Parametric Estimation for Processes Driven by

Infinite Dimensional Mixed Fractional Brownian Motion

B.L.S. Prakasa Rao

CR Rao Advanced Institute of Mathematics, Statistics

and Computer Science, Hyderabad 500046, India

Abstract: Parametric and nonparametric inference for stochastic processes
driven by a fractional Brownian motion were investigated in Mishura (2008) and
Prakasa Rao (2010) among others. Similar problems for processes driven by an
infinite dimensional fractional Brownian motion were studied in Prakasa Rao
(2004, 2013), Cialenco et al. (2009) and others. Parametric estimation for
processes driven by an infinite dimensional mixed fractional Brownian motion
is discussed in this article.

##  1 Introduction

Statistical inference for diffusion type processes satisfying stochastic
differential equations driven by Wiener processes has been studied earlier and
a comprehensive survey of various methods is given in Prakasa Rao (1999a) .
There has been a recent interest to study similar problems for stochastic
processes driven by a fractional Brownian motion to model processes having
long range dependence. 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 type process driven by a fractional Brownian
motion. This is a fractional analogue of the Ornstein-Uhlenbeck process driven
by a standard Wiener process. It is a continuous time first order auto-
regressive 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 $H\in[1/2,1).$ Such a process is the unique Gaussian process
satisfying the linear integral equation

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

They investigated the problem of estimation of the parameters $\theta$ and
$\sigma^{2}$ based on the observation $\\{X_{s},0\leq s\leq T\\}$ and proved
that the maximum likelihood estimator $\hat{\theta}_{T}$ is strongly
consistent as $T\rightarrow\infty.$ More general classes of stochastic
processes satisfying linear stochastic differential equations driven by a
fractional Brownian motion were studied and the asymptotic properties of the
maximum likelihood and the Bayes estimators for parameters involved in such
processes is investigated in Prakasa Rao (2003). Prakasa Rao (2010) gives a
comprehensive discussion on problems of estimation for processes driven by a
fractional Brownian motion.

Geometric Brownian motion driven by a standard Brownian motion has been widely
used for modeling fluctuations of share prices in a stock market using Black-
Scholes model. However efforts to model fluctuations in financial markets with
long range dependence through processes driven by a fractional Brownian motion
were not successful as it was noted that such a modeling creates arbitrage
opportunities contrary to the fundamental assumption of no arbitrage
opportunity for modeling rational market behaviour. Cheridito (2001) proposed
modeling through processes driven by a mixed fractional Brownian motion. It
was shown by Cheridito (2001) that a mixed fractional Brownian motion is a
semimartingale if and only if the Hurst index $H$ is either equal to
$\frac{1}{2}$ reducing the process to a Wiener process or $H\in(3/4,1)$.
Furthermore the probability measure generated by such a process is absolutely
continuous with respect to the probability measure generated by a Wiener
process if $H=1/2$ or $H\in(3/4,1).$ This in turn will lead to no arbitrage
opportunities for modeling financial market behaviour through processes driven
by a mixed fractional Brownian motion. This discussion is to motivate the
study of processes driven by a mixed fractional Brownian motion.

The problem of estimation of parameters for processes driven by processes
which are mixtures of independent Brownian and fractional Brownian motions
started from the works of Cheridito (2001), Rudomino-Dusyatska (2003) and more
recently in Prakasa Rao (2015a,b;2017a,b; 2018a,b; 2019, 2020, 2021a,b) among
others. Mixed fractional Brownian models were studied in Mishura (2008) and
Prakasa Rao (2010) . Cai et al. (2016) present a new approach via filtering
for analysis of mixed processes of type $\\{X_{t}=B_{t}+G_{t},0\leq t\leq
T\\}$ where $\\{B_{t},0\leq t\leq T\\}$ is a Brownian motion and
$\\{G_{t},0\leq t\leq T\\}$ is an independent Gaussian process. Statistical
analysis of mixed fractional Ornstein-Uhlenbeck process was investigated in
Chigansky and Kleptsyna (2019). Fractional Ornstein-Uhlenbeck type process
driven a mixed fractional Brownian motion has also been termed as “mixed
fractional Ornstein-Uhlenbeck process”’ in Marushkevych (2016). Large
deviations for drift parameter estimator of a mixed fractional Ornstein-
Uhlenbeck process were studied by Marushkevych (2016).

Huebner et al. (1995) initiated the study of parametric estimation for a class
of stochastic partial differential equations (SPDE) in the presence of white
noise or the driving force is an infinite dimensional Wiener process. These
results were extended to parabolic stoochastic partial differential equations
in Huebner and Rozovskii (1995). Prakasa Rao (2000) studied Bayes estimation
for stochastic partial differential equations in the white noise case. For
other results on parametric inference for SPDEs, see Prakasa Rao (2000, 2001,
2002a,b,2004, 2013). A comprehensive survey of results is given in Prakasa Rao
(2001,2002). Parameter estimation for a two-dimensional stochastic Navier-
Stokes equation driven by infinite dimensional fractional Brownian motion was
studied in Prakasa Rao (2013). Lototsky and Rozovsky (2017) give an extensive
survey of theory of SPDE and a discussion on parametric inference for such
processes. Cialenco and his coworkers obtained several results dealing with
parametric inference for SPDE based on continuous observation or discrete
sampling of the processes. Cialenco (2019) gives a survey of their results.

Our aim in this paper is to study parametric inference for processes driven by
infinite dimensional mixed fractional Brownian motion. As far as we are aware,
this problem has not been investigated earlier.

