A problem of drift parameter estimation is studied for a nonergodic weighted fractional Vasicek model defined as dXt=θ(μ+Xt)dt+dBta,b, t≥0, with unknown parameters θ>00$]]>, μ∈R and α:=θμ, whereas Ba,b:={Bta,b,t≥0} is a weighted fractional Brownian motion with parameters a>−1-1$]]>, |b|<1, |b|<a+1. Least square-type estimators (θ˜T,μ˜T) and (θ˜T,α˜T) are provided, respectively, for (θ,μ) and (θ,α) based on a continuous-time observation of {Xt,t∈[0,T]} as T→∞. The strong consistency and the joint asymptotic distribution of (θ˜T,μ˜T) and (θ˜T,α˜T) are studied. Moreover, it is obtained that the limit distribution of θ˜T is a Cauchy-type distribution, and μ˜T and α˜T are asymptotically normal.
Weighted fractional Vasicek modelparameter estimationstrong consistencyjoint asymptotic distributionYoung integral60G1560G2262F1262M0962M86Kuwait Foundation for the Advancement of SciencesPR18-16SM-04The Project was Funded by Kuwait Foundation for the Advancement of Sciences under project code: PR18-16SM-04. Introduction
The statistical inference of non-ergodic Itô-type diffusions has a long history. For motivation and further references, we refer the reader to Basawa and Scott [2], Dietz and Kutoyants [9], Jacod [17] and Shimizu [24]. On the one hand, the statistical analysis of equations driven by fractional Gaussian processes is obviously more recent. The development of stochastic calculus with respect to the fractional Gaussian processes allowed to study such models. On the other hand, the long range dependence property makes the fractional Gaussian processes important driving noises in modeling several phenomena arising from finance, economic, telecommunication networks and physics.
Let Ba,b:={Bta,b,t≥0} be a weighted fractional Brownian motion (wfBm) with parameters (a,b) such that a>−1-1$]]>, |b|<1 and |b|<a+1, that is, Ba,b is defined as a centered Gaussian process starting from zero with covariance
Ra,b(t,s)=EBta,bBsa,b=∫0s∧tua(t−u)b+(s−u)bdu,s,t≥0.
For a=0, −1<b<1, the wfBm is a fractional Brownian motion (fBm), up to a multiplicative constant 2b+1, with the Hurst parameter b+12. The process Ba,b was introduced in [5] as an extension of fBm. Moreover, it shares several properties with fBm, such as self-similarity, path continuity, behavior of increments, long-range dependence, nonsemimartingale, and others. But, unlike fBm, the wfBm does not have stationary increments for a≠0. For more details about the subject, we refer the reader to [5].
The purpose of this paper is to estimate jointly the drift parameters of the weighted fractional Vasicek (also called weighted fractional mean-reverting Ornstein–Uhlenbeck) process X:={Xt,t≥0} that is defined as the unique (pathwise) solution to
X0=0,dXt=θμ+Xtdt+dBta,b,t≥0,
where θ>00$]]> and μ∈R are considered as unknown parameters. When a=b=0, Ba,b is a standard Brownian motion, and in this case the model (2) with μ=0 was originally proposed by Ornstein and Uhlenbeck and then it was generalized by Vasicek, see [23].
In recent years, several researchers have been interested in studying statistical estimation problems for Gaussian Ornstein–Uhlenbeck processes. Estimation of the drift parameters in fractional-noise-driven Ornstein–Uhlenbeck processes is a problem that is both well-motivated by practical needs and theoretically challenging. In the finance context, our practical motivation to study this estimation problem is to provide tools to understand volatility modeling in finance. Indeed, any mean-reverting model in discrete or continuous time can be taken as a model for stochastic volatility. Let us mention some important results in this field where the volatility exhibits long-memory, which means that the volatility today is correlated to past volatility values with a dependence that decays very slowly. The authors of [6, 7] considered the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. More precisely they assumed that the dynamics of the volatility are described by the equation (2), where the driving process Ba,b is a standard fractional Brownian motion (corresponding to a=0) with the Hurst parameter H=b+12 greater than 1/2. On the other hand, the paper [16] on rough volatility contends that the short-time behavior indicates that the Hurst parameter H in the volatility is less than 1/2.
