# Drift Estimation for a Lévy-Driven Ornstein-Uhlenbeck Process with Heavy Tails

We consider the problem of estimation of the drift parameter of an ergodic Ornstein–Uhlenbeck type process driven by a Lévy process with heavy tails. The process is observed continuously on a long time interval [0,T], T→∞. We prove that the statistical model is locally asymptotic mixed normal and the maximum likelihood estimator is asymptotically efficient.

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04/07/2022

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## 1 Introduction, motivation, previous results

In his paper, we deal with an estimation of the drift parameter of an ergodic one-dimensional Ornstein–Uhlenbeck process driven by a Lévy process:

 Xt=X0−θ∫t0Xsds+Zt,t≥0. (1.1)

The process

is a one-dimensional Lévy process with known characteristics and with infinite variance. The process

is observed continuously on a long time interval , . The problem is to study asymptotic properties of the corresponding statistical model and to show that the maximum likelihood estimator of is asymptotically efficient in an appropriate sense. Although the continuous time observations are far from being realistic in applications, they are of theoretical importance since they can be considered as a limit of high frequency discrete models.

Since we deal with continuous observations, it is natural to assume that the Gaussian component of the Lévy process is not degenerate. In this case, the laws of observations corresponding to different values of are equivalent and the likelihood ratio has an explicit form.

There are a lot of papers devoted to inference for Lévy driven SDEs. Most of the literature treats the case of discrete time observations both in the high and low frequency setting. A general theory for the likelihood inference for continuously observed jump-diffusions can be found in Sørensen (1991).

A complete analysis of the drift estimation for continuously observed ergodic and non-ergodic Ornstein–Uhlenbeck process driven by a Brownian motion can be found in (Höpfner, 2014, Chapter 8.1).

For continuously observed square integrable Lévy driven Ornstein–Uhlenbeck processes, the local asymptotic normality (LAN) of the model and the asymptotic efficiency of the maximum likelihood estimator of the drift have been derived by Mai (2012, 2014) with the help of the theory of exponential families, see Küchler and Sørensen (1997).

High frequency estimation of a square integrable Lévy driven Ornstein–Uhlenbeck process with non-vanishing Gaussian component has been performed by Mai (2012, 2014). Kawai (2013)

studied the asymptotics of the Fisher information for three characterizing parameters of Ornstein–Uhlenbeck processes with jumps under low frequency and high frequency discrete sampling. The existence of all moments of the Lévy process was assumed.

Tran (2017) considered the ergodic Ornstein–Uhlenbeck process driven by a Brownian motion and a compensated Poisson process, whose drift and diffusion coefficients as well as its jump intensity depend on unknown parameters. He obtained the LAN property of the model in the high frequency setting.

We also mention the works by Hu and Long (2007, 2009a, 2009b); Long (2009); Zhang and Zhang (2013) devoted to the least-square estimation of parameters of the Ornstein–Uhlenbeck process driven by an -stable Lévy process.

There is vast literature devoted to parametric inference for discretely observed Lévy processes (see, e.g. a survey by Masuda (2015)) and Lévy driven SDEs. More results on the latter topic can be found e.g. in Masuda (2013); Ivanenko and Kulik (2014); Kohatsu-Higa et al. (2017); Masuda (2019); Uehara (2019); Clément and Gloter (2015); Clément et al. (2019); Clément and Gloter (2019); Nguyen (2018); Gloter et al. (2018) and the references therein.

In this paper, we fill the gap and analyse continuously observed ergodic Ornstein–Uhlenbeck process driven by a Lévy process with heavy regularly varying tails of the index , , in the presence of a Gaussian component. It turns out that the log-likelihood in this model is quadratic, however the model is not asymptotically normal and we prove only the local asymptotic mixed normality (LAMN) property. We refer to Le Cam and Yang (2000); Höpfner (2014) for the general theory of estimation for LAMN models.

The fact that the prelimiting log-likelihood is quadratic automatically implies that the maximum likelihood estimator is asymptotically efficient in the sense of Jeganathan’s convolution theorem and attains the local asymptotic minimax bound. Another feature of our model is that the asymptotic observed information has spectrally positive -stable distribution. This implies that the limiting law of the maximum likelihood estimator has tails of the order and hence finite moments of all orders.

