 (29), no.4, 20087, pp.163–169
OLEKSANDR MOKLYACHUK
SIMULATION OF RANDOM PROCESSES WITH
KNOWN CORRELATION FUNCTION WITH THE
HELP OF KARHUNEN-LOEVE DECOMPOSITION
A theorem is proved that allows to use approximations for construc-
tion of the Karhunen-Loeve model of stochastic process with known
correlation function.
1. Introduction
While modeling the financial state of the insurance company demand
flow is often considered as a Poisson process with the intensity generated
by the process eX(t), where X(t) is a centered Gaussian process.
Let X = {X(t), t ∈ T} be a centered (EX(t) = 0) Gaussian stochas-
tic process with correlation function B(t, s) = EX(t)X(s). To construct
a model of the process, we will use the method of simulation of stochastic
process with accuracy and reliability taken as parameters.
Consider a stochastic process X(t), t ∈ [0, T ] which can be represented
as a sum
X(t) =
∞∑
k=1
ξkfk(t)
which converges in the mean square. We will call the sum XN = XN(t),
t ∈ [0, T ],
XN(t) =
N∑
k=1
ξkfk(t)
the model of the process X(t).
2000 Mathematics Subject Classifications. 60G15, 65C20, 45B05.
Key words and phrases. Stochastic process, Karhunen-Loeve model, homogeneous
Fredholm equations of the second order, approximations
163
164 OLEKSANDR MOKLYACHUK
Let the stochastic process X(t) and all XN(t), N = 1, 2, . . . belong to a
Banach space B with the norm || · ||. Let two numbers α and δ (0 < α < 1,
δ > 0) be given. The model XN(t) approximates the stochastic process X(t)
with given reliability 1 − α and accuracy δ if for this model the following
inequality holds true:
P{||X(t)− XN || > δ} ≤ α (1)
As a result, to build a model of a stochastic process we have to find such N
that this inequality holds true for given α and δ.
Let us assume that we can establish the inequality:
P{||X(t) − XN(t)|| > δ} ≤ WN(δ),
where WN(δ), δ > 0 is a known monotone decreasing by N and δ function.
If N is such that WN(δ) ≤ α, then for all models XN ′ with N ′ ≥ N inequal-
ity (1) holds. This means that to build a model XN which approximates
the stochastic process X with given reliability 1 − α and accuracy δ in the
norm of space B, it is sufficient to find such N , minimal if it is possible,
that inequality WN (δ) ≤ α holds true.
2. The Karhunen-Loeve model
Consider method of modeling of stochastic processes based on the pro-
cesses decomposition using eigenfunctions of some integral equations.
Let [0, T ] be an interval in R, let X = {X(t), t ∈ [0, T ], EX(t) = 0}
be the mean square continuous Hilbert stochastic process, let B(t, s) =
EX(t)X(s), t, s ∈ [0, T ] be the correlation function of this process. The
function B(t, s) is positive semidefinite. As X(t) is mean square continuous,
then the function B(t, s) is continuous on [0, T ] × [0, T ].
Consider the homogenous Fredholm integral equation of the second type
ϕ(t) = λ
∫ T
0
B(t, s)ϕ(s)ds.
This equation has at most countable set of nonnegative eigenvalues. Let λ2
n,
n = 1, 2, . . . be eigenvalues of the equation, and let ϕn(t) be the correspond-
ing eigenfunctions. Let us also put λn in order 0 < λ1 ≤ λ2 ≤ . . . ≤ λn ≤
λn+1 ≤ . . .. It is known that ϕn(t) are continuous orthonormal functions
∫ T
0
ϕn(t)ϕm(t)dt = δn
m,
KARHUNEN-LOEVE MODEL WITH APPROXIMATIONS 165
where δn
m is the Kronecker delta.
