Continuous-time bilinear process; Fractional movement Brownian; Spectral representation; Itô's solution; Long memory property.
Abstract
The continuous-time bilinear (COBL) process has been used to model non linear and/or non Gaussian datasets. In this paper, the first-order continuous-time bilinear COBL(1,1) model driven by a fractional Brownian motion (fBm for short) process is presented. The use of fBm processes with certain Hurst parameter permits to obtain a much richer class of possibly long-range dependent property which are frequently observed in financial econometrics, and thus can be used as a power tool for modelling irregularly series having memory. So, the existence of Itô's solutions and there chaotic spectral representations for time-varying COBL(1,1) processes driven by fBm are studied. The second-order properties of such solutions are analyzed and the long-range dependency property are studied.
In discrete-time series analysis, the assumption of linearity and/or Gaussianity is frequently made. Unfortunately these assumption lead to models that fail to capture certain phenomena commonly observed in practice such as limit cycles, asymmetric distribution, leptokurtosis, etc. Motived by these deficiencies, non linear parametric modelling of time series has attracted considerable attention in recent years. Indeed, one of the most useful class of non-linear time series models is the bilinear specification obtained by adding to an Autoregressive moving average (ARMA) model one or more interaction components between the observed series and the innovations. However, it is observed that these models are not be able to give full information about some datasets exhibit unequally spaced observations and hence the resort to a continuous-time version is crucial. So, in this paper we consider a continuous-time bilinear (COBL) processes (Xt)t∈ℝ defined on some complete probability space Ω,A,P equipped with a filtration Att≥0 and subjected to be a solution of the following affine time-varying stochastic differential equation (SDE)
denoted hereafter COBL(1,1). The parameters α(t), μ(t), γ(t) and β(t) are differentiable complex deterministic functions subject to the following assumption
Condition 1
A1 For all T>t0, ∫toT|α(t)|dt<∞, ∫t0T|μ(t)|dt<∞, ∫t0T|γ(t)|2dt<∞. ∫t0T|β(t)|2dt<∞.
A2α(t),μ(t),β(t)∈ℂ and ℜeγt=0 and ℜeαt<0, for all t≥t0.
In Eq. (1), (Wht)t∈ℝ is a real fBm with Hurst parameter h∈]0,12[ defined on a basic given filtered stochastic probability space Ω,A,Att≥0,P, its covariance kernel is CovWht,Whs=κh2|t|2h+1+|s|2h+1−|t−s|2h+1, for all t,s≥0, where κh=Γ1−2hh2h+1πcosπ21−2h and admits a spectral representation Wh(t)=∫ℝϕt(λ)(iλ)−hdZ(λ) where ϕtλ=eitλ−1iλ and dZ. is a complex−valued Gaussian spectral measure defined on (Ω,A,P) with zero mean, variance E|dZλ|2=dGλ=dλ2π and where the principal value of 12π∫ℝϕt(λ)d(λ) is 0. Note that the initial state Xt0 is a random variable, defined on the same probability space Ω,A,P independent of σ(W(t),t0≤t≤T) such that EXt0=mt0 and VarXt0=Rt0<+∞.
It is well known that if h = 0, then the corresponding fBm reduces to the usual Brownian motion, otherwise, Whtt≥0 is neither a Markovian nor a semimartingales processes and hence the usual calculus cannot be used, so a different calculus is required. This non Markovian processes have not an independent stationary increments and are well suited for modelling data exhibiting a long-range dependency. For an in-depth detailed mathematical framework of the pertinent properties of fBm, we refer the reader to Mishura [1] and the references therein.
The SDEEq. (1) is called time-invariant when α(t), μ(t), γ(t) are complex deterministic constant functions, i.e., there is some constants complex α, μ, γ such that αt=α, μt=μ, γt=γ and for all t.
The SDEEq. (1) encompasses many commonly used models in the literature. Some examples among others are
First-order continuous-time autoregressive processes (CAR(1) for short): This classes of SDE may be obtained by assuming γt=0 for all t (see [2] and the reference therein).
Gaussian Ornstein-Uhlenbeck (OU) process: The Gaussian OU process is defined as dXt=μt−αtXtdt+βtdWht, with βt>0 for all t≥0. So it can be obtained from SDEEq. (1) by assuming γt=0 for all t (see [3] and the reference therein).
