Statistical inference for Vasicek-type model driven by Hermite processes
aa r X i v : . [ m a t h . P R ] O c t Statistical inference for Vasicek-type modeldriven by Hermite processes
Ivan Nourdin ∗ , T. T. Diu Tran † October 12, 2018
Abstract
Let Z denote a Hermite process of order q > and self-similarity parameter H ∈ ( , . This process is H -self-similar, has stationary increments and exhibits long-rangedependence. When q = 1 , it corresponds to the fractional Brownian motion, whereas it isnot Gaussian as soon as q > . In this paper, we deal with a Vasicek-type model drivenby Z , of the form dX t = a ( b − X t ) dt + dZ t . Here, a > and b ∈ R are considered asunknown drift parameters. We provide estimators for a and b based on continuous-timeobservations. For all possible values of H and q , we prove strong consistency and weanalyze the asymptotic fluctuations. Key words:
Parameter estimation, strong consistency, fractional Ornstein-Uhlenbeckprocess, Hermite Ornstein-Uhlenbeck processes, fractional Vasicek model, long-range depen-dence.
Our aim in this paper is to introduce and analyze a non-Gaussian extension of the fractionalmodel considered in the seminal paper [2] of Comte and Renault (see also Chronopoulouand Viens [3], as well as the motivations and references therein) and used by these authorsto model a situation where, unlike the classical Black-Scholes-Merton model, the volatilityexhibits long-memory. More precisely, we deal with the drift parameter estimation problemfor a Vasicek-type process X , defined as the unique (pathwise) solution to X = 0 , dX t = a ( b − X t ) dt + dZ q,Ht , t > , (1.1)where Z q,H is a Hermite process of order q > and Hurst parameter H ∈ ( , . Equivalently, X is the process given explicitly by X t = b (1 − e − at ) + Z t e − a ( t − s ) dZ q,Hs , (1.2)where the integral with respect to Z q,H must be understood in the Riemann-Stieltjes sense.In (1.1) and (1.2), parameters a > and b ∈ R are considered as (unknown) real parameters. ∗ Université du Luxembourg, Unité de Recherche en Mathématiques. E-mail: [email protected]. IN waspartially supported by the Grant F1R-MTH-PUL-15CONF (CONFLUENT) at Luxembourg University † Université du Luxembourg, Unité de Recherche en Mathématiques. E-mail: [email protected] Z q,H of order q > form a class of genuine non-Gaussian generalizationsof the celebrated fractional Brownian motion (fBm), this latter corresponding to the case q = 1 . Like the fBm, they are self-similar, have stationary increments and exhibit long-range dependence. Their main noticeable difference with respect to fBm is that they are not Gaussian. For more details about this family of processes, we refer the reader to Section 2.2.As we said, one main practical motivation to study this estimation problem is to providetools to understand volatility modeling in finance. Indeed, any mean-reverting model indiscrete or continuous time can be taken as a model for stochastic volatility. Classical textscan be consulted for this modeling idea; also see the research monograph [10]. Our paperproposes extensions in the type of tail weights for these processes. We acknowledge thatthere is a gap between the results we cover and their applicability. First, for any practicalproblems in finance, one must consider discrete-time observations, and one must then choose anobservation frequency, keeping in mind that very high frequencies (or infinitely high frequency,which is the case covered in this paper) are known to be inconsistent with models that have nojumps. This problem can be addressed by adding a term to account for microstructure noise insome contexts. Since this falls outside of the scope of this article, we omit any details. Second,for most financial markets, volatility is not observed. One can resort to proxies such as theCBOE’s VIX index for volatility on the S&P500 index. See the paper [4] for details aboutthe observation frequency which allows the use of the continuous-time framework in partialobservation. In this paper, and for most other authors, such as those for the aforementionedresearch monograph, these considerations are also out of scope.Our paper is relevant to the literature on parameter estimation for processes with Gaussianand non-Gaussian long-memory processes, including [1, 5, 6, 7, 8, 9, 12, 13, 14, 16, 27]. Inthe finance context, the highly cited paper [11] investigates the high-frequency behavior ofvolatility, drawing on ideas in the paper [21] on long-memory parameter estimation, and beforethis, the 1997 paper [14], and the 2001 paper [7]. The paper [11] on rough volatility contendsthat the short-time behavior indicates that the Hurst parameter H of volatility is less than .Most other authors point towards H being bigger than , as a model for long memory, whichfalls within our context, since Hermite processes are limited to this long-memory case. Thereare many other papers where long-memory noises are used to direct Ornstein-Uhlenbeck andother mean-reverting processes, where estimation is a main motivation. The reader can e.g.consult [8, 9] and the numerous references therein. These papers are always in the Gaussiancontext, which the current paper extends. A limited number of attempts have been made tocover non-Gaussian noises, notably [6, 23]. With the exception of the aforementioned papers[5, 6, 14, 27], all these papers, and the current paper, make no attempt to estimate the Hurstparameter H . This is a non-trivial task, which leads authors to consider restrictive self-similarframeworks, and makes quantitative estimator asymptotics difficult to obtain. We leave thisquestion out of the scope of this paper, mentioning only the paper [1] which, to the best ofour knowledge, is the only paper where H is jointly estimated for the fractional Ornstein-Uhlenbeck process, simultaneously with the process’s other parameters, though this paperdoes not include a rate of convergence in the asymptotics, and covers only the Gaussian case.Thus our paper covers new ground, and uncovers intriguing asymptotic behaviors which arenot visible in any of the prior literature.Let us now describe in more details the results we have obtained. Definition 1.1
Recall from (1.1)-(1.2) the definition of the Vasicek-type process X = ( X t ) t > driven by the Hermite process Z q,H . Assume that q > and H ∈ ( , are known, whereas > and b ∈ R are unknown. Suppose that we continuously observe X over the time interval [0 , T ] , T > . We define estimators for a and b as follows: b a T = (cid:18) α T H Γ(2 H ) (cid:19) − H , where α T = T R T X t dt − (cid:16) T R T X t dt (cid:17) , (1.3) b b T = 1 T Z T X t dt. In order to describe the asymptotic behavior of ( b a T , b b T ) when T → ∞ , we first need thefollowing proposition, which defines the σ { Z q,H } -measurable random variable G ∞ appearingin (1.8)-(1.9) below. Proposition 1.2
Assume either ( q = 1 and H > ) or q > . Fix T > , and let U T =( U T ( t )) t > be the process defined as U T ( t ) = R t e − T ( t − u ) dZ q,Hu . Eventually, define the randomvariable G T by G T = T q (1 − H )+2 H Z (cid:0) U T ( t ) − E [ U T ( t ) ]) dt. Then G T converges in L (Ω) to a limit written G ∞ . Moreover, G ∞ /B H,q is distributed ac-cording to the Rosenblatt distribution of parameter − q (1 − H ) , where B H,q = H (2 H − q ( H − )(4 H − × Γ(2 H + q (1 − H ))2 H + q (1 − H ) − , with H = 1 − − Hq . (1.4) (The definition of the Rosenblatt random variable is recalled in Definition 2.3.)
We can now describe the asymptotic behavior of ( b a T , b b T ) as T → ∞ . In the limits (1.8)and (1.9) below, note that the two components are (well-defined and) correlated, because G ∞ is σ { Z q,H } -measurable by construction. Theorem 1.3
Let X = ( X t ) t > be given by (1.1)-(1.2), where Z q,H = ( Z q,Ht ) t > is a Hermiteprocess of order q > and parameter H ∈ ( , , and where a > and b ∈ R are (unknown)real parameters. The following convergences take place as T → ∞ .1. [Consistency] ( b a T , b b T ) a . s . → ( a, b ) . [Fluctuations] They depend on the values of q and H . • (Case q = 1 and H < ) (cid:16) √ T { b a T − a } , T − H { b b T − b } (cid:17) law → (cid:18) − a H σ H H Γ(2 H ) N, a N ′ (cid:19) , (1.5) where N, N ′ ∼ N (0 , are independent and σ H is given by σ H = 2 H − H Γ(2 H ) vuutZ R Z R e − ( u + v ) | u − v − x | H − dudv ! dx. (1.6)3 (Case q = 1 and H = ) s T log T { b a T − a } , T (cid:8)b b T − b } ! → (cid:18) r aπ N, a N ′ (cid:19) , (1.7) where N, N ′ ∼ N (0 , are independent. • (Case q = 1 and H > ) (cid:16) T − H ) { b a T − a } , T − H (cid:8)b b T − b } (cid:17) law → (cid:18) − a H − H Γ(2 H ) (cid:16) G ∞ − ( B H ) (cid:17) , a B H (cid:19) , (1.8) where B H = Z ,H is the fractional Brownian motion and where the σ { B H } -measurablerandom variable G ∞ is defined in Proposition 1.2. • (Case q > and any H ) (cid:16) T q (1 − H ) { b a T − a } , T − H (cid:8)b b T − b } (cid:17) law → − a − q (1 − H ) H Γ(2 H ) G ∞ , a Z q,H ! , (1.9) where the σ { Z q,H } -measurable random variable G ∞ is defined in Proposition 1.2. As we see from our Theorem 1.3, strong consistency for b a T and b b T always holds, irre-spective of the values of q (and H ). That is, when one is only interested in the first orderapproximation for a and b , Vasicek-type model (1.1)-(1.2) displays a kind of universality withrespect to the order q of the underlying Hermite process. But, as point 2 shows, the situationbecomes different when one looks at the fluctuations, that is, when one seeks to constructasymptotic confidence intervals: they heavily depend on q (and H ). Furthermore, we high-light the dependence of two components in the limit (1.8) and (1.9), which is very different ofthe case q = 1 and H .The rest of the paper is structured as follows. Section 2 presents some basic results aboutmultiple Wiener-Itô integrals and Hermite processes, as well as some other facts which areused throughout the paper. The proof of Proposition 1.2 is then given in Section 3. Section4 is devoted to the proof of the consistency part of Theorem 1.3, whereas the fluctuations areanalyzed in Section 5. Let B = (cid:8) B ( h ) , h ∈ L ( R ) (cid:9) be a Brownian field defined on a probability space (Ω , F , P ) , thatis, a centered Gaussian family satisfying E [ B ( h ) B ( g )] = h h, g i L ( R ) for any h, g ∈ L ( R ) .For every q > , the q th Wiener chaos H q is defined as the closed linear subspace of L (Ω) generated by the family of random variables { H q ( B ( h )) , h ∈ L ( R ) , k h k L ( R ) = 1 } , where H q is the q th