Minimum $L_{1}$-norm Estimation for Fractional

Ornstein-Uhlenbeck Process Driven by a Gaussian Process

B.L.S. Prakasa Rao 111 2000 Mathematics Subject Classification: Primary 60G22,
62 M 09.
Keywords and phrases: Minimum $L_{1}$-norm estimation; Fractional Ornstein-
Uhlenbeck type process; Fractional Brownian motion; Gaussian Process.

CR Rao Advanced Institute of Mathematics, Statistics

and Computer Science, Hyderabad, India

Abstract: We investigate the asymptotic properties of the minimum $L_{1}$-norm
estimator of the drift parameter for fractional Ornstein-Uhlenbeck type
process driven by a general Gausssian process.

##  1 Introduction

Diffusion processes and diffusion type processes satisfying stochastic
differential equations driven by Wiener processes are used for stochastic
modeling in a wide variety of sciences such as population genetics, economic
processes, signal processing as well as for modeling sunspot activity and more
recently in mathematical finance. Statistical inference for diffusion type
processes satisfying stochastic differential equations driven by Wiener
processes have been studied earlier and a comprehensive survey of various
methods is given in Prakasa Rao (1999). There has been a recent interest to
study similar problems for stochastic processes driven by a fractional
Brownian motion to model processes involving long range dependence (cf.
Prakasa Rao (2010)). Le Breton (1998) studied parameter estimation and
filtering in a simple linear model driven by a fractional Brownian motion.
Kleptsyna and Le Breton (2002) studied parameter estimation problems for
fractional Ornstein-Uhlenbeck process. The fractional Ornstein-Uhlenbeck
process is a fractional analogue of the Ornstein-Uhlenbeck process, that is, a
continuous time first order autoregressive process $X=\\{X_{t},t\geq 0\\}$
which is the solution of a one-dimensional homogeneous linear stochastic
differential equation driven by a fractional Brownian motion (fBm)
$W^{H}=\\{W_{t}^{H},t\geq 0\\}$ with Hurst parameter $H.$ Such a process is
the unique Gaussian process satisfying the linear integral equation

(1. 1) $X_{t}=x_{0}+\theta\int_{0}^{t}X_{s}ds+\sigma W_{t}^{H},t\geq 0.$

They investigate the problem of estimation of the parameters $\theta$ and
$\sigma^{2}$ based on the observation $\\{X_{s},0\leq s\leq T\\}$ and study
the asymptotic behaviour of these estimators as $T\rightarrow\infty.$

In spite of the fact that maximum likelihood estimators (MLE) are consistent
and asymptotically normal and also asymptotically efficient in general, they
have some short comings at the same time. Their calculation is often
cumbersome as the expression for MLE involve stochastic integrals at times
which need good approximations for computational purposes. Further more MLE
are not robust in the sense that a slight perturbation in the noise component
will change the properties of MLE substantially. In order to circumvent such
problems, the minimum distance approach is proposed. Properties of the minimum
distance estimators (MDE) were discussed in Millar (1984) in a general frame
work. Kutoyants and Pilibossian (2000) studied the problem of minimum
$L_{1}$-norm estimation for the Ornstein-Uhlenbeck process. Prakasa Rao (2004)
investigated the problem of minimum $L_{1}$-norm estimation for the fractional
Ornstein-Uhlenbeck process driven by a fractional Brownian motion.

Our aim in this paper is to obtain the minimum $L_{1}$-norm estimator of the
drift parameter of a Ornstein-Uhlenbeck process driven by general Gaussian
processes and investigate the asymptotic properties of such estimators. El
Machkouri et al. (2015), Chen and Zhou (2020) and Lu (2022) study parameter
estimation for an Ornstein-Uhlenbeck process driven by a general Gaussian
process.

