Simulation of fractional Brownian motion with given reliability and accuracy in C([0, 1])
We present here an application of the results on simulation of weakly self-similar stationary increment φ-sub-Gaussian processes, obtained by Kozachenko, Sottinen and Vasylyk in [1], to the process of fractional Brownian motion.
Gespeichert in:
| Datum: | 2006 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2006
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/4457 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Simulation of fractional Brownian motion with given reliability and accuracy in C([0, 1]) / Y. Kozachenko, O. Vasylyk // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 55–62. — Бібліогр.: 4 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860066062564851712 |
|---|---|
| author | Kozachenko, Y. Vasylyk, O. |
| author_facet | Kozachenko, Y. Vasylyk, O. |
| citation_txt | Simulation of fractional Brownian motion with given reliability and accuracy in C([0, 1]) / Y. Kozachenko, O. Vasylyk // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 55–62. — Бібліогр.: 4 назв.— англ. |
| collection | DSpace DC |
| description | We present here an application of the results on simulation of weakly self-similar stationary increment φ-sub-Gaussian processes, obtained by Kozachenko, Sottinen and Vasylyk in [1], to the process of fractional Brownian motion.
|
| first_indexed | 2025-12-07T17:07:42Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 3–4, 2006, pp. 55–62
YURIY KOZACHENKO AND OLGA VASYLYK
SIMULATION OF FRACTIONAL BROWNIAN MOTION
WITH GIVEN RELIABILITY AND ACCURACY IN C([0, 1])1
We present here an application of the results on simulation of weakly
self-similar stationary increment ϕ-sub-Gaussian processes, obtained by
Kozachenko, Sottinen and Vasylyk in [1], to the process of fractional
Brownian motion.
1. Introduction
In this paper we consider simulation of fractional Brownian motion de-
fined on the interval [0, 1] with given reliability and accuracy in space
C([0, 1]).
We apply results obtained in [1] for centred second order Subϕ(Ω)-pro-
cesses defined on the interval [0, 1] with covariance function
R(t, s) =
1
2
(
t2H + s2H − |t − s|2H
)
.
The parameter H takes values in the interval (0, 1).
In order to construct a model of such process we used a series expansion
approach based on a series representation proved by Dzaparidze and van
Zanten [2] for the fractional Brownian motion B:
(1) Bt =
∞∑
n=1
sin(xnt)
xn
Xn +
∞∑
n=1
1 − cos(ynt)
yn
Yn, t ∈ [0, 1].
Here the Xn’s and the Yn’s are independent zero mean Gaussian random
variables with certain variances depending on H and n. The xn’s are the
positive real zeros of the Bessel function J−H of the first kind and the yn’s
are the positive real zeros of the Bessel function J1−H . The series in (1)
converge in mean square as well as uniformly on [0, 1] with probability 1.
Replacing the Xn’s and Yn’s by independent random variables from the
space Subϕ(Ω) we got series representation for ϕ-subGaussian random pro-
cesses with covariance function R. This representation was used for sim-
ulation of such processes with given reliability and accuracy in C([0, 1]).
Processes of fractional Brownian motion belong to the space Subϕ(Ω) with
1Invited lecture.
2000 Mathematics Subject Classification. Primary 60G18, 60G15, 68U20, 33C10.
Key words and phrases. Fractional Brownian motion, ϕ-sub-Gaussian processes, series
expansions, simulation.
55
56 YURIY KOZACHENKO AND OLGA VASYLYK
ϕ(x) = x2/2. So, in this paper we present some examples of simulation of
fractional Brownian motion with different values of parameter H.
2. Space Subϕ(Ω)
We need the following facts about the space Subϕ(Ω) of ϕ-sub-Gaussian
(or generalised sub-Gaussian) random variables.
Definition 2.1 ([3]). A continuous even convex function u = {u(x), x ∈ R}
is an Orlicz N-function if it is strictly increasing for x > 0, u(0) = 0,
u(x)
x
→ 0 as x → 0
and
u(x)
x
→ ∞ as x → ∞.
Proposition 2.2 ([3]). The function u is an Orlicz N-function if and only
if
u(x) =
|x|∫
0
l(u) du, x ∈ R,
where the density function l is nondecreasing, right continuous, l(u) > 0 as
u > 0, l(0) = 0 and l(u) → ∞ as u → ∞.