##  2 Properties of processes driven by a mfBm

Let $(\Omega,{\cal F},({\cal F}_{t}),P)$ be a stochastic basis satisfying the
usual conditions. The natural filtration of a stochastic process is understood
as the $P$-completion of the filtration generated by this process. Let
$\\{W_{t},t\geq 0\\}$ be a standard Wiener process and
$W^{H}=\\{W_{t}^{H},t\geq 0\\}$ be an independent normalized fractional
Brownian motion with Hurst parameter $H\in(0,1)$, that is, a Gaussian process
with continuous sample paths such that $W_{0}^{H}=0,E(W_{t}^{H})=0$ and

(2.1) $E(W_{s}^{H}W_{t}^{H})=\frac{1}{2}[s^{2H}+t^{2H}-|s-t|^{2H}],t\geq
0,s\geq 0.$

Let

$\tilde{W}_{t}^{H}=W_{t}+W_{t}^{H},t\geq 0.$

The process $\\{\tilde{W}_{t}^{H},t\geq 0\\}$ is called the mixed fractional
Brownian motion with Hurst index $H.$ We assume here after that Hurst index
$H$ is known and that $H\in(\frac{3}{4},1).$ Let us consider a stochastic
process $Y=\\{Y_{t},t\geq 0\\}$ defined by the stochastic integral equation

(2.2) $Y_{t}=\int_{0}^{t}C(s)ds+\tilde{W}_{t}^{H},t\geq 0$

where the process $C=\\{C(t),t\geq 0\\}$ is an $({\cal F}_{t})$-adapted
process. For convenience, we write the above integral equation in the form of
a stochastic differential equation

(2.3) $dY_{t}=C(t)dt+d\tilde{W}_{t}^{H},t\geq 0$

driven by the mixed fractional Brownian motion $\tilde{W}^{H}.$ Following the
recent works by Cai et al. (2016) and Chigansky and Kleptsyna (2019), one can
construct an integral transformation that transforms the mixed fractional
Brownian motion $\tilde{W}^{H}$ into a martingale $M^{H}.$ Let $g_{H}(s,t)$ be
the solution of the integro-differential equation

(2.4)
$g_{H}(s,t)+H\frac{d}{ds}\int_{0}^{t}g_{H}(r,t)|s-r|^{2H-1}sign(s-r)dr=1,0<s<t.$

Cai et al. (2016) proved that the process

(2.5) $M_{t}^{H}=\int_{0}^{t}g_{H}(s,t)d\tilde{W}_{s}^{H},t\geq 0$

is a Gaussian martingale with quadratic variation

(2.6) $<M^{H}>_{t}=\int_{0}^{t}g_{H}(s,t)ds,t\geq 0$

Let $w^{H}_{t}$ denote the quadratic variation $<M^{H}>_{t}$ over the interval
$[0,t].$ It is known that the natural filtration of the martingale $M^{H}$
coincides with that of the mixed fractional Brownian motion $\tilde{W}^{H}.$
Suppose that, for the martingale $M^{H}$ defined by the equation (2.5), the
sample paths of the process $\\{C(t),t\geq 0\\}$ are smooth enough in the
sense that the process

(2.7) $Q_{H}(t)=\frac{d}{d<M^{H}>_{t}}\int_{0}^{t}g_{H}(s,t)C(s)ds,t\geq 0$

is well defined. Define the process

(2.8) $Z_{t}=\int_{0}^{t}g_{H}(s,t)dY_{s},t\geq 0.$

As a consequence of the results in Cai et al. (2016), it follows that the
process $Z$ is a fundamental semimartingale associated with the process $Y$ in
the following sense. Theorem 2.1: Let $g_{H}(s,t)$ be the solution of the
equation (2.4). Define the process $Z$ as given in the equation (2.8). Then
the following relations hold.

(i) The process $Z$ is a semimartingale with the decomposition

(2.9) $Z_{t}=\int_{0}^{t}Q_{H}(t)d<M^{H}>_{s}+M^{H}_{t},t\geq 0$

where $M^{H}$ is the martingale defined by the equation (2.5).

(ii) The process $Y$ admits the representation

(2.10) $Y_{t}=\int_{0}^{t}\hat{g}_{H}(s,t)dZ_{s},t\geq 0$

where

(2.11) $\hat{g}_{H}(s,t)=1-\frac{d}{d<M^{H}>_{s}}\int_{0}^{t}g_{H}(r,s)dr.$

(iii) The natural filtrations $({\cal Y}_{t})$ and $({\cal Z}_{t})$ of the
processes $Y$ and $Z$ respectively coincide. Applying Corollary 2.9 in Cai et
al. (2016), it follows that the probability measures $\mu_{Y}$ and
$\mu_{\tilde{W}^{H}}$ generated by the processes $Y$ and $\tilde{W}^{H}$ on an
interval $[0,T]$ are absolutely continuous with respect to each other and the
Radon-Nikodym derivative is given by

(2.12)
$\frac{d\mu_{Y}}{d\mu_{\tilde{W}^{H}}}(Y)=\exp[\int_{0}^{T}Q_{H}(s)dZ_{s}-\frac{1}{2}\int_{0}^{T}[Q_{H}(s)]^{2}d<M^{H}>_{s}]$

which is also the likelihood function based on the observation $\\{Y_{s},0\leq
s\leq T.\\}$ Since the filtrations generated by the processes $Y$ and $Z$ are
the same, the information contained in the families of $\sigma$-algebras
$({\cal Y}_{t})$ and $({\cal Z}_{t})$ is the same and hence the problem of the
estimation of the parameters involved based on the observation $\\{Y_{s},0\leq
s\leq T\\}$ and $\\{Z_{s},0\leq s\leq T\\}$ are equivalent. Since the process
$\\{Z_{s},0\leq s\leq T\\}$ is driven by a martingale, it is convenient to
discuss asymptotic behaviour of the estimators through limit theorems
available for martingales. This explanation motivates the study of problem of
estimation through the process $Z$ instead of the original process $Y.$

##  3 Parametric estimation for SPDE driven by infinite dimensional mfBm

Kallianpur and Xiong (1995) discussed the properties of solutions of
stochastic partial differential equations (SPDE) driven by infinite
dimensional fractional Brownian motion. They indicate that SPDE’s are being
used for stochastic modelling, for instance, for the study of neuronal
behaviour in neurophysiology and in building stochastic models of turbulence.
The theory of SPDE’s is investigated in Ito (1984), Rozovskii (1990) and Da
Prato and Zabczyk (1992). Huebner et al. (1993) started the investigation of
maximum likelihood estimation of parameters of two types of SPDE’s and
extended their results for a class of parabolic SPDE’s in Huebner and
Rozovskii (1995). Asymptotic properties of Bayes estimators for such problems
were discussed in Prakasa Rao (2000). A short review and a comprehensive
survey of these results are given in Prakasa Rao (2001,2002). Our aim in this
section is to study the problems of parameter estimation for some SPDE driven
by an infinite dimensional mixed fractional Brownian motion. Stochastic PDE
with linear drift (absolutely continuous case) Let $U$ be a real separable
Hilbert space and $Q$ be a self-adjoint positive operator. Further suppose
that the operator $Q$ is nuclear. Then $Q$ admits a sequence of eigenvalues
$\\{q_{n},n\geq 1\\}$ with $0<q_{n}$ decreasing to zero as
$n\rightarrow\infty$ and $\sum_{n=0}^{\infty}q_{n}<\infty.$ In addition the
corresponding eigen vectors $\\{e_{n},n\geq 1\\}$ form an orthonormal basis in
$U.$ We define the infinite dimensional mixed fractional Brownian motion on
$U$ with covariance $Q$ as

