Parametric
estimation
I. MENDY
Parametric estimation for sub-fractional
Ornstein-Uhlenbeck process
Ibrahim MENDY
Université de Ziguinchor
CIMPA Abidjan March 17-28, 2014
Outline
Parametric
estimation
I. MENDY
1
Introduction
2
Preliminaries
Sub-fractional Brownian motion
Malliavin calculus associated with S H .
Young integral with S H
3
Main results
Asymptotic behavior of the least squares estimator
Asymptotic distribution of the estimator LSE
Outline
Parametric
estimation
I. MENDY
1
Introduction
2
Preliminaries
Sub-fractional Brownian motion
Malliavin calculus associated with S H .
Young integral with S H
3
Main results
Asymptotic behavior of the least squares estimator
Asymptotic distribution of the estimator LSE
Outline
Parametric
estimation
I. MENDY
1
Introduction
2
Preliminaries
Sub-fractional Brownian motion
Malliavin calculus associated with S H .
Young integral with S H
3
Main results
Asymptotic behavior of the least squares estimator
Asymptotic distribution of the estimator LSE
Introduction
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
X0 = 0, dXt = θXt dt + dStH , t ≥ 0
S H is a sub-fbm with Hurst index H >
1
2
(1.1)
and θ ∈ (−∞, ∞).
Introduction
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
• S H is a BM, θ̂ has been studied by LIPTSER and
SHIRYAEV 2001, KUTOYANTS 2004, BASAWA and
SCOTT 1983, DIETZ and KUTOYANTS 2003.
• S H is an α-stable Lévy motion in the equation (1.1), HU
and LONG 2007 studied θ̂
• S H FBM, θ̂ has been studied by LEBRETON 1998,
LEBRETON 2002, PRAKASA RAO 2008, HU and
NUALART 2010, BELFADI 2011.
Introduction
Parametric
estimation
I. MENDY
Introduction
Our goal is to study θ̂, in the case θ > 0, by using the least
squares estimator defined by
Preliminaries
Rt
θ̂t = R0t
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
0
Xs dXs
Xs2 ds
,
where the integral 0t Xs dXs is interpreted as a Young
integral.
This least squares estimator is obtained by the least
squares technique, that θ̂t minimizes
R
θ 7→
Z t
0
|Ẋs + θXs |2 ds.
(1.2)
Outline
Parametric
estimation
I. MENDY
Introduction
1
Introduction
2
Preliminaries
Sub-fractional Brownian motion
Malliavin calculus associated with S H .
Young integral with S H
3
Main results
Asymptotic behavior of the least squares estimator
Asymptotic distribution of the estimator LSE
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Definition
• BOJDECKY 2004 introduced and studied a rather
special class of self-similar Gaussian processes which
preserves many properties of fractional Brownian
motion.
• Sub-fBm with index H ∈ (0, 1) is mean zero Gaussian
process {StH , t ≥ 0} with S0H = 0 and the covariance
1
CH (t, s) = E[StH SsH ] = s2H + t 2H − [(s + t)2H + |s − t|2H ](2.1)
2
for all s, t ≥ 0.
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Properties
• S H is neither a semimartingale nor a Markov process
unless H = 21 , so many of the powerful techniques from
stochastic analysis are not available when dealing with
SH .
• The sub-fractional Brownian motion has properties
analogous to those of fractional Brownian motion
(self-similarity, long-range dependence, Hölder paths),
and satisfies the following estimates:
[(2 − 22H−1 ) ∧ 1]|t − s|2H ≤ E|StH − SsH |2
≤ [(2 − 22H−1 ) ∨ 1]|t − s|2H
(2.2)
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Properties
HS H is the closure of the linear span E generated by the
indicator function 1[0,t] , t ∈ [0, T ] with respect to the scalar
product
h1[0,t] , 1[0,s] iHSH = CH (t, s).
We know that the covariance of sub-fractional Brownian
motion can be written as
E(StH SsH ) =
Z tZ s
0
0
φH (u, v )dudv = CH (s, t)
(2.3)
where φH (u, v ) = H(2H − 1)[|u − v |2H−2 − (u + v )2H−2 ] and
1
2 < H < 1.
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Formulae (2.3) implies that
hϕ, ψiHSH =
Z tZ t
0
0
ϕu ψv φH (u, v )dudv
(2.4)
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
for any pair step functions ϕ and ψ on [0, T ].
Isometry between the Hilbert space HS H and L2 ([0, T ]).