Hermite polynomial ( H ( x ) = x , H ( x ) = x − , H ( x ) = x − x , and so on).The mapping I Bq ( h ⊗ q ) = H q ( B ( h )) can be extended to a linear isometry between L s ( R q ) (= the space of symmetric square integrable functions of R q , equipped with the modified norm4 q ! k · k L ( R q ) ) and the q th Wiener chaos H q . When f ∈ L s ( R q ) , the random variable I Bq ( f ) is called the multiple Wiener-Itô integral of f of order q ; equivalently, one may write I Bq ( f ) = Z R q f ( ξ , . . . , ξ q ) dB ξ . . . dB ξ q . (2.1)Multiple Wiener-Itô integrals enjoy many nice properties. We refer to [17] or [20] for acomprehensive list of them. Here, we only recall the orthogonality relationship, the isometryformula and the hypercontractivity property.First, the orthogonality relationship (when p = q ) or isometry formula (when p = q ) statesthat, if f ∈ L s ( R p ) and g ∈ L s ( R q ) with p, q > , then E [ I Bp ( f ) I Bq ( g )] = ( p ! (cid:10) f, g (cid:11) L ( R p ) if p = q if p = q. (2.2)Second, the hypercontractivity property reads as follows: for any q > , any k ∈ [2 , ∞ ) andany f ∈ L s ( R q ) , E [ | I Bq ( f ) | k ] /k ( k − q/ E [ | I Bq ( f ) | ] / . (2.3)As a consequence, for any q > and any k ∈ [2 , ∞ ) , there exists a constant C k,q > suchthat, for any F ∈ ⊕ ql =1 H l , we have E [ | F | k ] /k C k,q p E [ F ] . (2.4) We now give the definition and present some basic properties of Hermite processes. We referthe reader to the recent book [25] for any missing proof and/or any unexplained notion.
Definition 2.1
The Hermite process ( Z q,Ht ) t > of order q > and self-similarity parameter H ∈ ( , is defined as Z q,Ht = c ( H, q ) Z R q (cid:18) Z t q Y j =1 ( s − ξ j ) H − + ds (cid:19) dB ξ . . . dB ξ q , (2.5) where c ( H, q ) = s H (2 H − q ! β q ( H − , − H ) and H = 1 + H − q ∈ (cid:18) − q , (cid:19) . (2.6) (The integral (2.5) is a multiple Wiener-Itô integral of order q of the form (2.1).) The positive constant c ( H, q ) in (2.6) has been chosen to ensure that E [( Z q,H ) ] = 1 . Definition 2.2
A random variable with the same law as Z q,H is called a Hermite randomvariable of order q and parameter H . q = 1 is nothing but the fractional Brownian motion. It is theonly Hermite process to be Gaussian (and that one could have defined for H as well). Forother reasons, the value q = 2 is also special. Indeed, Hermite process of order 2 has attracteda lot of interest in the recent past (see [22] for a nice introduction and an overview), and haswon its own name: it is called the Rosenblatt process . Definition 2.3
A random variable with the same law as Z ,H is called a Rosenblatt randomvariable. Except for Gaussianity, Hermite processes of order q > share many properties with thefractional Brownian motion (corresponding to q = 1 ). We list some of them in the nextstatement. Proposition 2.4
The Hermite process Z q,H of order q > and Hurst parameter H ∈ ( , enjoys the following properties. • [Self-similarity] For all c > , ( Z q,Hct ) t > law = ( c H Z q,Ht ) t > . • [Stationarity of increments] For any h > , ( Z q,Ht + h − Z q,Hh ) t > law = ( Z q,Ht ) t > . • [Covariance function] For all s, t > , E [ Z q,Ht Z q,Hs ] = ( t H + s H − | t − s | H ) . • [Long-range dependence] P ∞ n =0 | E [ Z q,H ( Z q,Hn +1 − Z q,Hn )] | = ∞ . • [Hölder continuity] For any ζ ∈ (0 , H ) and any compact interval [0 , T ] ⊂ R + , ( Z q,Ht ) t ∈ [0 ,T ] admits a version with Hölder continuous sample paths of order ζ . • [Finite moments] For every p > , there exists a constant C p,q > such that E [ | Z q,Ht | p ] C p,q t pH for all t > . The Wiener integral of a deterministic function f with respect to a Hermite process Z q,H ,which we denote by R R f ( u ) dZ q,Hu , has been constructed by Maejima and Tudor in [15].Below is a very short summary of what will is needed in the paper about those integrals.The stochastic integral R R f ( u ) dZ q,Hu is well-defined for any f belonging to the space |H| offunctions f : R → R such that Z R Z R | f ( u ) f ( v ) || u − v | H − dudv < ∞ . We then have, for any f, g ∈ |H| , that E (cid:20) Z R f ( u ) dZ q,Hu Z R g ( v ) dZ q,Hu (cid:21) = H (2 H − Z R Z R f ( u ) g ( v ) | u − v | H − dudv. (2.7)Another important and useful property is that, whenever f ∈ |H| , the stochastic integral R R f ( u ) dZ q,Hu admits the following representation as a multiple Wiener-Itô integral of theform (2.1): Z R f ( u ) dZ q,Hu = c ( H, q ) Z R q (cid:18) Z R f ( u ) q Y j =1 ( u − ξ j ) H − + du (cid:19) dB ξ . . . dB ξ q , (2.8)with c ( H, q ) and H given in (2.6). 6 .4 Existing limit theorems To the best of our knowledge, only a few limit theorems have been already obtained in thelitterature for quadratic functionals of the Hermite process, see [5, 24, 27]. Here, we onlyrecall the following result from [24], which will be useful to study the fluctuations of ( b a T , b b T ) in Theorem 1.3. Proposition 2.5
Assume either q > or ( q = 1 and H > ), and let Y be defined as Y t = Z t e − a ( t − u ) dZ q,Hu , t > . (2.9) Then, as T → ∞ , T q (1 − H ) − Z T (cid:0) Y t − E [ Y t ] (cid:1) dt law → B H,q a − H − q (1 − H ) × R H ′ , (2.10) where R H ′ is distributed according to a Rosenblatt random variable of parameter H ′ = 1 − q (1 − H ) and B H,q is given by (1.4).
