1. No category

Columbia International Publishing
Contemporary Mathematics and Statistics
(2015) Vol. 3 No. 1 pp. 1-7
doi:10.7726/cms.2015.1001
Research Article
Haar Wavelet and Simulation of Stochastic Processes
Ievgen Turchyn1*
Received June 29 2013; Published online April 25 2015
© The author 2015. Published with open access at www.uscip.us
Abstract
A wavelet-based method for simulation of Gaussian processes with given accuracy and reliability in
𝐿𝑝 ([0, 𝑇]) is proposed.
Keywords: Gaussian processes; Simulation; Wavelets
1. Introduction
Applications of wavelets to stochastic processes have been extensively developed recently. In
particular, wavelet-based simulation and series expansions of stochastic processes have been
studied by Wornell (1993), Meyer et al. (1999), Whitcher (2001), Seleznjev and Staroverov (2002),
Ayache and Taqqu (2003), Pipiras (2004, 2005). Wavelet-based expansions of stochastic processes
in series with uncorrelated terms have been researched by Walter and Zhang (1994), Didier and
Pipiras (2008), Kozachenko et al. (2011).
We will prove a result about simulation of a Gaussian process with given accuracy and reliability
using a wavelet-based model. Simulation with given accuracy and reliability (see Kozachenko and
Pashko (1999)) is a special kind of simulation of stochastic processes. To simulate a random
process 𝑋(𝑡) with given accuracy 𝜀 and reliability 1 − 𝛿 means to build such a model 𝑋̂(𝑡) for the
process 𝑋(𝑡) that
P{∥ 𝑋(𝑡) − 𝑋̂(𝑡) ∥> 𝜀} ≤ 𝛿
(so simulation with given accuracy and reliability is a simulation with a certain prescribed rate of
convergence). This kind of simulation has substantial advantages over more traditional methods of
simulation of stochastic processes. For instance, traditional methods of simulation may produce
models which paths are quite different from paths of the process but simulation with given
__________________________________________________________________________________________________________________
*Corresponding e-mail: [email protected]
1 Oles Honchar Dnipropetrovsk National University, Dnipropetrovsk, Ukraine
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Ievgen Turchyn / Contemporary Mathematics and Statistics
(2015) Vol. 3 No. 1 pp. 1-7
accuracy and reliability will result in a model which paths are with high probability close to paths of
the process.
Our model of a process is obtained by a truncation of a wavelet-based expansion of a stochastic
process. We prove a theorem about simulation of a Gaussian process with given accuracy and
reliability in 𝐿𝑝 ([0, 𝑇]) using this wavelet-based model and the Haar wavelet. Analogous results
about simulation of Gaussian and sub-Gaussian processes with given accuracy and reliability by
means of a similar model have been obtained before in Kozachenko and Turchyn (2008) and
Turchyn (2011). Our result is valid for a wide class of Gaussian processes, this class includes certain
processes to which earlier theorems about simulation with given accuracy and reliability using
similar wavelet-based models are not applicable.
2. Preliminaries
Let 𝜑 ∈ 𝐿2 (ℝ) be such a function that the following assumptions hold:
i) ∑𝑘∈ℤ |𝜑̂(𝑦 + 2𝜋𝑘)|2 = 1 almost everywhere, where 𝜑̂(𝑦) is the Fourier transform of 𝜑,
𝜑̂(𝑦) = ∫ exp{−𝑖𝑦𝑥}𝜑(𝑥)𝑑𝑥;
ℝ
ii) There exists a function 𝑚0 ∈ 𝐿2 ([0,2𝜋]) such that 𝑚0 (𝑥) has period 2𝜋 and almost
everywhere
𝜑̂(𝑦) = 𝑚0 (𝑦⁄2)𝜑̂(𝑦⁄2);
iii) 𝜑̂(0) ≠ 0 and the function 𝜑̂(𝑦) is continuous at 0.
The function 𝜑(𝑥) is called 𝑓-wavelet. Let 𝜓(𝑥) be the inverse Fourier transform of the function
𝜓̂(𝑦) = 𝑚0 (𝑦⁄2 + 𝜋)exp{−𝑖𝑦⁄2}𝜑̂(𝑦⁄2).