Nelson's diffusion process: In the diffusion process of Nelson (see [4], Chapter 2), the time-varying volatility process may be defined as the second-order solution process Vtt≥0 of dVt=λtμt−Vtdt+γtVtdWht in which λt,μt and γt are positive deterministic functions. This SDE can be obtained easily from Eq. (1).
Geometric Brownian motion (GBM): This class of processes is defined as a ℝ–valued solution process Xtt≥0 of dXt=αtXtdt+γtXtdWht,t≥0. So it can be obtained from Eq. (1) by assuming βt=μt=0 for all t (see [5] and the reference therein).
It is worth noting that beside the above mentioned particular cases, the Eq. (1) may be extended to vectorial case, i.e., when X(t) is ℝd–valued process, so other particular models can be deduced.
2. THE SOLUTION PROCESSES OF COBL(1,1)
Let ℑh=ℑWh:=σWht,t≥t0 (resp ℑth:=σWhs,t0≤s≤t) be the σ-algebra generated by Whtt≥0 (resp. generated by Whs up to time t) and let L2ℑh=L2ℂ,ℑh,P (resp. L2ℑth) be the Hilbert space of nonlinear L2–functional of Whtt≥0. In this section, we are interested in solving the SDEEq. (1) in L2ℑth. As already pointed by several authors (see for instance [6] for further discussions), that there is no general theory for the solution of SDE driven by an fBm if h≠0. Nevertheless, recently some studies was investigated the existence of such solutions for various families of SDE driven by an fBm.
2.1. The Itô Approach
Our first approach is based on the Itô formula with respect to fBm and the general results on SDE to prove the uniqueness of the solution. First, we start by the fractional Itô's formula which is a powerful tool for dealing the solution. Consider the following SDE driven by fBm
dXt=at,Xtdt+bt,XtdWht,Xt0=X0
in which a.,., b.,. are known continuous functions that represents the drift and diffusion respectively of the SDEEq. (2) supposed to be smooth enough, and set Yt=Ut,Xt for some differentiable function U : ℝ → ℝ. Then Dai and Heyde [7] have shown that the Itô formula with respect to fBm is given by
dYt=∂U∂tt,Xt+at,w∂U∂xt,Xtdt+bt,w∂U∂xt,XtdWht.
Therefore, from the SDEEq. (2) and the Itô formula Eq. (3) we obtain
dYt=∂U∂tt,Xtdt+∂U∂xt,XtdXt
So, the Itô's solution of the SDEEq. (1) is given by
Theorem 2.1.
Under the assumption 1, the unique Itô's solution of SDEEq. (1) in L2ℑh is given by
where Φht,t0=exp∫t0tαsds+∫t0tγsdWhs with Φht0,t0=1 and the stochastic integral ∫t0tγ(s)dWh(s) is defined in Riemann's sense in probability.
Proof.
First it is no difficult to see that Φht,t0 is the unique solution of SDE
dΦht,t0=αtΦht,t0dt+γtΦht,t0dWht.
Now, set Φht,t0=expYt, Z(t)=X(0)+∫t0te−Y(s)μ(s)ds+∫t0te−Y(s)β(s)dWh(s) and let Xt=UYt,Zt, where U is the function defined by Ux,y=exy. The fractional Itô formula Eq. (3) and the expression Eq. (4) gives
If βt=0, then the Itô solution of SDEEq. (1) reduces to
X(t)=Φh(t,t0)X(t0)+∫t0tΦh−1(s,t0)μ(s)ds,t≥t0
and when γt=0 and βt≠0, this is provides a solution of Gaussian OU process, therefore if we are interested in non-Gaussian solution of Eq. (1), it is necessary to assume that |μt|2+|βt|2>0 and γt≠0.
Remark 2.