An example of interesting problem related to (2) is the statistical estimation of μ and θ when one observes the whole trajectory of X. In order to estimate the unknown parameters θ and μ when the whole trajectory of X defined in (2) is observed, we will consider the following least squares estimators (LSEs) proposed in [14]:
θ˜T=12TXT2−XT∫0TXsdsT∫0TXs2ds−∫0TXsds2
and
α˜T=XT∫0TXs2ds−12XT2∫0TXsdsT∫0TXs2ds−∫0TXsds2,
as statistics to estimate θ and α:=μθ, respectively. Furthermore, we can obtain a least squares-type estimator μ˜T for μ, that is, the statistic
μ˜T=α˜T/θ˜T=∫0TXs2ds−12XT∫0TXsds12TXT−∫0TXsds.
Let us mention that similar drift statistical problems for nonergodic Vasicek models were recently studied. The work [14] studied the case when the process Ba,b in (2) is replaced by a Gaussian process, and provided sufficient conditions on the driving Gaussian process in order to ensure the strong consistency and the asymptotic distribution of the estimators given by (3), (4) and (5). Note that we cannot apply directly this result here, because Ba,b verifies properties different of those given in [14].
Let us also describe what is known about the parameter estimation for the model (2) when Ba,b is a fBm, i.e., a=0, with the Hurst parameter H=b+12. Let BH:=BtH,t≥0 denote a fBm with the Hurst parameter H∈(0,1). Consider the following fractional Vasicek model driven by BH,
dXt=θμ+Xtdt+dBtH,X0=0,
where θ,μ∈R are unknown parameters. Notice that the process (6) is ergodic if θ<0, μ=0 and X0=∫−∞0e−θsdBsH. Otherwise, the process (6) is nonergodic if θ>00$]]>.
Now we recall several approaches to estimate the parameters of (6). For the maximum likelihood estimation approach, in general the techniques used to construct maximum likelihood estimators (MLEs) for the drift parameters of (6) are based on Girsanov transforms for fBm and depend on the properties of the deterministic fractional operators (determined by the Hurst parameter) related to the fBm. In general, the MLE is not easily computable. In particular, it relies on being able to constitute a discretization of an MLE. For a more recent comprehensive discussion via this method, we refer to [18].
A least squares approach has been also considered by several researchers to study statistical estimation problems for (6). Let us mention some works in this direction: in the case when θ<0, the statistical estimation for the parameters μ and θ based on continuous-time observations of {Xt,t∈[0,T]} as T→∞, has been studied by several papers, for instance [8, 3] and the references therein. When μ=0 in (6), the estimation of θ has been investigated by using least squares method as follows: the case of ergodic-type fractional Ornstein–Uhlenbeck processes, corresponding to θ<0, has been considered in [11, 15], and the case nonergodic fractional Ornstein–Uhlenbeck processes has been studied in [10, 12].
The paper is organized as follows. In Section 2 we analyze some pathwise properties of the Vasicek model (2). In Section 3 we prove the strong consistency of the estimators θ˜T, μ˜T and α˜T as T→∞. Section 4 is devoted to analyze the joint asymptotic distribution of the LSEs (θ˜T,μ˜T) and (θ˜T,α˜T) as T→∞.
Throughout the paper, we shall use notation C for various constants whose value is not important and may change from line to line and even in the same line.
Notations and auxiliary results
This section is devoted to study pathwise properties of the nonergodic weighted fractional Vasicek model (2). These properties will be needed in order to analyze the asymptotic behavior of the LSEs (θ˜T,μ˜T) and (θ˜T,α˜T).
Because (2) is linear, it is immediate to solve it explicitly; one then gets the following formula
Xt=μeθt−1+eθt∫0te−θsdBsa,b,t≥0,
where the integral with respect to Ba,b is understood in the Young sense (see Appendix).
Let us introduce the following processes, for every t≥0:
ζt:=∫0te−θsdBsa,b,Zt:=∫0te−θsBsa,bds,Σt:=∫0tXsds.
Thus, using (7), we can write
Xt=μeθt−1+eθtζt.
Furthermore, by (2),
Xt=μθt+θΣt+Bta,b.
Moreover, applying the formula (49), we have
ζt=e−θtBta,b+θ∫0te−θsBsa,bds=e−θtBta,b+θZt.
Suppose thata>−1-1$]]>,|b|<1and|b|<a+1. Then, we can rewrite the covarianceRa,b(t,s)ofBa,b, given in (1), as follows:Ra,b(t,s)=βa+1,b+1ta+b+1+sa+b+1−m(t,s),whereβ(c,d)=∫01xc−1(1−x)d−1dx,c>00$]]>,d>00$]]>, denotes the usual Beta function, and the functionm(t,s)is defined bym(t,s):=∫s∧ts∨tua(t∨s−u)bdu.There exists a constant C depending only ona,b, such that for everys,t≥0,EBta,b−Bsa,b2≤C(s∨t)a|t−s|b+1.