(3.1) $\tilde{\cal
W}_{Q}^{H}(t)=\sum_{n=0}^{\infty}\sqrt{q_{n}}e_{n}\tilde{W}_{n}^{H}(t)$

where $\tilde{W}_{n}^{H},n\geq 1$ are real independent mfBm’s with Hurst index
$H$ . Formal definition is given in the next section. Let $U=L_{2}[0,1]$ and
${\cal W}_{Q}^{H}$ be the infinite dimensional mfBm on $U$ with the Hurst
index $H$ and with the nuclear covariance operator $Q.$ Consider the process
$u_{\varepsilon}(t,x),0\leq x\leq 1,0\leq t\leq T$ governed by the stochastic
partial differential equation

(3.2) $du_{\varepsilon}(t,x)=(\triangle u_{\varepsilon}(t,x)+\theta
u_{\varepsilon}(t,x))dt+\varepsilon\;\;d\tilde{\cal W}_{Q}^{H}(t,x)$

where $\triangle=\frac{\partial^{2}}{\partial x^{2}}.$ Suppose that
$\varepsilon\rightarrow 0$ and $\theta\in\Theta\subset R.$ Suppose the initial
and the boundary conditions are given by

(3.3) $\displaystyle u_{\varepsilon}(0,x)$ $\displaystyle=$ $\displaystyle
f(x),f\in L_{2}[0,1]$ (3.4) $\displaystyle u_{\varepsilon}(t,0)$
$\displaystyle=$ $\displaystyle u_{\varepsilon}(t,1)=0,0\leq t\leq T.$

Let us consider a special covariance operator $Q$ with $e_{k}=\sin k\pi
x,k\geq 1$ and $\lambda_{k}=(\pi k)^{2},k\geq 1.$ Then $\\{e_{k}\\}$ is a
complete orthonormal system with the eigenvalues
$q_{i}=(1+\lambda_{i})^{-1},i\geq 1$ for the operator $Q$ and
$Q=(I-\triangle)^{-1}$. Guerra and Nualart (2008) proved an existence and
uniqueness theorem for solutions of multidimensional time dependent stochastic
differential equations driven by a multidimensional fractional Brownian motion
with Hurst index $H>\frac{1}{2}$ and a multidimensional standard Brownian
motion. Similar results were obtained by Mishura and Shevchenko (2011) and da
Silva and Erraoui (2018) under weaker conditions. Mishura et al. (2019) has
given sufficient conditions for the existence and uniqueness of a mild
solution $u_{\varepsilon}(t,x)$ for stochastic differential equation driven by
an infinite dimensional mfBm. We assume that sufficient conditions hold so
that there exists a unique square integrable solution $u_{\varepsilon}(t,x)$
of (3.2) under the conditions (3.3)-(3.4) and consider it as a formal sum

(3.5) $u_{\varepsilon}(t,x)=\sum_{i=1}^{\infty}u_{i\varepsilon}(t)e_{i}(x).$

It can be checked that the Fourier coefficient $u_{i\varepsilon}(t)$ satisfies
the stochastic differential equation

(3.6)
$du_{i\varepsilon}(t)=(\theta-\lambda_{i})u_{i\varepsilon}(t)dt+\frac{\varepsilon}{\sqrt{\lambda_{i}+1}}d\tilde{W}_{i}^{H}(t),\;0\leq
t\leq T$

with the initial condition

(3.7) $u_{i\varepsilon}(0)=v_{i},\;v_{i}=\int_{0}^{1}f(x)e_{i}(x)dx.$

Let $P_{\theta}^{(\varepsilon)}$ be the probability measure generated by
$u_{\varepsilon}$ when $\theta$ is the true parameter. Suppose $\theta_{0}$ is
the true parameter. Observe that the process $\\{u_{i\varepsilon}(t),0\leq
t\leq T\\}$ is a mixed fractional Ornstein-Uhlenbeck type process (cf.
Marushkevych (2016), Chigansky and Kleptsyna (2019), Cai et al. (2016)).
Following the notation given in the previous section, define

(3.8) $M_{i}^{H}(t)=\int_{0}^{t}g_{H}(s,t)d\tilde{W}_{i}^{H}(s),0\leq t\leq
T,$ (3.9)
$Q_{i\varepsilon}(t)=\frac{\sqrt{\lambda_{i}+1}}{\varepsilon}\frac{d}{dw_{t}^{H}}\int_{0}^{t}g_{H}(s,t)u_{i\varepsilon}(s)ds,t\in[0,T],$
(3.10)
$Z_{i\varepsilon}(t)=(\theta-\lambda_{i})\int_{0}^{t}Q_{i\varepsilon}(s)dw_{s}^{H}+M_{i}^{H}(t),0\leq
t\leq T.$

Observe that $M_{i}^{H}$ is a zero mean Gaussian martingale. Furthermore, it
follow that the process $\\{Z_{i\varepsilon}(t)\\}$ is a semimartingale and
the natural filtrations $({\cal Z}_{{i\varepsilon}_{t}})$ and $({\cal
U}_{{i\varepsilon}_{t}})$ of the processes $Z_{i\varepsilon}$ and
$u_{i\varepsilon}$ respectively coincide. Let $P_{i\theta}^{T,\varepsilon}$ be
the probability measure generated by the process $\\{u_{i\varepsilon}(t),0\leq
t\leq T\\}$ when $\theta$ is the true parameter. Let $\theta_{0}$ be the true
parameter. It follows, by the Girsanov type theorem, that

$\displaystyle\log\frac{dP_{i\theta}^{T,\varepsilon}}{dP_{i\theta_{0}}^{T,\varepsilon}}$
$\displaystyle=$
$\displaystyle\frac{\lambda_{i}+1}{\varepsilon^{2}}[(\theta-\theta_{0})\int_{0}^{T}Q_{i\varepsilon}(t)dZ_{i\varepsilon}(t)$
$\displaystyle\;\;\;\;-\frac{1}{2}\\{(\theta-\lambda_{i})^{2}-(\theta_{0}-\lambda_{i})^{2}\\}\int_{0}^{T}Q_{i\varepsilon}^{2}(t)dw_{t}^{H}].$

