Application of the theory of square-Gaussian processes to simulation of stochastic processes
In the paper the simulation of stochastic processes is considered. For this purpose the estimation for distribution of supremum of square-Gaussian processes is found. The theorems are proved that give the conditions under which the constructed model approximates stochastic process in Banach space w...
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| Цитувати: | Application of the theory of square-Gaussian processes to simulation of stochastic processes / Y. Kozachenko, I. Rozora // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 43–54. — Бібліогр.: 9 назв.— англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859723221502263296 |
|---|---|
| author | Kozachenko, Y. Rozora, I. |
| author_facet | Kozachenko, Y. Rozora, I. |
| citation_txt | Application of the theory of square-Gaussian processes to simulation of stochastic processes / Y. Kozachenko, I. Rozora // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 43–54. — Бібліогр.: 9 назв.— англ. |
| collection | DSpace DC |
| description | In the paper the simulation of stochastic processes is considered. For this purpose the estimation for distribution of supremum of square-Gaussian processes is found. The theorems are proved that give the conditions
under which the constructed model approximates stochastic process in Banach space with given accuracy and reliability. The obtained results can be widely used in actuarial science and financial mathematics.
|
| first_indexed | 2025-12-01T11:00:18Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 3–4, 2006, pp. 43–54
YURIY KOZACHENKO AND IRYNA ROZORA
APPLICATION OF THE THEORY OF SQUARE-GAUSSIAN
PROCESSES TO SIMULATION OF STOCHASTIC
PROCESSES1
In the paper the simulation of stochastic processes is considered. For this
purpose the estimation for distribution of supremum of square-Gaussian
processes is found. The theorems are proved that give the conditions
under which the constructed model approximates stochastic process in
Banach space with given accuracy and reliability. The obtained results
can be widely used in actuarial science and financial mathematics.
1. Introduction
We’ll consider some properties of square-Gaussian stochastic processes.
Namely, a theorem about large deviation of supremum of the process is
proved. This result is used for simulation of Gaussian stochastic process,
that is entered the system (filter) as input process, taking into account out-
put process, with given reliability and accuracy in Banach space C([0, T ]d).
The particular case in which the output process is equal to the derivative
of input process is also considered.
The theory of simulation of stochastic process is extensively used in theory
of actuarial science [7]. The obtained results can be also used in such way.
The paper consists of four parts. The first part is introduction, in the
second one the main definitions and properties of square-Gaussian process
are given. The third part is devoted to the distribution of supremum of
square-gaussian processes. In last part we construct the model for Gaussian
stochastic process which is entered the system and give conditions under
which this model approximates the process, taking into account output
process, with give accuracy and reliability in Banach space.
2. Square-Gaussian processes
Let (Ω,F , P ) be probability space and (T, ρ) be a compact metric space
with metric ρ.
Use the definitions which were given in the paper [6] .
Definition 1.[6] Let Ξ = {ξt, t ∈ T} be a family of jointed Gaussian
random variables, Eξt = 0 (for example, let ξt, t ∈ T, be a Gaussian random
process).
1Invited lecture.
2000 Mathematics Subject Classification. Primary 60G60; Secondary 68U20, 65C20.
Key words and phrases. Gaussian stochastic process, simulation, square-Gaussian
process.
43
44 YURIY KOZACHENKO AND IRYNA ROZORA
A space SGΞ(Ω) is called the space of square-Gaussian random variables
if any η ∈ SGΞ(Ω) can be represented in such way
(1) η = ξ̄T Aξ̄ − Eξ̄TAξ̄,
where ξ̄T = (ξ1, ξ2, . . . , ξn), ξk ∈ Ξ, k = 1, . . . , n, A is real-valued matrix,
or η ∈ SGΞ(Ω) can be represented as a limit in mean square of the
sequence of random variables from (1)
η = l.i.m.n→∞(ξ̄T
n Aξ̄n − Eξ̄T
n Aξ̄n).
Definition 2.[6] A stochastic process X = {X(t), t ∈ T} is called square-
Gaussian if for any t ∈ T a random variable X(t) belongs to the space
SGΞ(Ω).