The function 𝜓(𝑥) is called 𝑚-wavelet.
Let
𝜑𝑗𝑘 (𝑥) = 2𝑗/2 𝜑(2𝑗 𝑥 − 𝑘), 𝜓𝑗𝑘 (𝑥) = 2𝑗/2 𝜓(2𝑗 𝑥 − 𝑘), 𝑘 ∈ ℤ, 𝑗 = 0,1,2, …
It is known that the family of functions {𝜑0𝑘 , 𝜓𝑗𝑘 , 𝑗 = 0,1, … ; 𝑘 ∈ ℤ} is an orthonormal basis in 𝐿2 (ℝ)
(see, for example, Walter and Shen (2000)).
Any function 𝑓 ∈ 𝐿2 (ℝ) can be represented as
𝑓(𝑥) = ∑𝑘∈ℤ 𝛼0𝑘 𝜑0𝑘 (𝑥) + ∑∞
𝑗=0 ∑𝑘∈ℤ 𝛽𝑗𝑘 𝜓𝑗𝑘 (𝑥),
(2.1)
where 𝛼0𝑘 = ∫ℝ 𝑓(𝑥)𝜑0𝑘 (𝑥)𝑑𝑥, 𝛽𝑗𝑘 = ∫ℝ 𝑓(𝑥)𝜓𝑗𝑘 (𝑥)𝑑𝑥,
2
∑𝑘∈ℤ |𝛼0𝑘 |2 + ∑∞
𝑗=0 ∑𝑘∈ℤ |𝛽𝑗𝑘 | < ∞.
That is, series (2.1) converges in the norm of the space 𝐿2 (ℝ). Representation (2.1) is called wavelet
representation.
We need the following result.
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Ievgen Turchyn / Contemporary Mathematics and Statistics
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Theorem 2.1 Let 𝑋 = {𝑋(𝑡), 𝑡 ∈ ℝ} be a centered random process such that for all 𝑡 ∈ ℝ 𝐄|𝑋(𝑡)|2
is finite. Let 𝑅(𝑡, 𝑠) = 𝐄𝑋(𝑡)𝑋(𝑠) and there exists such a Borel function 𝑢(𝑡, 𝑦), 𝑡 ∈ ℝ, 𝑦 ∈ ℝ that
∫ℝ |𝑢(𝑡, 𝑦)|2 𝑑𝑦 < ∞ for all 𝑡 ∈ ℝ and
𝑅(𝑡, 𝑠) = ∫ℝ 𝑢(𝑡, 𝑦)𝑢(𝑠, 𝑦)𝑑𝑦.
Let {𝜑0𝑘 (𝑥), 𝜓𝑗𝑘 (𝑥), 𝑘 ∈ ℤ, 𝑗 = 0,1,2, … } be a wavelet system. Then the process 𝑋(𝑡) can be presented
as the following series which converges for all 𝑡 ∈ ℝ in mean square:
𝑋(𝑡) = ∑𝑘∈ℤ 𝜉0𝑘 𝛼0𝑘 (𝑡) + ∑∞
𝑗=0 ∑𝑘∈ℤ 𝜂𝑗𝑘 𝛽𝑗𝑘 (𝑡),
(2.2)
where
𝛼0𝑘 (𝑡) = ∫ℝ 𝑢(𝑡, 𝑦) ̅̅̅̅̅̅̅̅̅
𝜑0𝑘 (𝑦)𝑑𝑦,
(2.3)
̅̅̅̅̅̅̅̅̅
𝛽𝑗𝑘 (𝑡) = ∫ℝ 𝑢(𝑡, 𝑦)𝜓
𝑗𝑘 (𝑦)𝑑𝑦,
(2.4)
𝜉0𝑘 , 𝜂𝑗𝑘 are centered random variables such that
𝐄𝜉0𝑘 𝜉0𝑙 = 𝛿𝑘𝑙 , 𝐄𝜂𝑚𝑘 𝜂𝑛𝑙 = 𝛿𝑚𝑛 𝛿𝑘𝑙 , 𝐄𝜉0𝑘 𝜂𝑛𝑙 = 0 .