In the following we will use the next theorem
Theorem. (Karhunen-Loeve decomposition) Let X(t) = {X(t), t ∈
[0, T ]} be a mean square continuous centered Hilbert stochastic process with
the correlation function B = B(t, s). Then for all t ∈ [0, T ]
X(t) =
∞∑
n=1
ξnϕn(t),
with the mean square convergence, where ξn are centered orthonormal Gaus-
sian random variables: Eξn = 0, Eξnξm = δn
mλ−2
n .
Definition. We call the Karhunen-Loeve model of stochastic process
X(t) = {X(t), t ∈ [0, T ]} the stochastic process XN(t) = {XN(t), t ∈
[0, T ]}, where XN(t) =
∑N
n=1 ξnϕn(t).
3. Approximate solutions of homogenous Fredholm integral
equation of the second order
Only some types of integral equations can be solved in explicit forms.
This is the main problem while building the Karhunen-Loeve model of
stochastic process. That is why we have to use approximate methods to
find eigenfunctions and eigenvalues of the second order homogenous Fred-
holm integral operator.
To construct an approximation we will use the method based on the
formula of rectangles:
∫ b
a
f(x)dx = h
n∑
k=1
f(xk),
where h = (b − a)/n, xk = ξ + (k − 1)h a ≤ ξ ≤ a + h.
Consider the equation
g(x) = λ
∫ b
a
K(x, y)g(y)dy (2)
with given a, b and twice differentiable function K(x, y). Let us apply the
formula of mean ordinates to the integral
∫ b
a
K(x, y)g(y)dy:
∫ b
a
K(x, y)g(y)dy = h
n∑
k=1
K(x, xk)g(xk)
and use this result in (2):
166 OLEKSANDR MOKLYACHUK
g(x) − λh
n∑
k=1
K(x, xk)g(xk) = 0. (3)
We take xi from the mean ordinates formula, substitute them in the
equation (3), and receive the system of equations:
g(x1) − λh
∑n
k=1 K(x1, xk)g(xk) = 0
g(x2) − λh
∑n
k=1 K(x2, xk)g(xk) = 0
. . .
g(xn) − λh
∑n
k=1 K(xn, xk)g(xk) = 0
If we solve this system we will receive trivial solution because this sys-
tem is homogenous. But we can take the last equation off and get g(xk),
k = 1, . . . , n − 1 expressed through g(xn).
Lets find g(xn) now. We use the property of eigenfunctions which states
that ||g(x)|| = 1. For L2[a, b] this property if of the form∫ b
a
(g(x))2 dx = 1 (4)
After applying the mean ordinates formula to (4), we get
h
n∑
k=1
g(xk)
2 = 1
From this relation we can find
g(xn) =
√√√√1
h
−
n−1∑
k=1
g(xk).
Finally, we can obtain g(xk), k = 1, . . . , n − 1.
Now we substitute these expressions to (3) and, after some manipula-
tions, we get the approximation of eigenfunction that corresponds to the
eigenvalue λ.
g(x) = λh
n∑
k=1
K(x, xk)g(xk)
Let us estimate the error of the approximation of eigenfunctions. We
designate
In
m(t) = λmh
n∑
k=1
K(x, xk)gm(xk)
KARHUNEN-LOEVE MODEL WITH APPROXIMATIONS 167
According to the Runge rule, the error of this approximation is
Δm =
I2n
m − In
m
3
4. Estimation of Karhunen-Loeve model accuracy in the
space Lp(0, T )
Let us consider a Karhunen-Loeve model in the space Lp(0, T ). We will
designate τN (t) as the error expectation.
E(X(t) − XN (t))2 = τ 2
N (t)
To fulfill the required reliability 1−α and accuracy δ in the space Lp(0, T )
the model XN has to fulfill the next inequality:
P{(
∫ T
0
(X(t) − XN(t))pdt)
1
p > δ} ≤ α
The following inequality was proved in [3].
P{(
∫ T
0
(X(t) − XN (t))pdt)
1
p > δ} ≤ 2exp{− δ2
2τ 2
NT 2/p
},
if δ ≥ p1/2T 1/pτN .