Let $u_{\varepsilon}^{N}(t,x)$ be the projection of the solution
$u_{\varepsilon}(t,x)$ onto the subspace spanned by the eigen vectors
$\\{e_{i},1\leq i\leq N\\}.$ Then

(3.12) $u_{\varepsilon}^{N}(t,x)=\sum_{i=1}^{N}u_{i\varepsilon}(t)e_{i}(x)$

From the independence of the processes $\tilde{W}_{i}^{H},1\leq i\leq N$ and
hence of the processes $u_{i\varepsilon},1\leq i\leq N,$ it follows that the
Radon-Nikodym derivative, of the probability measure
$P_{\theta}^{N,T,\varepsilon}$ generated by the process
$u_{\varepsilon}^{N},0\leq t\leq T$ when $\theta$ is the true parameter with
respect to the probability measure $P_{\theta_{0}}^{N,T,\varepsilon}$
generated by the process $u_{\varepsilon}^{n},0\leq t\leq T$ when $\theta_{0}$
is the true parameter, is given by

$\displaystyle\log\frac{dP_{\theta}^{N,T,\varepsilon}}{dP_{\theta_{0}}^{N,T,\varepsilon}}(u_{\varepsilon}^{N})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\frac{\lambda_{i}+1}{\varepsilon^{2}}[(\theta-\theta_{0})\int_{0}^{T}Q_{i\varepsilon}(t)dZ_{i\varepsilon}(t)$
$\displaystyle\;\;\;-\frac{1}{2}\\{(\theta-\lambda_{i})^{2}-(\theta_{0}-\lambda_{i})^{2}\\}\int_{0}^{T}Q_{i\varepsilon}^{2}(t)dw_{t}^{H}].$

Furthermore the Fisher information is given by

$\displaystyle I_{N\varepsilon}(\theta)$ $\displaystyle=$ $\displaystyle
E_{\theta}[\frac{\partial\log\frac{dP_{\theta}^{N,T\varepsilon}}{dP_{\theta_{0}}^{N,T,\varepsilon}}}{\partial\theta}]^{2}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\frac{\lambda_{i}+1}{\varepsilon^{2}}E_{\theta}\\{\int_{0}^{T}Q_{i\varepsilon}^{2}(t)dw_{t}^{H}\\}.$

It is easy to check that the maximum likelihood estimator
$\hat{\theta}_{N,\varepsilon}$ of the parameter $\theta$ based on the
projection $u_{\varepsilon}^{N}$ of $u_{\varepsilon}$ is given by

(3.15)
$\hat{\theta}_{N,\varepsilon}=\frac{\sum_{i=1}^{N}(\lambda_{i}+1)\int_{0}^{T}Q_{i\varepsilon}(t)dZ_{i\varepsilon}(t)}{\sum_{i=1}^{N}(\lambda_{i}+1)\int_{0}^{T}Q_{i\varepsilon}^{2}(t)dw_{t}^{H}}.$

Suppose $\theta_{0}$ is the true parameter. It is easy to see that

(3.16)
$\varepsilon^{-1}(\hat{\theta}_{N,\varepsilon}-\theta_{0})=\frac{\sum_{i=1}^{N}\sqrt{\lambda_{i}+1}\int_{0}^{T}Q_{i\varepsilon}(t)dM_{i}^{H}(t)}{\sum_{i=1}^{N}(\lambda_{i}+1)\int_{0}^{T}Q_{i\varepsilon}^{2}(t)dw_{t}^{H}}.$

Observe that $M_{i},1\leq i\leq N$ are independent zero mean Gaussian
martingales with $<M_{i}>=w^{H},1\leq i\leq N.$ Theorem 3.1 : The maximum
likelihood estimator $\hat{\theta}_{N,\varepsilon}$ is strongly consistent,
that is,

(3.17)
$\hat{\theta}_{N,\varepsilon}\rightarrow\theta_{0}\;\;\mbox{a.s}\;\;[P_{\theta_{0}}]\;\mbox{as}\;\;\varepsilon\rightarrow
0$

provided

(3.18)
$\sum_{i=1}^{N}\int_{0}^{T}(\lambda_{i}+1)Q_{i\varepsilon}^{2}(t)dw_{t}^{H}\rightarrow\infty\;\;\mbox{a.s}\;\;[P_{\theta_{0}}]\;\;\mbox{as}\;\;\varepsilon\rightarrow
0.$

Proof : This theorem follows by observing that the process

(3.19)
$R_{\varepsilon}^{N}\equiv\sum_{i=1}^{N}\int_{0}^{T}\varepsilon\sqrt{\lambda_{i}+1}Q_{i\varepsilon}(t)dM_{t}^{H},T\geq
0$

is a local martingale with the quadratic variation process

(3.20)
$<R_{\varepsilon}^{N}>_{T}=\sum_{i=1}^{N}\int_{0}^{T}\varepsilon^{2}(\lambda_{i}+1)Q_{i\varepsilon}^{2}(t)dw_{t}^{H}$

and applying the Strong law of large numbers (cf. Liptser (1980); Prakasa Rao
(1999b), p. 61) under the condition (3.18) stated above.

Limiting distribution :

We now discuss the limiting distribution of the MLE
$\hat{\theta}_{N\varepsilon}$ as $\varepsilon\rightarrow 0.$ Theorem 3.2 :
Assume that the process $\\{R_{\varepsilon}^{N},\varepsilon\geq 0\\}$ is a
local continuous martingale and that there exists a norming function
$I_{\varepsilon}^{N},\varepsilon\geq 0$ such that

(3.21)
$\displaystyle(I_{\varepsilon}^{N})^{2}<R_{\varepsilon}^{N}>_{T}=(I_{\varepsilon}^{N})^{2}\sum_{i=1}^{N}\int_{0}^{T}\varepsilon^{2}(\lambda_{i}+1)Q_{i\varepsilon}^{2}(t)dw_{t}^{H}\rightarrow\eta^{2}\;\;\mbox{in
probability}\;\;\mbox{as}\;\;\varepsilon\rightarrow 0$

where $\eta$ is a random variable such that $P(\eta>0)=1.$ Then

(3.22)
$(I_{\varepsilon}^{N}R_{\varepsilon}^{N},(I_{\varepsilon}^{N})^{2}<R_{\varepsilon}^{N}>_{T})\rightarrow(\eta
Z,\eta^{2})\mbox{ in law}\;\;\mbox{as}\;\;\epsilon\rightarrow 0$

where the random variable $Z$ has the standard Gaussian distribution and the
random variables $Z$ and $\eta$ are independent. Proof : This theorem follows
as a consequence of the central limit theorem for local martingales (cf.
Theorem 1.49 ; Remark 1.47 , Prakasa Rao(1999b), p. 65). Observe that

