PARAMETRIC INFERENCE OF HIDDEN DISCRETE-TIME DIFFUSION
PROCESSES BY DECONVOLUTION

arXiv:1512.08193v2 [] 19 Dec 2016

SALIMA EL KOLEI AND FLORIAN PELGRIN

Abstract. We study a parametric approach for hidden discrete-time diffusion models based on contrast minimization and deconvolution. This approach leads to estimate a large class of stochastic
models with nonlinear drift and nonlinear diffusion. It can be applied, for example, for ecological and
financial state space models.
After proving consistency and asymptotic normality of the estimator, leading to asymptotic confidence
intervals, we provide a thorough numerical study, which compares many classical methods used in
practice (Monte Carlo Expectation Maximization Likelihood estimator and Bayesian estimators) to
estimate stochastic volatility models. We prove that our estimator clearly outperforms the Maximum
Likelihood Estimator in term of computing time, but also most of the other methods. We also show
that this contrast method is the most stable and also does not need any tuning parameter.

1

Introduction

This paper is motivated by the parametric estimation of hidden stochastic models of the form:

Yi = Xi + εi
(1)
Xi+1 = bθ0 (Xi ) + σθ0 (Xi )ηi+1 ,
where one observes Y1 ,· · · ,Yn , and where the random variables εi , ηi and Xi are unobserved. Notably
(Xi )i≥0 is a strictly stationary, ergodic process that depends on two measurable functions bθ0 and σθ0
and its stationary density is fθ0 , where θ0 belongs to Θ ⊂ Rp . The functions bθ0 , σθ0 and fθ0 are
known up to a finite dimensional parameter, θ0 , and the dependence with respect to θ0 is not required
to be the same in bθ0 and σθ0 . Finally, the innovations (ηi )i≥0 and the errors (εi )i≥0 are independent
and identically distributed (i.i.d.) random variables, the distribution of the innovations being known
for identifiability of the model.
In this work, we propose to estimate the parameters of the two functions bθ0 and σθ0 driving the
dynamics of the hidden variables (Xi )i≥0 . The principle of the estimation method goes as follows.
Taking that the stationary density, fθ0 , is known up to the finite dimensional parameter θ0 , our Mestimator consists in optimizing a contrast function that exploits a Fourier deconvolution strategy
in a parametric framework. In so doing, we exploit a ”Nadaraya-Watson approach” in the sense
that we estimate bθ0 (respectively, b2θ0 + σθ20 ) as ratio of an estimator of lθ0 = bθ0 fθ0 (respectively,
lθ0 = (b2θ0 + σθ20 )fθ0 ) and an estimator of fθ0 . Notably we provide an analytical expression of the
contrast function for a well-known example and characterize their main properties. Moreover we show
that this deconvolution-based estimator is consistent and asymptotically normally distributed for αmixing processes which leads to obtain confidence intervals in practice for many processes. Finally,
our Monte Carlo simulations show that our approach gives good results and is fast computing. All
the results are illustrated on the famous stochastic volatility model with discrete time version of CIR
1

2

SALIMA EL KOLEI AND FLORIAN PELGRIN

(Cox Ingersoll Ross, see [Cox et al., 1985]) process for the volatility and are compared with many
others methods used in the literature to estimate this model (Monte Carlo Expectation Maximisation,
Sequential Monte Carlo).
Our approach extends the previous work [El Kolei, 2013] where parametric estimation of models
of type of (1) is handled for constant volatility function (∀x, σθ0 (x) = σθ0 ) and where the estimator
proposed by the author is not adapted for stochastic process with nonlinear diffusion. As in the
previous work [El Kolei, 2013], this approach is the extension to the parametric framework of the
work [Comte et al., 2010] where the authors propose a non-parametric estimation procedure in the
case of discrete time stochastic models of the form of (1).1 Our aim consists in showing how their
procedure can be extended to a parametric framework and further by obtaining confidence intervals
which are useful in practice.
Applications. This class of parametric models includes, among others, the autoregressive model
with measurement errors, the autoregressive stochastic volatility model ([Taylor, 2005]), the discrete
time versions of well-known diffusion processes in finance ([Hull and White, 1990], [Heston, 1993])
and some families of stochastic processes: Vasicek, CIR, modified CIR and hyperbolic processes (see
[Cox et al., 1985] and [Genon-Catalot et al., 1999]).
Here, we focus on a stochastic volatility model of the form:
(

Ri+1 = exp



Xi+1
2



ξi+1 ,

√
Xi+1 = Xi + κ(µ − Xi )∆ + σ ∆Xi ηi+1

(2)

where ξi and ηi are centered gaussian random variables and ∆ the sampling interval. Hence, the
unobserved variance process Xi is driven by a mean reverting stochastic process which was introduced
in [Cox et al., 1985] to model the short term interest rates.
2 )−E[log(ξ 2 )] and ε
2
2
By applying a log-transformation Yi+1 = log(Ri+1
i+1 = log(ξi+1 )−E[log(ξi+1 )],
i+1
the SV model is a particular version of (1) since it can be written as



Yi = Xi + εi
√
Xi+1 = Xi + κ(µ − Xi )∆ + σ ∆Xi ηi+1

(3)

where εi follows a log chi-squared distribution.
From a practical point of view, the observed component (Yi )i≥1 stands for the log-return of an asset
price while the unobserved component stands for the volatility of this asset. The parameter κ is the
positive mean reverting parameter, µ is the positive long run parameter and σ the positive volatility
of the stochastic volatility process (Xi )i≥1 .
Organization of this paper. The paper is organized as follows. Section 2 presents the notations
and the model assumptions. Section 3 defines the deconvolution-based M-estimator and states all
of the theoretical properties. Some Monte Carlo simulations are discussed in Section 4 and some
concluding remarks are provided in the last section. All the proofs can be found in Appendix 5.

1See also Comte et Taupin in [Comte and Taupin, 2007].

PARAMETRIC INFERENCE

2

3

General setting and assumptions

In this section, we introduce some preliminary main notations and provide the assumptions of the
model (1).
2.1

Notations

∗
Subsequently,
R itx for any function v : R → R, we denote by v the Fourier transform of the function v:
= e v(x)dx, by ||v||2 its L2 (R)-norm, ||v||∞ its supremum norm, h·, ·i stands for the scalar
product in L2 (R) and “?” for the usual convolution product. Moreover, for any integrable and square1
integrable functions u, u1 , and u2 : we have (u∗ )∗ (x) = 2πu(−x)and hu1 , u2 i = 2π
hu∗1 , u∗2 i. Finally,
kAk denotes the Euclidean norm of a matrix A, Yi = (Yi , Yi+1 ) and yi = (yi , yi+1 ), Pn (respectively,
P)Pthe empirical (respectively, theoretical) expectation, that is, for any stochastic variable: Pn (X) =
n
1
i=1 Xi (respectively, P(X) = E[X]). Regarding the partial derivatives, for any function hθ , ∇θ hθ
n
is the vector of the partial derivatives of hθ with respect to (w.r.t) θ and ∇2θ hθ is the Hessian matrix
of hθ w.r.t θ.

v ∗ (t)

2.2

Assumptions

We consider the hidden discrete-time diffusion model (1). The assumptions are the following.
A0 θ0 belongs to the interior Θ0 of a compact set Θ, θ0 ∈ Θ ⊂ Rp .
A1 The errors (εi )i≥0
 are
 independent and identically distributed centered random variables with
finit variance, E ε21 = s2 . The density of ε1 , fε , belongs to L2 (R), and for all x ∈ R, fε∗ (x) 6= 0.
A2 The innovations (ηi )i≥0
 2are
 independent
 3 and identically distributed centered random variables
with unit variance E η1 = 1 and E η1 = 0.
A3 The Xi ’s are strictly stationary and ergodic with invariant density fθ0 .
A4 The sequences (Xi )i≥0 and (εi )i≥0 are independent. The sequence (εi )i≥0 and (ηi )i≥0 are
independent.
A5 On Θ0 , the functions θ 7→ bθ and θ 7→ σθ admit continuous derivatives with respect to θ up to
order 2.