Along the proof of Theorem 1.3, we will also make use of another result for the Gaussiancase ( q = 1 ), which we take from [16]. Proposition 2.6
Let Y be given by (2.9), with q = 1 and H ∈ (cid:0) , (cid:1) . Then, as T → ∞ , T − Z T (cid:0) Y t − E [ Y t ] (cid:1) dt law → a H σ H N, (2.11) where σ H is given by (1.6) and N ∼ N (0 , . Since T − R T (cid:0) Y t − E [ Y t ] (cid:1) dt (resp. T − H B HT ) belongs to the second (resp. first) Wienerchaos, we deduce from (2.11) and the seminal Peccati-Tudor criterion (see, e.g., [17, Theorem6.2.3]) that (cid:18) T − Z T (cid:0) Y t − E [ Y t ] (cid:1) dt, T − H B HT (cid:19) law → ( a H σ H N, N ′ ) , (2.12)where N, N ′ ∼ N (0 , are independent.Finally, in the critical case q = 1 and H = , we will need the following result, establishedin [13, Theorem 5.4]. Proposition 2.7
Let Y be given by (2.9), with q = 1 and H = . Then, as T → ∞ , ( T log T ) − Z T (cid:0) Y t − E [ Y t ] (cid:1) dt law → a N, (2.13) where N ∼ N (0 , . Similarly to (2.12) and for exactly the same reason, we actually have (cid:18) ( T log T ) − Z T (cid:0) Y t − E [ Y t ] (cid:1) dt, T − B T (cid:19) law → ( 2764 a N, N ′ ) , (2.14)where N, N ′ ∼ N (0 , are independent. 7 .5 A few other useful facts In this section, we let X be given by (1.2), with a > , b ∈ R and Z q,H a Hermite process oforder q > and Hurst parameter H ∈ ( , . We can write X t = h ( t ) + Y t , where h ( t ) = b (1 − e − at ) and Y t is given by (2.9). (2.15)The following limit, obtained as a consequence of the isometry property (2.7), will be usedmany times throughout the sequel: E [ Y T ] = H (2 H − Z [0 ,T ] e − a ( T − u ) e − a ( T − v ) | u − v | H − dudv = H (2 H − Z [0 ,T ] e − a u e − a v | u − v | H − dudv → H (2 H − Z [0 , ∞ ) e − a u e − a v | u − v | H − dudv = a − H H Γ(2 H ) < ∞ . (2.16)Identity (2.16) comes from (2 H − Z [0 , ∞ ) e − a ( t + s ) | t − s | H − dsdt = a − H (2 H − Z [0 , ∞ ) e − ( t + s ) | t − s | H − dsdt = a − H Γ(2 H ) , (2.17)see, e.g., Lemma 5.1 in Hu-Nualart [12] for the second equality. In particular, we note that E [ Y T ] = O (1) as T → ∞ . (2.18)Another simple but important fact that will be used is the following identity: Z T Y t dt = 1 a ( Z q,HT − Y T ) , (2.19)which holds true since Z T Y t dt = Z T (cid:18)Z t e − a ( t − u ) dZ q,Hu (cid:19) dt = Z T (cid:18)Z Tu e − a ( t − u ) dt (cid:19) dZ q,Hu = 1 a ( Z q,HT − Y T ) . We are now ready to prove Proposition 1.2.We start by showing that G T converges well in L (Ω) . In order to do so, we will check thatthe Cauchy criterion is satisfied. According to (2.8), we can write U T ( t ) = c ( H, q ) I q ( g T ( t, · )) ,where g T ( t, ξ , . . . , ξ q ) = Z t e − T ( t − v ) q Y j =1 ( v − ξ j ) H − + dv.