Since the process Ba,b has ((a+b+1)∧(b+1)−ε)-Hölder continuous paths for all ε∈(0,(a+b+1)∧(b+1)), and E(Bta,b)2=2βa+1,b+1ta+b+1, we can deduce from [10, Lemma 2.1] and [10, Lemma 2.2] the following result.
Assume thata>−1-1$]]>,|b|<1and|b|<a+1. Let Z and ζ be given by (8). Then for allε∈(0,(a+b+1)∧(b+1))the process ζ admits a modification with((a+b+1)∧(b+1)−ε)-Hölder continuous paths, still denoted ζ in the sequel.
Moreover,ZT⟶Z∞:=∫0∞e−θsBsa,bds,ζT⟶ζ∞:=θZ∞almost surely and inL2(Ω)asT→∞.
Also,limT→∞e−2θT∫0Te2θsζs2ds=12θζ∞2almost surely.
We will make use of the following two technical lemmas.
Suppose thata>−1-1$]]>,|b|<1and|b|<a+1. Then, almost surely, asT→∞,BTa,bTδ⟶0for allδ>a+b+12,\frac{a+b+1}{2},\end{array}\]]]>e−θTXT⟶μ+ζ∞,e−θT∫0TXsds⟶1θμ+ζ∞,e−θTT∫0TsXsds⟶1θμ+ζ∞,e−2θT∫0TXs2ds⟶12θμ+ζ∞2,e−θTTδ∫0T|Xs|ds⟶0for anyδ>0,0,\end{array}\]]]>e−θTT∫0T|Bta,bXt|dt⟶0ifa+b<1,whereζ∞is defined in Lemma2.
Let us prove (16). Let δ>a+b+12\frac{a+b+1}{2}$]]>. By the Borel–Cantelli lemma, it is sufficient to prove that, for any ε>00$]]>,
∑n≥0Psupn≤T≤n+1BTa,bTδ>ε<∞.\varepsilon \right)<\infty .\]]]>
Let q>00$]]> such that q(δ−a+b+12)>11$]]> and q(b+1)2>11$]]>. Applying Markov’s inequality, we obtain
Psupn≤T≤n+1BTa,bTδ>ε≤1εqEsupn≤T≤n+1BTa,bTδq≤1εqnqδEsupn≤T≤n+1BTa,bq.\varepsilon \right)& \displaystyle \le & \displaystyle \frac{1}{{\varepsilon ^{q}}}E\left[\underset{n\le T\le n+1}{\sup }{\left|\frac{{B_{T}^{a,b}}}{{T^{\delta }}}\right|^{q}}\right]\\ {} & \displaystyle \le & \displaystyle \frac{1}{{\varepsilon ^{q}}{n^{q\delta }}}E\left[\underset{n\le T\le n+1}{\sup }{\left|{B_{T}^{a,b}}\right|^{q}}\right].\end{array}\]]]>
Further, applying the Garsia–Rodemich–Rumsey Lemma (see [21, Lemma A.3.1]) for ψ(x)=xq, p(x)=xm+2q, with 0<m<q(b+1)2−1, we get for every n≤s,t≤n+1,
Bta,b−Bsa,bq≤Ct−sm∫nn+1∫nn+1Bua,b−Bva,bqu−vm+2dudv.
This together with (13) implies
Esupn≤s,t≤n+1Bta,b−Bsa,bq≤C∫nn+1∫nn+1(u∨v)qa2u−vq(b+1)2u−vm+2dudv≤Cnqa2∫01∫01x−yq(b+1)2−m−2dxdy≤Cnqa2.
Hence,
Esupn≤T≤n+1BTa,bq≤CEsupn≤T≤n+1BTa,b−Bna,bq+EBna,bq≤CEsupn≤s,t≤n+1Bta,b−Bsa,bq+nq(a+b+1)2≤Cnqa2+nq(a+b+1)2≤Cnq(a+b+1)2.
Then,
Psupn≤T≤n+1BTa,bTδ>ε≤Cεqnq(δ−a+b+12).\varepsilon \right)& \displaystyle \le & \displaystyle \frac{C}{{\varepsilon ^{q}}{n^{q(\delta -\frac{a+b+1}{2})}}}.\end{array}\]]]>
As a consequence, since q(δ−a+b+12)>11$]]>, the above series converges, which proves (16).
