MSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania MSTA54 10.15559/16-MSTA54 Research Article Large deviations for drift parameter estimator of mixed fractional Ornstein–Uhlenbeck process MarushkevychDmytrodmmar1992@gmail.com Laboratoire Manceau de Mathématiques, Faculté des Sciences et Techniques, Université du Maine, Avenue Olivier Messiaen, 72085, Le Mans, France 2016 76201632107117 552016 1652016 1652016 © 2016 The Author(s). Published by VTeX2016 Open access article under the CC BY license.

We investigate large deviation properties of the maximum likelihood drift parameter estimator for Ornstein–Uhlenbeck process driven by mixed fractional Brownian motion.

 Large deviations Ornstein–Uhlenbeck process mixed fractional Brownian motion maximum likelihood estimator 60G15 62F12 60G22 Introduction

Our purpose is to establish large deviations principle for the maximum likelihood estimator of drift parameter of the Ornstein–Uhlenbeck process driven by a mixed fractional Brownian motion: dXt=−ϑXtdt+dB˜t,t∈[0,T],T>0, 0,\]]]> where the initial state X0=0 , and the drift parameter ϑ is strictly positive. The process B˜ is a mixed fractional Brownian motion B˜t=Bt+BtH,t∈[0,T], where B=(Bt) is a Brownian motion, and BH=(BtH) is an independent fractional Brownian motion with Hurst exponent H∈(0,1] , that is, the centered Gaussian process with covariance function R(s,t)=EBtHBsH=12(t2H+s2H−|t−s|2H),s,t∈[0,T].

It is important to notice that the parameter H is considered to be known. The problem of Hurst parameter estimation is not considered in this work but presents a great interest for the future research.

Two following chapters contain information about maximum likelihood estimation procedure for the mixed fractional Ornstein–Uhlenbeck process and description of basic concepts of large deviations theory.

The formulation of our main results and their proofs are given in Section 4, whereas Section 5 contains auxiliary results.

 Maximum likelihood estimation procedure

The interest to mixed fractional Brownian motion was triggered by Cheridito [5]. Further, a mixed fractional Brownian motion and related models were comprehensively considered by Mishura [12]. Finally, the results of recent works of Cai, Kleptsyna, and Chigansky [4] and Chigansky and Kleptsyna [6] concerning the new canonical representation of mixed fractional Brownian motion present a great value for the purposes of this paper.

An interesting change in properties of a mixed fractional Brownian motion  B˜ occurs depending on the value of H. In particular, it was shown (see [5]) that B˜ is a semimartingale in its own filtration if and only if either H=12 or H∈(34,1] .

The main contribution of paper [4] is a novel approach to the analysis of mixed fractional Brownian motion based on the filtering theory of Gaussian processes. The core of this method is a new canonical representation of B˜ .

In fact, there is an integral transformation that changes the mixed fractional Brownian motion to a martingale. In particular (see [4]), let g(s,t) be the solution of the following integro-differential equation g(s,t)+Hdds∫0tg(r,t)|s−r|2H−1sign(s−r)dr=1,0<s<t≤T. Then the process Mt=∫0tg(s,t)dB˜s,t∈[0,T], is a Gaussian martingale with quadratic variation ⟨M⟩t=∫0tg(s,t)ds,t∈[0,T].

Moreover, the natural filtration of the martingale M coincides with that of the mixed fractional Brownian motion B˜ .

Further, to what has just been mentioned concerning the mixed fractional Brownian motion, an auxiliary semimartingale, appropriate for the purposes of statistical analysis, can be also associated to the corresponding Ornstein–Uhlenbeck process X defined by (1). In particular, for the martingale M defined by (3), the sample paths of the process X are smooth enough in the sense that the following process is well defined: Qt=dd⟨M⟩t∫0tg(s,t)Xsds. We also define the process Z=(Zt,t∈[0,T]) by Zt=∫0tg(s,t)dXs.

One of the most important results of [4] is that the process Z is the fundamental semimartingale associated to X in the following sense.

Let g(s,t) be the solution of (2), and the process Z be defined by (5). Then the following assertions hold:

Z is a semimartingale with the decomposition Zt=−ϑ∫0tQsd⟨M⟩s+Mt, where Mt is the martingale defined by (3).

X admits the representation Xt=∫0tgˆ(s,t)dZs,

where gˆ(s,t)=1−dd⟨M⟩s∫0tg(r,s)dr.

The natural filtrations (Xt) and (Zt) of X and Z, respectively, coincide.

In addition, it was shown by Chigansky and Kleptsyna [6] that the process Q admits the following representation: Qt=∫0tψ(s,t)dZs=12ψ(t,t)Zt+12∫0tψ(s,s)dZs,t∈[0,T], with ψ(s,t)=12(dtd⟨M⟩t+dsd⟨M⟩s).