A6 The function to estimate lθ := b2θ + σθ2 fθ belongs to L1 (R) ∩ L2 (R), is twice continuously
differentiable w.r.t θ ∈ Θ for any x and measurable w.r.t x for all θ in Θ. Each element of
∇θ lθ and ∇2θ lθ belongs to L1 (R) ∩ L2 (R).
A7 The application θ 7→ Pmθ admits a unique minimum and its Hessian matrix, denoted by Vθ ,
is non-singular in θ0 .
The compactness assumption A0 might be relaxed by assuming that θ0 is an element of the interior
of a convex parameter space Θ ∈ Rp . In this case, the statistical properties of the M-estimator can be
proved in the light of convex optimization arguments. Assumptions A1-A3 are quite standard when
considering estimation for convolution models. On the other hand, Assumption A3 implies that if
(Xi )i≥0 is an ergodic process then (Yi )i≥0 is stationary and ergodic since it is the sum of an ergodic
process and an i.i.d. noise process ([Dedecker et al., 2007]). Consequently Yi = (Yi , Yi+1 ) inherits
the ergodicity property. According to Assumption A4 the unknown density gθ0 of the Yi ’s is defined
to be fθ0 ? fε . It turns out that gθ∗0 = fθ∗0 fε∗ and thus fθ∗0 = gθ∗0 /fε∗ . Assumption A5 ensures some

4

SALIMA EL KOLEI AND FLORIAN PELGRIN

smoothness for the drift and diffusion functions. Assumption A6 is also quite usual in the literature
and serves for the construction and for asymptotic properties of our estimator.
3
3.1

Parametric deconvolution estimator

The contrast function

Definition 1 (Theoretical and empirical contrast functions). For any square integrable real function
v, we set
1 v ∗ (−x)
uv (x) =
,
2π fε∗ (x)
where we recall that fε∗ is the Fourier transform of the density of the observation noise. Let ϕ : R → R
be given by
(1) ϕ : x 7→ x, if σθ0 is a constant function of the hidden variable
(2) ϕ : x 7→ x2 − s2 , if σθ0 is not a constant function of the hidden variable
where s2 = E[ε21 ], and let us define the mapping m as
m· (·) : (θ, yi ) ∈ (Θ × R2 ) 7→ mθ (yi ) = ||lθ ||2 − 2ϕ (yi+1 ) u∗lθ (yi ).
Then, under Assumptions A1 up to A7, the contrast function is defined by:


E [mθ (Y1 )] := klθ k2 − 2E ϕ(Y2 )u∗lθ (Y1 ) ,

(4)

and its empirical couterparts is given by
n

Pn mθ =

1X
mθ (Yi ).
n

(5)

i=1

Remark 1. As said in the introduction, the case where the diffusion function σθ0 is a constant function
of the hidden variable has already been studied in [El Kolei, 2013]. Therefore, from now on, we focus
on the case 2 in Definition 1 and we refer to the aforementioned paper for the case 1 in Definition 1.
Definition 2 (Minimal contrast estimator). Suppose that Assumptions A0-A7 hold true then, the
minimum-contrast estimator θbn is defined as any solution of
θbn = arg min Pn mθ .
(6)
θ∈Θ

The existence of our estimator can be deduce from regularity properties of the function lθ and
compactness argument of the parameter space. See Appendix 5.2.1.
Remark 2. In this paper we consider the situation in which the observation noise variance is known.
This assumption, which is often not satisfied in practice, is necessary for the identifiability of the model
and so is a standard assumption for state-space models given in (1).
There is some restrictions on the distribution of the innovations in the Nadaraya-Watson approach. It
is known that the rate of convergence for estimating the function lθ is related to the rate of decreasing
of fε∗ . In particular, the smoother fε is, the slower the rate of convergence for estimating is. This
rate of convergence can be improved by assuming some additional regularity conditions on lθ (see
[Comte et al., 2010] and [Comte et al., 2006]).

PARAMETRIC INFERENCE

5

Remark 3. Let us explain the choice of the contrast function and how the strategy of deconvolution
works. The convergence of Pn mθ to Pmθ = E [mθ (Y1 )] as n tends to the infinity is provided by
the Ergodicity Theorem. Moreover, the limit E [mθ (Y1 )] of the contrast function can be explicitly
computed. Using (1) and Assumptions A1-A3, standard computations (see Appendix 5.1) lead to
E [mθ (Y1 )] = klθ k2 − 2 hlθ , lθ0 i = klθ − lθ0 k2 − klθ0 k2 ,
which is, obviously, minimal at point θ = θ0 .
3.2

Asymptotic properties

In this section we first show that our estimator is weakly consistent and asymptotically normally
distributed for mixing processes. To this aim, we further assume that for ϕ defined in (2) of Definition
1 the following assumptions hold true:
h
i
A8 (Local dominance): E supθ∈Θ ϕ(Yi+1 )u∗lθ (Yi ) < ∞.


2+δ
∗
< ∞.
A9 (Moment condition): For some δ > 0, E ϕ(Yi+1 )u∇θ lθ (Yi )
A10 (Hessian Local dominance): For some neighbourhood U of θ0 :


∗
E sup ϕ(Yi+1 )u∇2 lθ (Yi ) < ∞
θ∈U

θ

.
3.2.1 Asymptotic properties of the estimator: consistency and normality
The first result regards the (weak) consistency of our estimator.
Theorem 1. Consider the model (1) under the assumptions A0-A8, the estimator θ̂n defined by (6)
is weakly consistent:
θbn −→ θ0
as n → ∞ in Pθ0 − probability.
Sketch of proof. The main idea for proving the consistency of a M-estimator comes from the following
observation: if Pn mθ converges to Pmθ in probability, and if the true parameter solves the limit
minimization problem, then, the limit of the argminimum θbn is θ0 . By using an argument of uniform
convergence in probability and by compactness of the parameter space, we show that the argminimum
of the limit is the limit of the argminimum. A standard method to prove the uniform convergence
is to use the Uniform Law of Large Numbers (see Lemma 1 in Appendix 5.2.2). Combining these
arguments with the dominance argument (A8) give the consistency of our estimator, and then, the
Theorem 1. For further details see Appendix 5.2.2.

√
The second result states our estimator is n-consistent and asymptotically normally distributed.
Besides, we give in Corollary 1 the different terms of the asymptotic variance-covariance matrix. For
the CLT, we need some mixing properties (we refer the reader to [Dedecker et al., 2007] for a complete
review of mixing processes). Hence, in the following, we further assume that:
A11 The stochastic process Xi is α-mixing.