Notice that the convergence (17) is a direct consequence of (9) and (14).
On the other hand, using (49), (14) and (16) we have almost surely as T→∞,
e−θT∫0TeθsZsds=ZTθ−e−θTθ∫0TBsa,bds⟶ζ∞θ2.
Combining (9), (11) and (23), we get almost surely as T→∞,
e−θT∫0TXsds=e−θT∫0Tμ(eθs−1)+Bsa,b+θeθsZsds⟶1θμ+ζ∞,
which proves (18). Similarly, (49), (14), (16) and (23) imply that almost surely as T→∞,
e−θTT∫0TseθsZsds=ZTθ−e−θTθT∫0TsBsa,bds−e−θTθT∫0TeθsZsds⟶ζ∞θ2.
Combined with (9) and (11), a straightforward calculation as above, leads to (19).
Using similar arguments as in (23), we deduce that almost surely as T→∞,
e−2θT∫0Te2θsZsds⟶ζ∞2θ2.
Combining this latter convergence together with (9), (14), (15) and (16), we can deduce (20).
Using (17), we have almost surely as T→∞,
e−θTTδ∫0T|Xs|ds≤supt≥0Xteθte−θTTδ∫0Teθsds⟶0,
which implies (21).
Now we prove (22). Let a+b+12<δ<1,
e−θTT∫0T|Bta,bXt|dt≤supt≥0Bta,bXttδeθte−θTT∫0Ttδeθtdt≤supt≥0Bta,bXttδeθte−θTT1−δ∫0Teθtdt≤supt≥0Bta,bXttδeθt1θT1−δ⟶0
almost surely as T→∞, where we used that supt≥0Bta,bXttδeθt<∞ almost surely, thanks to (16) and (17). Thus the proof of (22) is done. □
Strong consistency
In this section we will prove the strong consistency of the estimators θ˜T, μ˜T and α˜T.
Suppose thata>−1-1$]]>,|b|<1and|b|<a+1. Letθ˜T,α˜Tandμ˜Tbe given by (3), (4) and (5), respectively. Then, almost surely, asT→∞,θ˜T⟶θ.Moreover, ifa+b<1,μ˜T⟶μ,and consequently,α˜T=μ˜Tθ˜T⟶α=μθalmost surely, asT→∞.
Combining (3) and the convergences (17), (20) and (21), we obtain
θ˜T=12e−θTXT2−e−θTXTe−θTT∫0TXsdse−2θT∫0TXs2ds−e−θTT∫0TXsds2⟶θ,almost surely, asT→∞.
Thus the convergence (24) is obtained.
Now we prove (25). It follows from (5) that μ˜T can be written as follows:
μ˜T=e−θTT∫0TXs2ds−XT2∫0TXsds×112e−θTXT−e−θTT∫0TXsds.
According to the convergences (17) and (21) we have, almost surely, as T→∞,
112e−θTXT−e−θTT∫0TXsds⟶2μ+ζ∞.
Therefore, it remains to prove
e−θTT∫0TXs2ds−XT2∫0TXsds⟶μ2μ+ζ∞
almost surely, as T→∞.
Using the formula (49) and the equation (2), we have
∫0TXs2ds−XT2∫0TXsds=∫0TXsdΣs−12μθT+θΣT+BTa,bΣT=∫0Tμθs+θΣs+Bsa,bdΣs−μθ2TΣT−θ2ΣT2−12BTa,bΣT=μθ∫0TsXsds+θ2ΣT2+∫0TBsa,bdΣs−μθ2TΣT−θ2ΣT2−12BTa,bΣT=μθ∫0TsXsds−μθ2TΣT+∫0TBsa,bdΣs−12BTa,bΣT=:IT+JT.
Moreover, by L’Hôpital’s rule and (17) we have
e−θTTIT=e−θTTμθ∫0TsXsds−μθ2TΣT⟶μ2μ+ζ∞
almost surely, as T→∞.
On the other hand, taking a+b+12<δ<1,
e−θTT|JT|=e−θTT∫0TBsa,bdΣs−12BTa,bΣT=e−θTT∫0TBsa,bXsds−12BTa,bΣT≤32sups≥0Bsa,bsδe−θTT1−δ∫0T|Xs|ds⟶0
almost surely, as T→∞, where we used (16) and (21).
Consequently, the convergence (26) is proved. Thus the desired results are obtained. □
Joint asymptotic distribution
In this section we analyze the joint asymptotic distribution of the LSEs (θ˜T,μ˜T) and (θ˜T,α˜T).
