ON A PROJECTION LEAST SQUARES ESTIMATOR FOR JUMP DIFFUSION PROCESSES HÉLÈNE HALCONRUYAND NICOLAS MARIE

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ON A PROJECTION LEAST SQUARES ESTIMATOR FOR JUMP DIFFUSION

PROCESSES

HÉLÈNE HALCONRUY†,⋄AND NICOLAS MARIE⋄

†Léonard de Vinci Pôle universitaire, Research Center,

12 avenue Léonard de Vinci, 92400, Courbevoie, France.

⋄Laboratoire Modal’X, Université Paris Nanterre,

200 avenue de la République, 92001, Nanterre, France.

Corresponding author: Nicolas MARIE (nmarie@parisnanterre.fr).

Abstract. This paper deals with a projection least squares estimator of the drift function of a jump

diﬀusion process Xcomputed from multiple independent copies of Xobserved on [0, T ]. Risk bounds

are established on this estimator and on an associated adaptive estimator. Finally, some numerical

experiments are provided.

Contents

1. Introduction 1

2. A projection least squares estimator of the drift function 3

2.1. The objective function 3

2.2. The projection least squares estimator and some related matrices 4

3. Risk bound on the projection least squares estimator 5

4. Model selection 8

5. Numerical experiments 9

Appendix A. Proofs 10

A.1. Proof of Lemma 1 10

A.2. Proof of Theorem 1 10

A.3. Proof of Theorem 2 13

Appendix B. Figures and tables 17

References 18

1. Introduction

Let Z= (Zt)t∈[0,T ]be the compound Poisson process deﬁned by

Zt:=

νt

n=1

ζn=Zt

0Z∞

−∞

zµ(ds, dz)

for every t∈[0, T ], where ν= (νt)t∈[0,T ]is a (usual) Poisson process of intensity λ > 0, independent

of the ζn’s which are i.i.d. random variables of probability distribution π, and µis the Poisson random

measure of intensity m(ds, dz) := λπ(dz)ds deﬁned by

µ([0, t]×dz) := |{s∈[0, t] : Zs−Zs−∈dz}| ;∀t∈[0, T ].

Key words and phrases. Projection least squares estimator ; Model selection ; Jump diﬀusion processes.

arXiv:2210.13164v2 [math.ST] 25 Jul 2023

2 HÉLÈNE HALCONRUY†,⋄AND NICOLAS MARIE⋄

In the sequel, Zis replaced by the centered martingale Z= (Zt)t∈[0,T ]deﬁned by

Zt:= Zt−Zt

0Z∞

−∞

zm(ds, dz) = Zt−cζλt

for every t∈[0, T ], where cζis the (common) expectation of the ζn’s. Now, let us consider the stochastic

diﬀerential equation

(1) Xt=x0+Zt

b(Xs)ds +Zt

σ(Xs)dBs+Zt

γ(Xs)dZs;t∈[0, T ],

where x0∈R,B= (Bt)t∈[0,T ]is a Brownian motion independent of Z,b∈C1(R)and its derivative is

bounded, and σ, γ :R→Rare bounded Lipschitz continuous functions such that infx∈Rσ(x)2∧γ(x)2>0.

Under these conditions on b,σand γ, Equation (1) has a unique (strong) solution X= (Xt)t∈[0,T ].

As for continuous diﬀusion processes, the major part of the estimators of the drift function in stochastic

diﬀerential equations driven by jump processes are computed from one path of the solution to Equation

(1) and converges when T→ ∞ (see Schmisser (2014), Gloter et al. (2018), Amorino et al. (2022), etc.).

The existence and the uniqueness of the ergodic stationary solution to Equation (1) is then required,

and obtained thanks to a restrictive dissipativity condition on b. For stochastic diﬀerential equations

driven by a pure-jump Lévy process, some authors have also studied estimation methods based on high

frequency observations, on a ﬁxed time interval, of one path of the solution (see Clément and Gloter

(2019,2020)).