In time-invariant case, with ℜeγ=0 and ℜeα<0, then the Itô solution of SDEEq. (1) can be written as
For any t≥t0, let −ξ(t)=∫t0tα(s)ds+∫t0tγ(s)dWh(s) and ηh(t)=∫t0tμ(s)ds+∫t0tβ(s)dWh(s), then the solution process Eq. (5) may be rewritten as
X(t)=e−ξtX(t0)+∫t0teξ(s)dηh(s),t≥t0
is the solution process of generalized Ornstein-Uhlenbeck (GOU) process driven by an fBm defined by dXt=−ξtXtdt+dηht,t≥t0,Xt0=X0.
2.2. The Frequency Approach
In this subsection, we discuss a second approach to solve the SDEEq. (1) based on the spectral representation. Indeed, it is now well known that for any regular second-order process Xtt≥t0 (i.e., Xt is ℑth–measurable not necessary stationary, belonging to L2ℑh) admits the so-called Wiener-Itô (or Stratonovich) spectral representation, i.e.,
where λ_(r)=(λ1,…,λr),Σλ_(r)=∑i=1rλi and dZ(λ_(r))=∏j=1rdZ(λi) (see [8] for more details). The representation Eq. (8) is unique up to the permutation of the arguments of the evolutionary transfer functions gtλ_r, r≥2 and gtλ_r∈L2Gh=L2ℂn,Bℂn,Gh for all t≥t0, with dGh(λ(r))=1(2π)r∏i=1r|λi|−2hdλ(r) and such that
∑r≥1r!∫ℝr|gt(λ_(r))|2dGh(λ_(r))<∞ for all t≥t0.
Let us recall here the so-called the diagram formula for Wiener–Itô representation Eq. (8) which play an important role in some subsequent proofs and that state that for all g and f defined on ℝ and on ℝr respectively such that g,f∈L2ℝ×L2rℝr, if f is symmetric then
The spectral representation of the solution process of SDEEq. (1) is given in the following theorem
Theorem 2.2.
Assume that the process (Xt)t≥t0 generated by the SDEEq. (1) has a regular second-order solution. Then, the evolutionary symmetrized transfer functions g˜tλrt≥t0,r∈ℕ of such solution are given by the symmetrization of the solution of the following first order ordinary differential equations
Noting that beside the condition Eq. (9) a necessary conditions for that the evolutionary transfer functions gtλ_r,r∈ℕ defined by Eq. (11) determines a second-order process are
∫ℝ|∫ℝgt(λ_r+1)|λr+1|−2hdλr+1|2|λr+1|−2hdλr<+∞ and ∫ℝ|gt(λ_r+1)||λr+1|−2hdλr+1<+∞
for all t≥t0. These conditions are extremely difficult to be verified, except in time-invariant case when an explicit formula for the transfer functions are given (see for instance [9]).
It is worth noting that if ℜeγt≠0, the SDEEq. (1) may be haven't a second-order solution, but it does if γ(t) is purely imaginary. So in what follows, we consider the particular SDE
dXt=αtXt+μtdt+iγtXtdWht,t≥t0,Xt0=X0
and assume that
A3. α(t),μ(t)∈ℂ,γ(t)∈ℝ and ℜe{α(t)}<0,γ(t)≠0 for all t≥t0.
Under the condition A3, the Itô's solution of Eq. (12) reduces to
X(t)=Φh(t,t0)X(t0)+∫t0tΦh−1(s,t0)μ(s)ds,
in which the function γ(t) is replaced by iγ(t). The spectral representation of Eq. (12) is given in the following lemma
Lemma 1.
Assume that the process (Xt)t≥t0 generated by the model Eq. (12) has a regular second-order solution. Then, the symmetrized evolutionary transfer functions g˜tλrt∈ℝ,r∈ℕ of such solution may be obtained by the symmetrization of the following functions
3. THE MOMENTS PROPERTIES AND THE SECOND-ORDER STRUCTURE
In this section, we analyze the spectrum, i.e., the second-order structure of the process Xtt≥t0 solution of the SDEEq. (1). For this purpose let Ψht,t0t≥t0 be the mean function of the process Φht,t0t≥t0, and set Wh(t,u,s,v)=h(2h+1)κh∫ut∫vsγ(v1)γ(v2)¯|v1−v2|2h−1dv2dv1,u≤t,v≤s. Then, we have
Lemma 3.