Suppose thata>−1-1$]]>,|b|<1and|b|<a+1. Then
The limit of the variance ofT−a2e−θT∫0TeθsdBsa,bexists asT→∞. More precisely,limT→∞ET−a2e−θT∫0TeθsdBsa,b2⟶Γ(b+1)θb+1.
For all fixeds≥0,limT→∞EBsa,bT−a2e−θT∫0TeθrdBra,b=0.
For all fixeds≥0, asT⟶∞EBsa,bBTa,bTa+b+12⟶0,EBTa,bTa+b+12T−a2e−θT∫0TeθrdBra,b⟶0.
Both convergences (27) and (28) are proved in [1]. Let us now prove the convergences given in (29). Fix s≥0. Using (12), and b−a−1<0 and change of variables x=u/T, we get
limT→∞EBsa,bBTa,bTa+b+12=limT→∞T−a−b−12βa+1,b+1sa+b+1+Ta+b+1−m(s,T)=limT→∞T−a−b−12βa+1,b+1Ta+b+1−∫sTua(T−u)bdu=limT→∞T−a−b−12−∫0Tua(T−u)bdu+∫0sua(T−u)bdu=limT→∞T−a−b−12−Ta+b+1∫01xa(1−x)bdx+∫0sua(T−u)bdu=limT→∞T−a−b−12∫0sua(T−u)bdu.
Furthermore,
T−a−b−12∫0sua(T−u)bdu=Tb−a−12∫0sua1−uTbdu≤Tb−a−12∫0suaduifb≥0,Tb−a−12∫0sua1−sTbduifb<0⟶0.
Thus, we deduce that, for all fixed s≥0,
limT→∞EBsa,bBTa,bTa+b+12=0.
In order to complete the proof, we show that
limT→∞EBTa,bTa+b+12T−a2e−θT∫0TeθrdBra,b=0.
Applying twice (49), we can write EBTa,bTa+b+12T−a2e−θT∫0TeθrdBra,b=T−a−b+12Ra,b(T,T)−θe−θT∫0TeθrRa,b(T,r)dr=T−a−b+12e−θT∫0Teθrβ(a+1,b+1)(a+b+1)ra+bdr−T−a−b+12e−θT∫0Teθr∂m(T,r)∂rdr=(a+b+1)β(a+1,b+1)T−a−b+12e−θT∫0Teθrra+bdr+T−a−b+12e−θT∫0Teθrra(T−r)bdr. Moreover, applying L’Hôspital’s rule, we get limT→∞T−a−b+12e−θT∫0Teθrra+bdr=limT→∞eθTTa+beθT(a+b+12)Ta+b−12+θTa+b+12=limT→∞1(a+b+12)T−b−12+θT1−b2=0, since |b|<1. Also, using change of variables x=r/T and y=T−r, we have
T−a−b+12e−θT∫0Teθrra(T−r)bdr=T−a−b+12e−θT∫0T/2eθrra(T−r)bdr+∫T/2Teθrra(T−r)bdr≤T−a−b+12e−θT/2∫0T/2ra(T−r)bdr+CT−b+12e−θT∫T/2Teθr(T−r)bdr=Tb+12e−θT/2∫01/2xa(1−x)bdr+CT−b+12∫0T/2e−θyybdy≤Tb+12e−θT/2β(a+1,b+1)+CT−b+12Γ(b+1)⟶0
as T→∞, thanks to a+1>00$]]> and b+1>00$]]>.
Consequently, combining (32), (34) and (35), we obtain (30). □
In order to investigate the asymptotic behavior in distribution of the estimators (θ˜T,μ˜T) and (θ˜T,α˜T), as T→∞, we will also need the following lemmas.
Suppose thata>−1-1$]]>,|b|<1and|b|<a+1. Let X be the process given by (2). Then we have for everyT>00$]]>,12XT2−XTT∫0TXtdt=θ∫0TXt2dt−1T∫0TXtdt2+μ+θZT∫0TeθtdBta,b+RT,whereZTis given in (8), and the processRTis defined byRT:=12μθT2+12(BTa,b)2−μBTa,b−(μθ)2T22−BTa,bT∫0TXtdt−θ∫0T(Bta,b)2dt+θ2∫0Te−θtBta,b∫0teθsBsa,bdsdt.Moreover, asT⟶∞,T−a2e−θTRT⟶0almost surely.