##  2 Minimum $L_{1}$-norm Estimation

Let $(\Omega,{\cal F},({\cal F}_{t}),P)$ be a stochastic basis satisfying the
usual conditions and the processes discussed in the following are $({\cal
F}_{t})$-adapted. Further the natural filtration of a process is understood as
the $P$-completion of the filtration generated by this process. We consider
centered a Gaussian process $G\equiv\\{G_{t},0\leq t\leq 1\\}.$ Let us
consider a stochastic process $\\{X_{t},t\in[0,1]\\}$ defined by the
stochastic integral equation

(2. 1) $X_{t}=x_{0}+\theta\int_{0}^{t}X(s)ds+\varepsilon G_{t},0\leq t\leq 1,$

where $\theta$ is an unknown drift parameters respectively. For convenience,
we write the above integral equation in the form of a stochastic differential
equation

(2. 2) $dX_{t}=\theta X(t)dt+\varepsilon dG_{t},X_{0}=x_{0},0\leq t\leq 1,$

driven by the Gaussian process $G.$ The class of Gaussian processes $G$
includes fractional Brownian motion, sub-fractional Brownian motion and
bifractional Brownian motion. We now consider the problem of estimation of the
parameter $\theta$ based on the observation of fractional Ornstein-Uhlenbeck
process $X=\\{X_{t},0\leq t\leq 1\\}$ satisfying the stochastic differential
equation

(2. 3) $dX_{t}=\theta X(t)dt+\varepsilon dG_{t},X_{0}=x_{0},0\leq t\leq 1$

where $\theta\in\Theta\subset R$ and study its asymptotic properties as
$\varepsilon\rightarrow 0.$ Let $x_{t}(\theta)$ be the solution of the above
differential equation with $\varepsilon=0.$ It is obvious that

(2. 4) $x_{t}(\theta)=x_{0}e^{\theta t},0\leq t\leq 1.$

Let

(2. 5) $S_{1}(\theta)=\int_{0}^{1}|X_{t}-x_{t}(\theta)|dt$

We define $\theta_{\varepsilon}^{*}$ to be a minimum $L_{1}$-norm estimator if
there exists a measurable selection $\theta_{\varepsilon}^{*}$ such that

(2. 6) $S_{1}(\theta_{\varepsilon}^{*})=\inf_{\theta\in\Theta}S_{1}(\theta).$

Conditions for the existence of a measurable selection are given in Lemma
3.1.2 in Prakasa Rao (1987). We assume that there exists a measurable
selection $\theta_{\varepsilon}^{*}$ satisfying the above equation. An
alternate way of defining the estimator $\theta_{\varepsilon}^{*}$ is by the
relation

(2. 7)
$\theta_{\varepsilon}^{*}=\arg\inf_{\theta\in\Theta}\int_{0}^{1}|X_{t}-x_{t}(\theta)|dt.$

We will now present some maximal inequalities for Gaussian processes out of
which one will be used in the sequel. Some Maximal Inequalities for Gaussian
processes: Let $G_{1}^{*}=\sup_{0\leq t\leq 1}|G_{t}|.$ If $G$ is a fractional
Brownian motion or a sub-fractional Brownian motion, maximal inequalities are
known and are reviewed in Prakasa Rao (2014, 2017, 2020) following self-
similarity for such processes. Maximal inequalities for general Gaussian
processes are surveyed in Li and Shao (2001). We now present some maximal
inequalities for general Gaussian processes $G.$ Let

$d_{G}(s,t)=[E(G_{t}-G_{s})^{2}]^{1/2},0\leq s,t\leq 1,$
$\rho(\varepsilon)=\sup_{t,s\in[0,1],|s-t|\leq\varepsilon}d_{G}(s,t)$

and

$Q(\delta)=\int_{0}^{\infty}\rho(\delta e^{-y^{2}})dy.$

Suppose that the function $\rho(.)$ is strictly increasing. Let
$\sigma^{2}=\sup_{0\leq t\leq 1}E(G_{t}^{2})$ and $Q^{-1}(.)$ denote the
inverse of the function $Q(.)$. The following result is due to Berman (1985).
Theorem 2.1: Under the conditions stated above, given a Gaussian process $G$
defined on the interval $[0,1],$ there exists a constant $C$ such that