Definition 2.3. Let u be an Orlicz N-function. The even function u∗ =
{u∗(x), x ∈ R} defined by the formula
u∗(x) = sup
y>0
(
xy − u(y)
)
, x ≥ 0,
is the Young-Fenchel transformation of the function u.
Proposition 2.4 ([3]). The function u∗ is an Orlicz N-function and for
x > 0
u∗(x) = xy0 − u(y0) if y0 = l−1(x).
Here l−1 is the generalised inverse function of l, i.e.
l−1(x) := sup{v ≥ 0 : l(v) ≤ x}.
Definition 2.5. An Orlicz N-function ϕ satisfies assumption Q if ϕ is qua-
dratic around the origin, i.e. there exist such constants x0 > 0 and C > 0
that ϕ(x) = Cx2 for |x| ≤ x0.
Definition 2.6. A zero mean random variable ξ belongs to the space
Subϕ(Ω), the space of ϕ-sub-Gaussian random variables, if there exists a
positive and finite constant a such that the inequality
E exp{λξ} ≤ exp{ϕ(aλ)}
holds for all λ ∈ R.
SIMULATION OF FRACTIONAL BROWNIAN MOTION 57
The space Subϕ(Ω) is a Banach space with respect to the norm
τϕ(ξ) = inf
{
a ≥ 0 : E exp{λξ} ≤ exp
{
ϕ(aλ)
}
, λ ∈ R
}
.
Definition 2.7. A stochastic process X = (Xt)t∈[0,1] is a Subϕ(Ω)-process
if it is a bounded family of Subϕ(Ω) random variables: Xt ∈ Subϕ(Ω) for
all t ∈ [0, 1] and
sup
t∈[0,1]
τϕ(Xt) < ∞.
The properties of random variables from the spaces Subϕ(Ω) were studied
in the book [4].
Remark 2.8. When ϕ(x) = x2
2
the space Subϕ(Ω) is called the space of sub-
Gaussian random variables and is denoted by Sub(Ω). Centred Gaussian
random variable ξ belongs to the space Sub(Ω), and in this case τϕ(ξ) is
just the standard deviation: (Eξ2)1/2. Also, if ξ is bounded, i.e. |ξ| ≤ c a.s.
then ξ ∈ Sub(Ω) and τϕ(ξ) ≤ c.
3. Simulation of Subϕ(Ω)-processes
Define a process Z = (Zt)t∈[0,1] by the expansion
(2) Zt =
∞∑
n=1
cn sin(xnt) ξn +
∞∑
n=1
dn
(
1 − cos(ynt)
)
ηn,
where
cn =
πH
√
2c
xH+1
n J1−H(xn)
, n = 1, 2, . . . ,(3)
dn =
πH
√
2c
yH+1
n J−H(yn)
, n = 1, 2, . . . ,(4)
c =
Γ(2H + 1) sin(πH)
π2H+1
,(5)
ξn, ηn, n = 1, 2, . . . , are independent identically distributed centred random
variables from the space Subϕ(Ω) with
Eξ2
n = Eη2
n = 1
and
τϕ(ξn) = τϕ(ηn) =: aϕ, n = 1, 2, . . . ;
xn is the nth positive real zero of the Bessel function J−H ; yn is the nth
positive real zero of J1−H ,
Jν(x) =
∞∑
n=0
(−1)n(x/2)ν+2n
Γ(n + 1)Γ(ν + n + 1)
.
58 YURIY KOZACHENKO AND OLGA VASYLYK
Here x > 0, ν �= −1,−2, . . . and Γ denotes the Euler Gamma function
Γ(z) =
∫ ∞
0
tz−1e−t dt.
We shall assume that the function ϕ(
√ · ) is convex.
Since ϕ-sub-Gaussian random variables are square integrable we have the
following direct consequence of the series representation (1) for fractional
Brownian motion.
Proposition 3.1. The series (2) converges in mean square and the covari-
ance function of the process Z is R.
Theorem 3.2 ([1]). The series (2) converges uniformly with probability
one and the process Z is almost surely continuous on [0, 1]. Moreover, if
Z is strongly self-similar with stationary increments then it is β-Hölder
continuous with any index β < H.
Consider the space C([0, 1]) equipped with the usual sup-norm.
Definition 3.3. The model Z̃ approximates the process Z with given reli-
ability 1 − ν, 0 < ν < 1, and accuracy δ > 0 in C([0, 1]) if
P
(
sup
t∈[0,1]
|Zt − Z̃t| > δ
)
≤ ν.