6

SALIMA EL KOLEI AND FLORIAN PELGRIN

Theorem 2. Consider the model (1) under the assumptions A0-A7, and suppose that the conditions
√
A8-A11 hold true. Then θ̂n defined by (6) is a n-consistent estimator of θ0 which satisfies:
√
L
n(θbn − θ0 ) → N (0, Σ(θ0 )) .
Sketch of proof. The asymptotic normality follows essentially from Central Limit Theorem for mixing
processes (see [Jones, 2004]). Thanks to the consistency, the proof is based on a moment condition of
the Jacobian vector of the function mθ (y) and on a local dominance condition of its Hessian matrix.
For further details, see Appendix 5.2.3.

The following corollary gives an expression of the variance-covariance matrix Σ(θ0 ) of Theorem 2
for the practical implementation:
Corollary 1. Under our assumptions, the variance-covariance matrix Σ(θ0 ) is given by:


+∞
X
0
Ω0 (θ0 ) + 2
Σ(θ0 ) = Vθ−1
Ω(θ0 ) = Vθ−1
Ωj−1 (θ0 ) Vθ−1
,
0
0
0
j=2

with

Ω0 (θ0 )

=

Ωj−1 (θ0 )

=

n h
0 i

4 E ϕ(Y2 )u∗∇θ lθ (Y1 ) ϕ(Y2 )u∗∇θ lθ (Y1 )


 

0 o
− E b2θ0 (X1 ) + σθ20 (X1 ) ∇θ lθ (X1 ) E b2θ0 (X1 ) + σθ20 (X1 ) ∇θ lθ (X1 )
n h


0 i
4 E b2θ0 (X1 ) + σθ20 (X1 ) ∇θ lθ (X1 ) b2θ0 (Xj ) + σθ20 (Xj ) ∇θ lθ (Xj )


 

0 o
−E b2θ0 (X1 ) + σθ20 (X1 ) ∇θ lθ (X1 ) E b2θ0 (X1 ) + σθ20 (X1 ) ∇θ lθ (X1 )
.

and the gradient ∇θ lθ is taken at point θ = θ0 ; furthermore, the Hessian matrix Vθ0 is given by:


[Vθ0 ]j,k




1≤j,k≤r

= 2

∂lθ ∂lθ
,
∂θk ∂θj


at point θ = θ0 .
j,k

Proof. See Appendix 5.2.4 for further details.
4



Application: the CIR process

We consider the following stochastic volatility model

Yi = Xi + εi
√
Xi+1 = Xi + κ(µ − Xi )∆ + σ ∆Xi ηi+1

(7)

where εi follows a log chi-squared distribution and ηi a gaussian distribution. This can be also seen
as a discrete version of the so called Heston model with independent noise between the log-returns Yi
and its volatility Xi . Here, we further assume that the variance process Xi is greater than zero. To
ensure this condition, we make the following assumption:

PARAMETRIC INFERENCE

7

2κ
2κθ
≥ 1 and c := 2 > 0,
2
σ
σ
which is known as the Feller condition (see [Cox et al., 1985]) and implies that the variance process
Xi is ergodic and ρ-mixing. Furthermore, the stationary distribution fθ for this process is the gamma
distribution γ(a, c) (see [Genon-Catalot et al., 1999]).
F

4.1

a :=

Minimum contrast estimator

In this case, the functions b· , σ· and l· are given by:

√
σθ (x) = σ x and lθ = b2θ (x) + σθ2 (x) γ(a, c)

bθ (x) = (1 − κ)x + κθ,

∀x ∈ R∗+ with θ = (κ, µ, σ). Using the Fourier transform of the Gamma and the log chi-squared
density, we have

fθ∗ (x)


=

ix
1−
c

−a

1
1
and fε∗ (x) = √ 2ix Γ( + ix) exp(−iCx),
2
π

with C the expectation of the logarithm of a chi-squared random variable, i.e. C = −1.27 (see
[Abramowitz and Stegun, 1992] and Appendix 5.3 for the expression of the Fourier transform). Next,
the Fourier transform of lθ is given by
"

−a−2 #




−a
a
ix
ix −a−1
ix −a
∗
+ iα2
lθ (x) = −α1
(a + 1) 1 −
1−
+ α3 1 −
.
c2
c
c
c
c
with α1 = (1 − κ)2 , α2 = 2 (1 − κ) κθ + σ 2 , α3 = (κθ)2 . Finally, the L2 -norm of lθ is given by:
Γ(2a + 3)
Γ(2a + 2)
+ 2α1 α2 2−(2a+2) c−2
2
Γ (a)
Γ2 (a)

Γ(2a + 1)
Γ(2a)
+ 2α1 α3 + α22 2−(2a+1) c−1
+ α2 α3 2−(2a) 2
Γ2 (a)
Γ (a)
Γ(2a − 1)
+α32 2−2a+1 c
.
Γ2 (a)
R
where Γ corresponds to the Gamma function given by Γ(z) = R+ tz−1 exp(−t)dt.
klθ k22 = α12 2−(2a+3) c−3

Proof. See Appendix 5.3.



Hence, the M-estimator solves:
(
θbn = arg min
θ∈Θ

klθ k22

)
n
2X
∗
−
Yi+1 ulθ (Yi )
n
i=1

(8)

8

SALIMA EL KOLEI AND FLORIAN PELGRIN

where:


ulθ (y) =

4.2



α
√  1
π


a
(a
c2



+ 1) 1 −

iy
c

−a−2 

+

iα2 ac



1−

iy
c

−a−1



+ α3 1 −

iy
c

−a 

2iβy Γ( 21 + iβy) exp(−iC̃y)





Others methods

Particle filters : EKF, APFS, APF and KSAPF. For the comparison with our contrast estimator given in (8), we use the following methods: the Extended Kalman Filter (EKF), the Auxiliary Particle filter (APF), the Auxiliary Particle filter with static parameter (APFS) and the Kernel Smoothing
Auxiliary Particle filter (KSAPF). We refer the reader to [Reif et al., 1999], [Pitt and Shephard, 1999],
[Doucet et al., 2001] and [Liu and West, 2001] for a complete revue of these methods.
In order to estimate the parameters with these methods, for the EKF, APF and KSAPF estimators we use the Kitagawa and al.’s approach (see [Doucet et al., 2001] chapter 10 p.189) in which the
parameters are supposed time-varying: θi+1 = θi + Gi+1 where Gi+1 is a centered Gaussian random
with a variance matrix Q supposed to be known. Hence, we consider the augmented state vector
X̃i+1 = (Xi+1 , θi+1 )0 where Xi+1 is the hidden state variable and θi+1 the unknown vector of parameters. Furthermore, for the numerical part we call APFS an Auxiliary Particle filter without
time-varying parameters.
The MCEM. From a theoretical point of view, the MLE is asymptotically efficient. However,
in practice since the states (X1 · · · , Xn ) are unobservable and since the model (7) is non Gaussian,
the likelihood is untractable. We have to use numerical methods to approximate it. In this section,
we illustrate the MCEM estimator which consists in approximating the likelihood and applying the
Expectation-Maximisation algorithm introduced by Dempster [Dempster et al., 1977] to find the parameter θ.

