Journal of Statistical Theory and Applications

Volume 17, Issue 4, December 2018, Pages 606 - 615

A Study of the First-Order Continuous-Time Bilinear Processes Driven by Fractional Brownian Motion

Authors
Abdelouahab Bibi1, Fateh Merahi2, *
1Department of Mathematics, UMC(1), Constantine, Algeria
2Department of Mathematics, UMC(1), Constantine, Algeria
*

Corresponding author. Email: merahfateh@yahoo.fr

Received 18 March 2017, Accepted 15 November 2017, Available Online 31 December 2018.
DOI
10.2991/jsta.2018.17.4.3How to use a DOI?
Keywords
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.

Copyright
© 2018 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. COBL(1,1) DRIVEN BY FRACTIONAL BROWNIAN MOTION

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 defined on some complete probability space Ω,A,P equipped with a filtration Att0 and subjected to be a solution of the following affine time-varying stochastic differential equation (SDE)

dX(t)=(α(t)X(t)+μ(t))dt+(β(t)+γ(t)X(t))dWh(t),tt0,X(t0)=X0
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<.

A2 α(t),μ(t),β(t) and eγt=0 and eαt<0, for all tt0.

In Eq. (1), (Wht)t is a real fBm with Hurst parameter h]0,12[ defined on a basic given filtered stochastic probability space Ω,A,Att0,P, its covariance kernel is CovWht,Whs=κh2|t|2h+1+|s|2h+1|ts|2h+1, for all t,s0, where κh=Γ12hh2h+1πcosπ212h and admits a spectral representation Wh(t)=ϕt(λ)(iλ)hdZ(λ) where ϕtλ=eitλ1iλ 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 12πϕt(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),t0tT) 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, Whtt0 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 SDE Eq. (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 SDE Eq. (1) encompasses many commonly used models in the literature. Some examples among others are

  1. 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).

  2. Gaussian Ornstein-Uhlenbeck (OU) process: The Gaussian OU process is defined as dXt=μtαtXt dt+βtdWht, with βt>0 for all t0. So it can be obtained from SDE Eq. (1) by assuming γt=0 for all t (see [3] and the reference therein).

  3. 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 Vtt0 of dVt=λtμtVtdt+γtVtdWht in which λt,μt and γt are positive deterministic functions. This SDE can be obtained easily from Eq. (1).

  4. Geometric Brownian motion (GBM): This class of processes is defined as a ℝ–valued solution process Xtt0 of dXt=αtXtdt+γtXtdWht,t0. 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 h=Wh:=σWht,tt0 (resp th:=σWhs,t0st) be the σ-algebra generated by Whtt0 (resp. generated by Whs up to time t) and let L2h=L2,h,P (resp. L2th) be the Hilbert space of nonlinear L2–functional of Whtt0. In this section, we are interested in solving the SDE Eq. (1) in L2th. 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 h0. 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 SDE Eq. (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=Utt,Xt+at,wUxt,Xtdt+bt,wUxt,XtdWht.
Therefore, from the SDE Eq. (2) and the Itô formula Eq. (3) we obtain
dYt=Utt,Xtdt+Uxt,XtdXt
So, the Itô's solution of the SDE Eq. (1) is given by

Theorem 2.1.

Under the assumption 1, the unique Itô's solution of SDE Eq. (1) in L2h is given by

X(t)=Φh(t,t0)X(t0)+t0tΦh1(s,t0)μ(s)ds+t0tΦh1(s,t0)β(s)dWh(s),tt0
where Φht,t0=expt0tαsds+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)+t0teY(s)μ(s)ds+t0teY(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

dXt=UxYt,ZtdYt+UyYt,ZtdZt=eYtZtdYt+eYtdZt=XtdYt+eYtdZt=Xtαtdt+γtdWht+eYteYtμt+eYtβtdWhtdt=αtXt+μtdt+γtXt+βtdWht.
and hence the result follows.

Remark 1.

If βt=0, then the Itô solution of SDE Eq. (1) reduces to

X(t)=Φh(t,t0)X(t0)+t0tΦh1(s,t0)μ(s)ds,tt0
and when γt=0 and βt0, 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 γt0.