Preliminaries
Parametric
estimation
I. MENDY
Introduction
The canonical Hilbert space HS H associate to S H satisfies:
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Lemma
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
1
L2 ([0, T ]) ⊂ L H ([0, T ]) ⊂ HS H ,
where H > 21 .
(2.5)
Outline
Parametric
estimation
I. MENDY
Introduction
1
Introduction
2
Preliminaries
Sub-fractional Brownian motion
Malliavin calculus associated with S H .
Young integral with S H
3
Main results
Asymptotic behavior of the least squares estimator
Asymptotic distribution of the estimator LSE
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Malliavin calculus associated with S H .
We refer to NUALART 2006 for detailed account these
notions.
The derivative operator D of a smooth cylindrical random
variables F = f (S H (ϕ1 ), ..., S H (ϕn )) is defined as the
HS H −valued random variable
DF =
n
X
∂f
j=1
∂xj
(S H (ϕ1 ), ..., S H (ϕn ))ϕj .
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Malliavin calculus associated with S H .
For any integer we denote by Dk ,p (k , p ≥ 1) the Sobolev
space defined as the closure of the space of smooth
random variables with respect to the norm
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
kF kpk ,p = E(|F |p ) +
k
X
j=1
kD j F kpp
L (Ω,H
⊗j
)
SH
.
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Malliavin calculus associated with S H .
Consider the adjoint δ of D in L2 (Ω, HS H ). Its domain is the
class of elements u ∈ L2 (Ω, HS H ) such that
E(hDF , uiHSH ) ≤ CkF kL2 (Ω) ,
Main results
Asymptotic
behavior of the
least squares
estimator
for any F ∈ D1,2 , and δ(u) is the element of L2 (Ω) given by
Asymptotic
distribution of the
estimator LSE
E(δ(u)F ) = E(hDF , uiHSH )
for any F ∈ D1,2 .
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Malliavin calculus associated with S H .
We will make use the notation
Z T
δ(u) =
0
us δSsH , u ∈ Dom(δ).
It is well-known that D1,2 (HS H ) is included in the domain of
δ.
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Malliavin calculus associated with S H .
Note that E(δ(u)) = 0 and the variance of δ(u) is given by
E(δ(u)2 ) = E(kuk2H H ) + E(hDu, (Du)∗ iHSH ⊗HSH ),
S
if u ∈ D1,2 (HS H ), where (Du)∗ is the adjoint of Du in the
Hilbert space HS H ⊗ HS H .
(2.6)
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Malliavin calculus associated with S H
We also need the commutativity relationship between D and
δ
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Z 1
Dδ(u) = u +
0
Dus δBs ,
(2.7)
if u ∈ D1,2 (HB ) and the process {Dus , s ∈ [0, 1]} belong to
the domain of δ.
Outline
Parametric
estimation
I. MENDY
Introduction
1
Introduction
2
Preliminaries
Sub-fractional Brownian motion
Malliavin calculus associated with S H .
Young integral with S H
3
Main results
Asymptotic behavior of the least squares estimator
Asymptotic distribution of the estimator LSE
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Young integral with S H
Let f , g : [0, T ] → R are Hölder continuous functions of
order α ∈ (0, 1) and β ∈ (0, 1) with α + β > 1.
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
• YOUNG 1936 proved that
the Riemman-Stiltjes (so
RT
called Young integral) 0 f (s)dg(s) exists.
• If α = β ∈ ( 21 , 1) and φ : R2 → R is a function of class
C 1 , the integrals
Z .
∂φ
Asymptotic
distribution of the
estimator LSE
0
and
∂f
Z .
∂φ
0
∂g
exist in Young sense.
(f (u), g(u))df (u)
(f (u), g(u))dg(u)
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Young integral with S H
and the following change variables holds:
φ(f (t), g(t)) = φ(f (0), g(0)) +
Z t
∂φ
0
Main results
Asymptotic
behavior of the
least squares
estimator
+
Z t
∂φ
0
Asymptotic
distribution of the
estimator LSE
for 0 ≤ t ≤ T .
∂g
∂f
(f (u), g(u))df (u)
(f (u), g(u))dg(u),
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Young integral with S H
As a consequence, if 12 < H < 1 and (ut , t ∈ [0, T ]) be a
process with Hölder paths of order α > 1 − H, the integral
RT
H
0 us dSs is well-defined as young integral.
Preliminaries
Parametric
estimation
I. MENDY
Young integral with S H
Introduction
Preliminaries
Suppose moreover that for any t ∈ [0, T ], ut ∈ D1,2 , and
Sub-fractional
Brownian motion
Z TZ T
Malliavin calculus
associated with
SH .