The specific structure of the process Q allows us to determine the likelihood function for (1), which according to Corollary 2.9 in [4] equals LT(ϑ,X)=dμXdμB˜(X)=exp(−ϑ∫0TQtdZt−12ϑ2∫0TQt2d⟨M⟩t), where μX and μB˜ are the probability measures induced by the processes X and B˜ , respectively. Thus, the score function for (1), that is, the derivative of the log-likelihood function from observations over the interval [0,T] is given by ΣT(θ)=−∫0TQtdZt−ϑ∫0TQt2d⟨M⟩t, which allows us to determine the maximum likelihood estimator for the drift parameter ϑ. Moreover, according to Theorem 2.9 in [6], which is also presented further, the maximum likelihood estimator is asymptotically normal.

Let g(s,t) be the solution of (2), and let the processes Q and Z be defined by (4) and (5), respectively. The maximum likelihood estimator of ϑ is given by ϑˆT(X)=−∫0TQtdZt∫0TQt2d⟨M⟩t. Since ϑ>0 0$]]>, this estimator is asymptotically normal at the usual rate: T(ϑˆT(X)−ϑ) →T→∞dN(0,2ϑ).

We will develop this result by proving the large deviation principle for the maximum-likelihood estimator (9).

 Large deviation principle

The large deviations principle characterizes the limiting behavior of a family of random variables (or corresponding probability measures) in terms of a rate function.

A rate function I is a lower semicontinuous function I:R→[0,+∞] such that, for all α∈[0,+∞) , the level sets {x:I(x)≤α} are closed subsets of R . Moreover, I is called a good rate function if its level sets are compacts.

We say that a family of real random variables (ZT)T>0 0}$]]> satisfies the large deviation principle with rate function I:R→[0,+∞] if for any Borel set Γ⊂R , −infx∈ΓoI(x)≤lim infT→∞1TlogP(ZT∈Γ)≤lim supT→∞1TlogP(ZT∈Γ)≤−infx∈Γ‾I(x), where Γo and Γ‾ denote the interior and closure of Γ, respectively. Note that a family of random variables can have at most one rate function associated with its large deviation principle (for the proof, we refer the reader to the book by Dembo and Zeitouni [7]). Moreover, it is obvious that if (ZT)T>0 0}$]]> satisfies the large deviation principle and a Borel set Γ⊂R is such that infx∈ΓoI(x)=infx∈Γ‾I(x), then limT→∞1TlogP(ZT∈Γ)=−infx∈ΓI(x).

We shall prove the large deviation principle for a family of maximum likelihood estimators (9) via a similar approach as that of [1] and [3] for an Ornstein–Uhlenbeck process and fractional Ornstein–Uhlenbeck process, respectively.

In order to prove the large deviations principle for the drift parameter estimator of mixed fractional Ornstein–Uhlenbeck process (1), the main tool is the normalized cumulant generating function of arbitrary linear combination of ∫0TQtdZt and ∫0TQt2d⟨M⟩t , LT(a,b)=1TlogE[exp(ZT(a,b))], where, for any (a,b)∈R2 , ZT(a,b)=a∫0TQtdZt+b∫0TQt2d⟨M⟩t.

Note that, for some (a,b)∈R2 , the expectation in (10) may be infinite. In fact, in order to establish a large deviation principle for ϑˆT it suffices to find the limit of LT(a,b) as T→∞ and apply the following lemma, which is a consequence of the Gärtner–Ellis theorem (Theorem 2.3.6 in [7]).

For a family of maximum likelihood estimators (ϑT)T>0 0}$]]>, let the function LT(a,b) be defined by (10), and, for each fixed value of x, let Δx denote the set of a∈R for which limT→∞LT(a,−xa) exists and is finite. If Δx is not empty for each value of x, then (ϑT)T>0 0}$]]> satisfies the large deviation principle with a good rate function I(x)=−infa∈ΔxlimT→∞LT(a,−xa).

Main results

The maximum likelihood estimator ϑˆT defined by (9) satisfies the large deviation principle with the good rate function I(x)=−(x+ϑ)24xifx<−ϑ3,2x+ϑifx≥−ϑ3.

As it was mentioned in the previous section, in order to establish the large deviation principle for ϑˆT and determine the corresponding good rate function, it is necessary to find the limit L(a,b)=limT→∞LT(a,b) and determine the set of (a,b)∈R2 for which this limit is finite.

For arbitrary φ∈R , consider the Doleans exponential of (φ+ϑ)∫0tQsdMs , Λφ(t)=exp((φ+ϑ)∫0tQsdMs−(φ+ϑ)22∫0tQs2d⟨M⟩s).