Remark 2.

In time-invariant case, with eγ=0 and eα<0, then the Itô solution of SDE Eq. (1) can be written as

X(t)=μtexpα(ts)+iγWh(t)Wh(s)ds+βtexpα(ts)dWh(s).

Remark 3.

For any tt0, 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)=eξtX(t0)+t0teξ(s)dηh(s),tt0
is the solution process of generalized Ornstein-Uhlenbeck (GOU) process driven by an fBm defined by dXt=ξtXtdt+dηht,tt0,Xt0=X0.

2.2. The Frequency Approach

In this subsection, we discuss a second approach to solve the SDE Eq. (1) based on the spectral representation. Indeed, it is now well known that for any regular second-order process Xttt0 (i.e., Xt is th–measurable not necessary stationary, belonging to L2h) admits the so-called Wiener-Itô (or Stratonovich) spectral representation, i.e.,

X(t)=gt(0)+r11r!rgt(λ_(r))eitΣλ_(r)j=1r(iλj)hdZ(λ_(r)).
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, r2 and gtλ_rL2Gh=L2n,Bn,Gh for all tt0, with dGh(λ(r))=1(2π)ri=1r|λi|2hdλ(r) and such that
r1r!r|gt(λ_(r))|2dGh(λ_(r))< for all tt0.
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,fL2×L2rr, if f is symmetric then
g(λ)dZ(λ)rf(λ_(r))dZ(λ_(r))=r+1g(λr+1)f(λ_(r))dZλ_(r+1)+r2πr1g(λr)¯f(λ_(r))dλrdZ(λ_(r1)).
The spectral representation of the solution process of SDE Eq. (1) is given in the following theorem

Theorem 2.2.

Assume that the process (Xt)tt0 generated by the SDE Eq. (1) has a regular second-order solution. Then, the evolutionary symmetrized transfer functions g˜tλrtt0,r of such solution are given by the symmetrization of the solution of the following first order ordinary differential equations

gt(1)(λ_r)=α(t)gt(0)+μ(t)+γ(t)2πgt(λ)|λ|2hdλ,r=0αtiΣλ_rgtλ_r+rδr=1βt+γtrgtλ_r1+12πgtλ_r+1|λr+1|2hdλr+1,r1
where the superscript (j) denotes j–fold differentiation with respect to t and where Σλ_r=i=1rλi.

Proof.

First, applying of the diagram formula for the nonlinear term XtdWhtdt we get

X(t)dWh(t)dt=gt(0)eitλ(iλ)hdZ(λ)+r=11r!r+1gt˜(λ_(r))eitΣλ_r+1t=1r+1(iλl)hdZ(λ_r+1)+r=11(r1)!r1eitΣλ_r112πgt(λ_r)|λr|2hdλrl=1r1(iλl)hdZ(λ_(r1)).

Second, we insert the spectral representation Eq. (8) of the process (Xt)tt0 and the last expression of XtdWht in the Eq. (1) the results follows.

Remark 4.

The existence and uniqueness of the solution Eq. (10) is ensured by general results on linear ordinary differential equations, so

gt(λ_(r))={ϕt(0)gto(0)+t0tϕs1(0)μ(s)+γ(s)12πgs(λ)|λ|2hdλds,r=0ϕt(λ)gto(λ)+t0tϕs1(λ)β(s)+γ(s)gs(0)+12πgs(λ_(2))|λ2|2hdλ2ds,r=1ϕt(λ_)(r)gro(λ_)(r)+t0tϕs1(λ_)(r)γ(s)β(s)+γ(s)rgs(λ_)(r1)+12πgs(λ_(r+1))|λr+1|2hdλr+1ds,r2
in which φtλ_r=expt0tα(s)iΣλ_rds.

Remark 5.

Noting that beside the condition Eq. (9) a necessary conditions for that the evolutionary transfer functions gtλ_r,r defined by Eq. (11) determines a second-order process are

|gt(λ_r+1)|λr+1|2hdλr+1|2|λr+1|2hdλr<+ and |gt(λ_r+1)||λr+1|2hdλr+1<+
for all tt0. 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 eγt0, the SDE Eq. (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,tt0,Xt0=X0
and assume that

A3. α(t),μ(t),γ(t) and e{α(t) }<0,γ(t)0 for all tt0.