(3.23)
$(I_{\varepsilon}^{N})^{-1}(\hat{\theta}_{N\varepsilon}-\theta_{0})=\frac{I_{\varepsilon}^{N}R_{\varepsilon}^{N}}{(I_{\varepsilon}^{N})^{2}<R_{\varepsilon}^{N}>}.$

Applying the above theorem, we obtain the following result. Theorem 3.3 :
Suppose the conditions stated in Theorem 3.2 hold. Then

(3.24)
$(I_{\varepsilon}^{N})^{-1}(\hat{\theta}_{N\varepsilon}-\theta_{0})\rightarrow\frac{Z}{\eta}\mbox{
in law}\;\;\mbox{as}\;\;\varepsilon\rightarrow 0$

where the random variable $Z$ has the standard Gaussian distribution and the
random variables $Z$ and $\eta$ are independent. Remarks : (i) If the random
variable $\eta$ is a constant with probability one, then the limiting
distribution of the maximum likelihood estimator is Gaussian with mean 0 and
variance $\eta^{-2}.$ Otherwise it is a mixture of the Gaussian distributions
with mean zero and variance $\eta^{-2}$ with the mixing distribution as that
of $\eta.$ (ii) Suppose that

(3.25) $\lim_{N\rightarrow\infty}\lim_{\varepsilon\rightarrow
0}\varepsilon^{2}I_{\epsilon}^{N}=I(\theta)$

exists and is positive. Since the sequence of Radon-Nikodym derivatives

$\\{\frac{dP_{\theta}^{N,T,\varepsilon}}{dP_{\theta_{0}}^{N,T,\varepsilon}},n\geq
1\\}$

form a non-negative martingale with respect to the filtration generated by the
sequence of random variables $\\{u_{\varepsilon}^{N},N\geq 1\\}$, it converges
almost surely to a random variable $\nu_{\varepsilon,\theta,\theta_{0}}$ as
$N\rightarrow\infty$ for every $\varepsilon>0.$ It is easy to see that the
limiting random variable is given by

(3.26) $\displaystyle\nu_{\varepsilon,\theta,\theta_{0}}(u_{\varepsilon})$
$\displaystyle=$
$\displaystyle\exp\\{\sum_{i=1}^{\infty}\frac{\lambda_{i}+1}{\varepsilon^{2}}[(\theta-\theta_{0})\int_{0}^{T}Q_{i\varepsilon}(t)dZ_{i\varepsilon}(t)$
$\displaystyle\;\;\;\;-\frac{1}{2}\\{(\theta-\lambda_{i})^{2}-(\theta_{0}-\lambda_{i})^{2}\\}\int_{0}^{T}Q_{i\varepsilon}^{2}(t)dw_{t}^{H}]\\}.$

Furthermore the sequence of random variables $u_{\varepsilon}^{N}(t)$ converge
in probability to the random variable $u_{\varepsilon}(t)$ as
$N\rightarrow\infty$ for every $\varepsilon>0.$ Hence, by Lemma 4 in Skorokhod
(1965, p. 100), it follows that the measures $P_{\theta}^{\varepsilon}$
generated by the processes $u_{\varepsilon}$for different values of $\theta,$
are absolutely continuous with respect to each other and the Radon-Nikodym
derivative of the probability measure $P_{\theta}^{\varepsilon}$ with respect
to the probability measure $P_{\theta_{0}}^{\varepsilon}$ is given by

$\displaystyle\frac{dP_{\theta}^{\varepsilon}}{dP_{\theta_{0}}^{\varepsilon}}(u_{\varepsilon})$
$\displaystyle=$
$\displaystyle\nu_{\varepsilon,\theta,\theta_{0}}(u_{\varepsilon})$
$\displaystyle=$
$\displaystyle\exp\\{\sum_{i=1}^{\infty}\frac{\lambda_{i}+1}{\varepsilon^{2}}[(\theta-\theta_{0})\int_{0}^{T}Q_{i\varepsilon}(t)dZ_{i\varepsilon}(t)$
$\displaystyle\;\;-\frac{1}{2}\\{(\theta-\lambda_{i})^{2}-(\theta_{0}-\lambda_{i})^{2}\\}\int_{0}^{T}Q_{i\varepsilon}^{2}(t)dw_{t}^{H}]\\}.$

It can be checked that the MLE $\hat{\theta}_{\varepsilon}$ of $\theta$ based
on $u_{\varepsilon}$ satisfies the likelihood equation

(3.28)
$\alpha_{\varepsilon}=\varepsilon^{-1}(\hat{\theta}_{\varepsilon}-\theta_{0})\beta_{\varepsilon}$

when $\theta_{0}$ is the true parameter where

(3.29)
$\alpha_{\varepsilon}=\sum_{i=1}^{\infty}\sqrt{\lambda_{i}+1}\int_{0}^{T}Q_{i\varepsilon}(t)dM_{i}^{H}(t)$

and

(3.30)
$\beta_{\varepsilon}=\sum_{i=1}^{\infty}(\lambda_{i}+1)\int_{0}^{T}Q^{2}_{i\varepsilon}(t)dw_{t}^{H}.$

One can obtain sufficient conditions for studying the asymptotic behaviour of
the estimator $\hat{\theta}_{\varepsilon}$ as in the finite projection case
discussed above. We omit the details.