Theorem 2.1 is a corollary from Theorem 1.1 in Kozachenko et al. (2011) which is obtained by
taking {𝜑0𝑘 , 𝜓𝑗𝑘 , 𝑗 = 0,1, … ; 𝑘 ∈ ℤ} as an orthonormal basis.
We will use the term “W-expansion of the process X(t)” for expansion (2.2).
3. Simulation
Lemma 3.1 Suppose that a random process 𝑋 = {𝑋(𝑡), 𝑡 ∈ ℝ} satisfies the conditions of Theorem 2.1,
the function 𝑢(𝑡, 𝑦) defined in Theorem 2.1 is differentiable with respect to y and satisfies the
following conditions:
𝐴(𝑡)
|𝑢𝑦′ (𝑡, 𝑦)| ≤ 1+|𝑦|𝛽 ,
(3.1)
where 𝛽 ∈ (1; 3/2),
𝐵(𝑡)
|𝑢(𝑡, 𝑦)| ≤ 1+|𝑦| .
(3.2)
Let 𝜑(𝑡) and 𝜓(𝑡) be the Haar 𝑓 -wavelet and 𝑚 -wavelet correspondingly. Then the following
inequalities hold for the coefficients 𝛼0𝑘 (𝑡) 𝑎𝑛𝑑 𝛽𝑗𝑘 (𝑡) in W-expansion (2.2) of the process X(t) with
respect to the Haar wavelet:
𝐵(𝑡)
|𝛼0𝑘 (𝑡)| ≤ 1+min{|𝑘|,|𝑘+1|} ,
(3.3)
|𝛽𝑗𝑘 (𝑡)| ≤ 4𝑘 𝛽 2𝑗(3/2−𝛽) , 𝑘 > 0 ,
(3.4)
𝐴(𝑡)
𝐴(𝑡)
|𝛽𝑗𝑘 (𝑡)| ≤ 4|𝑘+1/2|𝛽 2𝑗(3/2−𝛽) , 𝑘 < 0 ,
𝐴(𝑡)
|𝛽𝑗0 (𝑡)| ≤ 4 ∙23𝑗/2 .
(3.5)
(3.6)
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Ievgen Turchyn / Contemporary Mathematics and Statistics
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Proof. Let us prove inequality (3.4). We have for 𝑘 > 0:
|𝛽𝑗𝑘 (𝑡)| = |∫ 𝑢(𝑡, 𝑥) 2
𝑗 ⁄2
𝑗
𝜓(2 𝑥 − 𝑘)𝑑𝑥| = 2
𝑗 ⁄2
(𝑘+1⁄2)⁄2𝑗
≤2
𝑗 ⁄2
(𝑘+1⁄2)⁄2𝑗
∫
𝑘 ⁄ 2𝑗
|𝑢(𝑡, 𝑥) − 𝑢 (𝑡, 𝑥 +
𝑢(𝑡, 𝑥) 𝑑𝑥 − ∫
|∫
𝑘⁄2𝑗
ℝ
1
2𝑗+1
(𝑘+1)⁄2𝑗
(𝑘+1⁄2)⁄2𝑗
𝑢(𝑡, 𝑥) 𝑑𝑥|
)| 𝑑𝑥 ≤ sup |𝑢𝑥′ (𝑡, 𝑥)|/23𝑗⁄2+2 ,
𝑥∈𝐼
where 𝐼 = [𝑘/2𝑗 ; (𝑘 + 1/2)/2𝑗 ] . But it follows from (3.1) that
2𝑗𝛽
sup𝑥 ∈ 𝐼 |𝑢𝑥′ (𝑡, 𝑥)| ≤ 𝐴(𝑡) 𝑘 𝛽
and now we immediately obtain inequality (3.4).
Inequalities (3.3), (3.5) and (3.6) can be proved in a similar way.