Under the condition A3, the Itô's solution of Eq. (12) reduces to

X(t)=Φh(t,t0)X(t0)+t0tΦh1(s,t0)μ(s)ds,
in which the function γ(t) is replaced by (t). The spectral representation of Eq. (12) is given in the following lemma

Lemma 1.

Assume that the process (Xt)tt0 generated by the model Eq. (12) has a regular second-order solution. Then, the symmetrized evolutionary transfer functions g˜tλrt,r of such solution may be obtained by the symmetrization of the following functions

gt(λ(r))={φt(0)gt0(0)+t0tφs1(0)μ(s)+iγ(s)12πgs(λ) |λ|2hdλ)ds,r=0φt(λ_(r))gt0(λ(r))+it0tφs1(λ_(r))γ(s)rgs(λ(r1))+12πgs(λ(r+1))|λr+1|2hdλr+1ds,r1

Lemma 2.

In time-invariant case we obtain

g(λr)=g(λ(r))=1αμ+iγ2πgλ|λ|2hdλ if r=0iγαiλ_rrg(λ(r1))+12πg(λr+1)|λr+1|2hdλr+1 if r1
so, its symmetrized version may be written as
g˜(λ(r))=Symg(λ(r))=μ(iγ)r0expαuγ22khu2h+1j=1r1eiuλjiλjdu.

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 Xttt0 solution of the SDE Eq. (1). For this purpose let Ψht,t0tt0 be the mean function of the process Φht,t0tt0, and set Wh(t,u,s,v)=h(2h+1)κhutvsγ(v1)γ(v2)¯|v1v2|2h1dv2dv1,ut,vs. Then, we have

Lemma 3.

Under the conditions of 1, we have the following assertions

  1. Ψh(t,t0)=expt0tα(v1)dv1+h2h+1κ(h)2t0tt0tγ(v1)γ(v2)|v1v2|2h1dv1dv2 for tt0.

  2. EΦht,t0Φh1u,t0=Ψht,u for tu.

  3. EΦht,t0Φhs,t0¯=Ψht,t0Ψhs,t0¯expWht,t0,s,t0 for ts.

  4. EΦht,t0Φhs,t0¯Φh1v,t0¯=Ψht,t0Ψhs,v¯expWht,t0,s,v for tsv.

  5. EΦht,t0Φhs,t0¯Φh1u,t0Φh1v,t0¯=Ψht,uΨhs,v¯expWht,u,s,v for tsv.

Proof.

The assertions of the Lemma 3 follows upon observation that by using the expectation of exponential Gaussian process, we have

Ψht,t0=expt0tα(v1)dv1+12Et0tγ(v1)dWh(v1)2=expt0tα(v1)dv1+h(2h+1)κ(h)2t0tt0tγ(v1)γ(v2)|v1v2|2h1dv1dv2
and for tu
EΦht,t0Φh1u,t0=expt0tαv1dv1+12Et0tγ(v1)dWh(v1)2=expt0tαv1dv1+h2h+1κ(h)2ututγv1γv2|v1v2|2h1dv1dv2=Ψht,u.
and so on the rest are immediate.

Lemma 4.

Under the condition of Lemma 3, the mean function mht=EXttt0 is given by

mh(t)=Ψh(t,t0)m(t0)+t0tΨht,uμudu,tt0.

and the covariance function Rht,s=EXtmhtXsmhs¯ts is given by

Rht,s=Ψh(t,t0)Ψhs,t0¯expWht,t0,s,t0Rt0+Ψht,t0Ψhs,t0¯expWht,t0,s,t01|mt0|2+mt0t0sΨht,t0Ψhs,v¯expWht,t0,s,v1μv¯dv+mt0¯t0tΨhs,t0¯Ψht,uexpWht,u,s,t01μudu+t0tt0sΨht,uΨhs,v¯expWht,u,s,v1μv¯μudvdu+h2h+1κht0tt0sΨht,uΨhs,v¯expWht,u,s,vβv¯βu|uv|2h1dvdu.