Stochastic PDE with linear drift (singular case) :  Let $(\Omega,{\cal F},P)$
be a probability space and consider the process $u_{\varepsilon}(t,x),0\leq
x\leq 1,0\leq t\leq T$ governed by the stochastic partial differential
equation

(3.31) $du_{\varepsilon}(t,x)=\theta\;\triangle
u_{\varepsilon}(t,x)dt+\varepsilon(I-\triangle)^{-1/2}d\tilde{W}(t,x)$

where $\theta>0$ satisfying the initial and the boundary conditions

(3.32) $\displaystyle u_{\varepsilon}(0,x)$ $\displaystyle=$ $\displaystyle
f(x),\;0<x<1,\;f\in L_{2}[0,1],$ $\displaystyle u_{\varepsilon}(t,0)$
$\displaystyle=$ $\displaystyle u_{\varepsilon}(t,1)=0,\;0\leq t\leq T.$

Here $I$ is the identity operator, $\triangle=\frac{\partial^{2}}{\partial
x^{2}}$ as defined above and the process $\tilde{W}(t,x)$ is the cylindrical
infinite dimensional mfBm with $H\in[\frac{1}{2},1).$ Following the discussion
in the previous section, we assume the existence of a square integrable
solution $u_{\varepsilon}(t,x)$ for the equation (3.31) subject to the
boundary conditions (3.32). Then the Fourier coefficients
$u_{i\varepsilon}(t)$ of satisfy the stochastic differential equations

(3.33)
$du_{i\varepsilon}(t)=-\theta\lambda_{i}u_{i\varepsilon}(t)dt+\frac{\varepsilon}{\sqrt{\lambda_{i}+1}}d\tilde{W}_{i}^{H}(t),\;0\leq
t\leq T,$

with

(3.34) $u_{i\varepsilon}(0)=v_{i},v_{i}=\int_{0}^{1}f(x)e_{i}(x)dx.$

Let $u_{\varepsilon}^{(N)}(t,x)$ be the projection of $u_{\varepsilon}(t,x)$
onto the subspace spanned by $\\{e_{1},\cdots,e_{N}\\}$ in $L_{2}[0,1].$ In
other words

(3.35) $u_{\varepsilon}^{(N)}(t,x)=\sum_{i=1}^{N}u_{i\varepsilon}(t)e_{i}(x).$

Let $P_{\theta}^{(\varepsilon,N)}$ be the probability measure generated by
$u_{\varepsilon}^{(N)}$ on the subspace spanned by $\\{e_{1},\cdots,e_{N}\\}$
in $L_{2}[0,1].$ It can be shown that the measures
$\\{P_{\theta}^{(\varepsilon,N)},\theta\in\Theta\\}$ form an equivalent family
and

(3.36)
$\displaystyle\log\frac{dP_{\theta}^{(\varepsilon,N)}}{dP_{\theta_{0}}^{(\varepsilon,N)}}(u_{\varepsilon}^{(N)})$
$\displaystyle=$
$\displaystyle-\frac{1}{\varepsilon^{2}}\sum_{i=1}^{N}\lambda_{i}(\lambda_{i}+1)[(\theta-\theta_{0})\int_{0}^{T}Q_{i\varepsilon}(t)dZ_{i\varepsilon}(t)-\frac{1}{2}(\theta-\theta_{0})^{2}\lambda_{i}\int_{0}^{T}Q^{2}_{i\varepsilon}(t)dw_{t}^{H}].$

It can be checked that the MLE $\hat{\theta}_{\varepsilon,N}$ of $\theta$
based on $u_{\varepsilon}^{(N)}$ satisfies the likelihood equation

(3.37)
$\alpha_{\varepsilon,N}=-\varepsilon^{-1}(\hat{\theta}_{\varepsilon,N}-\theta_{0})\beta_{\varepsilon,N}$

when $\theta_{0}$ is the true parameter where

(3.38)
$\alpha_{\varepsilon,N}=\sum_{i=1}^{N}\lambda_{i}\sqrt{\lambda_{i}+1}\int_{0}^{T}Q_{i\varepsilon}(t)dM_{i}^{H}(t)$

and

(3.39)
$\beta_{\varepsilon,N}=\sum_{i=1}^{N}(\lambda_{i}+1)\lambda_{i}^{2}\int_{0}^{T}Q_{i,\varepsilon}^{2}(t)dw_{t}^{H}.$

Asymptotic properties of these estimators can be investigated as in the
previous example. We do not go into the details as the arguments are similar.
Remarks : One can study the local asymptotic mixed normality (LAMN) of the
family of probability measures generated by the log-likelihood ratio processes
by the standard arguments as in Prakasa Rao (1999b) and hence investigate the
asymptotic efficiency of the MLE using Hajek-Lecam type bounds.

##  4  Parametric estimation for stochastic parabolic equations driven by
infinite dimensional mfBm

We now extend some work of Cialenco et al. (2009) dealing with problems of
estimation in models more general than those discussed in the previous
section. We introduce some notation. Let ${\bf H}$ be a separable Hilbert
space with the inner product $(.,.)_{0}$ and with the corresponding norm
$||.||_{0}.$ Let $\Lambda$ be a densely defined linear operator on ${\bf H}$
with the property that there exists $c>0$ such that

$||\Lambda u||_{0}\geq c||u||_{0}$

for every $u$ in the domain of the operator $\Lambda.$ The operator powers
$\Lambda^{\gamma},\gamma\in R$ are well defined and generate the spaces ${\bf
H}^{\gamma}$ with the properties (i) for $\gamma>0,{\bf H}^{\gamma}$ is the
domain of $\Lambda^{\gamma},$ (ii) ${\bf H}^{0}={\bf H}$, and (iii) for
$\gamma<0,{\bf H}^{\gamma}$ is the completion of ${\bf H}$ with respect to the
norm $||.||_{\gamma}\equiv||\Lambda^{\gamma}.||_{0}$ (cf. Krein et al.
(1982)). The family of spaces $\\{{\bf H}^{\gamma},\gamma\in R\\}$ has the
following properties:

(i) $\Lambda^{\gamma}({\bf H}^{r})={\bf H}^{r-\gamma},\gamma,r\in R;$

(ii) For $\gamma_{1}<\gamma_{2},$ the space ${\bf H}^{\gamma_{2}}$ is densely
and continuously embedded into ${\bf H}^{\gamma_{1}}$, that is, ${\bf
H}^{\gamma_{2}}\subset{\bf H}^{\gamma_{1}}$ and there exists a constant
$c_{12}>0$ such that $||u||_{\gamma_{1}}\leq c_{12}||u||_{\gamma_{2}};$

(iii) for every $\gamma\in R$ and $m>0,$ the space ${\bf H}^{\gamma-m}$ is the
dual of the space ${\bf H}^{\gamma+m}$ with respect to the inner product in
${\bf H}^{\gamma},$ with duality $<.,.>_{\gamma,m}$ given by

$<u_{1},u_{2}>_{\gamma,m}=(\Lambda^{\gamma-m}u_{1},\Lambda^{\gamma+m}u_{2})_{0},\;\;u_{1}\in{\bf
H}^{\gamma-m},u_{2}\in{\bf H}^{\gamma+m}.$

Let $(\Omega,{\cal F},P)$ be a probability space and let
$\\{\tilde{W}_{j}^{H},j\geq 1\\}$ be a family of independent mixed fractional
Brownian motions on this space with the same Hurst index $H$ in $(0,1).$
Consider the SDE