Note that (1ψ(t,t)Qt)t≥0 is a Gaussian process whose mean and variance functions are bounded on [0,T] . Thus, Λφ satisfies the conditions of Girsanov’s theorem in accordance with Example 3 of paragraph 2 of Section 6 in [11], and we can apply a usual change of measures and consider the new probability Pφ defined by the local density dPφdP=Λφ(T)=exp((φ+ϑ)∫0TQtdMt−(φ+ϑ)22∫0TQt2d⟨M⟩t).

Observe that, due to (6), Λφ(T) can be rewritten in terms of the fundamental semimartingale dPφdP=Λφ(T)=exp((φ+ϑ)∫0TQtdZt+(φ+ϑ)ϑ∫0TQt2d⟨M⟩t−(φ+ϑ)22∫0TQt2d⟨M⟩t)=exp((φ+ϑ)∫0TQtdZt−φ2−ϑ22∫0TQt2d⟨M⟩t). Consequently, we can rewrite LT(a,b) as LT(a,b)=1TlogE[exp(ZT(a,b))]=1TlogEφ[exp(ZT(a,b))Λφ(T)−1]=1TlogEφexp((a−φ−ϑ)∫0TQtdZt+12(2b−ϑ2+φ2)∫0TQt2d⟨M⟩t). Given an arbitrary real number φ, we can choose φ=a−ϑ . Then LT(a,b)=1TlogEφexp(12(2b−ϑ2+(a−ϑ)2)∫0TQt2d⟨M⟩t) or, denoting μ=−12(2b−ϑ2+(a−ϑ)2) , LT(a,b)=1TlogEφexp(−μ∫0TQt2d⟨M⟩t). As it was mentioned before, the expectation in (13) can be infinite for some combinations of μ and φ. Our purpose is to determine the set of (φ,μ)∈R2 for which this expectation and limit (12) are finite. According to Girsanov’s theorem, under Pφ , the process Mt−(φ+ϑ)∫0tQsd⟨M⟩s=Zt−φ∫0tQsd⟨M⟩s has the same distribution as M under P . Consequently, applying the inverse integral transformation (7) to (14), we get that, under Pφ , the process Xt−φ∫0tXsds is a mixed fractional Brownian motion.

Under the new probability measure Pφ , the process X is a mixed fractional Ornstein–Uhlenbeck process with drift parameter −φ . Consequently, in order to find the limit of (13) as T→∞ , we can apply Lemma 1, which is presented in Section 5. Thus, we have the equality L(a,b)=−φ2−φ24+μ2=−12(a−ϑ+ϑ2−2b), and convergence (12) holds for μ>−φ22 -\frac{{\varphi }^{2}}{2}$]]>, which gives ϑ2−2b>0 0$]]>.

For x∈R , denote the function Lx(a)=L(a,−xa)=−12(a−ϑ+ϑ2+2xa) defined on the set Δx={a∈R|ϑ2+2xa>0}. 0\big\}.\]]]> Then, according to (11), the rate function I(x) for the maximum likelihood estimator ϑˆT can be found as I(x)=−infa∈ΔxLx(a). Consequently, straightforward calculations of this infimum finish the proof of the theorem.  □

Observe that the rate function I(x) does not depend on the parameter H. Hence, ϑˆT shares the same large deviation principles as those established by Florens-Landais and Pham [8] for a standard Ornstein–Uhlenbeck process and by Bercu, Coutin, and Savy [3] for a fractional Ornstein–Uhlenbeck process (see also [2, 9]).

 Auxiliary results

We can observe that the following lemma plays a key role in the proof of Theorem 3.

For a mixed fractional Ornstein–Uhlenbeck process X with drift parameter ϑ, we have the following limit: KT(μ)=1TlogEexp(−μ∫0TQt2d⟨M⟩t)→ϑ2−ϑ24+μ2,T→∞, for all μ>−ϑ22 -\frac{{\vartheta }^{2}}{2}$]]>.

We shall prove the lemma using an approach similar to that in [6]. Denote Vt=∫0tψ(s,s)dZs . Then, according to (8), we can rewrite dZt=−ϑQtd⟨M⟩t+dMt=−ϑ2ψ(t,t)Ztd⟨M⟩t−ϑ2Vtd⟨M⟩t+dMt=−ϑ2Ztdt−ϑ2Vt1ψ(t,t)dt+1ψ(t,t)dWt, where Wt is a Brownian motion. Consequently, we get dVt=ψ(t,t)dZt=−ϑ2ψ(t,t)Ztdt−ϑ2Vtdt+ψ(t,t)dWt.