Acknowledgements: The authors thank the DAAD exchange programme Eastern Partnership for financial support. A.G. thanks Friedrich Schiller University Jena for hospitality.

## 2 Setting and the main result

Consider a stochastic basis , being right-continuous. Let be a Lévy process with the characteristic triplet and the Lévy–Itô decomposition

 Zt=σWt+bt+∫t0∫|z|≤1z~N(dz,ds)+∫t0∫|z|>1zN(dz,ds), (2.1)

where is a standard one-dimensional Brownian motion, is a Poissonian random measure on with the Lévy measure satisfying , is the compensated Poissonian random measure, and .

For , let be an Ornstein–Uhlenbeck type process being a solution of the SDE

 Xt=X0−θ∫t0Xsds+Zt,t≥0, (2.2)

where is an unknown parameter. The initial value

is a random variable whose distribution does not depend on

. Note that has an explicit representation

 Xt=X0e−θt−∫t0e−θ(t−s)dZs,t≥0, (2.3)

see, e.g. (Applebaum, 2009, Sections 4.3.5 and 6.3) and (Sato, 1999, Section 17).

Let be the space of real-valued càdlàg functions equipped with Skorokhod topology and Borel -algebra . The space is Polish, and coincides with the -algebra generated by the coordinate projections. We define a (right-continuous) filtration consisting of -algebras

 Gt:=⋂s>tσ(ωr:r≤s,ω∈D),t≥0. (2.4)

For each , the process induces a measure on the path space . Let

 PθT=Pθ∣∣GT (2.5)

be a restriction of to the -algebra .

In order to establish the equivalence of the laws and , , we have to make the following assumption.

A: The Brownian component of is non-degenerate, i.e. .

###### Proposition 2.1.

Let A hold true. Then for each , any

 PθT∼Pθ0T, (2.6)

and the likelihood ratio is given by

 LT(θ0,θ)=dPθTdPθ0T=exp(−θ−θ0σ2∫T0ωsdm(θ0)s−(θ−θ0)22σ2∫T0ω2sds), (2.7)

where

 m(θ0)t=ωt−ω0+θ0∫t0ωsds−bt−∑s≤tΔωsI(|Δωs|>1)−∫t0∫|x|≤1x(μ(dx,ds)−ν(dx)ds) (2.8)

is the continuous local martingale component of under the measure , and the random measure

 μ(dx,ds)=∑sI(Δωs≠0)δ(Δωs,s)(dx,ds) (2.9)

is defined by the jumps of .

###### Proof.

See (Jacod and Shiryaev, 2003, Theorem III-5-34). ∎

Consider a family of statistical experiments

 (D,GT,{PθT}θ>0)T>0. (2.10)

Our goal is to establish local asymptotic mixed normality (LAMN) of these experiments under the assumption that the process has heavy tails. We make the following assumption.

A: The Lévy measure has a regularly varying heavy tail of the order , i.e.

 H(R):=∫|z|>Rν(dz)∈RV−α,R>0. (2.11)

In other words, and there is a positive function slowly varying at infinity such that

 H(R)=l(R)Rα,R>0. (2.12)

For the tail we construct an absolutely continuous strictly decreasing function

 ~H(R)=α∫∞RH(z)zdz,R>0, (2.13)

such that by Karamata’s theorem, see e.g. (Resnick, 2007, Theorem 2.1 (a)),

 limR→∞~H(R)H(R)=1. (2.14)

We introduce the monotone increasing continuous scaling defined by the relation

 1φT:=~H−1(1T), (2.15)

where is the (continuous) inverse of . It is easy to see that .

###### Remark 2.2.

We make use of the absolutely continuous strictly decreasing function just for convenience in order to avoid technicalities connected with the inversion of càdlàg functions. For instance it holds for the generalized inverse , see (Bingham et al., 1987, Chapter 1.5.7).

###### Example 2.3.

Let the jump part of the process be an -stable Lévy process, i.e. for and , , let

 ν(dz)=(c−|z|1+αI(z<0)+c+z1+αI(z>0))dz. (2.16)

Then

 H(R) =~H(R)=c−+c+αRα, (2.17) ~H−1(T) =(αc−+c+)1/α1T1/α,

and

 φT=(c−+c+α)1/α1T1/α. (2.18)

The main result is the LAMN property of our model.