Let c̃n and d̃n be the approximated values of the cn and dn, respectively.
Let
|c̃n − cn| ≤ γc
n,
|d̃n − dn| ≤ γd
n,
n = 1, . . . .
The errors γc
n and γd
n are assumed to be known. Let x̃n and ỹn be approxi-
mations of the corresponding zeros xn and yn with error bounds
|x̃n − xn| ≤ γx
n,
|ỹn − yn| ≤ γy
n.
The error bounds γx
n and γy
n are also assumed to be known.
Then, the model of the process Z we define as follows
(6) Z̃t =
N∑
n=1
(
c̃n sin(x̃nt) ξn + d̃n
(
1 − cos(ỹnt)
)
ηn
)
.
The following theorem contains the main result of the paper [1].
SIMULATION OF FRACTIONAL BROWNIAN MOTION 59
Theorem 3.4 ([1]). Let b and α be such that 0 < b < α < H. Denote
γ0 =
√
γappr + γcut,
γα =
√
γappr
α + γcut
α ,
β = min
{
γ0,
γα
2α
}
,
where
γcut = a2
ϕ
∞∑
n=N+1
(
c2
n + 4d2
n
)
,
γappr = a2
ϕ
N∑
n=1
{(
cnγx
n + γc
n
)2
+
(
dnγy
n + 2γd
n
)2
}
,
γcut
α = 22−2αa2
ϕ
∞∑
n=N+1
(
c2
nx2α
n + d2
ny2α
n
)
,
γappr
α = 23−2αa2
ϕ
N∑
n=1
{
x2α
n (γc
n)2 + y2α
n (γd
n)2
+ 23−2α
(
(c̃n)2(γx
n)2α
(
(xn + x̃n)2α
22α
+ 1
)
+
+ (d̃n)2(γy
n)2α
(
(yn + ỹn)
2α
22α
+ 1
))}
.
Let l be the density of ϕ.
The model Z̃, defined by (6), approximates the separable process Z, defined
by (2), with given reliability
1 − ν, 0 < ν < 1,
and accuracy δ > 0 in C([0, 1]) if the following three inequalities are satis-
fied:
(7) γ0 < δ,
(8)
βγ0
γα
<
δ
2α(exp{ϕ(1)} − 1)α
,
(9) 2 exp
{
−ϕ∗
(
δ
γ0
−1
)}(
1
2b(1− b
α
)
(
γαδ
βγ0
) b
α
l−1
(
δ
γ0
−1
)
+1
)2
b
≤ ν.
60 YURIY KOZACHENKO AND OLGA VASYLYK
4. Simulation of fractional Brownian motion
Let us assume that the constants cn and dn and the zeros xn and yn are
correctly calculated. In case of sub-Gaussian random processes we have the
following corollary from the Theorem 3.4.
Corollary 4.1. Suppose that there is no approximation error, i.e.
γc
n = γd
n = γx
n = γy
n = 0.
If the process Z is sub-Gaussian then conditions of the Theorem 3.4 are
satisfied if
(10) N ≥ max
⎧⎨
⎩
(
aϕ
δ
√
5c
2H
)1/H
+ 1;
22− 4
H 5
1
H
π
⎫⎬
⎭
and
(11) 2μ exp
⎧⎨
⎩−1
2
⎛
⎝ δNH
aϕ
√
5c
2H
−1
⎞
⎠
2⎫⎬
⎭N14 ≤ ν,
where
μ = π22
22
H
−45−
8
H
(
H
c
) 6
H
(
δ
aϕ
) 12
H
.
Recall that in the sub-Gaussian case we have ϕ(x) = x2
2
and that centered
Gaussian random variables belong to the space Sub(Ω). Parameters α and b
have to be optimized, but here α = H
2
and b = H
4
.
If in the series representation (2) the ξn and ηn, n = 1, 2, . . . , are inde-
pendent identically distributed centered Gaussian random variables with
Eξ2
n = Eη2
n = 1, n = 1, 2, . . . ,
then Z is a process of fractional Brownian motion. In this case aϕ = 1.
Using Corollary 4.1 we construct a model Z̃ of the fractional Brownian
motion Z, such that Z̃ approximates Z with given reliability 1 − ν = 0.99
and accuracy δ = 0.01 in C([0, 1]). In this paper we present examples of
such models for three values of parameter H .
• H1 = 3
4
. In this case we have: c = 0.0537337; μ = 97700.7.