8s a result, we can write, thanks to [19, identity (3.25)],
Cov( U S ( s ) , U T ( t ) )= c ( H, q ) q X r =1 (cid:18) qr (cid:19) (cid:26) q ! k g S ( s, · ) ⊗ r g T ( t, · ) k + r ! (2 q − r )! k g S ( s, · ) e ⊗ r g T ( t, · ) k (cid:27) , implying in turn that E [ G T G S ]= ( ST ) q (1 − H )+2 H Z [0 , Cov( U S ( s ) , U T ( t ) ) dsdt = c ( H, q ) ( ST ) q (1 − H )+2 H q X r =1 (cid:18) qr (cid:19) q ! Z [0 , k g S ( s, · ) ⊗ r g T ( t, · ) k dsdt + c ( H, q ) ( ST ) q (1 − H )+2 H q X r =1 (cid:18) qr (cid:19) r ! (2 q − r )! Z [0 , k g S ( s, · ) e ⊗ r g T ( t, · ) k dsdt. To check the Cauchy criterion for G T , we are thus left to show the existence, for any r ∈{ , . . . , q } , of lim S,T →∞ ( ST ) q (1 − H )+2 H Z [0 , k g S ( s, · ) ⊗ r g T ( t, · ) k dsdt (3.1)and lim S,T →∞ ( ST ) q (1 − H )+2 H Z [0 , k g S ( s, · ) e ⊗ r g T ( t, · ) k dsdt. (3.2)Using that R R ( u − x ) H − + ( v − x ) H − + du = c H | v − u | H − with c H a constant depending onlyon H and whose value can change from one line to another, we have (cid:0) g S ( s, · ) ⊗ r g T ( t, · ) (cid:1) ( x , . . . , x q − r )= c H Z s Z t | v − u | (2 H − r e − S ( s − u ) e − T ( t − v ) q − r Y j =1 ( u − x j ) H − + 2 q − r Y j = q − r +1 ( v − y j ) H − + dudv. Now, let σ, γ be two permutations of S q − r , and write g S ( s, · ) ⊗ σ,r g T ( t, · ) to indicate thefunction ( x , . . . , x q − r ) (cid:0) g S ( s, · ) ⊗ r g T ( t, · ) (cid:1) ( x σ (1) , . . . , x σ (2 q − r ) ) . We can write, for some integers a , . . . , a satisfying a + a = a + a = q − r (and whoseexact value is useless in what follows), (cid:10) g S ( s, · ) ⊗ σ,r g T ( t, · ) , g S ( s, · ) ⊗ γ,r g T ( t, · ) (cid:11) = c H Z s Z t Z s Z t | v − u | (2 H − r | z − w | (2 H − r | u − w | (2 H − a ×| u − z | (2 H − a | v − w | (2 H − a | u − z | (2 H − a × e − S ( s − u ) e − T ( t − v ) e − S ( s − w ) e − T ( t − z ) dudvdwdz.
9e deduce that ( ST ) q (1 − H )+2 H Z [0 , (cid:10) g S ( s, · ) ⊗ σ,r g T ( t, · ) , g S ( s, · ) ⊗ γ,r g T ( t, · ) (cid:11) dsdt = c H ( ST ) q (1 − H )+2 H Z [0 , (cid:18)Z s Z t Z s Z t | v − u | (2 H − r | z − w | (2 H − r ×| u − w | (2 H − a | u − z | (2 H − a | v − w | (2 H − a | v − z | (2 H − a × e − S ( s − u ) e − T ( t − v ) e − S ( s − w ) e − T ( t − z ) dudvdwdz (cid:17) dsdt = c H ( ST ) q (1 − H )+2 H Z [0 , (cid:18)Z s Z t Z s Z t | v − u − t + s | (2 H − r | z − w + t − s | (2 H − r ×| u − w | (2 H − a | u − z + t − s | (2 H − a | v − w − t + s | (2 H − a ×| v − z | (2 H − a e − Su e − T v e − Sw e − T z dudvdwdz (cid:17) dsdt = c H S q (1 − H )(1+ a − q ) T q (1 − H )(1+ a − q ) × Z [0 , (cid:18)Z Ss Z T t Z Ss Z T t (cid:12)(cid:12)(cid:12) vT − uS − t + s (cid:12)(cid:12)(cid:12) (2 H − r (cid:12)(cid:12)(cid:12) zT − wS + t − s (cid:12)(cid:12)(cid:12) (2 H − r ×| u − w | (2 H − a (cid:12)(cid:12)(cid:12) uS − zT + t − s (cid:12)(cid:12)(cid:12) (2 H − a (cid:12)(cid:12)(cid:12) vT − wS − t + s (cid:12)(cid:12)(cid:12) (2 H − a ×| v − z | (2 H − a e − u e − v e − w e − z dudvdwdz (cid:17) dsdt. It follows that lim
S,T →∞ ( ST ) q (1 − H )+2 H Z [0 , (cid:10) g S ( s, · ) ⊗ σ,r g T ( t, · ) , g S ( s, · ) ⊗ γ,r g T ( t, · ) (cid:11) dsdt exists whatever r and a , . . . , a such that a + a = a + a = q − r . Note that this limit isalways zero, except when r = 1 , a = a = q − and a = a = 0 , in which case it is given by c H Z [0 , | t − s | H − dtds × Z R | u − w | (2 H − q − e − ( u + w ) dudw ! < ∞ . Since g S ( s, · ) e ⊗ r g T ( t, · ) = 1(2 q − r )! X σ ∈ S q − r g S ( s, · ) ⊗ σ,r g T ( t, · ) the existence of the two limits (3.1)-(3.2) follow, implying in turn the existence of G ∞ .Now, let us check the claim about the distribution of G ∞ . Let e Y t = U ( t ) , that is, e Y t = R t e − ( t − u ) dZ q,Hu , t > . By a scaling argument, it is straightforward to check that ( e Y tT ) t > = T H ( U T ( t )) t > for any fixed T > . As a result, T q (1 − H ) − Z T ( e Y t − E [ e Y t ]) dt = T q (1 − H ) Z ( e Y tT − E [ e Y tT ]) dt law = G T . Using (2.10), we deduce that G T /B H,q converges in law to the Rosenblatt distribution ofparameter − q (1 − H ) , hence the claim. 10 Proof of the consistency part in Theorem 1.3
The consistency part of Theorem 1.3 is directly obtained as a consequence of the followingtwo propositions.