$P(\sup_{0\leq t\leq 1}|G_{t}|>\varepsilon)\leq
C(Q^{-1}(1/\varepsilon))^{-1}\exp(-\frac{\varepsilon^{2}}{2\sigma^{2}}).$

for all $\epsilon>0.$ Consider a Gaussian process $G$ on the interval $[0,1]$
and let $D=\sup_{0\leq s,t\leq 1}d_{G}(s,t).$ Let $N(\varepsilon,d_{G},1)$
denote the minimum number of closed intervals of length $2\varepsilon>0,$
needed to cover the interval $[0,1]$ with mid-points in $[0,1]$ in the semi-
metric $d_{G}.$ We say that the Gaussian process $G$ is pairwise non-
degenerate if for every $\epsilon>0$ there exist closed intervals of length
$2\varepsilon>0,$ in the semi-metric $d_{G}$ needed to cover the Interval
$[0,1].$ If the Gaussian process is pairwise non-degenerate, then the semi-
metric $d_{G}$ is a metric. The following result is due to Marcus and Rosen
(2006). Theorem 2.2: Suppose $G$ is a Gaussian process which is pairwise non-
degenerate and continuous in mean square. If

$\int_{0}^{D}\sqrt{\log N(\epsilon,d_{G},1)}d\epsilon<\infty,$

then the process $G$ has continuous sample paths and there exists a positive
constant $C$ such that

$E[\sup_{0\leq s\leq 1}|G(s)|]\leq C\int_{0}^{D/2}\sqrt{\log
N(\epsilon,d_{G},1)}d\epsilon<\infty.$

The following result is due to Borovkov et al. (2017). Theorem 2.3: Suppose
$G$ is a Gaussian process on the interval $[0,1]$ which is pairwise non-
degenerate and there exists constants $0\leq C_{1}\leq C_{2}$ and $H\in(0,1)$
such that

$C_{1}|t-s|^{H}\leq d_{G}(s,t)\leq C_{2}|t-s|^{H},t,s\in[0,1].$

Then

$\frac{C_{1}}{5\sqrt{H}}\leq E[\sup_{0\leq t\leq
1}G_{t}]\leq\frac{(16.3)C_{2}}{\sqrt{H}}.$

Another useful maximal inequality for Gaussian process is the following result
due to Nourdin (2012).

Theorem 2.4: Suppose $G$ is a centered and continuous Gaussian process on the
interval $[0,1]$. Let $\sigma^{2}=\sup_{0\leq t\leq 1}E[G_{t}^{2}].$ Suppose
that $0<\sigma^{2}<\infty.$ Then $m=E[\sup_{0\leq t\leq 1}G_{t}]$ is finite
and for all $x>m,$

$P(\sup_{0\leq t\leq 1}G_{t}\geq x)\leq\exp(-\frac{(x-m)^{2}}{2\sigma^{2}}).$

From the fact that $G$ is centered Gaussian process, it follows that

$P(\sup_{0\leq t\leq 1}|G_{t}|\geq x)\leq
2\;\exp(-\frac{(x-m)^{2}}{2\sigma^{2}})$

under the conditions stated in Theorem 2.4.

##  3 Consistency of the estimator:

Let $\theta_{0}$ denote the true parameter. For any $\delta>0,$ define

(3. 1)
$g(\delta)=\inf_{|\theta-\theta_{0}|>\delta}\int_{0}^{1}|x_{t}(\theta)-x_{t}(\theta_{0})|dt.$

Note that $g(\delta)>0$ for any $\delta>0.$ Theorem 3.1: Suppose $G$ is a
centered and continuous Gaussian process on the interval $[0,1]$. Let
$\sigma^{2}=\sup_{0\leq t\leq 1}E[G_{t}^{2}]$ and $m=E[\sup_{0\leq t\leq
1}G_{t}].$ Then there exists a positive constant $C$ such that for every
$\delta>0,$

$P_{\theta_{0}}^{(\varepsilon)}\\{|\theta_{\varepsilon}^{*}-\theta_{0}|>\delta\\}=O(e^{-C[g(\delta)]^{2}\varepsilon^{-2}}).$