Using similar arguments as in [14, Lemma 3.1] we obtain (36). On the other hand, the convergence (37) is a direct consequence of (16), (18) and |b|<1. □
Suppose thata>−1-1$]]>,|b|<1and|b|<a+1. Let F be anyσ{Bta,b,t≥0}-measurable random variable such thatP(F<∞)=1. Then asT→∞,F,T−a2e−θT∫0TeθtdBta,b⟶LawF,Γ(b+1)θb+1N2,andBTa,bTa+b+12,F,T−a2e−θT∫0TeθtdBta,b⟶Law2β(a+1,b+1)N1,F,Γ(b+1)θb+1N2,whereN1∼N(0,1),N2∼N(0,1)andBa,bare independent.
The convergence (38) is proved in [10, Lemma 2.4]. Now we prove (39). Using similar arguments as in the proof of [13, Lemma 7] it suffices to prove that for every positive integer d, and for all fixed s1,…,sd≥0, (BTa,bTa+b+12,Bs1a,b,…,Bsda,b,T−a2e−θT∫0TeθtdBta,b) converges in distribution to
2β(a+1,b+1)N1,Bs1a,b,…,Bsda,b,Γ(b+1)θb+1N2
as T→∞. Moreover, since the left-hand side in this latter convergence is a Gaussian vector, it is sufficient to establish the convergence of its covariance matrix. Combining this with Lemma 4, the desired result is obtained. □
Recall that if X∼N(m1,σ1) and Y∼N(m2,σ2) are two independent random variables, then X/Y follows a Cauchy-type distribution. For a motivation and further references, we refer the reader to [22], as well as [19]. Notice also that if N∼N(0,1) is independent of Ba,b, then N is independent of ζ∞, since ζ∞=θ∫0∞e−θsBsa,bds is a functional of Ba,b.
Suppose thata>−1-1$]]>,|b|<1and|b|<a+1. Suppose thatN1∼N(0,1),N2∼N(0,1)andBa,bare independent. Then asT→∞,eθT(θ˜T−θ)⟶Law2Γ(b+1)θb−1N2μ+ζ∞.Moreover, ifa+b<1, then asT→∞,T1−a−b2μ˜T−μ⟶Law2β(a+1,b+1)θN1,T1−a−b2α˜T−α⟶Law2β(a+1,b+1)N1.Also, asT→∞,eθT(θ˜T−θ),T1−a−b2μ˜T−μ⟶Law2Γ(b+1)θb−1N2μ+ζ∞,2β(a+1,b+1)θN1,eθT(θ˜T−θ),T1−a−b2α˜T−α⟶Law2Γ(b+1)θb−1N2μ+ζ∞,2β(a+1,b+1)N1.
First we prove (40). From (3) and (36) we can write
T−a2eθTθ˜T−θ=μ+θZTT−a2e−θT∫0TeθtdBta,b+T−a2e−θTRTe−2θT∫0TXt2dt−1T∫0TXtdt2=T−a2e−θT∫0TeθtdBta,bμ+ζ∞×μ+ζ∞μ+θZTe−2θT∫0TXt2dt−1T∫0TXtdt2+T−a2e−θTRTe−2θT∫0TXt2dt−1T∫0TXtdt2=:aT×bT+cT.
Lemma 6 yields, as T→∞,
aT⟶LawΓ(b+1)θb+1N2μ+ζ∞,
whereas (20), (21) and (14) imply that bT⟶2θ almost surely as T→∞. On the other hand, by (20), (21) and (37), we obtain that cT⟶0 almost surely as T→∞.
Combining all these facts together with (38), we get (40).
Using (3) and (5), a straightforward calculation shows that θ˜T and μ˜T verify
θ˜Tμ˜TT=θ˜Tμ˜TXTXTT=XT−θ˜T∫0TXtdt.
Combining (46) with (2), we obtain T1−a−b2μ˜T−μ=1θ˜T−eθTθ˜T−θe−θTTa+b+12∫0TXtdt−μT1−a−b2eθTeθTθ˜T−θ+BTa,bTa+b+12=:1θ˜TDT+BTa,bTa+b+12. Using (21), (24), (40) and Slutsky’s theorem, we obtain DT⟶0 in probability as T→∞. This together with (24) implies (41).
Further, according to (46) and (2) we can write
T1−a−b2α˜T−α=−eθTθ˜T−θe−θTTa+b+12∫0TXtdt+BTa,bTa+b+12,
which proves (42), by using (21), (24), (40), and Slutsky’s theorem.