Now, consider Xi:= I(x0, Bi,Zi)for every i∈ {1, . . . , N}, where I(.)is the Itô map associated to

Equation (1) and (B1,Z1),...,(BN,ZN)are N∈N∗independent copies of (B, Z). The estimation of

the drift function bfrom a continuous-time or a discrete-time observation of (X1, . . . , XN)is a functional

data analysis problem already investigated in the parametric and in the nonparametric frameworks for

continuous diﬀusion processes (see Ditlevsen and De Gaetano (2005), Picchini and Ditlevsen (2011),

Delattre et al. (2013), Comte and Genon-Catalot (2020b), Denis et al. (2021), Marie and Rosier (2023),

etc.). Up to our knowledge, no such estimator of the drift function has been already proposed for jump

diﬀusion processes. So, our paper deals with a projection least squares estimator b

bmof bcomputed from

X1, . . . , XN, which means that b

bmis minimizing the objective function

τ7−→ γN(τ) := 1

i=1 ZT

τ(Xi

s)2ds −2ZT

τ(Xi

s)dXi

on a m-dimensional function space Sm. Precisely, risk bounds are established on b

bmand on the adaptive

estimator b

bcm, where

bm= arg min

m∈c

MN{−∥b

bm∥2

N+ pen(m)}

with c

MN⊂ {1, . . . , N},

pen(m) := ccal

N;∀m∈N∗

and ccal >0is a constant to calibrate in practice.

In Section 2, a detailed deﬁnition of the projection least squares estimator of bis provided. Section 3

deals with a risk bound on b

bmand Section 4 with a risk bound on the adaptive estimator b

bcm. Finally,

some numerical experiments are provided in Section 5. The proofs (resp. tables and ﬁgures) are post-

poned to Appendix A (resp. Appendix B).

Notations and basic deﬁnitions:

•cζn:= E(ζn

1)for every n∈N∗.

•Consider d∈N∗. The j-th component of any x∈Rdis denoted by xjor [x]j.

ON A PROJECTION LEAST SQUARES ESTIMATOR FOR JUMP DIFFUSION PROCESSES 3

•For every k∈N∗,∥.∥k,d is the norm on Rddeﬁned by

∥x∥k,d := 



j=1 |xj|k



1/k

;∀x∈Rd.

•The spectral norm on the space Md(R)of the d×dreal matrices is denoted by ∥.∥op:

∥A∥op := sup

x∈Rd:∥x∥2,d=1 ∥Ax∥2,d ;∀A∈ Md(R).

2. A projection least squares estimator of the drift function

2.1. The objective function. Assume that the probability distribution of Xshas a density ps(x0, .)

with respect to Lebesgue’s measure for every s∈(0, T ], that s7→ ps(x0, x)belongs to L1([0, T ], dt)for

every x∈Rwhich legitimates to consider the density function fTdeﬁned by

fT(x) := 1

TZT

ps(x0, x)ds ;∀x∈R,

and that Z∞

−∞

b(x)4fT(x)dx < ∞.

Remark 1. Assume that band πsatisfy the following additional conditions:

(1) The function bbelongs to the Kato class

K2:= (φ:R→R: lim

δ→0sup

x∈RZδ

0Z∞

−∞ |φ(x+y) + φ(x−y)|s1/2(|y|+s1/2)−3dyds = 0).

(2) The Lévy measure πλ(.) := λπ(.)has a density θwith respect to Lebesgue’s measure. Moreover,

there exists α∈(0,2) such that z∈R7→ θ(z)|z|1+αis bounded, and if α= 1, then

Zr<|z|<R

zθ(z)dz = 0 ; ∀R > r > 0.

By Chen et al. (2017), Theorem 1.1 and the remark p. 126, l. 5-7, in Amorino and Gloter (2021), for

every s∈(0, T ], the probability distribution of Xshas a density ps(x0, .)with respect to Lebesgue’s

measure, and there exist two constants cp,mp>0, not depending on s, such that

(2) ps(x0, x)⩽cps−1/2exp −mp

(x−x0)2

s+s

(s1/2+|x−x0|)1+α;∀x∈R.