Proposition 4.1
Let X be given by (1.1)-(1.2) with a > , b ∈ R , q > and H ∈ ( , . As T → ∞ , one has T Z T X t dt → b a.s. (4.1) Proof . We use (1.2) to write T Z T X t dt = bT Z T (1 − e − at ) dt + 1 T Z T Y t dt. Since it is straightforward that bT R T (1 − e − at ) dt → b , we are left to show that T R T Y t dt → almost surely.By (2.19), one can write, for any integer n > , E "(cid:18) n Z n Y t dt (cid:19) a n (cid:0) E [( Z q,Hn ) ] + E [ Y n ] (cid:1) = O ( n H − ) , where the last equality comes from the H -selfsimilarity property of Z q,H as well as (2.18).Since T R T Y t dt belongs to the q th Wiener chaos, it enjoys the hypercontractivity property(2.3). As a result, for all p > − H and λ > , ∞ X n =1 P (cid:18)(cid:12)(cid:12)(cid:12) n Z n Y t dt (cid:12)(cid:12)(cid:12) > λ (cid:19) λ p ∞ X n =1 E (cid:20)(cid:12)(cid:12)(cid:12) n Z n Y t dt (cid:12)(cid:12)(cid:12) p (cid:21) cst( p ) λ p ∞ X n =1 E (cid:20)(cid:16) n Z n Y t dt (cid:17) (cid:21) p/ cst( p ) λ p ∞ X n =1 n − (1 − H ) p < ∞ . We deduce from the Borel-Cantelli lemma that n R n Y t dt → almost surely as n → ∞ .Finally, fix T > and let n = ⌊ T ⌋ be its integer part. We can write T Z T Y t dt = 1 n Z n Y t dt + 1 T Z Tn Y t dt + (cid:18) T − n (cid:19) Z n Y t dt. (4.2)We have just proved above that n R n Y t dt tends to zero almost surely as n → ∞ . We nowconsider the second and third terms in (4.2). We have, almost surely as T → ∞ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) T − n (cid:19) Z n Y t dt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) − nT (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) n Z n Y t dt (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) n Z n Y t dt (cid:12)(cid:12)(cid:12)(cid:12) → , and (cid:12)(cid:12)(cid:12)(cid:12) T Z Tn Y t dt (cid:12)(cid:12)(cid:12)(cid:12) n Z n +1 n | Y t | dt.
11o conclude, it remains to prove that n R n +1 n | Y t | dt → almost surely as n → ∞ . Using(2.18) we have, for all fixed λ > , P (cid:26) n Z n +1 n | Y t | dt > λ (cid:27) λ E (cid:20)(cid:18) n Z n +1 n | Y t | dt (cid:19) (cid:21) λ n Z n +1 n Z n +1 n p E [ Y s ] q E [ Y t ] dsdt = O ( n − ) . Hence, as n → ∞ , the Borel-Cantelli lemma applies and implies that n R n +1 n | Y t | dt goes tozero almost surely. This completes the proof of (4.1). Proposition 4.2
Let X be given by (1.1)-(1.2) with a > , b ∈ R , q > and H ∈ ( , . As T → ∞ , one has T Z T X t dt → b + a − H H Γ(2 H ) a.s. (4.3) Proof . We first use (1.2) to write T Z T X t dt = 1 T Z T h ( t ) dt + 2 T Z T h ( t ) Y t dt + 1 T Z T Y t dt. We now study separately the three terms in the previous decomposition. More precisely wewill prove that, as T → ∞ , T Z T h ( t ) dt → b , (4.4) T Z T h ( t ) Y t dt → a.s. (4.5) T Z T Y t dt → a − H H Γ(2 H ) a.s. , (4.6)from which (4.3) follows immediately.First term. By Lebesgue dominated convergence, one has T Z T h ( t ) dt = Z h ( T t ) dt = b Z (1 − e − aT t ) dt → b , that is, (4.4) holds.Second term. First, we claim that T − H Z T h ( t ) Y t dt law → ba Z q,H . (4.7)Indeed, let us decompose: Z T h ( t ) Y t dt = b Z T (1 − e − at ) Y t dt = b Z T Y t dt − b Z T e − at Y t dt. Z T e − at Y t dt = Z T e − at (cid:18)Z t e − a ( t − s ) dZ q,Hs (cid:19) dt = Z T (cid:18)Z Ts e − a (2 t − s ) dt (cid:19) dZ q,Hs = 12 a Z T ( e − a (2 T − s ) − e − as ) dZ q,Hs = 12 a (cid:18) e − aT Y T − Z T e − as dZ q,Hs (cid:19) → − a Z ∞ e − as dZ q,Hs in L (Ω) as T → ∞ . (4.8)The announced convergence (4.7) is a consequence of (2.19), (4.8) and the selfsimilarity of Z q,H . Now, relying on the Borel-Cantelli lemma and the fact that R T h ( t ) Y t dt enjoys thehypercontractivity property, it is not difficult to deduce from (4.7) that (4.5) holds.Third term. Firstly, let us write, as T → ∞ , T Z T E [ Y t ] dt = H (2 H −