###### Theorem 2.4.

Let A and A hold true. Then the family of statistical experiments (2.10) is locally asymptotically mixed normal at each , namely for each

 Law(lnL(θ0,θ0+φTu)∣∣Pθ0T)→N√S(α/2)2σ2θ0u−12S(α/2)2σ2θ0u2,T→∞, (2.19)

where is a standard Gaussian random variable and is an independent spectrally positive -stable random variable with the Laplace transform

 Ee−λS(α/2)=e−Γ(1−α2)λα/2,λ≥0. (2.20)

Theorem 2.4 is based on the following key result.

###### Theorem 2.5.

Let A and A hold true. Then for each

 Law(φ2T∫T0X2sds∣∣Pθ0T)→S(α/2)2θ0, (2.21)

where is a random variable with the Laplace transform (2.20).

###### Corollary 2.6.

Let A and A hold true. Then for each

 Law(φT∫T0XsdWs,φ2T∫T0X2sds∣∣Pθ0T)→(N√S(α/2)2θ0,S(α/2)2θ0). (2.22)

Proposition 2.1 and Theorem 2.4 allow us to establish asymptotic distribution of the maximum likelihood estimator of . Moreover, the special form of the likelihood ratio guarantees that is asymptotically efficient.

###### Corollary 2.7.

1. Let A hold true. Then the maximum likelihood estimator of satisfies

 (2.23)

2. Let A and A hold true. Then

 Law(^θT−θ0φT∣∣Pθ0T)→σ√2θ0⋅N√S(α/2),T→∞. (2.24)

The maximum likelihood estimator is asymptotically efficient in the sense of the convolution theorem and the local asymptotic minimax theorem for LAMN models, see (Höpfner, 2014, Theorems 7.10 and 7.12).

###### Remark 2.8.

It is instructive determine the tails of the random variable : for each

 limsupx→+∞x−αlnP(|N|√S(α/2)>x)<0, (2.25)

and in particular all moments of the r.h.s. of (2.24) are finite.

The proof of all the results formulated above will be given in Section 4 after necessary preparations made in the next Section.

## 3 Auxiliary results

We decompose the Lévy process into a compound Poisson process with heavy jumps, and the rest. For definiteness, let for , be a non-decreasing function.

Denote

 ηTt =∫t0∫|z|>RTzN(dz,ds), (3.1) ξTt =σWt+∫t0∫|z|≤RTz~N(dz,ds), bT =b+∫1<|z|≤RTzν(dz), ZTt =Zt−ηTt=ξTt+bTt.

For each , the process is a compound Poisson process with intensity , the iid jumps occurring at arrival times , such that

 P(|JTk|≥z)=H(z)H(RT),z≥RT, (3.2) P(τTk+1−τTk>u)=e−H(RT)u,u≥0.

Denote also by the Poisson counting process of ; it is a Poisson process with intensity .

We decompose the Ornstein–Uhlenbeck process into a sum

 Xt =XTt+XηTt, (3.3) XTt :=X0e−θt+∫t0e−θ(t−s)dZTs, XηTt :=∫t0e−θ(t−s)dηTs.

Since and , , by Potter’s bounds (see, e.g. (Resnick, 2007, Proposition 2.6 (ii))) for each there are constants such that for

 cεuα+ε ≤H(u)≤Cεuα−ε, (3.4) cεu1α+ε ≤φu≤Cεu1α−ε.

The following Lemma gives useful asymptotics of the truncated moments of the Lévy measure .

###### Lemma 3.1.

1. For and any there is such that

 ∫1<|z|≤R|z|ν(dz)≤C(ε)R1−α+ε. (3.5)

2. For there is such that

 ∫1<|z|≤R|z|ν(dz)≤C. (3.6)

3. For and any there is such that

 ∫1<|z|≤Rz2ν(dz)≤C(ε)R2−α+ε. (3.7)
###### Proof.

To prove the first inequality we integrate by parts and note that for any

 ∫1<|z|≤R|z|ν(dz)=−∫(1,R]zdH(z) =−zH(z)∣∣R1+∫(1,R]H(z)dz (3.8) ≤H(1)+Cε∫R1dzzα−ε.