Proof.

From the Itô's solution Eq. (5), we can obtain

mh(t)=EXt=EΦht,t0Xt0+t0tEΦht,t0Φh1u,t0μudu=Ψht,t0mt0+t0tΨht,uμudu.

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

EXtXs¯=EΦht,t0Φhs,t0¯E|Xt0|2+mt0t0sEΦht,t0Φhs,t0¯Φh1v,t0¯μv¯dv+mt0¯t0tEΦhs,t0¯Φht,t0Φh1u,t0μudu+t0tt0sEΦht,t0Φhs,t0¯Φh1u,t0Φh1v,t0¯μv¯μudvdu+h2h+1κht0tt0sEΦht,t0Φhs,t0¯Φh1u,t0Φh1v,t0¯βv¯βu|uv|2h1dvdu,
In other hand
mh(t)mh(s)¯=Ψh(t,t0)Ψh(s,t0)¯|m(t0)|2+m(t0)t0sΨht,t0Ψhs,v¯μv¯dv+mt0¯t0tΨhs,t0¯Ψhs,uμudu+t0tt0sΨht,uΨhs,v¯μv¯μudvdu,
the fact that Rht,s=EXtXs¯mhtmhs¯the expression for Rh(t,s) follows.

Lemma 5.

Consider the time-invariant process Xttt0 generated by SDE Eq. (1). Then under the condition 1, the mean and covariance functions of the solution process Xttt0 are given by

mh=μ0Khudu,Rh|τ|=|μ|200Khu1Khu2¯expγ22κhWτhu1,u21du1du2+|β|2h2h+1κh00Khu1Khu2¯expγ22κhWτhu1,u2du1du2,
where
Wτhu1,u2=|τ|2h+1|τu1|2h+1|τ+u2|2h+1+|τu1+u2|2h+1,
and Kht=expαtγ22κht2h+1.

Proof.

Straightforward and hence omitted.

Corollary 1.

Consider the time-invariant version of the SDE Eq. (12), then limτ+Rτcτδ=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. R|τ|dτ=.

Proof.

First we have

expγ22κh|τ|2h+1|τu1|2h+1|τ+u2|2h+1+|τu1+u2|2h+1=expγ22κh|τ|2h+11|1u1τ|2h+1|1+u2τ|2h+1+|1+u2u1τ|2h+1,
and
1u1τ2h+1=12h+1u1τ+2h+12h2u12τ2+τ+1+u2τ2h+1=1+2h+1u2τ+2h+12h2u22τ2+τ+1+u2u1τ2h+1=1+2h+1u2u1τ+2h+12h2(u2u1)2τ2+τ+.

Let δ=2h1, it is clear 0<δ<1 because 0<h<12, then we have

limτ+expγ22κhWτhu1,u21τδ=limτ+expγ22κhh2h+1u1u2τ2h11τ2h1=γ22h2h+1u1u2.

It follows that

limτ+Rττδ=|μ|200Khu1Khu2¯limτ+τδexpγ22κhWτhu1,u21du1du2=γ22κhh2h+1|μ|20u1Khu1du10u1Khu2du2¯=γ22κhh2h+1|μ|2|0uKhudu|2=c<,

Hence, the process (Xt)t0 generated by the SDE Eq. (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 SDE Eq. (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)r0Khu1eiuλjiλjdu,r0.

Then using the representation Eq. (8) we can obtain the following approximation

Xt=g0+gλ1eitλ1dZλ1+2gλ2eitλ_2dZλ2+ξt=X(1)(t)+X(2)(t)+ξ(t),
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=μiγ0Khu1eiuλ1iλ1du and gλ1,λ2=μiγ20Khuj=121eiuλ2iλ2du

It can be shown that

Chs,u=EXtg0(Xt+sg0(Xt+ug0=EX1tX1t+sX2t+u+EX1tX2t+sX1t+u+EX2tX1t+sX1t+u+O1.