4.3

Numerical Results

In this section we present some Monte Carlo simulations using the model (7). For the analysis
we consider the following “true parameter” θ0 = (κ0 , µ0 , σ02 ) = (4, 0.03, 0.4) which is consistent with
empirical applications of daily data (see [Do, 2005]). Thus, we have sampled the trajectory of the Xi ,
and conditionally to the trajectory, we have sampled the variables Yi with a variance noise s2ε = 0.1 2
The numerical illustration goes as follows: we work with a number of observations n equal to 1000,
we first compare all the methods proposed in term of computing time, then we run N = 100 estimates
for each method and we compare the performance of our estimator with others methods by computing
the Mean Square Error (MSE) defined as:
2 For the simulation (see Appendix 5.3) we take f ∗ (x) =
ε

C̃ = βC where C = −1.27.

√1 2iβx Γ( 1
2
π

+ iβx) exp(−iC̃x) with β =

p
2s2ε /π 2 and

PARAMETRIC INFERENCE

p N
1 XX i
MSE =
(θ̂j − θ0i )2 ,
N

9

(9)

i=1 j=1

where p corresponds to the dimension of the vector of parameters.
Then, we illustrate the statistical properties of our contrast estimator, by computing the confidence
intervals for different number of observations. Finally, we study the influence of the signal noise ratio
since the performance of our estimator depends on the regularity of lθ∗ and fε∗ (see Remark 2).
For particles methods, we take a number of particles M equal to 5000. Note that for the Bayesian
procedure (APF, APFS and KSAPF) we need a prior on θ, and this only at the first step. The prior
for θ is taken to be the Uniform law and conditionally to θ the distribution of X1 is its stationary law:


p(θ) = U(3, 5) × U(0.02, 0.04) × U(0.3, 0.5)
fθ (X1 ) = γ (θ)

For the KSAPF, we take a bandwidth h = 0.1 and for the APF we take a matrix Q = [10−3 , 0.1.10−4 , 10−4 ]I3
with I3 the identity matrix in R3 (see Section 4.3.1 for the definition of the matrix Q).
4.3.1 Computing time
Table 1. Comparison of the CPU for Particle filters estimators, MCEM estimator
and Contrast estimator.
APF
KSAPF
APFS
EKF Contrast MCEM
CPU (sec) 105.1695 93.8846 192.2166 0.2
20.4074 217430
This comparison illustrates the numerical complexity of the MCEM. Therefore, in the following, we
only compare our contrast estimator with the particles estimators.

4.3.2 Parameter estimates
We illustrate by boxplots the different estimates (see Figures [1] up to [3]). In Table [2], we compute
the MSE defined in (9) for each method and with a number of MC equal to N = 100 and the CPU
for a number of observations n = 1000 (see Table [1]).
We note that for all parameters, the EKF estimator is very bad since the stochastic volatility model
(7) is strongly nonlinear, and its corresponding boxplots have the largest dispersion meaning that this
filter is not stable and not appropriated to estimate this model. Among particle filters, the KSAPF
and the APF are the best estimators although the dispersion is huge for the mean reversion parameter
κ and the volatility parameter σ.
Besides, the APFS is less efficient than the others particle filters. Our estimator is stable and performs
the others when one compare the MSE. From a computational point of view, all particles filters have
an equivalent CPU. Our contrast estimator is fast and its implementation is straightforward and the
MSE is the smallest (see Table [1]).

10

SALIMA EL KOLEI AND FLORIAN PELGRIN

‘
Table 2. Comparison of the M SE(θ) for Particle filters and Contrast estimator.
APF KSAPF APFS EKF Contrast
MSE(θ) 0.189
0.166
0.205 0.43
0.124

Figure 1. Boxplot of the parameter κ. True value equal to 4.

Figure 2. Boxplot of the parameter µ. True value equal to 0.03.

PARAMETRIC INFERENCE

11

Figure 3. Boxplot of the parameter σ 2 . True value equal to 0.4.
4.3.3 Confidence Interval of the minimum contrast estimator
To illustrate the statistical properties of our contrast estimator, we compute the confidence intervals
with the confidence level 1 − α equal to 0.95 for N = 100 estimators. The coverage corresponds to the
number of times for which the true parameter θ0i , i = 1, · · · , p belongs to the confidence interval. The
results are illustrated in Figure [4]. We note that the coverage converges to 95% for a small number
of observations and as expected, the confidence interval decreases with the number of observations.
Note that of course a MLE confidence interval would be smaller since the MLE is efficient but the
corresponding computing time would be huge (see Table [1]).

Figure 4. Coverage with respect to the number of observations n = 500 up to 3000
for N = 100 estimators.

12

SALIMA EL KOLEI AND FLORIAN PELGRIN

4.3.4 Ratio signal-noise for the contrast estimator
We denote by r = [s2ε /σ 2 ] the ratio signal-noise and in Table (3) we compare the MSE for different
r and different number of observations n for the contrast estimator. We note that the MSE decreases
with the number of observations and is smaller for small ratio-signal-noise. As explained in Section
3.1 and see [Comte et al., 2010] for more details, the rate of convergence of our approach depends on
the regularity of the noise density fε . And, in particular, the smoother the noises are, the slower the
rate of convergence is. For the CIR model, the density of the noises and the function lθ are ordinary
smooth, so we are in a favourable case.
Table 3. Ratio signal-noise for the estimation of the CIR model
Mean(µ̂n ) Mean(κ̂n ) Mean(σ̂n2 ) MSE
n = 500 and r = 0.1
0.0315
3.88
0.401
0.14
n = 500 and r = 1
0.0303
3.89
0.405
0.16
n = 1000 and r = 0.1
0.0312
3.76
0.401
0.11
n = 1000 and r = 1
0.0308
3.83
0.41
0.18

4.4

Summary and Conclusions

In this paper we have proposed a new method to estimate hidden nonlinear diffusion process. This
method is based on a deconvolution strategy and leads to consistent and asymptotically normal estimator. We have numerically studied the performance of our estimator for the CIR process widely used
in many domains and we were able to construct confidence interval (see Figure [4]). As the boxplots [1]
up to [3] show, only Contrast, APF, and KSAPF estimators are comparable. Indeed EKF and APFS
estimators are biased and their MSE are bad, especially for the EKF method since the CIR process is
nonlinear. Furthermore, if one compares the MSE of the particle filters, the KSAPF estimator is the
best method. Among particles filters, it is clearly known that the APFS is less efficient than the APF
filter since the parameters are not time-varying and so the only randomness is made at the first step
by the prior law and not in each propagation step.
Then, the Contrast, APF, and KSAPF methods lead to unbiased and not so much varying estimator.
We emphasize that our estimator performs the others in a MSE aspect (see Table [1]). Most importantly, our estimator can be constructed without any arbitrary parameters choice, is straightforward
to implement, fast and allows to construct confidence interval.

References
[Abramowitz and Stegun, 1992] Abramowitz, M. and Stegun, I. A., editors (1992). Handbook of mathematical functions
with formulas, graphs, and mathematical tables. Dover Publications Inc., New York. Reprint of the 1972 edition.
[Comte et al., 2010] Comte, F., Lacour, C., and Rozenholc, Y. (2010). Adaptive estimation of the dynamics of a discrete
time stochastic volatility model. J. Econometrics, 154(1):59–73.
[Comte et al., 2006] Comte, F., Rozenholc, Y., and Taupin, M.-L. (2006). Penalized contrast estimator for adaptive
density deconvolution. Canad. J. Statist., 34(3):431–452.