Let us now prove (43). By (45) and (48) we have
eθT(θ˜T−θ),T1−a−b2μ˜T−μ=aT×bT+cT,1θ˜TDT+BTa,bTa+b+12=aT×bT,1θ˜TBTa,bTa+b+12+cT,1θ˜TDT=1θ˜T2θ2aT,BTa,bTa+b+12+1θ˜TaT×(bT×θ˜T−2θ2),0+cT,1θ˜TDT.
By the above convergences and Slutsky’s theorem we deduce, as T→∞,
1θ˜TaT×(bT×θ˜T−2θ2),0⟶0,cT,1θ˜TDT⟶0in probability.
Therefore, using (39), we obtain as T→∞,
1θ˜T2θ2aT,BTa,bTa+b+12=1θ˜T2θ2e−θT∫0TeθtdBta,bμ+ζ∞,BTa,bTa+b+12⟶Law2Γ(b+1)θb−1N2μ+ζ∞,2β(a+1,b+1)θN1,
which completes the proof of (43). Finally, following the same arguments as in the proof of (43) we obtain (44). □
Appendix: Young integral
In this section, we briefly recall some basic elements of the Young integral (see [25]), which are helpful for some of the arguments we use. For any α∈[0,1], we denote by Hα([0,T]) the set of α-Hölder continuous functions, that is, the set of functions f:[0,T]→R such that
|f|α:=sup0≤s<t≤T|f(t)−f(s)|(t−s)α<∞.
We also set |f|∞=supt∈[0,T]|f(t)|, and we equip Hα([0,T]) with the norm
‖f‖α:=|f|α+|f|∞.
Let f∈Hα([0,T]), and consider the operator Tf:C1([0,T])→C0([0,T]) defined as
Tf(g)(t)=∫0tf(u)g′(u)du,t∈[0,T].
It can be shown (see, e.g., [20, Section 3.1]) that, for any β∈(1−α,1), there exists a constant Cα,β,T>00$]]> depending only on α, β and T such that, for any g∈C1([0,T]),
∫0·f(u)g′(u)duβ≤Cα,β,T‖f‖α‖g‖β.
We deduce that, for any α∈(0,1), any f∈Hα([0,T]) and any β∈(1−α,1), the linear operator Tf:C1([0,T])⊂Hβ([0,T])→Hβ([0,T]), defined as Tf(g)=∫0·f(u)g′(u)du, is continuous. By density, it extends (in a unique way) to an operator defined on Hβ. As a consequence, if f∈Hα([0,T]), if g∈Hβ([0,T]) and if α+β>11$]]>, then the (so-called) Young integral ∫0·f(u)dg(u) is (well) defined as being Tf(g).
The Young integral obeys the following formula. Let f∈Hα([0,T]) with α∈(0,1), and g∈Hβ([0,T]) with β∈(0,1). If α+β>11$]]>, then ∫0.gudfu and ∫0.fudgu are well-defined as Young integrals, and for all t∈[0,T],
ftgt=f0g0+∫0tgudfu+∫0tfudgu.
Acknowledgments
We thank the two anonymous reviewers for their very careful reading and suggestions, which have led to significant improvements in the presentation of our results.