Example 1. Consider a family of Gaussian centered stochastic processes
ξ1(t), ξ2(t), . . . , ξn(t), t ∈ T. Let a matrix A(t) be symmetric. Then
X(t) = ξ̄T (t)A(t)ξ̄(t) −Eξ̄T (t)A(t)ξ̄(t),
where ξ̄T (t) = (ξ1(t), ξ2(t), . . . , ξn(t)), is square-Gaussian stochastic process.
Some properties of square-Gaussian stochastic processes can be found in
papers [3,4].
Under N(u) we denote the least numbers of closed balls of radius u cov-
ering the set T with respect to metric ρ.
Let X = {X(t, t ∈ T)} be a square-Gaussian process. A function
σ(h), h > 0, is monotonically increasing continuous σ(h) → 0 as h → 0
and
sup
ρ(t,s)≤h
(Var (X(t) − X(s)))
1
2 ≤ σ(h).
Define
ε0 = inf
t∈T
sup
s∈T
ρ(t, s), t0 = σ(ε0),
γ0 = sup
t∈T
(VarX(t))1/2,
Under σ(−1)(h) we will understand a general inverse function to the function
σ(h).
Theorem 1.[3] Let X(t) = {X(t), t ∈ T} be separable square-Gaussian
stochastic process. Consider increasing function r(u) ≥ 0, u ≥ 1 such that
r(u) → ∞ as u → ∞ and the function r(exp{t}) is convex. If the condition∫ t0
0
r(N(σ(−1)(u)))du < ∞,
holds true then for all integer M = 1, 2, . . ., 0 < p < 1 and u such that
(2) 0 < u <
1 − p√
2
min
{
1
γ0
,
1
t0pM−1
}
,
SIMULATION OF STOCHASTIC PROCESSES 45
the following inequality is satisfied
(3) P
{
sup
t∈T
|X(t)| > x
}
≤ W (p, x),
where
W (p, x) = 2
(
R
(u
√
2γ0
1 − p
))1−p
· A(p)
×
(
1 − pM−1u
√
2t0
1 − p
)−p/2
exp
{
−pMu
√
2t0
2(1 − p)
− ux
}
,
the function R(s) = (1 − |s|)− 1
2 exp{− |s|
2
} and
A(p) = r(−1)
(
1
t0pM
∫ t0pM
0
r(N(σ(−1)(v))) dv
)
.
3. An estimation of supremum distribution of
square-Gaussian stochastic processes
We consider the space T = [0, T ]d, d ≥ 1, with respect to metric ρ(t, s) =
max
1≤i≤d
|ti−si|. Let X = {X(t), t ∈ T} be square-Gaussian stochastic process.
In the case when σ(h) = C · hα, α ∈ (0, 1], where C > 0, the constants ε0
and t0 are equal to
ε0 = inf
t∈T
sup
s∈T
ρ(t, s) =
T
2
, t0 = σ(ε0) = C
(
T
2
)α
.
The following theorem holds true.
Theorem 2. Let X(t), t ∈ [0, T ]d, be separable square-Gaussian stochastic
process and
sup
ρ(t,s)≤h
(Var (X(t) − X(s)))
1
2 ≤ σ(h) = C · hα, α ∈ (0, 1], C > 0.
If for integer M > 1, x > 0
(4) x >
√
2γ0Md
α
max
{
1;
((T
2
)α
C
1
γ0
) 1
M−1}
,
then the next estimation holds
P
{
sup
t∈T
|X(t)| > x
}
≤ 21+de
(M+1)d
α exp
{
− x√
2γ0
}
×
( αx√
2γ0Md
)Md/α(
1 +
2x√
2γ0
)1/2
,(5)
where γ0 = sup
t∈T
(VarX(t))
1
2 .
46 YURIY KOZACHENKO AND IRYNA ROZORA
Proof. The proof of this theorem follows from theorem 1.
Notice that since σ(h) = C · hα, then σ(−1)(h) =
(
h
C
)1/α
.