(4.1) $du(t)+({\cal A}_{0}+\theta{\cal A}_{1})u(t)dt=\sum_{j\geq
1}g_{j}(t)d\tilde{W}_{j}^{H}(t),0\leq t\leq T,u(0)=u_{0}$

where ${\cal A}_{0},{\cal A}_{1}$ are linear operators, $g_{j},j\geq 1$ are
non-random and $\theta\in\Theta\subset R.$ The equation (4.1) is said to be
diagonalizable if the operators ${\cal A}_{0},{\cal A}_{1}$ have the same
system of eigenfunctions $\\{h_{j},j\geq 1\\}$ such that $\\{h_{j},j\geq 1\\}$
is an orthonormal basis in ${\bf H}$ and each $h_{j}$ belongs to
$\cap_{\gamma\in R}{\bf H}^{\gamma}.$ It is called $(m,\gamma)$-parabolic for
some $m\geq 0,\gamma\in R,$ if (i) the operator ${\cal A}_{0}+\theta{\cal
A}_{1}$ is uniformly bounded from ${\bf H}^{\gamma+m}$ to ${\bf H}^{\gamma-m}$
for every $\theta\in\Theta$, that is, there exists $C_{1}>0$ such that

(4.2) $||({\cal A}_{0}+\theta{\cal A}_{1})v||_{\gamma-m}\leq
C_{1}||v||_{\gamma+m},\;\;\theta\in\Theta,v\in H^{\gamma+m};$

and
(ii)there exists a $\delta>0$ and $C\in R$ such that,

(4.3) $-2<({\cal A}_{0}+\theta{\cal
A}_{1})v,v>_{\gamma,m}+\delta||v||^{2}_{\gamma+m}\leq
C||v||^{2}_{\gamma},v\in{\bf H}^{\gamma+m},\theta\in\Theta.$

If the equation (4.1) is $(m,\gamma)$-parabolic, then the condition (ii)
implies that

$<(2{\cal A}_{0}+2\theta{\cal
A}_{1}+CI)v,v)_{\gamma,m}\geq\delta||v||^{2}_{\gamma+m}$

where $I$ is the identity operator. The Cauchy-Schwartz inequality and the
continuous embedding of ${\bf H}^{\gamma+m}$ into ${\bf H}^{\gamma}$ will
imply that

$||(2{\cal A}_{0}+2\theta{\cal
A}_{1}+CI)v||_{\gamma}\geq\delta_{1}||v||_{\gamma}$

for some $\delta_{1}>0$ uniformly in $\theta\in\Theta.$ Let us choose
$\Lambda=[2{\cal A}_{0}+2\theta_{0}{\cal A}_{1}+CI]^{1/(2m)}$ for some fixed
$\theta_{0}\in\Theta.$ If the operator ${\cal A}_{0}+\theta{\cal A}_{1}$ is
unbounded, we say that ${\cal A}_{0}+\theta{\cal A}_{1}$ has order $2m$ and
$\Lambda$ has order 1. If the equation (4.1) is $(m,\gamma)$-parabolic and
diagonalizable, we will assume that the operator $\Lambda$ has the same
eigenfunctions as the operators ${\cal A}_{0}$ and ${\cal A}_{1}.$ This is
justified by the comments made above. Suppose the equation (4.1) is
diagonalizable and there exists eigenvalues $\\{\rho_{j},j\geq
1\\},\\{\nu_{j},j\geq 1\\}$ such that

${\cal A}_{0}h_{j}=\rho_{j}h_{j}\;\;\mbox{and}\;\;{\cal
A}_{1}h_{j}=\nu_{j}h_{j}.$

Without loss of generality, we can also assume that there exists
$\\{\lambda_{j},j\geq 1\\}$ such that

$\Lambda h_{j}=\lambda_{j}h_{j}.$

Following the arguments in Cialenco et al. (2009), it can be shown that the
equation (4.1) is $(m,\gamma)$-parabolic if and only if there exists
$\delta>0,C_{1}>0$ and $C_{2}\in R$ such that, for all $j\geq
1,\theta,\in\Theta,$

(4.4) $|\rho_{j}+\theta\nu_{j}|\leq C_{1}\lambda_{j}^{2m}$

and

(4.5) $-2(\rho_{j}+\theta\nu_{j})+\delta\lambda_{j}^{2m}\leq C_{2}.$

As the conditions in (4.4) and (4.5) do not depend on $\gamma,$ we conclude
that a diagonalizable equation (4.1) is $(m,\gamma)$-parabolic for some
$\gamma$ if and only if it is $(m,\gamma)$-parabolic for every $\gamma.$ Here
after we will say that the equation (4.1) is $m$-parabolic. We will assume
that the equation (4.1) is diagonalizable and fix the basis $\\{h_{j},j\geq
1\\}$ in ${\bf H}$ consisting of the eigenfunctions of ${\cal A}_{0}$, ${\cal
A}_{1}$ and $\lambda$. Recall that set of eigenfunctions is the same for all
the three operators. Since $h_{j}$ belongs to every ${\bf H}^{\gamma},$ and
since $\cap_{\gamma}{\bf H}^{\gamma}$ is dense in $\cup_{\gamma}{\bf
H}^{\gamma},$ every element $f$ of $\cup_{\gamma}{\bf H}^{\gamma},$ has a
unique expansion $\sum_{j\geq 1}f_{j}h_{j}$ where $f_{j}=<f,h_{j}>_{0,m}$ for
suitable $m.$

The functional structure described above follows the work in Cialenko et al.
(2009).

