A Multiplicative Wavelet-based Model for Simulation of a Random Process

We consider a random process $Y(t)=\exp\{X(t)\}$, where $X(t)$ is a centered second-order process which correlation function $R(t,s)$ can be represented as $\int_{\mathbb{R}} u(t,y)\overline{u(s,y)} dy.$ A multiplicative wavelet-based representation is found for $Y(t)$. We propose a model for simulation of the process $Y(t)$ and find its rates of convergence to the process in the spaces $C([0,T])$ and $L_p([0,T])$ for the case when $X(t)$ is a strictly sub-Gaussian process.


Introduction
Simulation of random processes is a wide area nowadays, there exist many methods for simulation of stochastic processes (see e.g. [1,2]).
But there exists one substantial problem: for most of traditional methods of simulation of random processes it is difficult to measure the quality of approximation of a process by its model in terms of "distance" between paths of the process and the corresponding paths of the model. Therefore models for which such distance can be estimated are quite interesting.
There exists a concept for simulation by such models which is called simulation with given accuracy and reliability. Simulation with given accuracy and reliability is considered, for example, in [3,4].
Simulation with given accuracy and reliability can be described in the following way. An approximationX(t) of a random process X(t) is built.
The random processX(t) is called a model of X(t). A model depends on certain parameters. The rate of convergence of a model to a process is given by a statement of the following type: if numbers δ (accuracy) and ε (1 − ε is called reliability) are given and the parameters of the model satisfy certain restrictions (for instance, they are not less than certain lower bounds) then Many such results have been proved for the cases when the norm in (1) is the L p norm or the uniform norm. But simulation with given accuracy and reliability has been developed so far almost only for processes which one-dimensional distributions have tails which are not heavier than Gaussian tails (e.g. for sub-Gaussian processes).
We consider a random process Y (t) = exp{X(t)} and a scaling function φ(x) with the corresponding wavelet ψ(x), where X(t) is a centered second-order process such that its correlation function R(t, s) can be represented as R(t, s) = R u(t, λ)u(s, λ)dλ.
We prove that where ξ 0k , η jl are random variables, a 0k (t), b jl (t) are functions that depend on X(t) and the wavelet.
We take as a model of Y (t) the procesŝ Let us consider the case when X(t) is a sub-Gaussian process. Note that the class of processes Y (t) = exp{X(t)}, where X(t) is a sub-Gaussian process, is a rich class which includes many processes which one-dimensional distributions have tails heavier than Gaussian tails, e.g. when X(t) is a Gaussian process the one-dimensional distributions of Y (t) are lognormal.
We describe the rate of convergence ofŶ (t) to a sub-Gaussian process in such a way: if ε ∈ (0; 1) and δ > 0 are given and the parameters N 0 , N, M j are big enough then P sup A similar statement which characterizes the rate of convergence ofŶ (t) is also proved for the case when (2) is replaced by the If the process X(t) = ln Y (t) is Gaussian then the modelŶ (t) can be used for computer simulation of Y (t).
One of the merits of our model is its simplicity. Besides, it can be used for simulation of processes which one-dimensional distributions have tails which are heavier than Gaussian tails.

Auxiliary facts
A random variable ξ is called sub-Gaussian if there exists such a constant The class of all sub-Gaussian random variables on a standard probability space {Ω, B, P } is a Banach space with respect to the norm τ (ξ) = inf{a ≥ 0 : E exp{λξ} ≤ exp{λ 2 a 2 /2}, λ ∈ R}.
A centered Gaussian random variable and a random variable uniformly distributed on [−b, b] are examples of sub-Gaussian random variables.
A sub-Gaussian random variable ξ is called strictly sub-Gaussian if For any sub-Gaussian random variable ξ and A family ∆ of sub-Gaussian random variables is called strictly sub-Gaussian if for any finite or countable set I of random variables ξ i ∈ ∆ and for any A stochastic process X = {X(t), t ∈ T} is called sub-Gaussian if all the random variables X(t), t ∈ T, are sub-Gaussian. We call a stochastic process X = {X(t), t ∈ T} strictly sub-Gaussian if the family {X(t), t ∈ T} is strictly sub-Gaussian. Any centered Gaussian process is strictly sub-Gaussian.
Details about sub-Gaussian random variables and processes can be found in [5].
We will use wavelets (see [6] for details) for an expansion of a stochastic process. Namely, we use a scaling function φ(x) of an MRA and the corresponding wavelet ψ(x). Set We require orthonormality of the system {φ(· − k), k ∈ Z}. We denote byf the Fourier transform of a function f ∈ L 2 (R).
The following statement is crucial for us.
Let φ(x) be a scaling function, ψ(x) -the corresponding wavelet. Then the process X(t) can be presented as the following series which converges for any t ∈ R in L 2 (Ω): where ξ 0k , η jl are centered random variables such that Definition. Condition RC holds for stochastic process X(t) if it satisfies the conditions of Theorem 2.1, u(t, ·) ∈ L 1 (R) ∩ L 2 (R) and inverse Fourier transformũ x (t, x) of function u(t, x) with respect to x is a real function.
Suppose that X(t) is a process which satisfies the conditions of Theorem 2.1. Let us consider the following approximation (or model) of X(t): where ξ 0k , η jl , a 0k (t), b jl (t) are defined in Theorem 2.1.
Approximation of Gaussian and sub-Gaussian processes by model (8) has been studied in [7] and [8].
is a Gaussian process then we can take as ξ 0k , η jl in (8) independent random variables with distribution N(0; 1).