PARAMETRIC INFERENCE

13

[Comte and Taupin, 2007] Comte, F. and Taupin, M.-L. (2007). Adaptive estimation in a nonparametric regression
model with errors-in-variables. Statist. Sinica, 17(3):1065–1090.
[Cox et al., 1985] Cox, J. C., Ingersoll, Jr., J. E., and Ross, S. A. (1985). A theory of the term structure of interest rates.
Econometrica, 53(2):385–407.
[Dedecker et al., 2007] Dedecker, J., Doukhan, P., Lang, G., León R., J. R., Louhichi, S., and Prieur, C. (2007). Weak
dependence: with examples and applications, volume 190 of Lecture Notes in Statistics. Springer, New York.
[Dempster et al., 1977] Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete
data via the EM algorithm. J. Roy. Statist. Soc. Ser. B, 39(1):1–38. With discussion.
[Do, 2005] Do, B. (2005). Estimating the Heston stochastic volatility model from option prices by filtering. A simulation
exercise.
[Doucet et al., 2001] Doucet, A., de Freitas, N., and Gordon, N., editors (2001). Sequential Monte Carlo methods in
practice. Statistics for Engineering and Information Science. Springer-Verlag, New York.
[El Kolei, 2013] El Kolei, S. (2013). Parametric estimation of hidden stochastic model by contrast minimization and
deconvolution. Metrika, 76(8):1031–1081.
[Fan et al., 1990] Fan, J., Truong, Y. K., and Wang, Y. (1990). Nonparametric function estimation involving errors-invariables.
[Genon-Catalot et al., 1999] Genon-Catalot, V., Jeantheau, T., and Laredo, C. (1999). Parameter estimation for discretely observed stochastic volatility models. Bernoulli, 5(5):855–872.
[Hansen and Horowitz, 1997] Hansen, B. E. and Horowitz, J. L. (1997). Handbook of econometrics, vol. 4 robert f. engle
and daniel l. mcfadden, editors elsevier science b. v., 1994. Econometric Theory, 13(01):119–132.
[Hayashi, 2000] Hayashi, F., editor (2000). Econometrics. Princeton University Press, Princeton NJ.
[Heston, 1993] Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications
to Bond and Currency Options. Review of Financial Studies, 6(2):327–43.
[Hull and White, 1990] Hull, J. and White, A. (1990). Pricing interest-rate-derivative securities. Review of Financial
Studies, 3(4):573–92.
[Jones, 2004] Jones, G. L. (2004). On the Markov chain central limit theorem. Probab. Surv., 1:299–320.
[Liu and West, 2001] Liu, J. and West, M. (2001). Combined parameter and state estimation in simulation-based filtering. in arnaud doucet, nando de freitas, and neil gordon, editors. Sequential Monte Carlo Methods in Practice.
[Newey and McFadden, 1994] Newey, W. K. and McFadden, D. (1994). Large sample estimation and hypothesis testing.
In Handbook of econometrics, Vol. IV, volume 2 of Handbooks in Econom., pages 2111–2245. North-Holland, Amsterdam.
[Pitt and Shephard, 1999] Pitt, M. K. and Shephard, N. (1999). Filtering via simulation: auxiliary particle filters. J.
Amer. Statist. Assoc., 94(446):590–599.
[Reif et al., 1999] Reif, K., Günther, S., Yaz, E., and Unbehauen, R. (1999). Stochastic stability of the discrete-time
extended Kalman filter. IEEE Trans. Automat. Control, 44(4):714–728.
[Taylor, 2005] Taylor, S. (2005). Financial returns modelled by the product of two stochastic processes, a study of daily
sugar prices., volume 1, pages 203–226. Oxford University Press.

5
5.1

Appendix

The contrast function

The procedure: The limit E [mθ (Y1 )] of the contrast function can be explicitly computed. Using
(1) and Assumptions A1-A3, we obtain:





E (Y22 − s2ε )u∗lθ (Y1 ) = E X22 + 2X2 ε2 + ε22 − s2ε u∗lθ (Y1 )


= E X22 u∗lθ (Y1 ) by assumption A1



= E b2θ0 (X1 ) + σθ20 (X1 )η22 + 2bθ0 (X1 )σθ0 (X1 )η2 u∗lθ (Y1 )



= E b2θ0 (X1 ) + σθ20 (X1 ) u∗lθ (Y1 ) by assumption A2,

14

SALIMA EL KOLEI AND FLORIAN PELGRIN

Using Fubini’s Theorem and (1), it follows that:


Z


 ∗
 2
iY1 y
2
2
2
e
ulθ (z)dz
E bθ0 (X1 ) + σθ0 (X1 ) ulθ (Y1 ) = E bθ0 (X1 ) + σθ0 (X1 )


Z

1 1 iY1 z
2
2
∗
= E bθ0 (X1 ) + σθ0 (X1 )
e
(lθ (−z)) dy
2π fε∗ (z)
Z h
i 1

1
E b2θ0 (X1 ) + σθ20 (X1 ) ei(X1 +ε1 )z ∗ (lθ (−z))∗ dz
=
2π
fε (z)
 iε z 
Z
1



E e
1
=
E b2θ0 (X1 ) + σθ20 (X1 ) eiX1 z (lθ (−z))∗ dy
∗
2π
f (z)

 ε
Z

1
iX1 z
∗
2
2
=
e
(lθ (−z)) dz
E bθ0 (X1 ) + σθ0 (X1 )
2π


1  2
=
E bθ0 (X1 ) + σθ20 (X1 ) ((lθ (−X1 ))∗ )∗
2π


= E b2θ0 (X1 ) + σθ20 (X1 ) lθ (X1 ) .
Z


=
b2θ0 (x) + σθ20 (x) fθ0 (x) b2θ (x) + σθ2 (x) fθ (x)dx
= hlθ , lθ0 i .
Then,
E [mθ (Y1 )] = klθ k2 − 2 hlθ , lθ0 i = klθ − lθ0 k2 − klθ0 k2 .
5.2

Proofs

For the reader convenience we split the proof of Theorems 1 and 2 into three parts: in Subsection
5.2.1, we give the proof of the existence of our contrast estimator defined in (6). In Subsection 5.2.2,
we prove the consistency, that is, the Theorem 1. Then, we prove the asymptotic normality of our
estimator in Subsection 5.2.3, that is, the Theorem 2. The Subsection 5.2.4 is devoted to Corollary 1.
Recall from Remark 1 that we only made the proof for the function ϕ defined by (2) in Definition
1 and we refer to [El Kolei, 2013] for the proof in the case (1) of Definition 1.

5.2.1

Proof of the existence and measurability of the M-Estimator

By assumption, the function θ 7→ klθ k22 is continuous. Moreover, lθ∗ and then u∗lθ (x) =

1
2π

R

eixy

lθ∗ (−y)
fε∗ (y) dy

are continuous w.r.t θ. In particular, the function mθ (yi ) = klθ k22 − 2ϕ(yi+1 )u∗lθ (yi ) is continuous w.r.t
P
θ, for ϕ : x 7→ x2 − s2ε . Hence, the function Pn mθ = n1 ni=1 mθ (Yi ) is continuous w.r.t. θ belonging
to the compact subset Θ. So, there exists θ̃ that belongs to Θ such that:
inf Pn mθ = Pn mθ̃ . 