It was proved in [2] that N(u) on [0, T ]d is satisfied the inequality
N(σ(−1)(u)) ≤
(
T
2σ(−1)(u)
+ 1
)d
=
(
T
2
(C
u
)1/α
+ 1
)d
.
Let’s consider a function r(u) = uβ − 1, β ∈ (0, α
d
) for which all condi-
tions of theorem 1 are satisfied. It’s easy to see that r(−1)(u) = (u + 1)1/β.
Since 0 < p < 1 and t0 = C
(
T
2
)α
then T
2
(
C
pM t0
)1/α
> 1. Hence, as
0 < u < t0p
M
N(σ(−1)(u)) ≤
(
T
(C
u
)1/α
)d
.
We estimate now A(p) in such way:
A(p) =
( 1
t0pM
∫ t0pM
0
[(T
2
(C
u
)1/α
+ 1
)dβ − 1
]
du + 1
)1/β
≤
( 1
t0pM
∫ t0pM
0
[
T
(C
u
)1/α]dβ
du
)1/β
= 2d
( α
α − dβ
)1/β
p−Md/α.
Let’s find minimum of A(p) with respect to β.
min
β∈(0, α
d
)
( α
α − dβ
)1/β
= lim
β→0
( 1
1 − dβ/α
)1/β
= ed/α.
Therefore, from (3) of theorem 1 follows the inequality
W (p, x) ≤ 21+ded/αp−Md/α
(
R
(u
√
2γ0
1 − p
))1−p
×
[(
1 − pM−1u
√
2t0
1 − p
)− 1
2
exp
{
−pM−1u
√
2t0
2(1 − p)
}]p
e−ux.(6)
We remind that increasing function R(s) is equal to
R(s) = (1 − |s|)−1/2 exp{−|s|
2
}.
If t0p
M−1 < γ0 then from (6) follows that
W (p, x) ≤ 21+ded/αp−Md/αR
(u
√
2γ0
1 − p
)
exp{−ux}.
The minimum of right side with respect to u is reached at the point
umin =
1
z
− 1
z + 2x
, where z =
√
2γ0
1 − p
.
SIMULATION OF STOCHASTIC PROCESSES 47
The point umin is satisfied condition (2).
If substitute umin in W (p, x) we obtain
W (p, x) ≤ 21+ded/αp−Md/α exp
{
−x(1 − p)√
2γ0
}(
1 +
2x(1 − p)√
2γ0
)1/2
.
Since 0 < p < 1 then
W (p, x) ≤ 21+ded/αp−Md/α exp
{
−x(1 − p)√
2γ0
}(
1 +
2x√
2γ0
)1/2
.
Let’s find minimum of right expression with respect to p. We obtain
p =
√
2γ0Md
xα
.
From p ∈ (0, 1) follows that
x >
√
2γ0Md
α
,
and this is holds true from conditions of theorem. Then
P
{
sup
t∈T
|X(t)| > x
}
≤ 21+ded/α exp
{
− x√
2γ0
+
Md
α
}
×
( αx√
2γ0Md
)Md
α
(
1 +
2x√
2γ0
)1/2
.
In the proof of the theorem we suppose that t0p
M−1 < γ0, but it follows
from relationship (4). Then the theorem is proved.
In the case when T = [0, T ] the following corollary is carried out.
Corollary 1. Let X(t), t ∈ [0, T ], be separable square-Gaussian stochastic
process for which
sup
ρ(t,s)≤h
(Var (X(t) − X(s)))
1
2 ≤ σ(h) = C · hα, α ∈ (0, 1], C > 0.
If for integer M > 1 and x > 0
x >
√
2γ0M
α
max
{
1;
((T
2
)α
C
1
γ0
) 1
M−1}
,
then
P
{
sup
t∈T
|X(t)| > x
}
≤ 4e
M+1
α exp
{
− x√
2γ0
}
×
( αx√
2γ0M
)M/α(
1 +
2x√
2γ0
)1/2
,
where γ0 = sup
t∈[0,T ]
(VarX(t))
1
2 .