A multiplicative representation
We will obtain a multiplicative representation for a wide class of stochastic processes.
Theorem 3.1. Suppose that a random process Y (t) can be represented as where the process X(t) satisfies the conditions of Theorem 2.1. Then the equality holds, where product (9) converges in probability for any fixed t and ξ 0k , η jl , The statement of the theorem immediately follows from Theorem 2.1.
Remark 3.1. It was shown in [7] that any centered second-order wide-sense stationary process X(t) which has the spectral density satisfies the conditions of Theorem 2.1. The process Y (t) = exp{X(t)} can be represented as product (9) and therefore the class of processes which satisfy the conditions of Theorem 3.1 is wide enough.
It is natural to approximate a stochastic process Y (t) = exp{X(t)} which satisfies the conditions of Theorem 3.1 by the model is a Gaussian process then we can use the modelŶ (t) for computer simulation of Y (t), taking as ξ 0k , η jl in (10) independent random variables with distribution N(0; 1).

Simulation with given relative accuracy and reliability in C([0, T ])
Let us study the rate of convergence in C([0, T ]) of model (10) to a process Y (t). We will need several auxiliary facts.
, t ∈ R} be a centered stochastic process which satisfies the requirements of Theorem 2.1, T > 0, φ be a scaling function, ψ be the corresponding wavelet, the functionφ(y) be absolutely continuous on any interval, the function u(t, y) be absolutely continuous with respect to y for any fixed t, there exist the derivatives u ′ Let the processX(t) be defined by (8), there exist functions v(y) and w(y) such that a 0k (t) and b jl (t) are defined by equalities (6) and (7), . holds.
We omit the proof due to its triviality.
We will denote the norm in L 2 (Ω) as · 2 below.
Using inequality (25), simple properties of metric entropy (see [5], Lemma 3.2.1, p. 88) and the inequality Since ε 0 ≤ C U T we obtain It is easy to check using Lemma 4.3 that under the conditions of the theorem the inequality holds. It follows from (23) and (29) that and therefore using (28) we obtain Now the statement of the theorem follows from (19), (24) and (30).
Example 4.1. Let us consider a function u(t, λ) = t/(1 + t 2 + λ 2 ) 4 and an arbitrary Daubechies wavelet (with the corresponding scaling function φ and the wavelet ψ). We will use the notations and consider the stochastic process where ξ 0k , η jl (k, l ∈ Z, j = 0, 1, . . .) are independent uniformly distributed Then the following inequalities hold for the coefficients a 0k (t), b jl (t) in expansion (5) of the process X(t): The proof of inequalities (31)-(34) is analogous to the proof of similar inequalities for the coefficients of expansion (5) of a stationary process in [7].
Definition. We say that a modelŶ (t) approximates a stochastic process Y (t) with given accuracy δ and reliability 1 − ε (where ε ∈ (0; 1)) in x m be the root of the equation then the modelŶ (t) defined by (10) approximates Y (t) with given accuracy δ Proof. We will use the following notations: τ (X(t)), τ (∆X(t)), We will denote the norm in L p ([0, T ]) as · p .
An application of Cauchy-Schwarz inequality yields: We will need two auxiliary inequalities. Using the power mean inequality where r ≥ 1, and setting a = e c , b = 1 we obtain (e c + 1) r ≤ 2 r−1 (e cr + 1).