Definition 3.1 Let a random process 𝑋 = {𝑋(𝑡), 𝑡 ∈ ℝ} satisfy the conditions of Theorem 2.1. We call
the following process a model of 𝑋(𝑡):
𝑀𝑗 −1
𝑁0 −1
𝑋̂(𝑡) = ∑𝑘=−(𝑁
𝜉 𝛼 (𝑡) + ∑𝑁−1
𝑗=0 ∑𝑘=−(𝑀
0 −1) 0𝑘 0𝑘
𝑗 −1)
𝜂𝑗𝑘 𝛽𝑗𝑘 (𝑡),
(3.7)
where 𝜉0𝑘 , 𝜂𝑗𝑘 are random variables from expansion (2.2), 𝛼0𝑘 (𝑡) and 𝛽𝑗𝑘 (𝑡) are calculated using
formulae (2.3) and (2.4), 𝑁0 > 1, 𝑁 > 1, 𝑀𝑗 > 1.
Definition 3.2 Suppose that 0 < 𝛿 < 1, 𝜀 > 0. We say that the model 𝑋̂(𝑡) defined by (3.7)
approximates a process 𝑋(𝑡) with given reliability 1 − 𝛿 and accuracy 𝜀 in 𝐿𝑝 ([0, 𝑇]) if
1/𝑝
T
P{(∫0 |𝑋(𝑡) − 𝑋̂(𝑡)|𝑝 𝑑𝑡)
> 𝜀} ≤ 𝛿.
We will need the following fact which is a modification of Theorem 5.1 from Kozachenko and
Turchyn (2008).
Theorem 3.1 Suppose that a Gaussian random process 𝑋 = {𝑋(𝑡), 𝑡 ∈ ℝ} satisfies the conditions of
Theorem 2.1, 𝑋̂(𝑡) is its model defined by (3.7), 0 < 𝛿 < √2𝑒 −𝑝/2 , 𝑝 ≥ 1. If
𝜀2
sup 𝐄|𝑋(𝑡) − 𝑋̂(𝑡)|2 ≤ 2𝑇 2/𝑝 ln(
𝑡∈[0,𝑇]
√2/𝛿)
,
then the model 𝑋̂(𝑡) approximates the process 𝑋(𝑡) with reliability 1 − 𝛿 and accuracy 𝜀 in 𝐿𝑝 ([0, 𝑇]).
Now we can formulate and prove the main result of the paper.
Theorem 3.2 Suppose that a centered Gaussian stochastic process 𝑋 = {𝑋(𝑡), 𝑡 ∈ ℝ} satisfies the
conditions of Lemma 3.1, 𝐴(𝑡), 𝐵(𝑡), 𝛽 are defined in Lemma 3.1, 𝑇 > 0,
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Ievgen Turchyn / Contemporary Mathematics and Statistics
(2015) Vol. 3 No. 1 pp. 1-7
𝐴̃(𝑇) = sup 𝐴(𝑡) < ∞,
𝑡∈[0,𝑇]
𝐵̃(𝑇) = sup 𝐵(𝑡) < ∞,
𝑡∈[0,𝑇]
the function 𝑢(𝑡, 𝑦) defined in Theorem 2.1 is continuous with respect to t, 𝑢(𝑡,⋅) ∈ 𝐿1 (ℝ) ∩ 𝐿2 (ℝ) and
inverse Fourier transform 𝑢̃(𝑡, 𝑦) of the function 𝑢(𝑡, 𝑦) with respect to 𝑦 is a real function. Let 𝜑(𝑡)
and 𝜓(𝑡) be the Haar 𝑓-wavelet and 𝑚-wavelet respectively. Let 𝑝 > 1, 𝜀 > 0, 𝛿 ∈ (0; √2𝑒 −𝑝/2 ). If
𝑁0 > 4 +
6
(𝐵̃(𝑇))2 ,
𝜀1
(3.8)
3(𝐴̃(𝑇))2
)1/(2𝛽−1)
8(2𝛽−1)𝜀1 (1−𝑞)
3
2
𝑀𝑗 > + (
𝑁 > max{
ln(𝐿(1−𝑞))
ln𝑞
7
(𝑗 = 0,1, . . . , 𝑁 − 1 ),
𝜀
1
, −log 8 (3 (𝐴̃(𝑇))
2 )} ,
(3.9)
(3.10)
where
𝜀1 =
𝜀2
2𝑇 2/𝑝 ln(√2/𝛿)
,
𝑞 = 4𝛽−3/2 ,
2 2𝛽
̃
𝐿 = 8𝜀1 /(3(𝐴(𝑇)) (2 + 22𝛽−1 /(2𝛽 − 1) + 2𝛽/(2𝛽 − 1))),
then the model 𝑋̂(𝑡) defined by (3.7) approximates the process 𝑋(𝑡) with given reliability 1 − 𝛿 and
accuracy 𝜀 in 𝐿𝑝 ([0, 𝑇]) .