θ∈Θ

PARAMETRIC INFERENCE

5.2.2

15

Proof of the Consistency

For the consistency of our estimator, we need to use the uniform convergence given in the following
Lemma. Let us consider the following quantities:
n

P n hθ =

1X
hθ (Yi );
n

Pn Sθ =

i=1

n

n

i=1

i=1

1X
1X 2
∇θ hθ (Yi ) and Pn Hθ =
∇θ hθ (Yi )
n
n

where hθ is real function from Θ × Y with value in R.
Lemma 1. Uniform Law of Large Numbers (ULLN)(see [Newey and McFadden, 1994] for the proof.)
Let (Yi ) be an ergodic stationary process and suppose that:
(1) For all y ∈ R, θ 7→ hθ (y) is continuous and for all θ ∈ Θ, y 7→ hθ (y) is measurable.
(2) There exists a function (called the dominating function) s : R → R such that for all θ ∈ Θ
|hθ (y)| ≤ s(y) and E[s(Y1 )] < ∞. Then:

sup |Pn hθ − Phθ | → 0

in probability as n → ∞.

θ∈Θ

Moreover, Phθ is a continuous function of θ.

By assumption for all x, θ 7→ lθ (x) is continuous and for all θ, x 7→ lθ (x) is measurable which implies
the continuity and the measurability of the function Pn mθ on the compact subset Θ. Furthermore,
the local dominance assumption (A8) implies that E [supθ∈Θ |mθ (Yi )|] is finite. Indeed,

|mθ (yi )| =

klθ k22 − 2ϕ(yi+1 )u∗lθ (yi )

≤ klθ k22 + 2 ϕ(yi+1 )u∗lθ (yi ) .
with ϕ the function defined in (2) in Definition 1.
2
As klθ k22 ishcontinuous on the compact
i subset Θ, supθ∈Θ klθ k2 is finite. Therefore, E [supθ∈Θ |mθ (Yi )|]

is finite if E supθ∈Θ ϕ(Yi+1 )u∗lθ (Yi ) is finite. Lemma 1 gives us the uniform convergence in probability of the contrast function: for any ε > 0,




lim P sup |Pn mθ − Pmθ | ≤ ε

n→+∞

= 1.

θ∈Θ

Combining the uniform convergence with Theorem 2.1 p. 2121 chapter 36 in [Hansen and Horowitz, 1997]
yields the weak (convergence in probability) consistency of the estimator.


16

SALIMA EL KOLEI AND FLORIAN PELGRIN

5.2.3 Proof of the asymptotic normality
Consider the model (1) under the assumptions A0-A7. The proof of the asymptotic normality
results from assumptions A8-A11 is a straighforward application of [Hayashi, 2000] (Propostion 7.8.
p. 472 and [Jones, 2004]). Furthermore, for the CLT, we need to recall some mixing properties (we
refer the reader to [Dedecker et al., 2007] for a complete revue of mixing processes).
Let Kθ (x, dy) be a Markov transition kernel on a general space (X , B(X ))) and let Kθn (x, dy) denotes
the n step Markov transition corresponding to Kθ . Then, for k ∈ N, x ∈ X and a measurable set A:
Kθn (x, dy) = Pθ (Xn+k ∈ A|Xk = x)
Let M : R → R be a nonnegative function and γ : N → Z+ be a nonnegative decreasing function
such that:

where ||.||V T

(C)
||Kθn (x, .) − fθ (.)||V T ≤ M (x)γ(n)
denotes the total variation norm.

Remark 4. X is geometrically ergodic if (C) holds with γ(n) = tn for some t < 1. X is uniform
ergodic if (C) holds with M bounded and γ(n) = tn for some t < 1. X is polynomial ergodic of order
m where m ≥ 0 if (C) holds with γ(n) = n−m .
The proof of the asymptotic normality is based on the following Lemma:
Lemma 2. [see [Hayashi, 2000] and [Jones, 2004] for the proof.]
Suppose that the conditions of the consistency hold. Suppose further that:
(1) Yi is α-mixing.
(2) (Moment condition): for some δ > 0 and for each j ∈ {1, · · · , p} :
"
#
∂mθ (Y1 ) 2+δ
E
<∞
∂θj
.
δ
P
(3) Assumption (C) holds such that E[M (X1 )] < ∞ and γ(n) satisfies n γ(n) 2+δ < ∞ with δ
defined in (2).
(4) (Hessian Local condition): for some neighbourhood U of θ0 and for j, k ∈ {1, · · · , p}


∂ 2 mθ (Y1 )
< ∞.
E sup
∂θj ∂θk
θ∈U
Then, θbn defined in Eq.(6) is asymptotically normal with asymptotic covariance matrix given by:
Σ(θ0 ) = Vθ−1
Ω(θ0 )Vθ−1
0
0
where Vθ0 is the Hessian of the application Pmθ given in Eq.(4).
Proof. It just remains to check that the conditions (2) and (4) of Lemma 2 hold under our assumptions.

PARAMETRIC INFERENCE

17

Moment condition: as the function lθ is twice continuously differentiable w.r.t θ, for all yi ∈ R2 , the

application mθ (yi ) : θ ∈ Θ 7→ mθ (yi ) = ||lθ ||22 − 2ϕ(yi+1 )u∗lθ (yi ) is twice continuously differentiable
for all θ ∈ Θ and its first derivatives are given by:
∇θ mθ (yi ) = ∇θ ||lθ ||22 − 2ϕ(yi+1 )∇θ u∗lθ (yi ).
∂lθ
By assumption, for each j ∈ {1, · · · , p}, ∂θ
∈ L1 (R), therefore one can apply the Lebesgue differj
entiation Theorem and Fubini’s Theorem to obtain :



∇θ mθ (yi ) = ∇θ ||lθ ||22 − 2ϕ(yi+1 )u∗∇θ lθ (yi ) .

(10)

Then, for some δ > 0:
|∇θ mθ (yi )|2+δ =

∇θ ||lθ ||22 − 2ϕ(yi+1 )u∗∇θ lθ (yi )

≤ C1 ∇θ ||lθ ||22

2+δ

2+δ

+ C2 ϕ(yi+1 )u∗∇θ lθ (yi )

2+δ

,

(11)

where C1 and C2 are two positive constants. By assumption, the function ||lθ ||22 is twice continuously
differentiable w.r.t θ. Hence, θ 7→ ∇θ ||lθ ||22 is continuous on the compact subset Θ and the first term
of equation (11) is finite. The second term is finite by the moment assumption (A9).
2

lθ
Hessian Local dominance: for j, k ∈ {1, · · · , p}, ∂θ∂j ∂θ
∈ L1 (R), the Lebesgue differentiation Theorem
k

gives:
∇2θ mθ (yi ) = ∇2θ ||lθ ||22 − 2ϕ(yi+1 )u∗∇2 lθ (yi ),
θ

and, for some neighbourhood U of θ0 :

E sup
θ∈U

∇2θ mθ (Yi )


≤ sup
θ∈U

∇2θ ||lθ ||22


+ 2E sup
θ∈U

ϕ(Yi+1 )u∗∇2 lθ (Yi )
θ


.

The first term of the above equation is finite by continuity and compactness argument and the second
term is finite by the Hessian local dominance assumption (A10).