From condition (10) follows that
N ≥ max {101.5, 0.270008} .
And condition (11) gives us N ≥ 6832.5.
So, we take N = 6833. As a result of simulation we have the
model, which approximates fractional Brownian motion with para-
meter H = 3
4
with given reliability 0.99 and accuracy 0.01 in C([0, 1])
(see Figure 1).
SIMULATION OF FRACTIONAL BROWNIAN MOTION 61
Figure 1. Model of fractional Brownian motion with parameter H = 3
4
• H2 = 7
8
. In this case c = 0.0264278; μ = 50832.8.
From conditions (10) and (11) follows:
N ≥ max {57.8064, 0.336974}
and N ≥ 1054.22.
The model with N = 1055 for fractional Brownian motion with
parameter H = 7
8
is presented on Figure 2.
Figure 2. Model of fractional Brownian motion with parameter H = 7
8
• H3 = 8
9
. In this case we have: c = 0.0234107 and μ = 47369.5.
From condition (10) follows that
N ≥ max {54.8061, 0.344046} .
From condition (11) we have N ≥ 863.771.
For N = 864 we obtained the model presented on Figure 3.
We can see that, as it was expected, N decreases when value of H in-
creases, so the closer is H to 1 the smoother curve we get.
All calculations and simulation were made using software Mathematica.
Acknowledgments
The authors were supported by the European Union project Tempus Tacis
NP 22012-2001 and the NATO Grant PST.CLG.980408.
62 YURIY KOZACHENKO AND OLGA VASYLYK
Figure 3. Model of fractional Brownian motion with parameter H = 8
9
References
1. Kozachenko, Yu., Sottinen, T., Vasylyk, O., Simulation of Weakly Self-
Similar Stationary Increment Subϕ(Ω)-Processes: A Series Expansion Approach,
Methodology and Computing in Applied Probability, 7, (2005), 379–400.
2. Dzhaparidze, K. O. and van Zanten, J. H., A series expansion of frac-
tional Brownian motion, Probability Theory and Related Fields, 130, (2004),
39–55.
3. Krasnoselskii, M. A. and Rutitskii, Ya. B. , Convex Functions in the Orlicz
spaces, “Fizmatiz”, Moscow, (1958).
4. Buldygin, V. V. and Kozachenko, Yu. V., Metric Characterization of Random
Variables and Random Processes, American Mathematical Society, Providence,
RI, (2000).
Department of Probability Theory and Math. Statistics, Mechanics
and Mathematics faculty, Taras Shevchenko Kyiv National University,
Volodymyrska 64, Kyiv, Ukraine
E-mail address: yvk@univ.kiev.ua
E-mail address: vasylyk@univ.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4457 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T17:07:42Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kozachenko, Y. Vasylyk, O. 2009-11-11T15:20:16Z 2009-11-11T15:20:16Z 2006 Simulation of fractional Brownian motion with given reliability and accuracy in C([0, 1]) / Y. Kozachenko, O. Vasylyk // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 55–62. — Бібліогр.: 4 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4457 We present here an application of the results on simulation of weakly self-similar stationary increment φ-sub-Gaussian processes, obtained by Kozachenko, Sottinen and Vasylyk in [1], to the process of fractional Brownian motion. en Інститут математики НАН України Simulation of fractional Brownian motion with given reliability and accuracy in C([0, 1]) Article published earlier |
| spellingShingle | Simulation of fractional Brownian motion with given reliability and accuracy in C([0, 1]) Kozachenko, Y. Vasylyk, O. |
| title | Simulation of fractional Brownian motion with given reliability and accuracy in C([0, 1]) |
| title_full | Simulation of fractional Brownian motion with given reliability and accuracy in C([0, 1]) |
| title_fullStr | Simulation of fractional Brownian motion with given reliability and accuracy in C([0, 1]) |
| title_full_unstemmed | Simulation of fractional Brownian motion with given reliability and accuracy in C([0, 1]) |
| title_short | Simulation of fractional Brownian motion with given reliability and accuracy in C([0, 1]) |
| title_sort | simulation of fractional brownian motion with given reliability and accuracy in c([0, 1]) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4457 |
| work_keys_str_mv | AT kozachenkoy simulationoffractionalbrownianmotionwithgivenreliabilityandaccuracyinc01 AT vasylyko simulationoffractionalbrownianmotionwithgivenreliabilityandaccuracyinc01 |