Hence (3.5) follows for any and (3.6) is obtained if we choose . The estimate (3.7) is obtained analogously to (3.5). ∎

The next Lemma will be used to determine the tail behaviour of the product of any two independent normalized jumps , .

###### Lemma 3.2.

Let and

be two independent random variables with the probability distribution function

 P(UR>x)=P(VR>x)=¯FR(x)=H(xR)H(R),R≥1,x≥1. (3.9)

Then for each there is such that for all and all

 P(URVR>x)≤C(ε)xα−ε. (3.10)
###### Proof.

Recall that Potter’s bounds (Resnick, 2007, Proposition 2.6 (ii)) imply that for each there is such that for each and

 ¯FR(x)=H(xR)H(R)≤C0(ε)xα−ε. (3.11)

Moreover,

 ¯FR(x) ≡1,x∈[0,1]. (3.12)

For we write

 P(URVR>x) =∫∞1∫∞x/udFR(v)dFR(u) (3.13) =(∫x1+∫∞x)¯FR(x/u)dFR(u)=I(1)R(x)+I(2)R(x).

Then

 I(2)R(x) =∫∞x¯FR(x/u)dFR(u)≤∫∞xdFR(u)≤¯FR(x)≤C0(ε)xα−ε. (3.14)

Eventually,

 I(1)R(x) ≤C0(ε)xα−ε∫x1uα−εdFR(u)=−C0(ε)xα−ε∫x1uα−εd¯FR(u) (3.15) =−C0(ε)xα−εuα−ε¯FR(u)∣∣x1+(α−ε)C0(ε)xα−ε∫x1uα−1−ε¯FR(u)du ≤C0(ε)xα−ε+(α−ε)C0(ε)2xα−ε∫x1uα−1−εuα−εdu ≤C0(ε)xα−ε+(α−ε)C0(ε)2xα−εlnx ≤C(ε)xα−2ε

for some . ∎

###### Remark 3.3.

A finer tail asymptotics of products of iid non-negative Pareto type random variables can be found in (Rosiński and Woyczyński, 1987, Theorem 2.1) and (Jessen and Mikosch, 2006, Lemma 4.1 (4)). In Lemma 3.2, however, we establish rather rough estimates which are valid for the families of iid random variables .

The following useful Lemma will be used to determine the conditional distribution of the interarrival times of the compound Poisson process .

###### Lemma 3.4.

Let and let be a Poisson process, be its arrival times, . Then for each , and

 P(τj+k−τj≤s|NT=m)=P(σk≤sT),s∈[0,1], (3.16)

where is a -distributed random variable with the density

 f(m)σk(u)=m!(k−1)!(m−k)!uk−1(1−u)m−k,u∈[0,1],m≥1, 1≤k≤m. (3.17)
###### Proof.

It is well known that the conditional distribution of the arrival times , given that , coincides with the distribution of the order statistics obtained from

samples from the population with uniform distribution on

, see (Sato, 1999, Proposition 3.4).

Let for brevity . The joint density of , is well known, see e.g. (Balakrishnan and Nevzorov, 2003, Chapter 11.10):

 f(m)τj,τj+k(u,v) =cj,k,m⋅uj−1(v−u)k−1(1−v)m−j−kI(0≤u

and consequently

 f(m)τj+k−τj,τj(u,v)=cj,k,m⋅vj−1uk−1(1−u−v)m−j−k,u,v,u+v∈[0,1]. (3.19)

Hence, the probability density of the difference is obtained by integration w.r.t. ,

 f(m)τj+k−τj(u) =ck,j,m⋅uk−1∫1−u0vj−1(1−u−v)m−j−kdv (3.20) v=(1−u)z=cj,k,m⋅uk−1⋅(1−u)m−k∫10zj−1(1−z)m−j−kdz.

Recalling the definition of the Beta-function, we get

 ∫10zj−1(1−z)m−j−kdz=(j−1)!(m−j−k)!(m−k)!, (3.21)

which yields the desired result. ∎

###### Lemma 3.5.

Let A hold true and be the scaling defined in (2.15). Then for any

 φ2T[ηT]Td→S(α/2),T→∞, (3.22)

where is a spectrally positive -stable random variable with the Laplace transform (2.20).

