1. Introduction
Statistical inference for stochastic equations is a main research direction in probability theory and its applications. When the noise is a standard Brownian motion or a Lévy process, such problems have been extensively studied. Some surveys and complete literature for this direction could be found in Bishwal [
1], Iacus [
2], Kutoyants [
3], Liptser and Shiryaev [
4], Prakasa Rao [
5], and the references therein. However, in contrast to the extensive studies on semimartingale types, other statistical inferences associated with some Gaussian processes are very limited, and a common denominator in all these works is that it is assumed that the equation admits only an unknown parameter. Let us consider the parameter estimates of the Vasicek-type model driven by a Gaussian process
G:
where
,
are two parameters.
When
and
G is a fractional Brownian motion with Hurst index
, the question has been studied by many authors. We mention the works of Berzin et al. [
6], Es-Sebaiy [
7], Es-Sebaiy and Nourdin [
8], Hu and Nualart et al. [
9,
10], Kleptsyna and Le Breton [
11], Prakasa Rao [
12], and the references therein for results on parameter estimation of stochastic equations driven by the fractional Brownian motion (fBm). When
G is not a fractional Brownian motion, the research for this question is very limited. For
and
G a sub-fractional Brownian motion, Mendy [
13] considered the least squares estimation of
and studied the consistency and asymptotic behavior. For
and
G a Gaussian process, El Machkouri et al. [
14] showed the strong consistency and the asymptotic distribution of the least squares estimator
of
based on the properties of
G, and as some examples, the authors also studied the three Vasicek-type models driven by fractional Brownian motion, sub-fractional Brownian motion, and bi-fractional Brownian motion, respectively.
Motivated by these above results and for simplicity, in this paper, we consider the least squares estimation of Equation (
1) when
G is a sub-fractional Brownian motion
with Hurst index
and both
and
are unknown. That is, the parameter estimation of the so-called Vasicek-type model driven by sub-fractional Brownian motion:
where
is a sub-fractional Brownian motion and
,
are two unknown parameters. On the other hand, there exists still a practical motivation for studying the parameter estimation, that is to provide optional tools to understand volatility modeling in finance. In fact, any mean-reverting model in continuous or discrete observations can be regarded as a model for stochastic volatility. We can consult the research monograph [
15] for this modeling idea. Since stochastic volatility is not observed for many financial markets and the sub-fractional Brownian motion is a process without ergodicity, the discussions on the parameter estimation based on discrete observations are beyond the scope of this article. For the sake of simplicity, we focus on tackling the least squares estimation of Equation (
2) based on the so-called continuous observations.
The so-called sub-fractional Brownian motion (sub-fBm in short)
with index
is introduced by Bojdecki et al. [
16], which arises from occupation time fluctuations of branching particle systems with the Poisson initial condition. It is a mean zero Gaussian process with
and:
for all
. For
,
coincides with the standard Brownian motion
B. Sub-fBm
is neither a semimartingale nor a Markov process unless
. The sub-fBm has many properties analogous to those of fractional Brownian motion such as self-similarity, long/short-range dependence, and Hölder paths. However, it has no stationary increments. Moreover, it admits the estimates:
More works for sub-fractional Brownian motion can be found in Bojdecki Y et al. [
17,
18], Li and Xiao [
19], Shen and Yan [
20], Sun and Yan [
21,
22], Tudor [
23,
24,
25,
26], Yan et al. [
27,
28], and the references therein. On the other hand, in contrast to the extensive studies on fractional Brownian motion, there has been little systematic investigation on other self-Gaussian processes. The main reason for this is the complexity of dependence structures, and in general, these Gaussian processes have no stationary increments and the representation based on Wiener integral with respect to a Brownian motion. Therefore, it seems interesting to study the asymptotic behavior associated with other self-Gaussian processes.
Now, we consider Equation (
2) with
and
. Clearly, we have:
for all
, and the trajectory of
X is
-Hölder continuous for all
(see
Section 3). As an immediate result, we see that the Young integral
is well defined for all
. Let now the system Equation (
2) be observed continuously, and let
H be known. By using the least squares method due to Hu and Nualart [
10], the least squares estimators of
and
can be motivated by minimizing the contrast function:
Minimizing the above contrast function
, we introduce estimators of
and
as follows:
and:
where the stochastic integral
is a Young integral for
. Our main statement is as follows:
• The least squares estimators
and
are strong consistent, and we have:
and:
in distribution, as
T tends to infinity, where
are mutually independent,
,
, and:
This paper is organized as follows. In
Section 2, we present some preliminaries for sub-fBm. In
Section 3, we prove the consistence of
and
. In
Section 4, we investigate the asymptotic distribution of estimators
and
.