The Gaussian vector ζt=(Zt,Vt)T is a solution of the linear system of the Itô stochastic differential equation dζt=−ϑ2A(t)ζtdt+b(t)dWt, where A(t)=11ψ(t,t)ψ(t,t)1andb(t)=1ψ(t,t)ψ(t,t). Moreover, KT(μ) in (15) can be rewritten as KT(μ)=1TlogEexp(−μ∫0TQt2d⟨M⟩t)=1TlogEexp(−μ4∫0T(ψ(t,t)Zt+Vt)2d⟨M⟩t)=1TlogEexp(−μ4∫0T(ψ(t,t)Zt+1ψ(t,t)Vt)2dt)=1TlogEexp(−μ4∫0TζtTR(t)ζtdt), where R(t)=ψ(t,t)111ψ(t,t). By the Cameron–Martin-type formula from Section 4.1 of [10], KT(μ)=−μ4T∫0Ttr(Γ(t)R(t))dt, where Γ(t) is the solution of the equation Γ(t)˙=−ϑ2A(t)Γ(t)−ϑ2Γ(t)AT(t)−μ2Γ(t)R(t)Γ(t)+B(t) with B(t)=b(t)bT(t) and initial condition Γ(0)=0 .

We shall search solution of (16) as the ratio Γ(t)=Ψ1−1(t)Ψ2(t) , where Ψ1(t) and Ψ2(t) are the solutions of the following equation system: Ψ˙1(t)=ϑ2Ψ1(t)A(t)+μ2Ψ2(t)R(t),Ψ˙2(t)=Ψ1(t)B(t)−ϑ2Ψ2(t)AT(t), with initial conditions Ψ1(0)=I and Ψ2(0)=0 . From the first equation of (17) we get Ψ1−1(t)Ψ˙1(t)=ϑ2A(t)+μ2Γ(t)R(t), and since tr A(t)=2 , it follows that μ2tr(Γ(t)R(t))=tr(Ψ1−1(t)Ψ˙1(t))−ϑ. Since Ψ˙1(t)=Ψ1(t)(Ψ1−1(t)Ψ˙1(t)) , by Liouville’s formula we have −μ4T∫0Ttr(Γ(t)R(t))dt=−12T∫0Ttr (Ψ1−1(t)Ψ˙1(t))dt+ϑ2=−12TlogdetΨ1(T)+ϑ2. In order to calculate limT→∞1TlogdetΨ1(T) , define the matrix J=0110 and note that A(t)T=JA(t)J , R(t)=JA(t) , and B(t)=A(t)J . Setting Ψ˜2(t)=Ψ2(t)J , from (17) we obtain the following equation system: Ψ˙1(t)=ϑ2Ψ1(t)A(t)+μ2Ψ2(t)A(t),Ψ˜˙2(t)=Ψ1(t)A(t)−ϑ2Ψ˜2(t)A(t), with initial conditions Ψ1(0)=I and Ψ˜2(0)=0 . When ϑ22+μ>0 0$]]>, the coefficient matrix of system (18) ϑ2μ21−ϑ2 has two real eigenvalues ±λ with λ=ϑ24+μ2 and eigenvectors v±=ϑ2±λ1. Denote a±=ϑ2±λ=ϑ2±ϑ24+μ2 . Diagonalizing system (18), we get Ψ1(t)=a+Υ1(t)+a−Υ2(t), where Υ1(t) and Υ2(t) are the solutions of the equations Υ˙1(t)=λΥ1(t)A(t),Υ˙2(t)=−λΥ2(t)A(t), with initial conditions Υ1(0)=12λI and Υ2(0)=−12λI . Denote the matrix M(T)=Υ2−1(T)Υ1(T) , which is the solution of the equation M˙(t)=λ(A(t)M(t)+M(t)A(t)) subject to initial condition M(0)=−I . Then 1TlogdetΨ1(T)=1Tlogdet(a+Υ1(T)+a−Υ2(T))=1Tlogdet(a−Υ2(T))+1Tlogdet(I+a+a−M(T))=1Tlogdet(a−Υ2(T))+1Tlog(1+(a+a−)2detM(T)+a+a−tr M(T))=1Tlogdet(a−Υ2(T))+1Tlog(1+(a+a−)2detM(T))+1Tlog(1+a+a−tr M(T)1+(a+a−)2detM(T)). Applying Liouville’s formula to (19), we get 1Tlogdet(a−Υ2(t))+1Tlog(1+(a+a−)2detM(T))=1Tlog((a−2λ)2exp(−2λT))+1Tlog(1+(a+a−)2exp(4λT))→2λ as T→∞ . Thus, in order to prove that limit (15) holds, we should show that 1Tlog(1+a+a−tr M(T)1+(a+a−)2exp(4λT))→0,T→∞. Given (20), by Theorem 3 in [13] we have |tr M(T)|≤22exp(2λT). Thus, limit (21) holds, which finishes the proof of the lemma.  □

 Acknowledgments

My deepest gratitude is to my advisor Marina Kleptsyna, who proposed me to consider the problem presented in this paper and whose advices and help were very important. I also would like to thank the reviewers whose remarks and advices allowed me to significantly improve this article.

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