Nonparametric drift estimation from diffusions with correlated brownian motions

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arXiv:2210.13173v2 [math.ST] 11 Jul 2023

Nonparametric drift estimation from diﬀusions with correlated brownian motions

Fabienne COMTEa, Nicolas MARIEb,∗

aUniversit´e Paris Cit´e, CNRS, MAP5 UMR 8145, F-75006 Paris, France

bLaboratoire Modal’X, Universit´e Paris Nanterre, Nanterre, France

Abstract

In the present paper, we consider that Ndiﬀusion processes X1,...,XNare observed on [0,T], where Tis ﬁxed and

Ngrows to inﬁnity. Contrary to most of the recent works, we no longer assume that the processes are independent.

The dependency is modeled through correlations between the Brownian motions driving the diﬀusion processes. A

nonparametric estimator of the drift function, which does not use the knowledge of the correlation matrix, is proposed

and studied. Its integrated mean squared risk is bounded and an adaptive procedure is proposed. Few theoretical tools

to handle this kind of dependency are available, and this makes our results new. Numerical experiments show that the

procedure works in practice.

Keywords: Correlated Brownian motions, Diﬀusion processes, Model selection, Projection least squares estimator.

2020 MSC: Primary 62G07, Secondary 62M05

1. Introduction

We start by describing our model. Consider the diﬀusion process X=(Xt)t∈[0,T], deﬁned by

Xt=x0+Zt

b(Xs)ds +Zt

σ(Xs)dBs;t∈[0,T],(1)

where x0∈R,B=(Bt)t∈[0,T]is a Brownian motion, b:R→Ris a Lipschitz continuous function, and σ:R→Ris

a bounded Lipschitz continuous function. Now, let B1,...,BNbe N∈N/{0}copies of Bsuch that

E(Bi

sBk

t)=Ri,k(s∧t) ; ∀i,k∈ {1,...,N},∀s,t∈[0,T],(2)

where R=(Ri,k)i,kis a correlation matrix. Note that, thanks to the (stochastic) integration by parts formula, the depen-

dence condition (2) on B1,...,BNimplies that, for every i,k∈ {1,...,N},dhBi,Bkit=Ri,kdt, with Ri,i=1. Finally,

consider Xi:=I(x0,Bi) for every i∈ {1,...,N}, where I(.) is the Itˆo map associated to Equation (1). In the present

paper, we consider that these Ndiﬀusion processes X1,...,XNare observed on [0,T], where Tis ﬁxed and Ngrows

to inﬁnity, and our aim is to estimate nonparametrically the drift function b(.).

In the case of independent Brownian motions, that is R=IN(the N×Nidentity matrix), projection least squares

estimator have been studied in Comte and Genon-Catalot [7] for continuous time observations, in Denis el al. [14] for

discrete time (with small step) observations with a classiﬁcation purpose in the parametric setting, and in Denis et al.

[15] in the nonparametric context, for instance. Marie and Rosier [20] propose a kernel based Nadaraya-Watson esti-

mator of the drift function b, with bandwidth selection relying on the Penalization Comparison to Overﬁtting (PCO)

criterion recently introduced by Lacour et al. [18]. Still in the case R=IN, Comte and Marie [12] investigate the

properties of the projection least squares estimator of the drift when Bis a fractional Brownian motion.

Dependency is often encountered in recent works in the context of stochastic systems of Ninteracting particles, with