48 YURIY KOZACHENKO AND IRYNA ROZORA
4. Accuracy and reliability of simulation of Gaussian
stochastic processes which can be represented in the form
of series
Consider the same space T = [0, T ]d, d > 1 with metric ρ(t, s) = max
1≤i≤d
|ti−
si|, where t, s are vectors from T. Let ξ = {ξ(t), t ∈ T} be centered Gauss-
ian stochastic process and
(7) ξ(t) =
∞∑
n=0
ξnfn(t),
where the functions fn(t), n ≥ 0, are continuous and such that for all t ∈ T
∞∑
n=0
f 2
n(t) < ∞,
ξn, n = 0, 1, 2, . . . , are independent Gaussian random variables, Eξn =
0, Eξ2
n = 1. Since
Eξ2(t) =
∞∑
n=0
f 2
n(t) < ∞,
then the series
∞∑
n=0
ξnfn(t) converges with probability one (see, for example,
[8]).
Consider such situation: Let Σ be some system(filter, device), which is
intended for transformation of signals (functions) fn(t). The function which
has to be transformed is called input function on system; the transformed
function is called output function or reaction on input function. Under gn(t)
we will define output function. More information about filter can be found
in [9].
Remark 1. In particular case gn(t) = zn · fn(t). It means that transforma-
tion doesn’t change the shape of signal.
In particular case can be also considered the situation when gn(t) = f ′
n(t).
If input process on the system Σ is ξ(t) =
∞∑
n=0
ξnfn(t), then output process
is η(t) =
∞∑
n=0
ξngn(t). Suppose that for all t ∈ T the series
∞∑
n=0
g2
n(t) con-
verges. i’s sufficient condition for convergence with probability one of the
series η(t) =
∞∑
n=0
ξngn(t).
Definition 3. The process ξ̃N(t) is called the model of the process ξ(t), t ∈
T if
ξ̃N(t) =
N∑
k=0
ξkfk(t), t ∈ T.
SIMULATION OF STOCHASTIC PROCESSES 49
Let’s define the difference between the process and the model under
ξN(t) = ξ(t) − ξ̃N(t) =
∞∑
k=N+1
ξkfk(t), t ∈ T.
In the same way ηN(t) can be defined:
ηN(t) =
∞∑
k=N+1
ξkgk(t), t ∈ T.
We’ll investigate conditions under which the model ξ̃N(t) approximates
ξ(t) with given accuracy and reliability in Banach space C([0, T ]d) taking
into account the process η(t). For this purpose the relationship ξ2(t)+η2(t)
can be analyzed. If generalize this case we can consider a semi-positive
quadratic form
X(x, y) = a · x2 + 2c · x · y + b · y2,
where a, b, c are such that a > 0, ab − c2 > 0.
For convenience,under XN(t) we’ll define a quadratic form which is de-
fined on the processes ξN(t), ηN(t) :
XN(t) = X(ξN(t), ηN(t)) = a · (ξN(t))2 + 2c · ξN(t) · ηN(t) + b · (ηN(t))2.
Stochastic process XN (t) is equal to
(8) XN(t) =
∞∑
k=N+1
∞∑
n=N+1
ξkξnφkn(t),
where
(9) φkn(t) = afk(t)fn(t) + c(fk(t)gn(t) + gk(t)fn(t)) + bgk(t)gn(t).
the function φkn(t) is symmetric with respect to k and n. Hence, φkn(t) =
φnk(t).
Denote
X̄N(t) = XN(t) − EXN(t).
Definition 4. The model ξ̃N(t) approximates stochastic process ξ(t) on in-
put of the system, taking into account output process, with given reliability
1 − ν, ν ∈ (0, 1) and accuracy δ > 0 in Banach space C([0, T ]d), if
P
{
sup
t∈T
|X̄N(t)| > δ
}
≤ ν.
Notice that XN(t) − EXN(t) = X̄N(t), t ∈ [0, T ]d, is square-Gaussian
stochastic process.
The next additional assertion is proved.