Proof. It follows from (3.8)–(3.10) that the following inequalities hold:
sup ∑𝑘:|𝑘|≥𝑁0 |𝛼0𝑘 (𝑡)|2 ≤
𝜀1
,
3
(3.11)
2
sup ∑𝑁−1
𝑗=0 ∑𝑘:|𝑘|≥𝑀𝑗 |β𝑗𝑘 (𝑡)| ≤
𝜀1
,
3
(3.12)
2
sup ∑∞
𝑗=𝑁 ∑𝑘∈ℤ |β𝑗𝑘 (𝑡)| ≤
𝜀1
.
3
(3.13)
𝑡 ∈[0,𝑇]
𝑡 ∈[0,𝑇]
𝑡 ∈[0,𝑇]
We will prove inequality (3.13) (inequalities (3.11) and (3.12) can be proved similarly). Let us take
an arbitrary fixed 𝑡 ∈ [0, 𝑇]. Using estimates (3.4)–(3.6) it is easy to obtain the inequalities
(𝐴(𝑡))2
(𝐴(𝑡))2
1
1
∞
∞
−1
2
∑∞
𝑗=𝑁 ∑𝑘∈ℤ |β𝑗𝑘 (𝑡)| ≤ ∑𝑗=𝑁(16∙22𝑗(3/2−𝛽) ∑𝑘=1 𝑘 2𝛽 + 16∙22𝑗(3/2−𝛽) ∑𝑘=−∞ |𝑘+1/2|2𝛽 +
≤
(𝐴̃(𝑇))2
(𝑆1 𝐶𝛽
16
(𝐴(𝑡))2
)
16∙23𝑗
1
+ 𝑆2 𝐶𝛽 + 7∙8𝑁−1 ) = 𝐺,
where
1
1
𝑆1 = ∑∞
𝑘=1 𝑘 2𝛽 ≤ 2𝛽−1 + 1,
1
22𝛽−1
2𝛽
𝑆2 = ∑−1
+ 2𝛽−1 ,
𝑘=−∞ |𝑘+1/2|2𝛽 ≤ 2
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Ievgen Turchyn / Contemporary Mathematics and Statistics
(2015) Vol. 3 No. 1 pp. 1-7
1
𝐶𝛽 = ∑∞
𝑗=𝑁 4 𝑗(3/2−𝛽) .
It follows from (3.10) that 𝐺 ≤ 𝜀1 ⁄3 and therefore (3.13) is true.
Finally, we deduce from (3.11)–(3.13) that
sup 𝐄|𝑋(𝑡) − 𝑋̂(𝑡)|2
𝑡∈[0,𝑇]
∞
2
2
= sup (∑𝑘:|𝑘|≥𝑁0 |𝛼0𝑘 (𝑡)|2 + ∑𝑁−1
𝑗=0 ∑𝑘:|𝑘|≥𝑀𝑗 |𝛽𝑗𝑘 (𝑡)| + ∑𝑗=𝑁 ∑𝑘∈ℤ |𝛽𝑗𝑘 (𝑡)| ) ≤ 𝜀1
𝑡∈[0,𝑇]
and applying Theorem 3.1 we obtain the assertion of the theorem.
4. Conclusions
There has been proved a theorem about simulation of a Gaussian stochastic process with given
accuracy and reliability using a new wavelet-based model. This result holds for a wide class of
Gaussian processes.
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