2. Preliminaries
In this section, we briefly recall some basic definitions and results of sub-fBm. Throughout this paper, we assume that
is arbitrary, but fixed, and let
be a one-dimensional sub-fBm with Hurst index
H and defined on
.
can be written as a Volterra process, and it is also possible to construct a stochastic calculus of variations with respect to the Gaussian process
, which will be related to the Malliavin calculus. Some surveys and complete literature for Malliavin calculus of the Gaussian process could be found in Alòs et al. [
29], Nualart [
30], and Tudor [
25,
26].
Recall that a mean zero Gaussian process
with Hurst index
is called the sub-fractional Brownian motion (sub-fBm) if
and the covariance:
for all
. Consider the kernel
by:
where
denotes the Erdély–Kober-type fractional integral operator defined by:
for all measurable functions
,
,
. Some basic properties of this fractional integral can be found in Samko et al. [
31]. By using the kernel
, we have the Wiener integral representation (in distribution) of sub-fBm
as follows:
for some standard Brownian motion, where:
Let
be the family of elementary functions
of the form:
and let
be the completion of the linear space
with respect to the inner product:
When
, we can characterize
as:
with
. When
, we have:
and
, and:
As usual, we define the linear mapping
on
by:
for all
. Then, the linear mapping is an isometry from
to the Gaussian space generated by
, and it can be extended to
and:
for any
, which is called the Wiener integral with respect to
, denoted by:
for any
. If the Wiener integral
is well defined for every
, we then can define the integral:
for any
satisfying:
Thus, we can call Equation (
12) the indefinite Wiener integral. Denote by
the set of smooth functionals of the form:
where
(
f and all its derivatives are bounded) and
. Denote by
and
the Malliavin derivative and divergence integral operator associated with sub-fractional Brownian motion
, respectively. Then, we have:
We denote by
the closure of
with respect to the norm:
for
. The divergence integral
is the adjoint of derivative operator
and:
for
. We will use the notation:
to express the Skorohod integral of an adapted process
u, and the indefinite Skorohod integral is defined as
. Clearly, the divergence integral is closed in
.
Finally, we recall Young’s integration and some results established in Bertoin [
32] and Föllmer [
33]. A Borel function
f on
is said to be of bounded
p-variation with
if:
where the supremum is taken over all partitions
of
. The estimates Equation (
4) and the normality imply that the sub-fractional Brownian motion
admits almost surely a bounded
-variation on any finite interval for any sufficiently small
. That is, we have:
for all
and
. The definition of
p-variation for processes is slightly different. We say that the continuous adapted process
Z has a locally-bounded
p-variation if there exists an increasing sequence of stopping times
such that
, a.s., as
and
has a bounded
p-variation for all
n. It is easy to prove that if
Y is an adapted continuous process, such that for
P-a.s.
and all positive
, the function
has a bounded
p-variation on
, then the process
Y has a locally-bounded
p-variation.
Let
X and
Y be two adapted continuous processed with locally-bounded
p and
q variations, respectively, such that
, then one can define (see, for example, Bertoin [
32]):
as the limit in probability of a Riemann sum, which generalizes the usual integral when
X or
Y are semimartingales, and
Z has a locally-bounded
p-variation. Moreover, Bertoin [
32] showed that
has a locally-bounded
q-variation and:
provided
is an adapted continuous process with locally-bounded
q-variation.
Lemma 1 (Föllmer [
33])
. Let U and V be two continuous adapted processes with locally-bounded p-variation (). Then, and have locally-bounded two-variations, and It’s formula:holds for all . In particular, we have the integration by parts formula:for all . Corollary 1. Let . If u is a continuous adapted process with bounded q-variations with , then Young’s integral:is well-defined and:for all . Corollary 2 (Alós et al. [
29])
. Let . If u is a continuous adapted process with bounded q-variations with and , we then have:for all .