Under the conditions of 1, we have the following assertions
Ψh(t,t0)=exp∫t0tα(v1)dv1+h2h+1κ(h)2∫t0t∫t0tγ(v1)γ(v2)|v1−v2|2h−1dv1dv2 for t≥t0.
EΦht,t0Φh−1u,t0=Ψht,u for t≥u.
EΦht,t0Φhs,t0¯=Ψht,t0Ψhs,t0¯expWht,t0,s,t0 for t≥s.
EΦht,t0Φhs,t0¯Φh−1v,t0¯=Ψht,t0Ψhs,v¯expWht,t0,s,v for t≥s≥v.
EΦht,t0Φhs,t0¯Φh−1u,t0Φh−1v,t0¯=Ψht,uΨhs,v¯expWht,u,s,v for t≥s≥v.
Proof.
The assertions of the Lemma 3 follows upon observation that by using the expectation of exponential Gaussian process, we have
Since Wht independent of Xt0, then EΦht,t0Xt0=EΦht,t0EXt0=Ψht,t0mht0. In order to evaluate the expression of Rht,s we use the Itô's solution Eq. (5) to obtain
the fact that Rht,s=EXtXs¯−mhtmhs¯the expression for Rh(t,s) follows.
Lemma 5.
Consider the time-invariant process Xtt≥t0 generated by SDEEq. (1). Then under the condition 1, the mean and covariance functions of the solution process Xtt≥t0 are given by
Consider the time-invariant version of the SDEEq. (12), then limτ→+∞Rτcτ−δ=1 for some constant c and 0<δ<1, this means that the solution process exhibits long range dependence. In this case the dependence between Xt and Xt+τ decays slowly as τ→+∞ and. ∫ℝR|τ|dτ=∞.
Hence, the process (Xt)t≥0 generated by the SDEEq. (12) with time-invariant parameters is a long memory process.
3.1. Third-Order Structure of COBL(1,1) Process
For the sake of convenience and simplicity, we shall consider the time-invariant version of the SDEEq. (1). Moreover, we assume the process solution admits the spectral representation Eq. (8) in which the symmetrized version of transfer functions gλ_r may be written as
g(λ(r))=μ(iγ)r∫0∞Khu1−e−iuλjiλjdu,∀r≥0.
Then using the representation Eq. (8) we can obtain the following approximation
where ξt is a second-order stationary process which it is orthogonal to the first two terms. The symmetrized transfer functions g˜λ1 and g˜λ2 are given by
gλ1=μiγ∫0∞Khu1−e−iuλ1iλ1du and gλ1,λ2=μiγ2∫0∞Khu∏j=121−e−iuλ2iλ2du
By taking Fourier transforms (omitting the terms of O(1)), the bispectral density function fλ1,λ2 can be shown to be fλ1,λ2=2(2π)2Sλ1,λ2+Sλ2,−λ1−λ2+Sλ1,−λ1−λ2 where Sλ1,λ2=gλ1gλ2g−λ1,−λ2. It is clear from the above that the bispectrum is zero for all frequencies λ1 and λ2 if and only if the process is linear (γ=0) (and Gaussian).
4. CONCLUSION
This paper describes some basic probabilistic properties of COBL process driven by an (f)Bm. Our main aim was focused firstly on the existence of the solution in time-frequency domain and secondary to prove that the use of fBm as innovation we led to a long-range dependency property.
COMPETING INTEREST
Stochastic differential equation (SDE), Estimation of SDE, Asymptotic Inference.
ACKNOWLEDGMENTS
We would like to thank Prof. M. Ahsanullah Editor-in-Chief of the journal, for his attention, encouragement and valuable advice. Also, we are very grateful to an anonymous referee for reading the paper very carefully and making many constructive remarks and suggestions.
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TY - JOUR
AU - Abdelouahab Bibi
AU - Fateh Merahi
PY - 2018
DA - 2018/12/31
TI - A Study of the First-Order Continuous-Time Bilinear Processes Driven by Fractional Brownian Motion
JO - Journal of Statistical Theory and Applications
SP - 606
EP - 615
VL - 17
IS - 4
SN - 2214-1766
UR - https://doi.org/10.2991/jsta.2018.17.4.3
DO - 10.2991/jsta.2018.17.4.3
ID - Bibi2018
ER -