Proof: Let $||.||$ denote the $L_{1}$-norm. Then

$\displaystyle
P_{\theta_{0}}^{(\varepsilon)}\\{|\theta_{\varepsilon}^{*}-\theta_{0}|>\delta\\}$
$\displaystyle=$ $\displaystyle
P_{\theta_{0}}^{(\varepsilon)}\\{\inf_{|\theta-\theta_{0}|\leq\delta}||X-x(\theta)||>\inf_{|\theta-\theta_{0}|>\delta}||X-x(\theta)||\\}$
$\displaystyle\leq$ $\displaystyle
P_{\theta_{0}}^{(\varepsilon)}\\{\inf_{|\theta-\theta_{0}|\leq\delta}(||X-x(\theta_{0})||+||x(\theta)-x(\theta_{0})||)$
$\displaystyle\;\;\;\;\;>\inf_{|\theta-\theta_{0}|>\delta}(||x(\theta)-x(\theta_{0})||-||X-x(\theta_{0})||)\\}$
$\displaystyle=$ $\displaystyle
P_{\theta_{0}}^{(\varepsilon)}\\{2||X-x(\theta_{0})||>\inf_{|\theta-\theta_{0}|>\delta}||x(\theta)-x(\theta_{0})||\\}$
$\displaystyle=$ $\displaystyle
P_{\theta_{0}}^{(\varepsilon)}\\{||X-x(\theta_{0})||>\frac{1}{2}g(\delta)\\}.$

Since the process $X$ satisfies the stochastic differential equation (2.1), it
follows that

$\displaystyle X_{t}-x_{t}(\theta_{0})$ $\displaystyle=$ $\displaystyle
x_{0}+\theta_{0}\int_{0}^{t}X_{s}ds+\varepsilon G_{t}-x_{t}(\theta_{0})$
$\displaystyle=$
$\displaystyle\theta_{0}\int_{0}^{t}(X_{s}-x_{s}(\theta_{0}))ds+\varepsilon
G_{t}$

since $x_{t}(\theta)=x_{0}e^{\theta t}.$ Let $U_{t}=X_{t}-x_{t}(\theta_{0}).$
Then it follows from the equation given above that

(3. 4) $U_{t}=\theta_{0}\int_{0}^{t}U_{s}\;\;ds+\varepsilon G_{t}.$

Let $V_{t}=|U_{t}|=|X_{t}-x_{t}(\theta_{0})|.$ The relation given above
implies that

(3. 5)
$V_{t}=|X_{t}-x_{t}(\theta_{0})|\leq|\theta_{0}|\int_{0}^{t}V_{s}ds+\varepsilon|G_{t}|.$

Applying the Gronwall-Bellman Lemma, it follows that

(3. 6) $\sup_{0\leq t\leq 1}|V_{t}|\leq\varepsilon e^{|\theta_{0}|}\sup_{0\leq
t\leq 1}|G_{t}|.$

Hence

$\displaystyle
P_{\theta_{0}}^{(\varepsilon)}[||X-x(\theta_{0})||>\frac{1}{2}g(\delta)]$
$\displaystyle\leq$ $\displaystyle P[\sup_{0\leq t\leq
1}|G_{t}|>\frac{e^{-|\theta_{0}|}g(\delta)}{2\varepsilon}]$ $\displaystyle=$
$\displaystyle P[G_{1}^{*}>\frac{e^{-|\theta_{0}|}g(\delta)}{2\varepsilon}].$

Let $m=E[\sup_{0\leq t\leq 1}G(t)].$ Applying the maximal inequalities for
Gaussian processes given in Theorem 2.4, we get that, for fixed $\delta>0,$ we
can choose $\epsilon$ sufficiently small so that
$\frac{e^{-|\theta_{0}|}g(\delta)}{2\epsilon}>m.$ For such $\varepsilon,$

$\displaystyle
P_{\theta_{0}}^{(\varepsilon)}[|\theta_{\varepsilon}^{*}-\theta_{0}|>\delta]$
$\displaystyle\leq$ $\displaystyle
2\exp(-\frac{((e^{-|\theta_{0}|}g(\delta)/2\epsilon)-m)^{2}}{2\sigma^{2}})$
$\displaystyle=$ $\displaystyle O(e^{-C[g(\delta)]^{2}\varepsilon^{-2}})$

for some positive constant $C$ independent of $\varepsilon.$ Remarks:  As a
consequence of the result obtained above, it follows that