∗Corresponding author. Email address: n.marie@parisnanterre.fr

Preprint submitted to Journal of Multivariate Analysis July 12, 2023

recent nonparametric drift estimators proposals in Della Maestra and Hoﬀmann [13], Belomestny et al. [3] or Comte

and Genon-Catalot [9]. These kinds of models are related to physics. We rather have in mind economic or ﬁnancial

models. For instance, in Duellmann et al. [16], the authors consider a portfolio of Nhomogenous ﬁrms such that the

asset value Xi

tat time tof the i-th ﬁrm is modeled by Merton’s model (see [21]) dXi

t=µXi

tdt +σXi

tdBi

twith Xi

0=X0,

which corresponds to (1) with b(x)=µxand σ(x)=σx. Intending to capture the dependency between the ﬁrms, they

also assume that dBi

t=√ρdWt+p1−ρdWi

t, where Wis a common systematic risk factor,Wiis a ﬁrm-speciﬁc risk

factor and ρ∈[0,1]. This corresponds to a particular matrix R, precisely Ri,k=ρfor i,k(and Ri,i=1), so that one

single parameter ρrepresents the so-called asset correlation. This model has been considered in e.g. Bush et al. [4],

for the more mathematical purpose of studying the limit of the empirical distribution of the Xi

t’s (see also references

therein). Our extension from speciﬁc geometric Brownian motion to general nonparametric diﬀusion (1), and from

one single correlation parameter to a general matrix representation, is therefore standard in both respects. This context

has nevertheless never been considered before up to our knowledge. Let us emphasize that our aim is not to estimate

R, but to exhibit conditions on it such that b(.) can be estimated with performance near of the independent setting.

In our framework, Tis ﬁxed, and Nis large. Our results are nonasymptotic, but the idea is that Ngrows to in-

ﬁnity. We ﬁx a subset Iof Rand build a collection of projection least squares estimators of bI=b1Iwhere Iis

compact or not. The estimators are deﬁned by their coeﬃcients on an orthonormal basis of L2(I), ϕ1,...,ϕm, result-

ing from a standard least squares computation. Precisely, we consider the estimator of the drift function bminimizing

the objective function γN(τ)

τ7−→ γN(τ) :=1

i=1 ZT

τ(Xi

s)2ds −2ZT

τ(Xi

s)dXi

s!(3)

on the m-dimensional function space Sm=span{ϕ1,...,ϕm}. The ﬁrst part of γN(τ) involves a quantity

kτk2

N:=1

i=1ZT

τ(Xi

s)2ds,

which is considered as the squared empirical norm of the function τ. These estimators are the same as in Comte and

Genon-Catalot [7], but their study is made signiﬁcantly more diﬃcult by the dependency context. We do not have at

our disposal any coupling method nor any transformation leading to a simpler system; in particular, applying R−1/2to

the system does not bring any simpliﬁcation because of a ”widespreading” of the components of the process. Tropp’s

deviation inequalities used in the independent context (see Tropp [24], Matrix Chernov Inequality, Theorem 1.1 and

Matrix Bernstein Inequality, Theorem 1.4), which allow to consider the empirical norm and its expectation (an integral

norm, thus) as equivalent with high probability, no longer apply. Martingale properties still are useful, and we turn to

Azuma’s matrix deviation inequality (see Tropp [24], Theorem 7.1), which however requires to set sparsity conditions

on R(see Assumption 3). This equivalence property between empirical and weighted integral norms is the key of the

rigorous study of the risk of the drift estimator, and the correlation matrix is therefore at the heart of the proofs.

The plan of the paper is the following. A ﬁrst parametric example motivates the model and the way of estimating

a drift parameter in Section 2. The general nonparametric drift estimator is deﬁned in Section 3 and a risk bound on

a ﬁxed projection space is proved. Adaptive estimation is studied in Section 4 and the whole procedure is illustrated

through simulations in Section 5. Lastly, proofs are gathered in Section 6.

2. Preliminary motivation and example in the parametric framework

This preliminary section deals with the geometric model described in the introduction, in the parametric frame-

work, in order to motivate our investigations. Similarly to Duellmann et al. [16], consider Nrisky assets of same

nature and of prices processes X1, . . . , XNobserved on the time interval [0,T]. Since these assets are of same nature,

to model their prices by dXi

t=µXi

tdt +σXi

tdBi

twith µ∈Rand σ > 0 not depending on i∈ {1,...,N}is realistic, but

it is also very realistic to consider that B1,...,BNmay be dependent, through the correlation matrix described above.