P
Young integral with
SH
0
0
!
2H−2
|Ds ut ||t − s|
dsdt < ∞
= 1.
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Then, by the same argument as in ALOS and NUALART
2003 we have
Z t
0
us dSsH
Z t
=
0
us δSsH
Z tZ t
+
0
0
Ds ur φH (s, r )dsdr .
(2.8)
Preliminaries
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Young integral with S H
In particular, when u is a non-random Hölder continuous
function of order α > 1 − H, we obtain
Z t
0
us dSsH =
Z t
0
us δSsH .
(2.9)
Outline
Parametric
estimation
I. MENDY
Introduction
1
Introduction
2
Preliminaries
Sub-fractional Brownian motion
Malliavin calculus associated with S H .
Young integral with S H
3
Main results
Asymptotic behavior of the least squares estimator
Asymptotic distribution of the estimator LSE
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Parametric
estimation
The linear equation (1.1) has the following explicit solution:
I. MENDY
Introduction
Xt = e
θt
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
0
e−θs dSsH , t ≥ 0
(3.1)
Let us introduce the following process
Young integral with
SH
Main results
Z t
Z t
Yt =
0
e−θs dSsH , t ≥ 0.
By using the equation (1.1) and (3.1) we can write the least
square estimator θ̂t defined in (1.2) as follows
Rt
0
Xs dSsH
θ̂t = θ + R t
2
0 Xs ds
R t sθ
e Ys dSsH
= θ + R0t
2
2θs Y ds
s
0e
(3.2)
Main resultats
Parametric
estimation
I. MENDY
Introduction
Consistency of the estimator LSE
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
The following theorem proves the strong consistency of the
LSE θ̂t .
Theorem
Assume H ∈ ( 12 , 1), then
θ̂t → θ almost surely, as t → ∞.
Main resultats
Parametric
estimation
I. MENDY
Consistency of the estimator LSE
Introduction
Preliminaries
For the proof of theorem we need the following two lemmas.
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Lemma
Suppose that H ∈ ( 12 , 1). Then
i) For all ε ∈ (0, H), the process Y admits a modification
with (H − ε)-Hölder continuous paths, still denoted Y in
the sequel.
ii) Yt → Y∞ =
t → ∞.
R ∞ −θr
dSsH almost surely and in L2 (Ω) as
0 e
Main resultats
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Consistency of the estimator LSE
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Lemma
Suppose that H ∈ ( 12 , 1). Then, as t → ∞
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
e−2θt
Z t
0
Xs2 ds = e−2θt
Z t
0
e2θs Ys2 ds →
2
Y∞
almost surely.
2θ
Outline
Parametric
estimation
I. MENDY
Introduction
1
Introduction
2
Preliminaries
Sub-fractional Brownian motion
Malliavin calculus associated with S H .
Young integral with S H
3
Main results
Asymptotic behavior of the least squares estimator
Asymptotic distribution of the estimator LSE
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Main resultats
Parametric
estimation
I. MENDY
Introduction
Asymptotic distribution of the estimator LSE
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Theorem
Assume H > 12 . Then, as t → ∞,
law
eθt (θ̂t − θ) →
2θ`(H)
C(1),
2]
E[Y∞
where C(1) the standardRCauchy distribution and `(H) is the
limiting variance of e−θt 0t eθs dSsH .
Main resultats
Parametric
estimation
I. MENDY
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Young integral with
SH
Main results
Asymptotic distribution of the estimator LSE
In order to prove Theorem we need the following two
lemmas.
Lemma
Assume H > 12 . Let F be any σ(S H )−mesurable random
variable such that P(F < ∞) = 1. Then, as t → ∞,
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
F, e
−θt
!
Z t
e
0
θs
dSsH
law
→ (F , `(H)N)
where N ∼ N (0, 1) is independent
of S H and `(H) is the
R t θs H
−θt
limiting variance of e
0 e dSs .
Main resultats
Parametric
estimation
I. MENDY
Asymptotic distribution of the estimator LSE
Introduction
Preliminaries
Sub-fractional
Brownian motion
Malliavin calculus
associated with
SH .
Lemma
Assume H > 12 . Then, as t → ∞,
Young integral with
SH
θt
e− 2
Main results
Asymptotic
behavior of the
least squares
estimator
Asymptotic
distribution of the
estimator LSE
Z t
0
δSsH e−θs
Z s
0
δSrH eθr → 0 in L2 (Ω)
(3.3)
and
e
− θt
2
Z t
dse
0
−θs
Z s
0
dreθr φH (s, r ) → 0
(3.4)