50 YURIY KOZACHENKO AND IRYNA ROZORA
Lemma 1. Let the series
∞∑
k,n=N+1
φ2
kn(t) be convergent for any t ∈ T.
Define
Δkn(t, s) = φkn(t) − φkn(s).
Then for the processes X̄N (t), XN(t) the following relationships hold true
EXN(t) =
∞∑
k=N+1
φkk(t),
(10) Var X̄N(t) = 2
∞∑
k=N+1
∞∑
n=N+1
φ2
kn(t),
(11) Var (XN(t) − XN(s)) = 2
∞∑
k=N+1
∞∑
n=N+1
Δ2
kn(t, s),
where the functions φkn(t) are from (9).
Proof. From (8) it follows that
XN(t) =
∞∑
k=N+1
∞∑
n=N+1
φkn(t)ξkξn,
ξk, k ≥ 0, are independent Gaussian random variables with zero expectation
and variance 1. Then
EXN(t) =
∞∑
k=N+1
φkk(t).
Find now E(XN(t))2.
E(XN(t))2 =
∞∑
k=N+1
∞∑
n=N+1
∞∑
k′=N+1
∞∑
n′=N+1
φkn(t)φk′n′(t)Eξkξnξk′ξn′
From equality
Eξkξnξk′ξn′ = EξkξnEξk′ξn′ + Eξkξk′Eξnξn′ + Eξkξn′Eξk′ξn,
and from φkn(t) = φnk(t) follows that
E(XN(t))2 =
∞∑
k=N+1
∞∑
n=N+1
(
φkk(t)φnn(t) + 2φ2
kn(t)
)
.
The relationship (10) we obtain from VarXN(t) = Var (X̄N (t)).
Let’s find EXN(t)XN(s).
EXN(t)XN(s) =
∞∑
k=N+1
∞∑
n=N+1
(φkk(t)φnn(s) + 2φkn(t)φkn(s)) .
SIMULATION OF STOCHASTIC PROCESSES 51
The formula (11) is obtained from
Var (X̄N(t) − X̄N(s)) = Var X̄N (t) + Var X̄N(s)
− 2EXN(t)XN(s) + 2EXN(t)EXN(s).
The lemma is proved.
Denote skn = sup
t∈T
|φkn(t)|, then
(12) sup
t∈T
(Var X̄N (t))
1
2 ≤
(
2
∞∑
k=N+1
∞∑
n=N+1
s2
kn
)1
2
.
The following theorem holds true.
Theorem 3. Let ξ(t), t ∈ [0, T ]d, be separable Gaussian stochastic process
for which
(13) sup
ρ(t,s)≤h
|φkn(t) − φkn(s)| ≤ dknh
α, α ∈ (0, 1],
and
2
∞∑
k=N+1
∞∑
n=N+1
d2
kn = C2(N) < ∞,
whereφkn(t) are from (9).
The model ξ̃N(t) approximates separable Gaussian process ξ(t), taking
into accout output process, with given accuracy δ > 0 and reliability 1 −
ν, ν ∈ (0, 1), if for N ≥ 1 the conditions are fulfilled
δ >
√
2γ0(N)Md
α
max
{
1;
((T
2
)α C(N)
γ0(N)
) 1
M−1}
,
21+de
(M+1)d
α exp
{
− δ√
2γ0(N)
}( αδ√
2γ0(N)Md
)Md/α(
1 +
2δ√
2γ0(N)
)1/2
< ν,
where M > 1 is arbitrary integer number, γ0(N) =
(
2
∞∑
k=N+1
∞∑
n=N+1
s2
kn
) 1
2
.
Proof. Since the process X̄N(t) is square-Gaussian then it can be used the
results of theorem 2. From theorem condition (13) and from (11) follows
52 YURIY KOZACHENKO AND IRYNA ROZORA
that
sup
ρ(t,s)≤h
(Var (X(t) − X(s)))
1
2 = sup
ρ(t,s)≤h
(
2
∞∑
k=N+1
∞∑
n=N+1
(φkn(t) − φkn(s))2
)1
2
≤
(
2
∞∑
k=N+1
∞∑
n=N+1
d2
kn
) 1
2
hα
= C(N)hα = σN (h), α ∈ (0, 1].