Let us compute the quadratic risk of the least squares estimator b

θNof θ=µ−σ2/2 in this special case. Since we can

write that Xi

t=x0exp(Yi

t) with Yi

t=θt+σBi

tfor every i∈ {1,...,N}and t∈[0,T], we set

θN=1

i=1

T=θ+σ

i=1

Then,

E(|b

θN−θ|2)=σ2

N2T2

i=1

E(|Bi

T|2)+X

i,k

E(Bi

TBk

T)

=σ2

NT +σ2

N2TX

i,k

Ri,k=σ2

NT 1+1

i,k

Ri,k.

This means that the rate of convergence of b

θNis of order

V:=1

N+1

N2X

i,k

Ri,k.

We note that if Ri,k=ρfor all i,k, then the estimator is not consistent. This would be the same if all the coeﬃcients

of Rwere positive and only bounded by a constant ρ > 0. However, if we set a sparsity condition by saying that R

is block-diagonal with blocks of size (less than) k0, and if we assume that all nonzero coeﬃcients are equal to (or

bounded by) ρ, then Pi,kRi,k6k0ρN. So, k0ρis the loss in risk due to dependency, while the rate remains O(1/N).

Referring to the ﬁrms model of Duellmann et al. [16] and Bush et al.[4], this means for instance that for a large N,

dependent ﬁrms have to be grouped as several independent sets aggregated in the global model.

Another way to model the dependency with few parameters is to assume that

dBi

t=√adWi

t+√1−adWi+1

where W1,...,WN+1are independent Brownian motions and a∈[0,1]. This is a way of saying that each ﬁrm is

correlated to the following one in the list. In that case,

Ri,i+1=Ri,i−1=pa(1 −a),Ri,i=1,Ri,k=0 for |k−i|>1;

and then

V=1

N 1+2 1−1

N!pa(1 −a)!

has order O(1/N). Note that this matrix is sparse in the sense of Assumption 3 below.

Our purpose is to show that, at least for some special dependence schemes on B1,...,BN, the variance term of the

projection (nonparametric) least squares estimator of b(.) introduced in the following section is at most of order

N1+1

i,k|Ri,k|.

It is noteworthy that the estimator b

θNis the maximum likelihood estimator (MLE) when X1,...,XNare independent,

while the MLE in our dependent setting would involve – and thus require the complete knowledge of – the matrix R

(more speciﬁcally, its inverse). In the present strategy, the knowledge of Ris not required, which is interesting and

may justify a loss of eﬃciency.

3. A projection least squares estimator of the drift function

3.1. The objective function

Set NT:=[NT ]+1 and let fTbe the density function deﬁned by

fT(x) :=1

TZT

ps(x0,x)ds ;∀x∈R,

where ps(x0, .) is the density with respect to Lebesgue’s measure of the probability distribution of Xsfor every

s∈(0,T]. Let us consider the projection space Sm:=span{ϕ1,...,ϕm}, where ϕ1,...,ϕNTare continuous func-

tions from Iinto Rsuch that (ϕ1, . . . , ϕNT) is an orthonormal family in L2(I,dx), and I⊂Ris a non-empty interval.

We recall now that the objective function τ∈ Sm7→ γN(τ) is deﬁned by (3), where m∈ {1,...,NT}. We choose a

contrast which is the same as in the independent case. Note that for the nonparametric estimation of the drift function

from Nobserved paths, even in the independent case, least squares and maximum likelihood strategies do not match.

Indeed, the likelihood would involve weights σ(Xi

s)−2inside all integrals. In the dependent case, there would also be

the matrix R−1to take into account. Even if both σ(.) and Rcan be considered as known, it is interesting not to need

them to compute the drift estimator. In particular, the step to discrete time high frequency data is then much simpler.