$P_{\theta_{0}}^{(\varepsilon)}\\{|\theta_{\varepsilon}^{*}-\theta_{0}|>\delta\\}\rightarrow
0\;\;\mbox{as}\;\;\epsilon\rightarrow 0$

for every $\delta>0.$ Hence the minimum norm $L_{1}$-estimator
$\theta_{\varepsilon}^{*}$ is weakly consistent for estimating the parameter
$\theta_{0}.$

##  4 Asymptotic distribution of the estimator:

We will now study the asymptotic distribution if any of the estimator
$\theta_{\varepsilon}^{*}$ after suitable scaling. It can be checked that

(4. 1) $X_{t}=e^{\theta_{0}t}\\{x_{0}+\int_{0}^{t}e^{-\theta_{0}s}\varepsilon
dG_{s}\\}$

or equivalently

(4. 2) $X_{t}-x_{t}(\theta_{0})=\varepsilon
e^{\theta_{0}t}\int_{0}^{t}e^{-\theta_{0}s}dG_{s}.$

Let

(4. 3) $Y_{t}=e^{\theta_{0}t}\int_{0}^{t}e^{-\theta_{0}s}dG_{s}.$

Note that $\\{Y_{t},0\leq t\leq 1\\}$ is a Gaussian process and can be
interpreted as the ”derivative” of the process $\\{X_{t},0\leq t\leq 1\\}$
with respect to $\varepsilon.$ We obtain that, $P$-a.s.,

(4. 4) $Y_{t}e^{-\theta_{0}t}=\int_{0}^{t}e^{-\theta_{0}s}dG_{s}$

The integral with respect to the process $G$ is interpreted as Young integral
(cf. El Machkouri et al. (2015)). In particular it follows that the random
variable $Y_{t}e^{-\theta_{0}t}$ and hence $Y_{t}$ has the normal distribution
with mean zero and further more, for any $0\leq t,s\leq 1,$

$\displaystyle Cov(Y_{t},Y_{s})$ $\displaystyle=$ $\displaystyle
e^{\theta_{0}t+\theta_{0}s}E[\int_{0}^{t}e^{-\theta_{0}u}dG_{u}\int_{0}^{s}e^{-\theta_{0}v}dG_{v}]$
$\displaystyle=$ $\displaystyle
e^{\theta_{0}t+\theta_{0}s}\int_{0}^{t}\int_{0}^{s}e^{-\theta_{0}(u+v)}dudv$
$\displaystyle=$ $\displaystyle R(t,s)\;\;\mbox{(say)}.$

In particular

(4. 6) $Var(Y_{t})=R(t,t).$

Observe that $\\{Y_{t},0\leq t\leq 1\\}$ is a zero mean Gaussian process with
$Cov(Y_{t},Y_{s})=R(t,s).$ Let

(4. 7)
$\zeta=\arg\inf_{-\infty<u<\infty}\int_{0}^{1}|Y_{t}-utx_{0}e^{\theta_{0}t}|dt.$

Theorem 4.1: As $\varepsilon\rightarrow 0,$ the random variable
$\varepsilon^{-1}(\theta_{\varepsilon}^{*}-\theta_{0})$ converges in
probability to a random variable whose probability distribution is the same as
that of $\zeta$ under $P_{\theta_{0}}.$ Proof: Let
$x^{\prime}_{t}(\theta)=x_{0}te^{\theta t}$ and let

(4. 8) $Z_{\varepsilon}(u)=||Y-\varepsilon^{-1}(x(\theta_{0}+\varepsilon
u)-x(\theta_{0}))||$

and

(4. 9) $Z_{0}(u)=||Y-ux^{\prime}(\theta_{0})||.$

Further more, let

(4. 10)
$A_{\varepsilon}=\\{\omega:|\theta_{\varepsilon}^{*}-\theta_{0}|<\delta_{\varepsilon}\\},\delta_{\varepsilon}=\varepsilon^{\tau},\tau\in(\frac{1}{2},1),L_{\varepsilon}=\varepsilon^{\tau-1}.$