From (12) follows that
sup
t∈T
(Var X̄N(t))
1
2 ≤
(
2
∞∑
k=N+1
∞∑
n=N+1
d2
kn
) 1
2
= γ0(N).
If we substitute the obtained relationships in inequalities from theorem 2,
the theorem will be proved.
In particular case T = [0, T ] the following corollary holds true.
Corollary 2. Let ξ(t), t ∈ [0, T ], be separable Gaussian stochastic process
for which
sup
|t−s|≤h
|φkn(t) − φkn(s)| ≤ dknh
α, α ∈ (0, 1],
and
2
∞∑
k=N+1
∞∑
n=N+1
d2
kn = C2(N) < ∞,
where φkn(t) are from (9).
The model ξ̃N(t) approximates separable Gaussian process ξ(t), taking
into account output process, with given accuracy δ > 0 and reliability 1 −
ν, ν ∈ (0, 1), if for N the next inequalities are satisfied
δ >
√
2γ0(N)M
α
max
{
1;
((T
2
)α C(N)
γ0(N)
) 1
M−1}
,
4e
(M+1)
α exp
{
− δ√
2γ0(N)
}( αδ√
2γ0(N)M
)M/α(
1 +
2δ√
2γ0(N)
)1/2
< ν,
where M > 1 is arbitrary integer number, γ0(N) =
(
2
∞∑
k=N+1
∞∑
n=N+1
s2
kn
) 1
2
.
Example 2. Let ξ = {ξ(t), t ∈ [0, T ]}, be centered Gaussian process
which can be represented in the form (7), where the functions fn(t), n ≥ 0,
SIMULATION OF STOCHASTIC PROCESSES 53
are continuously differentiable and for all t ∈ [0, T ]
∞∑
n=0
(f ′
n(t))2 < ∞ and
∞∑
n=0
|f ′
n(t)| < ∞.
Consider the case when output process is equal to η(t) = ξ′(t), t ∈ [0, T ].
There exists derivative of stochastic process ξ′(t) =
∞∑
n=0
f ′
n(t)ξn in mean
square.
The difference between the process and the model is
ξN(t) = ξ(t) − ξ̃N(t) =
∞∑
k=N+1
ξkfk(t), t ∈ T
and the process ηN (t) is equal to
ηN (t) =
∞∑
k=N+1
ξkf
′
k(t), t ∈ T.
Let’s construct a semi-positive quadratic form XN(t), which is defined on
the processes ξN(t), ηN(t)
XN(t) = X(ξN(t), ηN(t)) = a · (ξN(t))2 + 2c · ξN(t) · ηN(t) + b · (ηN(t))2.
The process XN(t) can be represented in the form
(14) XN(t) =
∞∑
k=N+1
∞∑
n=N+1
ξkξnφkn(t),
where
(15) φkn(t) = afk(t)fn(t) + c(fk(t)f
′
n(t) + f ′
k(t)fn(t)) + bf ′
k(t)f
′
n(t).
Then it can be used corollary 2 for stochastic process (14) which gives the
conditions under which the model approximates separable Gaussian process,
taking into account its derivative, with given accuracy and reliability. It’s
shown in the next theorem.
Theorem 4. Let ξ(t), t ∈ [0, T ], be separable Gaussian stochastic process
for which
(16) sup
|t−s|≤h
|φkn(t) − φkn(s)| ≤ dknh
α, α ∈ (0, 1],
and
2
∞∑
k=N+1
∞∑
n=N+1
d2
kn = C2(N) < ∞,
where φkn(t) are from (15).