We calculate EX1tX1t+sX2t+u, and the other terms can be obtained by symmetry. First we observe that

EX1tX1t+sX2t+u=E2gλ1gλ2eitλ1+it+sλ2dZλ22gλ3,λ4eit+uλ3+λ4dZλ3,λ4=2!2symgλ1gλ2eitλ1+it+sλ2symgλ1,λ2eit+uλ1+λ2¯dFλ2=22gλ1gλ2gλ1,λ2symeisλ1eiuλ1+λ2dλ1λ2(2π)2=1(2π)22gλ1gλ2gλ1,λ2eisλ1eiuλ1+λ2dλ1λ2+2gλ1gλ2gλ1,λ2eisλ2eiuλ1+λ2dλ1λ2=1(2π)22gλ1gλ2gλ1,λ2ei(su)λ1iuλ2dλ1λ2+2gλ1gλ2gλ1,λ2eiuλ1+isuλ2dλ1λ2.

Moreover we have

EX1tX2t+sX1t+u=E2gλ1gλ2eitλ1+it+uλ2dZλ22gλ3,λ4eit+sλ3+λ4dZλ3,λ4=2!2symgλ1gλ2eitλ1+it+uλ2symgλ1,λ2eit+sλ1+λ2¯dFλ2=22gλ1gλ2gλ1,λ2symeiuλ1eisλ1+λ2dλ1λ2(2π)2=1(2π)22gλ1gλ2gλ1,λ2eiuλ1eisλ1+λ2dλ1λ2+2gλ1gλ2gλ1,λ2eiuλ2eisλ1+λ2dλ1λ2=1(2π)22gλ1gλ2gλ1,λ2eiusλ1isλ2dλ1λ2+2gλ1gλ2gλ1,λ2eisλ1+iusλ2dλ1λ2.

It remains to compute EX2tX1t+sX1t+u, then

EX2tX1t+sX1t+u=E2gλ3gλ4eit+sλ3+it+uλ4Zdλ3,dλ42gλ1,λ2eitλ1+λ2Zdλ2=2!2symgλ1gλ2eit+sλ1+it+uλ2symgλ1,λ2eitλ1+λ2¯dFλ2=22gλ1gλ2gλ1,λ2)symeisλ1+iuλ2dλ1λ2(2π)2=1(2π)22gλ1gλ2gλ1,λ2eisλ1+iuλ2dλ1λ2+2gλ1gλ2gλ1,λ2eiuλ1+isλ2dλ1λ2.

Hence

Chs,u=22gλ1gλ2gλ1,λ2symeisuλ1uλ2+eeiusλ1sλ2+eisλ1+uλ2dλ1λ2(2π)2.

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,λ1λ2+Sλ1,λ1λ2 where Sλ1,λ2=gλ1gλ2gλ1,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.

REFERENCES

1.Y. S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Vol. 1929, 2008.
2.T. E. Duncun and B. Pasik-Duncun, Parameter identification for some linear systems with fractional Brownian motion, in 15th Triennial World Congress, Barcelona, IFAC Proceedings, Vol. 35, No. 1, 2002, pp. 319-324.
3.G. Shen, X. Yin, and L. Yan, Acta Mathematica Scientia, Vol. 36B, No. 2, 2016, pp. 394-408.
4.A. Swishchuk, Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities, World Scientific Books, World Scientific Publishing Co. Pte. Ltd, Singapore, 2013.
5.C. Bendr and P. Parczewski, Bernoulli, Vol. 16, No. 2, 2010, pp. 389-417.
6.T. E. Duncan, Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems, Springer, Boston, 2006, pp. 97-108.
7.W. Dai and C.C. Heyde, J. Appl. Math. Stoch. Anal., Vol. 9, No. 4, 1996, pp. 439-448.
8.A. Bibi and F. Merahi, Int. J. Stat. Probab., Vol. 4, No. 3, 2015, pp. 150-160.
9.E. Iglói and G. Terdik, Bilinear stochastic systems with fractional Brownian motion input, Ann. Appl. Probab., Vol. 9, No. 1, 1999, pp. 46-77.
Journal
Journal of Statistical Theory and Applications
Volume-Issue
17 - 4
Pages
606 - 615
Publication Date
2018/12/31
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.2018.17.4.3How to use a DOI?
Copyright
© 2018 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

Cite this article

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  -