So,

•fTis well-deﬁned and even bounded, which is crucial in Section 4. Indeed, since −1/2<1−(1 +

α)/2<1/2, for every x∈R,

0⩽fT(x)⩽cp

T ZT

s−1/2ds +ZT

s1−(1+α)/2ds!=cp

T2T1/2+T2−(1+α)/2

2−(1 + α)/2<∞.

•If there exists a constant cb>0and ε∈(0, α)as close as possible to 0such that |b(x)|⩽

cb(1 + |x|)(α−ε)/4for every x∈R, then

b(x)4ps(x0, x) =

x→±∞,s→0+Os−1/2

|x−x0|1+ε,

which leads to Z∞

−∞

b(x)4fT(x)dx < ∞.

4 HÉLÈNE HALCONRUY†,⋄AND NICOLAS MARIE⋄

Now, let us consider the objective function γN(.)deﬁned by

γN(τ) := 1

i=1 ZT

τ(Xi

s)2ds −2ZT

τ(Xi

s)dXi

for every τ∈ Sm, where m∈ {1, . . . , NT},NT:= [N T ]+1,Sm:= span{φ1, . . . , φm},φ1, . . . , φNTare

continuous functions from Iinto Rsuch that (φ1, . . . , φNT)is an orthonormal family in L2(I, dx), and

I⊂Ris a non-empty interval. For any τ∈ Sm,

E(γN(τ)) = 1

TZT

E(|τ(Xs)−b(Xs)|2)ds −1

TZT

E(b(Xs)2)ds

=Z∞

−∞

(τ(x)−b(x))2fT(x)dx −Z∞

−∞

b(x)2fT(x)dx.

Then, the closer τis to b, the smaller E(γN(τ)). For this reason, the estimator of bminimizing γN(.)is

studied in this paper.

2.2. The projection least squares estimator and some related matrices. Consider

J:=

j=1

θjφjwith θ1, . . . , θm∈R.

Then,

∇γN(J) = 1

i=1 2

ℓ=1

θℓZT

φj(Xi

s)φℓ(Xi

s)ds −2ZT

φj(Xi

s)dXi

s!!j∈{1,...,m}

= 2( b

Ψm(θ1, . . . , θm)∗−b

Xm)

where

Ψm:= 1

i=1 ZT

φj(Xi

s)φℓ(Xi

s)ds!j,ℓ∈{1,...,m}

and

Xm:= 1

i=1 ZT

φj(Xi

s)dXi

s!j∈{1,...,m}

The symmetric matrix b

Ψmis positive semideﬁnite because

u∗b

Ψmu=1

i=1 ZT

0



j=1

ujφj(Xi

s)



ds ⩾0

for every u∈Rm. If in addition b

Ψmis invertible, it is positive deﬁnite, and then

(3) b

bm=

j=1 b

θjφjwith b

θm:= (b

θ1,...,b

θm)∗=b

Ψ−1

is the only minimizer of γN(.)on Sm, called the projection least squares estimator of b.

Remarks:

(1) b

Ψm= (⟨φj, φℓ⟩N)j,ℓ, where

⟨φ, ψ⟩N:= 1

i=1 ZT

φ(Xi

s)ψ(Xi

s)ds

for every continuous functions φ, ψ :R→R.

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ONAPROJECTIONLEASTSQUARESESTIMATORFORJUMPDIFFUSIONPROCESSESHÉLÈNEHALCONRUY†,⋄ANDNICOLASMARIE⋄†LéonarddeVinciPôleuniversitaire,ResearchCenter,12avenueLéonarddeVinci,92400,Courbevoie,France.⋄LaboratoireModal’X,UniversitéParisNanterre,200avenuedelaRépublique,92001,Nanterre,France.Correspondingauthor:Ni...

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