Since the strategy works in the independent case, we can hope that if the correlations are not too strong, then the

strategy remains relevant.

Remark. 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 more τis close to b, the smaller E(γN(τ)). For this reason, the estimator of bminimizing γN(.) is studied in

this paper.

3.2. The projection least squares estimator and some related matrices

In this section, mis a ﬁxed integer in {1,...,NT}. We consider the estimator

bm:=arg min

τ∈Sm

γN(τ) (4)

of b, if it exists and is unique. Since Sm=span{ϕ1,...,ϕm}, there exist msquare integrable random variablesb

θ1,...,b

θm

such that

bm=

j=1b

θjϕj.

Then,

∇γN(b

bm)=1

i=12

ℓ=1b

θℓZT

ϕj(Xi

s)ϕℓ(Xi

s)ds −2ZT

ϕj(Xi

s)dXi

sj∈{1,...,m}

Let

Ψm:=1

i=1ZT

ϕj(Xi

s)ϕℓ(Xi

s)dsj,ℓ∈{1,...,m}

and

Xm:=1

i=1ZT

ϕj(Xi

s)dXi

sj∈{1,...,m}

Therefore, by (4) and if b

Ψmis invertible, necessarily

Θm:=(b

θ1,...,b

θm)∗=b

Ψ−1

Xm,

where M∗denotes the transpose of the matrix M.

Remarks:

1. We can write b

Ψm=(hϕj, ϕℓiN)j,ℓ, where

hϕ, ψiN:=1

i=1ZT

ϕ(Xi

s)ψ(Xi

s)ds

for every measurable functions ϕand ψfrom Rinto itself.

2. The following useful decomposition holds: b

Xm=(hb, ϕjiN)∗

j+b

Em, where

Em:=1

i=1ZT

σ(Xi

s)ϕj(Xi

s)dBi

s

∗

j∈{1,...,m}

Let us introduce the two following deterministic matrices related to the previous random ones:

•Ψm:=E(b

Ψm)=(hϕj, ϕℓifT)j,ℓ, where h., .ifTis the scalar product in L2(I,fT(x)dx).

•Ψm,σ :=NT E(b

Emb

E∗

m).

Note that under the following assumption, Comte and Genon-Catalot established in [7] (see Lemma 1) that Ψmis

invertible.

Assumption 1. The ϕj’s satisfy the three following conditions:

1. (ϕ1,...,ϕm)is an orthonormal family of L2(I,dx).

2. The ϕj’s are bounded, continuously derivable, and have bounded derivatives.

3. There exist x1,...,xm∈I such that det[(ϕj(xℓ))j,ℓ],0.

Let us conclude this section with the following suitable bound on the trace of Ψ−1/2

mΨm,σΨ−1/2

m. To that aim, we deﬁne

the following quantity associated with the basis:

L(m) :=1∨sup

x∈I

j=1

ϕj(x)2.

Lemma 1. Under Assumption 1, for σbelonging to L2(R,fT(x)dx)but possibly unbounded,

trace(Ψ−1/2

mΨm,σΨ−1/2

m)6c1L(m)kΨ−1

mkop 1+1

i,k|Ri,k|(5)

with

c1=Z∞

−∞

σ(x)2fT(x)dx.

If in addition σis bounded, then

trace(Ψ−1/2

mΨm,σΨ−1/2

m)6mkσk2

∞1+1

i,k|Ri,k|.(6)

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摘要：

arXiv:2210.13173v2[math.ST]11Jul2023NonparametricdriftestimationfromdiﬀusionswithcorrelatedbrownianmotionsFabienneCOMTEa,NicolasMARIEb,∗aUniversit´eParisCit´e,CNRS,MAP5UMR8145,F-75006Paris,FrancebLaboratoireModal’X,Universit´eParisNanterre,Nanterre,FranceAbstractInthepresentpaper,weconsiderthatNdiﬀu...

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Nonparametric drift estimation from diffusions with correlated brownian motions

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