Transforming this inequality, we will obtain
2exp{− δ2
2τ 2
NT 1/p
}} ≤ α ⇒ δ2
2τ 2
NT 2/p
≥ − ln
α
2
Hence that, (1) holds true if
τ 2
N ≤ δ2
2(− ln α
2
)T 2/p
and
τ 2
N ≤ δ2
T 2/pp
Let us now pay more attention to expectation τN
τ 2
N = sup
0≤t≤T
E|X(t) − XN(t)|2 =
= sup
0≤t≤T
E
⎛
⎝ ∞∑
k=1
ξk
ϕk(t)√
λk
−
N∑
k=1
ξk
ϕ̂k(t)√
λ̂k
⎞
⎠
2
=
168 OLEKSANDR MOKLYACHUK
= sup
0≤t≤T
⎛
⎝ N∑
k=1
⎛
⎝ ϕ̂k(t)√
λ̂k
− ϕk(t)√
λk
⎞
⎠
2
+
∞∑
k=N+1
ϕ2
k(t)
λk
⎞
⎠ ≤
≤
N∑
k=1
sup
0≤t≤T
⎛
⎝ ϕ̂(k)√
λ̂k
− ϕ(k)√
λk
⎞
⎠
2
+
∞∑
k=N+1
sup0≤t≤T (ϕ2
k(t))
λk
Now we can apply approximations errors for eigenfunctions and eigen-
values mentioned above
τ2
N <
N∑
k=1
(3|
√
λ̂k + η −
√
λ̂k| sup0≤t≤T ϕ̂k(t) +
√
λ̂k sup0≤t≤T |I2n
k (t) − In
k (t)|)2
9λ̂k(λ̂k − η)
+
∞∑
k=N+1
sup0≤t≤T (ϕ2
k(t))
λk
It is also known that supt∈(0,T )ϕ(t) < C, where C is a constant, and we
can build a sequence μk such that ∀k 1/λk < 1/μk, where
∑∞
k=1 μk is finite.
The results obtained can be summarized in a theorem
Theorem. Stochastic process XN approximates stochastic process X
with reliability 1 − α and accuracy δ in the space Lp(0, T ),
P{(
∫ T
0
(X(t) − XN (t))2dt)
1
p > δ} ≤ α,
if N fulfills the next condition:
N∑
k=1
(3|
√
λ̂k + η −
√
λ̂k| sup0≤t≤T ϕ̂k(t) +
√
λ̂k sup0≤t≤T |I2n
k (t) − In
k (t)|)2
9λ̂k(λ̂k − η)
+
+
∞∑
k=N+1
sup0≤t≤T (ϕ2
k(t))
λk
< min
{
ε2
2(− ln α
2
)T 1/p
,
δ2
T 2/pp
}
,
where λ̂k is the approximation of k-th eigenvalue of the equation
ϕk(t) = λ
∫ b
a
K(t, s)ϕk(s)ds,
KARHUNEN-LOEVE MODEL WITH APPROXIMATIONS 169
η is the error of approximation of this eigenvalue, ϕk(t) is the corresponding
eigenfunction, and In
k is the n-th approximation of the eigenfunction ϕk(t).
This theorem allows us to build a Karhunen-Loeve model with given
accuracy and reliability in the space Lp(0, T ).
References
1. Kozachenko Yu.V., Pashko A.O., Modeling of shochastic processes. Kyiv
university, Kyiv, (1999).
2. Ferlan A.F., Sizikov V.S., Integral equations: methods, algorythms, pro-
grams. Naukova dumka, Kyiv, (1986).
3. Rita Giuliano Antonini, Yuriy V. Kozachenko, Antonina M. Tegza, Ac-
curacy of simulation in Lp of Gaussian random processes, Visnik of Kyiv
university, vol 5, (2002), 7–14.
Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: olgerdoutlander@ukr.net
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