1) 1 T Z T dt Z t Z t dudve − au e − av | u − v | H − = H (2 H − Z dt Z T t Z T t dudve − au e − av | u − v | H − −→ H (2 H − Z ∞ Z ∞ dudve − au e − av | u − v | H − = a − H H Γ(2 H ) . (4.9)To conclude the proof of (4.6), we are thus left to show that : T Z T ( Y t − E [ Y t ]) dt → a.s. (4.10)First, we claim that, as n ∈ N ∗ goes to infinity, G n := 1 n Z n ( Y t − E [ Y t ]) dt → a.s. (4.11)Indeed, for all fixed λ > and p > we have, by the hypercontractivity property (2.4) for G n belonging to a finite sum of Wiener chaoses, P {| G n | > λ } λ p E [ | G n | p ] cst ( p ) λ p E [ G n ] p/ . If ( q > and H > ) or q > , combining (2.10) with, e.g., [18, Lemma 2.4] leads to sup T > E (cid:20)(cid:18) T q (1 − H ) − Z T ( Y t − E [ Y t ]) dt (cid:19) (cid:21) < ∞ (4.12)(note that one could also prove (4.12) directly), implying in turn that P {| G n | > λ } = O ( n − pq (1 − H ) ); choosing p so that pq (1 − H ) > leads to P ∞ n =1 P {| G n | > λ } < ∞ , andso our claim (4.11) follows from Borel-Cantelli lemma.13f q = 1 and H < , the same reasoning (but using this time (2.11) instead of (2.10)) leadsexactly to the same conclusion (4.11).Now, fix T > and consider its integer part n = ⌊ T ⌋ . One has G T = G n + 1 T Z Tn ( Y t − E [ Y t ]) dt + (cid:18) T − n (cid:19) Z n ( Y t − E [ Y t ]) dt. (4.13)We have just proved above that G n tends to zero almost surely as n → ∞ . We now considerthe third term in (4.13). We have, using (4.11): (cid:12)(cid:12)(cid:12)(cid:12) T − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z n ( Y t − E [ Y t ]) dt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) − nT (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) n Z n ( Y t − E [ Y t ]) dt (cid:12)(cid:12)(cid:12)(cid:12) | G n | → a.s . Finally, as far as the second term in (4.13) is concerned, we have (cid:12)(cid:12)(cid:12)(cid:12) T Z Tn ( Y t − E [ Y t ]) dt (cid:12)(cid:12)(cid:12)(cid:12) n Z n +1 n | Y t − E [ Y t ] | dt. To conclude, it thus remains to prove that, as n → ∞ , F n := 1 n Z n +1 n | Y t − E [ Y t ] | dt → a.s . (4.14)By hypercontractivity and (2.16), one can writeVar ( Y t ) cst ( q )( E [ Y t ]) cst ( q ) a − H H Γ(2 H ) . Thus, sup t Var ( Y t ) < ∞ , and it follows that E [ F n ] = 1 n Z n +1 n Z n +1 n E (cid:2)(cid:12)(cid:12) Y t − E [ Y t ] (cid:12)(cid:12)(cid:12)(cid:12) Y s − E [ Y s ] (cid:12)(cid:12)(cid:3) dsdt = O ( n − ) . Hence P ∞ n =1 P {| F n | > λ } P ∞ n =1 1 λ E [ F n ] < ∞ for all λ > , and Borel-Cantelli lemma leadsto (4.14) and concludes the proof of (4.6). We now turn to the proof of the part of Theorem 1.3 related to fluctuations. We start withthe fluctuations of b b T , which are easier compared to b a T .Fluctuations of b b T . Using first (2.15) and then (2.18) and (2.19), we can write T − H (cid:8)b b T − b } = T − H (cid:26) T Z T Y t dt − bT Z T e − at dt (cid:27) = Z q,HT aT H + O ( T − H ) , (5.1)which will be enough to conclude, see the end of the present section.Fluctuations of b a T . As a preliminary step, we first concentrate on the asymptotic behavior,as T → ∞ , of the random quantity ℓ T := α T − a − H H Γ(2 H ) , α T is given by (1.3). Since X t = h ( t ) + Y t , see (2.15), we have ℓ T = A T + B T + 2 C T + D T − E T − E T F T − F T , where A T = 1 T Z T ( Y t − E [ Y t ]) dt, B T = 1 T Z T E [ Y t ] dt − a − H H Γ(2 H ) C T = 1 T Z T Y t h ( t ) dt, D T = 1 T Z T h ( t ) dt, E T = 1 T Z T Y t dt, F T = 1 T Z T h ( t ) dt. We now treat each of these terms separately.