Observe that the random variable
$u_{\varepsilon}^{*}=\varepsilon^{-1}(\theta_{\varepsilon}^{*}-\theta_{0})$
satisfies the equation

(4. 11)
$Z_{\varepsilon}(u_{\varepsilon}^{*})=\inf_{|u|<L_{\varepsilon}}Z_{\varepsilon}(u),\omega\in
A_{\varepsilon}.$

Define

(4. 12) $\zeta_{\varepsilon}=\arg\inf_{|u|<L_{\varepsilon}}Z_{0}(u).$

Observe that, with probability one,

$\displaystyle\sup_{|u|<L_{\varepsilon}}|Z_{\varepsilon}(u)-Z_{0}(u)|$
$\displaystyle=$
$\displaystyle|||Y-ux^{\prime}(\theta_{0})-\frac{1}{2}\varepsilon
u^{2}x^{\prime\prime}(\tilde{\theta})||-||Y-ux^{\prime}(\theta_{0})|||$
$\displaystyle\leq$
$\displaystyle\frac{\varepsilon}{2}L_{\varepsilon}^{2}\sup_{|\theta-\theta_{0}<\delta_{\varepsilon}}\int_{0}^{T}|x^{\prime\prime}(\theta)|dt$
$\displaystyle\leq$ $\displaystyle C\varepsilon^{2\tau-1}.$

Here $\tilde{\theta}=\theta_{0}+\alpha(\theta-\theta_{0})$ for some
$\alpha\in(0,1).$ Note that the last term in the above inequality tends to
zero as $\varepsilon\rightarrow 0.$ Further more the process
$\\{Z_{0}(u),-\infty<u<\infty\\}$ has a unique minimum $u^{*}$ with
probability one. This follows from the arguments given in Theorem 2 of
Kutoyants and Pilibossian (1994). In addition, we can choose the interval
$[-L,L]$ such that

(4. 14) $P_{\theta_{0}}^{(\varepsilon)}\\{u_{\varepsilon}^{*}\in(-L,L)\\}\geq
1-\beta(g(L))^{-1}$

and

(4. 15) $P\\{u^{*}\in(-L,L)\\}\geq 1-\beta(g(L))^{-1}$

where $\beta>0.$ Note that $g(L)$ increases as $L$ increases. The processes
$Z_{\varepsilon}(u),u\in[-L,L]$ and $Z_{0}(u),u\in[-L,L]$ satisfy the
Lipschitz conditions and $Z_{\varepsilon}(u)$ converges uniformly to
$Z_{0}(u)$ over $u\in[-L,L].$ Hence the minimizer of $Z_{\varepsilon}(.)$
converges to the minimizer of $Z_{0}(u).$ This completes the proof. Remarks :
We have seen earlier that the process $\\{Y_{t},0\leq t\leq T\\}$ is a zero
mean Gaussian process with the covariance function $Cov(Y_{t},Y_{s})=R(t,s)$
for $0,t,s\leq 1.$ Recall that

(4. 16)
$\zeta=\arg\inf_{-\infty<u<\infty}\int_{0}^{1}|Y_{t}-utx_{0}e^{\theta_{0}t}|dt.$

It is not clear what the distribution of the random variable $\zeta$ is. It
depends on the Gaussian process $G.$ Observe that for every $u,$ the integrand
in the above integral is the absolute value of a Gaussian process
$\\{J_{t},0\leq t\leq 1\\}$ with the mean function
$E(J_{t})=-utx_{0}e^{\theta_{0}t}$ and the covariance function
$Cov(J_{t},J_{s})=R(t,s)$ for $0\leq s,t\leq 1.$ It is easy to extend the
results to any Gaussian process defined on any interval $[0,T]$ for any fixed
$T>0.$ Acknowledgment: This work was supported by the INSA Senior Scientist
fellowship at the CR Rao Advanced Institute of Mathematics, Statistics and
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CR Rao Advanced Institute of Mathematics, Statistics and Computer Science,
Hyderabad, India.
e-mail<EMAIL_ADDRESS>