5.2.4 Proof of Corollary 1
By replacing ∇θ mθ (Y1 ) by its expression (10), we have:


Ω0 (θ) = Var ∇θ ||lθ ||22 − 2ϕ(Y2 )u∗∇θ lθ (Y1 )


= 4Var ϕ(Y2 )u∗∇θ lθ (Y1 )
h h

0 i

 
0 i
= 4 E ϕ(Y2 )2 u∗∇θ lθ (Y1 ) u∗∇θ lθ (Y1 ) − E ϕ(Y2 )u∗∇θ lθ (Y1 ) E ϕ(Y2 )u∗∇θ lθ (Y1 ) .
Furthermore, by Eq.(1) and by independence of the centered noise (ε2 ) and (η2 ), we have:

18

SALIMA EL KOLEI AND FLORIAN PELGRIN





E ϕ(Y2 )u∗∇θ lθ (Y1 ) = E (b2θ0 + σθ20 )(X1 )u∗∇θ lθ (Y1 ) .
Using Fubini’s Theorem and Eq.(1) we obtain:


Z

 2
iY1 z
2
2
∗
2
u∇θ lθ (z)dz
E (bθ0 + σθ0 )(X1 )u∇θ lθ (Y1 ) = E (bθ0 + σθ0 )(X1 ) e


Z
1 1 iY1 z
2
2
∗
= E (bθ0 + σθ0 )(X1 )
e
(∇θ lθ ) (−z)dz
2π fε∗ (z)
Z h
i 1
1
=
E (b2θ0 + σθ20 )(X1 )ei(X1 +ε1 )z ∗ (∇θ lθ )∗ (−z)dz
2π
fε (z)
 iε z 
Z
1


E e
1
=
E (b2θ0 + σθ20 )(X1 )eiX1 z (∇θ lθ )∗ (−z)dz,
∗
2π
fε (z)
so that


E (b2θ0 + σθ20 )(X1 )u∗∇θ lθ (Y1 ) =
=
=
=
=

1
2π

Z

i 1
h
E (b2θ0 + σθ20 )(X1 )ei(X1 +ε1 )z ∗ (∇θ lθ )∗ (−z)dz
fε (z)
 iε z 
Z
1


E e
1
E (b2θ0 + σθ20 )(X1 )eiX1 z (∇θ lθ )∗ (−z)dz
∗
2π
f (z)
 ε

Z
1
E (b2θ0 + σθ20 )(X1 ) eiX1 z (∇θ lθ )∗ (−z)dz
2π

1  2
E (bθ0 + σθ20 )(X1 ) ((∇θ lθ )∗ (−X1 ))∗
2π

E (b2θ0 + σθ20 )(X1 )∇θ lθ (X1 ) .
(12)

Hence,
Ω0 (θ) = 4 (P2 − P1 ) ,
where

 
0
P1 = E (b2θ0 + σθ20 )(X1 )∇θ lθ (X1 ) E (b2θ0 + σθ20 )(X1 )∇θ lθ (X1 ) ,
h
0 i

P2 = E ϕ(Y2 )2 u∗∇θ lθ (Y1 ) u∗∇θ lθ (Y1 ) .
Calculus of the covariance matrix of Corollary (1): By replacing (∇θ mθ (Y1 )) by its expression (10)
we have:

Ωj−1 (θ) = Cov ∇θ ||lθ ||22 − 2ϕ(Y2 )u∗∇θ lθ (Y1 ), ∇θ ||lθ ||22 − 2ϕ(Yj+1 )u∗∇θ lθ (Yj ) ,

= 4Cov ϕ(Y2 )u∗∇θ lθ (Y1 ), ϕ(Yj+1 )u∗∇θ lθ (Yj ) ,
h


0 i
∗
∗
∗
∗
= 4 E ϕ(Y2 )u∇θ lθ (Y1 )ϕ(Yj+1 )u∇θ lθ (Yj ) − E ϕ(Y2 )u∇θ lθ (Y1 ) E ϕ(Yj+1 )u∇θ lθ (Yj ) .
By using Eq.(12) and the stationary property of the Yi , one can replace the second term of the
above equation by:

PARAMETRIC INFERENCE

19

 

0
E (b2θ0 + σθ20 )(X1 )∇θ lθ (X1 ) E (b2θ0 + σθ20 )(X1 )∇θ lθ (X1 ) .
Furthermore, by using Eq.(1) we obtain:





E ϕ(Y2 )ϕ(Yj+1 )u∗∇θ lθ (Y1 )u∗∇θ lθ (Yj ) = E (b2θ0 + σθ20 )(X1 )(b2θ0 + σθ20 )(Xj )u∗∇θ lθ (Y1 )u∗∇θ lθ (Yj )


+ E (b2θ0 + σθ20 )(X1 ) (ηj+1 + εj+1 ) u∗∇θ lθ (Y1 )u∗∇θ lθ (Yj ) (13)


(14)
+ E (b2θ0 + σθ20 )(Xj ) (η2 + ε2 ) u∗∇θ lθ (Y1 )u∗∇θ lθ (Yj )


(15)
+ E (η2 + ε2 ) (ηj+1 + εj+1 ) u∗∇θ lθ (Y1 )u∗∇θ lθ (Yj ) .

By independence of the centered noise, the term (13), (14) and (15) are equal to zero. Now, if we use
Fubini’s Theorem we have:





E (b2θ0 + σθ20 )(X1 )(b2θ0 + σθ20 )(Xj )u∗∇θ lθ (Y1 )u∗∇θ lθ (Yj ) = E (b2θ0 + σθ20 )(X1 )(b2θ0 + σθ20 )(Xj )∇θ lθ (X1 )∇θ lθ (Xj ) .
(16)
Hence, the covariance matrix is given by:



Ωj−1 (θ) = 4 E (b2θ0 + σθ20 )(X1 )(b2θ0 + σθ20 )(Xj ) (∇θ lθ (X1 )) (∇θ lθ (Xj ))0
−E



(b2θ0

+

  2
0
E (bθ0 + σθ20 )(X1 ) (∇θ lθ (X1 ))

!

σθ20 )(X1 ) (∇θ lθ (X1 ))



 
0 
= 4 C̃j−1 − E (b2θ0 + σθ20 )(X1 ) (∇θ lθ (X1 )) E (b2θ0 + σθ20 )(X1 ) (∇θ lθ (X1 ))


= 4 C̃j−1 − P1 .
Finally, we obtain: Ω(θ) = Ω0 (θ)+2

P∞

j>1 Ωj−1 (θ)



with Ω0 (θ) = 4 (P2 − P1 ) and Ωj−1 (θ) = 4 C̃j−1 − P1 .

Expression of the Hessian matrix Vθ : We have:

Pmθ = ||lθ ||22 − 2 hlθ , lθ0 i .

(17)

For all θ in Θ, the application θ 7→ Pmθ is twice differentiable w.r.t θ on the compact subset Θ.
And for j ∈ {1, · · · , p}:

20

SALIMA EL KOLEI AND FLORIAN PELGRIN




∂lθ
∂lθ
, lθ − 2
, lθ
∂θj
∂θj 0


∂lθ
= 2
, lθ − lθ0 ,
∂θj
= 0 at the point θ0 ,

∂Pm
(θ) = 2
∂θj



and for j, k ∈ {1, · · · , p}:
∂ 2 Pm
(θ) = 2
∂θj ∂θk


 
∂ 2 lθ
∂lθ ∂lθ
, lθ − lθ0 +
,
∂θj θk
∂θk ∂θj
j,k


∂lθ ∂lθ
= 2
,
at the point θ0 .