54 YURIY KOZACHENKO AND IRYNA ROZORA
The model ξ̃N(t) approximates separable Gaussian process ξ(t), taking
into account output process, with given accuracy δ > 0 and reliability 1 −
ν, ν ∈ (0, 1), if for N the next inequalities are satisfied
δ >
√
2γ0(N)M
α
max
{
1;
((T
2
)α C(N)
γ0(N)
) 1
M−1}
,
4e
(M+1)
α exp
{
− δ√
2γ0(N)
}( αδ√
2γ0(N)M
)M/α(
1 +
2δ√
2γ0(N)
)1/2
< ν,
where M > 1is arbitrary integer number, γ0(N) =
(
2
∞∑
k=N+1
∞∑
n=N+1
s2
kn
) 1
2
.
References
1. Kozachenko, Yu., Rozora, I., Simulation of Gaussian stochastic processes, Ran-
dom Oper. and Stoch. Equ., 11, (2003), No. 3, 275–296.
2. Buldygin, V. V. and Kozachenko, Yu. V., Metric Characterization of Random
Variables and Random Processes. American Mathematical Society, Providence,
RI (2000).
3. Kozachenko, Yu. V. and Moklyachuk, O. M., Large deviation probabilities for
square-Gaussian stochastic processes, Extremes, 2(3), (1999), 269–293.
4. Kozachenko, Yu. V. and Moklyachuk, O. M., Square-Gaussian stochastic
processes, Theory of Stochastic Processes, 6(22), (2000), no. 3-4, 98-121.
5. Kozachenko Yu., Rozora I., Simulation of Gaussian Stochastic Fields, Theory of
Stochastic Processes, 10 (26), (2004), no.1-2,48-60.
6. Kozachenko, Yu.V. and Stus, O. V., General strictly pregaussian stochastic
processes, Theory of Probab. and Mathem. Statist, 58, (1998), 61–75.
7. Tougels, Jef and Sundt, Bjørn, Encyclopedia of Actuarial Science. Wiley
(2004),ISBN:0-470-84676-3.
8. M. Loev, Probability theory. D.van Nostrand, Princeton, NJ (1960).
9. Gikhman, I.I. and Skhorokhod, A.V., An Introduction to the theory of stochastic
processes, Moscow, Nauka (1977).
Department of Probability Theory and Mathematical Statistics, Kyiv
National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: yvk@univ.kiev.ua
Department of Applied Statistics, Faculty of Cybernetics, Kyiv Na-
tional Taras Shevchenko University, Kyiv, Ukraine
E-mail address: irozora@bigmir.net
|
| id | nasplib_isofts_kiev_ua-123456789-4456 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-01T11:00:18Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kozachenko, Y. Rozora, I. 2009-11-11T15:19:34Z 2009-11-11T15:19:34Z 2006 Application of the theory of square-Gaussian processes to simulation of stochastic processes / Y. Kozachenko, I. Rozora // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 43–54. — Бібліогр.: 9 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4456 In the paper the simulation of stochastic processes is considered. For this purpose the estimation for distribution of supremum of square-Gaussian processes is found. The theorems are proved that give the conditions under which the constructed model approximates stochastic process in Banach space with given accuracy and reliability. The obtained results can be widely used in actuarial science and financial mathematics. en Інститут математики НАН України Application of the theory of square-Gaussian processes to simulation of stochastic processes Article published earlier |
| spellingShingle | Application of the theory of square-Gaussian processes to simulation of stochastic processes Kozachenko, Y. Rozora, I. |
| title | Application of the theory of square-Gaussian processes to simulation of stochastic processes |
| title_full | Application of the theory of square-Gaussian processes to simulation of stochastic processes |
| title_fullStr | Application of the theory of square-Gaussian processes to simulation of stochastic processes |
| title_full_unstemmed | Application of the theory of square-Gaussian processes to simulation of stochastic processes |
| title_short | Application of the theory of square-Gaussian processes to simulation of stochastic processes |
| title_sort | application of the theory of square-gaussian processes to simulation of stochastic processes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4456 |
| work_keys_str_mv | AT kozachenkoy applicationofthetheoryofsquaregaussianprocessestosimulationofstochasticprocesses AT rozorai applicationofthetheoryofsquaregaussianprocessestosimulationofstochasticprocesses |