ReferencesAlsenafi, A., Al-Foraih, M., Es-Sebaiy, K.: Least squares estimation for non-ergodic weighted fractional Ornstein-Uhlenbeck process of general parameters (2020). Preprint. arXiv:2002.06861Basawa, I.V., Scott, D.J.: Asymptotic Optimal Inference for Non-Ergodic Models. Lecture Notes in Statist., vol. 17. Springer, New York (1983). MR0688650Bajja, S., Es-Sebaiy, K., Viitasaari, L.: Least squares estimator of fractional Ornstein-Uhlenbeck processes with periodic mean. J. Korean Stat. Soc.46(4), 608–622 (2017). MR3718150. https://doi.org/10.1016/j.jkss.2017.06.002Belfadli, Es-Sebaiy K, R., Ouknine, Y.: Parameter estimation for fractional Ornstein-Uhlenbeck processes: non-ergodic case. Front. Sci. Eng.1(1), 1–16 (2011)Bojdecki, T., Gorostiza, L., Talarczyk, A.: Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems. Electron. Commun. Probab.12, 161–172 (2007). MR2318163. https://doi.org/10.1214/ECP.v12-1272Chronopoulou, A., Viens, F.: Estimation and pricing under long-memory stochastic volatility. Ann. Finance8, 379–403 (2012). MR2922802. https://doi.org/10.1007/s10436-010-0156-4Chronopoulou, A., Viens, F.: Stochastic volatility and option pricing with long-memory in discrete and continuous time. Quant. Finance12, 635–649 (2012). MR2909603. https://doi.org/10.1080/14697688.2012.664939Dehling, H., Franke, B., Woerner, J.H.C.: Estimating drift parameters in a fractional Ornstein Uhlenbeck process with periodic mean. Stat. Inference Stoch. Process.1–14 (2016). MR3619570. https://doi.org/10.1007/s11203-016-9136-2Dietz, H.M., Kutoyants, Y.A.: Parameter estimation for some non-recurrent solutions of SDE. Stat. Decis.21, 29–46 (2003). MR1985650. https://doi.org/10.1524/stnd.21.1.29.20321El Machkouri, M., Es-Sebaiy, K., Ouknine, Y.: Least squares estimator for non-ergodic Ornstein-Uhlenbeck processes driven by Gaussian processes. J. Korean Stat. Soc.45, 329–341 (2016). MR3527650. https://doi.org/10.1016/j.jkss.2015.12.001El Onsy, B., Es-Sebaiy, K., Viens, F.: Parameter Estimation for a partially observed Ornstein-Uhlenbeck process with long-memory noise. Stochastics89(2), 431–468 (2017). MR3590429. https://doi.org/10.1080/17442508.2016.1248967Es-Sebaiy, K., Alazemi, F., Al-Foraih, M.: Least squares type estimation for discretely observed non-ergodic Gaussian Ornstein-Uhlenbeck processes. Acta Math. Sci.39(4), 989–1002 (2019). MR4066516. https://doi.org/10.1007/s10473-019-0406-0Es-Sebaiy, K., Nourdin, I.: Parameter estimation for α-fractional bridges. Springer Proc. Math. Stat.34, 385–412 (2013). MR3070453. https://doi.org/10.1007/978-1-4614-5906-4_17Es-Sebaiy, K., Sebaiy M, Es.: Estimating drift parameters in a non-ergodic Gaussian Vasicek-type model. Statistical Methods & Applications1–28 (2020). https://doi.org/10.1007/s10260-020-00528-4Hu, Y., Nualart, D., Zhou, H.: Parameter estimation for fractional Ornstein-Uhlenbeck processes of general Hurst parameter. Stat. Inference Stoch. Process.1–32 (2017). MR3918739. https://doi.org/10.1007/s11203-017-9168-2Gatheral, J., Jaisson, T., Rosenbaum, M.: Volatility is rough. Quant. Finance18(6), 933–949 (2018). MR3805308. https://doi.org/10.1080/14697688.2017.1393551Jacod, J.: Parametric inference for discretely observed non-ergodic diffusions. Bernoulli12, 383–401 (2006). MR2232724. https://doi.org/10.3150/bj/1151525127Kleptsyna, M.L., Le Breton, A.: Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Stat. Inference Stoch. Process.5(3), 229–248 (2002). MR1943832. https://doi.org/10.1023/A:1021220818545Marsaglia, G.: Ratios of normal variables and ratios of sums of uniform variables. J. Am. Stat. Assoc.60, 193–204 (1965). MR0178490. https://doi.org/10.1080/01621459.1965.10480783Nourdin, I.: Selected Aspects of Fractional Brownian Motion. Bocconi & Springer Series, vol. 4. Springer/Bocconi University Press, Milan (2012). MR3076266. https://doi.org/10.1007/978-88-470-2823-4Nualart, D.: The Malliavin calculus and related topics (Vol. 1995). Springer, Berlin (2006). MR1344217. https://doi.org/10.1007/978-1-4757-2437-0Pham-Gia, T., Turkkan, N., Marchand, E.: Density of the ratio of two normal random variables and applications. Commun. Stat., Theory Methods35(9), 1569–1591 (2006). MR2328495. https://doi.org/10.1080/03610920600683689Vasicek, O.: An equilibrium characterization of the term structure. J. Financ. Econ.5(2), 177–188 (1977). https://doi.org/10.1016/0304-405X(77)90016-2Shimizu, Y.: Notes on drift estimation for certain non-recurrent diffusion from sampled data. Stat. Probab. Lett.79, 2200–2207 (2009). MR2572052. https://doi.org/10.1016/j.spl.2009.07.015Young, L.C.: An inequality of the Hölder type connected with Stieltjes integration. Acta Math.67, 251–282 (1936). MR1555421. https://doi.org/10.1007/BF02401743