###### Proof.

The process is a compound Poisson process with the Lévy measure with the tail

 HT(u)=∫∞uνT(dz)=H(√uφT∨RT),u>0. (3.23)

The Laplace transform of has the cumulant

 KT(λ):=lnEe−λφ2T[ηT]T =−T∫∞0(e−λu−1)dHT(u),λ≥0. (3.24)

Integrating by parts yields

 KT(λ)=−T(e−λu−1)HT(u)∣∣∞0−λT∫∞0e−λuHT(u)du. (3.25)

Since the first summands on the r.h.s. of (3.25) vanish, it is left to evaluate the integral term. Taking into account (2.15), namely that , we write for any

 KT(λ) =−λT∫∞0e−λuHT(u)du (3.26) =−λT∫u00e−λuH(√uφT∨RT)du−λH(1/φT)~H(1/φT)1H(1/φT)∫∞u0e−λuH(√uφT∨RT)du =−I(1)T(λ)−I(2)T(λ).

It is evident that . Moreover for due to (Resnick, 2007, Proposition 2.4), the convergence

 limT→∞H(√uφT∨RT)H(1φT)=1uα/2 (3.27)

holds uniformly on each half-line , , and thus for each

 limT→∞I(2)T(λ)=λ∫∞u0e−λuuα/2du. (3.28)

Further we estimate

 I(1)T(λ)≤2λTφ2T∫√u0/φT0yH(y)dy. (3.29)

Note that is integrable at by the definition of the Lévy measure, , and the integration by parts. Eventually by Karamata’s theorem (Resnick, 2007, Theorem 2.1 (a))

 I(1)T≤2λH(1/φT)~H(1/φT)⋅φ2TH(1/φT)⋅∫√u0/φT0yH(y)dyu0φ2TH(√u0φT)⋅u0φ2T⋅H(√u0φT)→2λ2−αu1−α20,T→∞. (3.30)

Hence choosing sufficiently small and letting we obtain convergence of to the cumulant of a spectrally positive stable random variable

 limT→∞KT(λ)=−λ∫∞0e−λuuα/2du=−Γ(1−α2)λα/2. (3.31)

###### Lemma 3.6.

For any and any

 φ2T|XTT|2d→0,T→∞, (3.32) φ2T∫T0|XTs|2dsd→0,T→∞. (3.33)
###### Proof.
 |XTs|2≤3|X0|2e−2θs+3e−2θs∣∣∫s0eθrdξTr∣∣2+3b2Te−2θs∣∣∫s0eθrdr∣∣2=a1(s)+a2(s)+a3(s). (3.34)

By the Itô isometry and Lemma 3.1, for any we estimate for each

 φ2TEa2(s) =φ2T⋅32θ(σ2+∫|z|≤RTz2ν(dz))⋅e−2θs(e2θs−1)≤C1⋅T−2α+ε+ρ(2−α+ε). (3.35)

Analogously, Lemma 3.1 yields

 b2T≤{C2T2ρ(1−α+ε),α∈(0,1],C2,α∈(1,2). (3.36)

and hence for each

 φ2T⋅a3(s)≤C3max{1,T2ρ(1−α+ε)}⋅T−2α+ε→0. (3.37)

Finally, for

 φ2Ta1(s)≤C4|X0|2⋅T−2α+ε→0a.s.\ as  T→+∞. (3.38)

For any we can choose sufficiently small such that the bounds in (3.35) and (3.37) and (3.38) converge to 0 as which gives (3.32). Integrating these inequalities w.r.t.  results in an additional factor on the r.h.s. of these estimates, and convergence to still holds true for sufficiently small. ∎

###### Lemma 3.7.

For any and any

 φ2T∫T0XηTs−dηTsd→0,T→∞. (3.39)
###### Proof.

The Ornstein–Uhlenbeck process as well as its integral w.r.t.  can be written explicitly in the form of sums:

 XηTt =∞∑j=1JTje−θ(t−τTj)I[τTj,∞)(t), (3.40) XηTτTk− =k−1∑j=1JTje−θ(τTk−τTj),k≥1, ∫T0XηTs−dηTs =NTT∑k=1XηTτTk−JTk=NTT∑k=1JTkk−1∑