Term B T . Recall from (2.16) that, as T → ∞ , E [ Y T ] = H (2 H − Z [0 ,T ] e − a ( u + v ) | u − v | H − dudv → H (2 H − Z [0 , ∞ ) e − a ( u + v ) | u − v | H − dudv = a − H H Γ(2 H ) . As a result, | B T | = (cid:12)(cid:12)(cid:12)(cid:12) T Z T E [ Y t ] dt − a − H H Γ(2 H ) (cid:12)(cid:12)(cid:12)(cid:12) H (2 H − T Z T dt Z [0 , ∞ ) \ [0 ,t ] dudv e − a ( u + v ) | u − v | H − H (2 H − T Z T dt Z ∞ t dv e − av Z ∞ du e − au { v > u } ( v − u ) H − H (2 H − T Z ∞ dt Z ∞ t dv e − av Z v du u H − = 2 HT Z ∞ dt Z ∞ t dv e − av v H − = 2 HT Z ∞ e − av v H dv = O ( 1 T ) . Term C T . We can write C T = 1 T Z T Y t h ( t ) dt = bT Z T (1 − e − at ) Y t dt = b (cid:18) T Z T Y t dt − T Z T e − at Y t dt (cid:19) . But T Z T e − at Y t dt = 1 T Z T e − at (cid:18)Z t e − a ( t − s ) dZ q,Hs (cid:19) dt = 1 T Z T e as (cid:18)Z Ts e − at dt (cid:19) dZ q,Hs = 12 aT (cid:18)Z T e − as dZ q,Hs − e − aT Y T (cid:19) . Using (2.18) and R T e − as dZ q,Hs → R ∞ e − as dZ q,Hs in L (Ω) , we deduce that C T = b E T + O ( 1 T ) . Term D T . It is straightforward to check that 15 T = 1 T Z T h ( t ) dt = b T Z T (1 − e − at ) dt = b + O ( 1 T ) . Term E T . Thanks to (2.18) and (2.19), we have E T = 1 T Z T Y T dt = 1 a T ( Z q,HT − Y T ) = Z q,HT a T + O ( 1 T ) . Since Z q,HT law = T H Z q,H by selfsimilarity, we deduce E T = ( Z q,HT ) a T + O ( T H − ) . Term F T . Similarly to D T , it is straightforward to check that F T = 1 T Z T h ( t ) dt = bT Z T (1 − e − at ) dt = b + O ( 1 T ) . Combining everything together, we eventually obtain that ℓ T = A T − ( Z q,HT ) a T + O ( T − ) . (5.2)Fluctuations of ( b a T , b b T ) . • We consider first the case ( q = 1 and H > ) or ( q > . Since Z q,H satisfies the scaling property, we can write (cid:16) T q (1 − H ) A T , T − H Z q,HT (cid:17) law = (cid:16) a − q (1 − H ) − H G aT , Z q,H (cid:17) , and we deduce from the L -convergence of G T (see Proposition 1.2) that (cid:16) T q (1 − H ) A T , T − H Z q,HT (cid:17) law → (cid:16) a − q (1 − H ) − H G ∞ , Z q,H (cid:17) . (5.3)On the other hand, a Taylor expansion yields T q (1 − H ) { b a T − a } = T q (1 − H ) a "(cid:18) a H ℓ T H Γ(2 H ) (cid:19) − H − = − a H H Γ(2 H ) T q (1 − H ) A T − T q (1 − H ) − ( Z q,HT ) a ! + o (1) , implying in turn (cid:16) T q (1 − H ) { b a T − a } , T − H (cid:8)b b T − b } (cid:17) = − a H H Γ(2 H ) " T q (1 − H ) A T − T q − − H ) ( T − H Z q,HT ) a , T − H Z q,HT a ! + o (1) ,
16o that, from (5.3), (cid:16) T q (1 − H ) { b a T − a } , T − H (cid:8)b b T − b } (cid:17) law → (cid:18) − a − q (1 − H ) H Γ(2 H ) G ∞ , Z q,H a (cid:19) if q > (cid:16) − a H − H Γ(2 H ) ( G ∞ − ( B H ) ) , B H a (cid:17) if q = 1 and H > , as claimed. • Assume now that q = 1 and H < , and let us write B H instead of Z ,H for simplicity.We deduce from (5.2) and T − ( B HT ) = T H − ( B H ) that √ T ℓ T = √ T A T + o (1) , so that (cid:16) √ T { b a T − a } , T − H (cid:8)b b T − b } (cid:17) = (cid:18) − a H H Γ(2 H ) √ T A T , T − H B HT a (cid:19) + o (1) , implying in turn by (2.12) that (cid:16) √ T { b a T − a } , T − H (cid:8)b b T − b } (cid:17) → (cid:18) − a H σ H H Γ(2 H ) N, N ′ a (cid:19) , where N, N ′ ∼ N (0 , are independent, as claimed. • Finally, we consider the case q = 1 and H = . We deduce again from (5.2) and T − ( B / T ) = T − ( B / ) that s T log T ℓ T = s T log T A T + o (1) , so that, using (2.14), s T log T { b a T − a } , T (cid:8)b b T − b } ! = − a √ π s T log T A T , T − B HT a ! + o (1) → (cid:18) r aπ N, N ′ a (cid:19) , where N, N ′ ∼ N (0 , are independent. Acknowledgment . We gratefully acknowledge an anonymous referee for a very construc-tive report, which led to a significant improvement of the paper, especially its link with thefinancial literature.
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