∂θk ∂θj
j,k

5.3



M-estimator using the example in Section 4

√
Expression of fε∗ : consider the random variable ε = ε−C
with ε = log(X 2 ) where X is standard Gaussian
V

random variable, C = E[log(X 2 )] and V = V[log(X 2 )]. The Fourier transform of ε is given by:


iC
E [exp (iεy)] = exp − √ y E [exp (iεy)]
V
 


2iy
iC
√
= exp − √ y E X V
V


 2
Z +∞ 2iy
iC
1
x
√
= exp − √ y √
x V exp −
dx
2
2π −∞
V
Using a change of variable z =

x2
2 ,

we get:


iC
E [exp (iεy)] = exp − √
V

iC
≡ exp − √
V


y

y

iy
√

2 V
√
π

+∞

Z

z

iy
√
V

− 21 −z

e

dz

0

iy
√

2 V
√ Γ
π



1
iy
+√
2
V


.

Then
fε∗


iC
= exp − √
V

iC
≡ exp − √
V


y

y

iy
√

2 V
√
π
iy
√

+∞

Z

2 V
√ Γ
π

z

iy
√
V

− 12 −z

e

dz

0



1
iy
+√
2
V


.

The CIR process: taking that ηi+1 ∼ N (0, 1) and εi has a (log-) Chi-squared probability density function,

if the Feller’s condition holds true (a =

2κθ
σ2

≥ 1) and c =

2κ
σ2

> 0, then the volatility process Xi is

PARAMETRIC INFERENCE

21

stationary ergodic and ρ − mixing. The stationary distribution fθ is the gamma distribution γ(a, c)
(see [Genon-Catalot et al., 1999]). On the other hand, the functions bθ , σθ and lθ are given by:
√
bθ (x) = (1 − κ)x + κθ and σθ (x) = σ x

b2θ (x) + σθ2 (x) γ(a, c)


=
(1 − κ)2 x2 + 2x (1 − κ) κθ + σ 2 + (κθ)2 fθ (x),

= α1 x2 + α2 x + α3 fθ (x).

lθ (x) =

where θ = (κ, µ, σ) and α1 = (1 − κ)2 , α2 = 2 (1 − κ) κθ + σ 2 , α3 = (κθ)2 .
Therefore
lθ∗ (t) = E[eitX lθ (X)]
= α1 E[X 2 eitX ] + α2 E[XeitX ] + α3 E[eitX ] with X ∼ Γ(a, c)
∂f ∗
∂2f ∗
= −α1 2θ (t) − iα2 θ (t) + α3 fθ∗ (t).
∂t
∂t
−a
After replacing fθ∗ (t) by 1 − itc
, we obtain:
"
lθ∗ (x) = −α1

#






−a
ix −a−2
ix −a−1
ix −a
a
(a + 1) 1 −
1−
+ α3 1 −
.
+ iα2
c2
c
c
c
c

It follows that for all θ, the squared norme of lθ is given by:
Z
2
2
klθ k =
b2θ + σθ2 (x) Γ2 (a, c)dx
Z

=
β1 x4 + β2 x3 + β3 x2 + β4 x + β5 Γ2 (a, c)dx,

where β1 = α12 , β2 = 2α1 α2 , β3 = 2α1 α3 + α22 , β4 = 2α2 α3 , β3 = α32 . Finally, using the non-centered
moments of a Gamma-distributed random variable, E[X r ] = Γ(a+r)
Γ(a)cr , we get:
Z
β1

4 2

c2a −2cx 2a−2
e
x
Γ2 (a)
Z
(2c)2a+3 −(2c)x (2a+3)−1
Γ(2a + 3)
= β1 2−(2a+3) c−3
e
x
dx
Γ2 (a)
Γ(2a + 3)
Z
−(2a+3) −3 Γ(2a + 3)
= β1 2
c
Γ(2a + 3, 2c)dx
Γ2 (a)
Γ(2a + 3)
= β1 2−(2a+3) c−3
Γ2 (a)

x Γ (a, c)dx = β1

Z

x4

22

SALIMA EL KOLEI AND FLORIAN PELGRIN

and
Z

Γ(2a + 2)
β2 2−(2a+2) c−2
Γ2 (a)

Z

Γ(2a + 2)
,
Γ2 (a)
Z
Z
Γ(2a + 1)
2 2
−(2a+1) −1 Γ(2a + 1)
β3 x Γ (a, c)dx = β3 2
c
Γ(2a + 1, 2c)dx = β3 2−(2a+1) c−1
,
Γ2 (a)
Γ2 (a)
Z
Γ(2a)
β4 xΓ2 (a, c)dx = β4 2−(2a) 2 ,
Γ (a)
Z
Γ(2a − 1)
β5 Γ2 (a, c)dx = β5 2−2a+1 c
.
Γ2 (a)

β2

3 2

x Γ (a, c)dx =

Γ(2a + 2, 2c)dx = β2 2−(2a+2) c−2

and the expression of the contrast function (8) is obtained. It is worth noting that the function u∗lθ (y)
defined in Definition 1 must be approximated numerically by using standard quadrature methods or
Fast Fourier Transform.

5.3.1 Proof of Theorems 1 and 2 for the CIR process

Mixing property: under the Feller’s condition, the volatility process Xt is ρ-mixing and so αmixing by using the strong Markov property.
Regularity conditions: for the CIR process, the function lθ is given by the following polynomial
function x 7→ (α1 x2 + α2 x + α3 )fθ (x) with α1 = (1 − κ)2 , α2 = 2 (1 − κ) κθ + σ 2 , α3 = (κθ)2 and for
all x this function is smooth w.r.t. θ ∈ Θ. Hence, it remains to prove the moment condition and the
local dominance to apply Theorem 2.
Since the function lθ is polynomial w.r.t θ belonging to the compact subset Θ, all the derivatives exist
and in particular supθ∈Θ lθ and supθ∈Θ ∇2θ lθ are finite. Furthermore, by combining the compactness
argument and as the Fourier transform f∗ satisfies (see [Fan et al., 1990]):

|fε∗ (x)|

=

√

 
 π 
1
2 exp − |x| 1 + O
,
2
|x|

|x| → ∞,

which means that fε is ordinary-smooth in its terminology, we obtain:
 

∗ (Y )

E
sup
ϕ(Y
)u
<∞

2
1
θ∈Θ
l

θ


 
2+δ
E ϕ(Y2 )u∗∇θ lθ (Y1 )
<∞







E supθ∈U ϕ(Y2 )u∗∇2 l (Y1 ) < ∞
θ θ

for some δ > 0,
for some neighbourhood U of θ0 .

PARAMETRIC INFERENCE

23

.
CREST, Ecole Nationale de la Statistique et de l’Analyse de l’Information (Ensai), Campus de KerLann, rue Blaise Pascal, BP 37203, 35172 Bruz cedex, France.
E-mail address, Salima El Kolei: salima.el kolei@ensai.fr
Ecole des Hautes Etudes Commerciales (EDHEC Business School), 24, avenue Gustave Delory CS
50411, 59057 Roubaix Cedex 1-France.
E-mail address, Florian Pelgrin: florian.pelgrin@edhec.edu