Definition : The infinite dimensional mixed fractional Brownian motion
$\tilde{W}^{H}$ is an element of $\cup_{\gamma\in R}{\bf H}^{\gamma}$ with the
expansion

(4.6) $\tilde{W}^{H}(t)=\sum_{j\geq 1}\tilde{h}_{j}\tilde{W}_{j}^{H}(t).$

Definition : The solution of the diagonalizable equation

(4.7) $du(t)+({\cal A}_{0}+\theta{\cal A}_{1})u(t)dt=d\tilde{W}^{H}(t),0\leq
t\leq T,u(0)=u_{0},$

with $u_{0}\in{\bf H},$ is defined to be a random process $u(t),0<t\leq T,$
with values in $\cup_{\gamma}{\bf H}^{\gamma}$ and has an expansion

(4.8) $u(t)=\sum_{j\geq 1}h_{j}u_{j}(t)$

where

(4.9)
$u_{j}(t)=(u_{0},h_{j})_{0}e^{-(\theta\nu_{j}+\rho_{j})t}+\int_{0}^{t}e^{-(\theta\nu_{j}+\rho_{j})(t-s)}\;\;d\tilde{W}_{j}^{H}(s).$

Let

(4.10) $\mu_{j}(\theta)=\theta\nu_{j}+\rho_{j},j\geq 1.$

In view of (4.5), we get that there exists a positive integer $J$ such that

(4.11) $\mu_{j}(\theta)>0\;\;\mbox{for}\;\;j\geq J$

if the equation (4.1) is $m$-parabolic and diagonalizable. Theorem 4.1 :
Suppose that $H\geq\frac{1}{2}$ and the equation (4.1) is $m$-parabolic and
diagonalizable. Further suppose that there exists a positive real number
$\gamma$ such that

(4.12) $\sum_{j\geq 1}(1+|\mu_{j}(\theta)|)^{-\gamma}<\infty.$

Then, for every $t>0,\tilde{W}^{H}(t)\in L_{2}(\Omega,{\bf
H}^{-m\gamma})\;\mbox{and}\;u(t)\in L_{2}(\Omega,{\bf
H}^{-m\gamma+m\min(2H,1)}).$ Proof : The condition (3) implies that
$\lim_{j\rightarrow\infty}|\mu_{j}|=\infty,$ and hence the operators ${\cal
A}_{0}+\theta{\cal A}_{1}$ and $\Lambda$ are unbounded. The parabolicity
assumption and the equations (4.4) and (4.5) imply that, for sufficently large
$j,$

$1+|\mu_{j}(\theta)|^{m}\leq C_{2}\lambda_{j}^{2m}$

uniformly in $\theta\in\Theta.$ Furthermore

$E||\tilde{W}^{H}(t)||^{2}_{-m\gamma}=(t^{2H}+t)\sum_{j\geq
1}\lambda_{j}^{-2m\gamma}\leq C_{2}(t^{2H}+t)\sum_{j\geq
1}(1+\mu_{j}(\theta)|^{-\gamma}<\infty.$

From the definition of the mixed fractional Brownian motion and the fact that
the component fractional Brownian motion and the Brownian motion are
independent and centered, it follows that

$\displaystyle E[u_{j}^{2}(t)]$ $\displaystyle=$ $\displaystyle
H(2H-1)e^{-2\mu_{j}(\theta)t}\int_{0}^{t}\int_{0}^{t}e^{\mu_{j}(\theta)(s_{1}+s_{2})}|s_{1}-s_{2}|^{2H-2}ds_{1}ds_{2}+e^{-2\mu_{j}(\theta)t}\int_{0}^{t}e^{2\mu_{j}(s)}ds$
$\displaystyle=$ $\displaystyle J_{1}(t)+J_{2}(t)\;\;\mbox{(say).}$

by the properties of fractional Brownian motion (cf. Prakasa Rao (2010)) and
Brownian motion. It can be checked that

$\lim_{j\rightarrow\infty}|\mu_{j}(\theta)|^{2H}J_{1}(t)=H(2H-1)\int_{0}^{\infty}x^{2H-2}e^{-x}dx=H(2H-1)\Gamma(2H-1)$

and

$\lim_{j\rightarrow\infty}|\mu_{j}(\theta)|J_{2}(t)=\frac{1}{2},$

Hence

$\lim_{j\rightarrow\infty}|\mu_{j}(\theta)|^{\min(2H,1)}E(u_{j}^{2}(t))<\infty.$

and

$\sum_{j=1}^{\infty}(1+|\mu_{j}(\theta)|)^{-\gamma+\min(2H,1)}E(u_{j}^{2}(t))<\infty.$

Maximum likelihood estimation : Consider the diagonalizable equation

(4.14) $du(t)+({\cal A}_{0}+\theta{\cal
A}_{1})u(t)dt=d\tilde{W}^{H}(t),u(0)=0,0\leq t\leq T$

with

(4.15) $u(t)=\sum_{j\geq 1}h_{j}u_{j}(t)$

as given by (4.9). Suppose the processes $\\{u_{i}(t),0\leq t\leq
T\\},i=1,\dots,N$ can be observed continuously over the interval $[0,T].$ The
problem is to estimate the parameter $\theta$ using these observed paths over
the interbal $[0,T].$. Note that $\mu_{j}(\theta)=\rho_{j}+\nu_{j}\theta,$
where $\rho_{j}$ and $\nu_{j}$ are the eigenvalues of ${\cal A}_{0}$ and
${\cal A}_{1}$ respectively. Furthermore each process $u_{j}$ is a mixed
fractional Ornstein-Uhlenbeck process satisfying the stochastic differential
equation

(4.16)
$du_{j}(t)=-\mu_{j}(\theta)u_{j}(t)dt+d\tilde{W}_{j}^{H}(t),u_{j}(0)=0,0\leq
t\leq T.$

Since the processes $\\{\tilde{W}_{j}^{H},j\geq 1\\}$ are independent, it
follows that the processes $\\{u_{j},1\leq j\leq N\\}$ are independent.
Following the notation introduced above, let

(4.17)
$M_{j}^{H}(t)=\int_{0}^{t}g_{H}(s,t)d\tilde{W}_{j}^{H}(s),Q_{j}(t)=\frac{d}{dw_{j}^{H}(t)}\int_{0}^{t}g_{H}(s,t)u_{j}(s)ds$

and

(4.18) $Z_{j}(t)=\int_{0}^{t}g_{H}(s,t)du_{j}(s)$

for $j=1,\dots,N.$ Applying the Girsanov-type formula, it can be shown that
the measure generated by the process $(u_{1},\dots,u_{N})$ is absolutely
continuous with respect to the measure generated by the process
$(\tilde{W}_{1}^{H},\dots,\tilde{W}_{N}^{H})$ and their Radon-Nikodym
derivative is given by

(4.19)
$\exp(-\sum_{j=1}^{N}\mu_{j}(\theta)\int_{0}^{T}Q_{j}(s)dZ_{j}(s)-\sum_{j=1}^{N}\frac{[\mu_{j}(\theta)]^{2}}{2}\int_{0}^{T}Q_{j}^{2}(s)dw_{H}(s)).$

Maximizing this function with respect to the parameter $\theta,$ we get the
maximum likelihood estimator

$\hat{\theta}_{N}=-\frac{\sum_{j=1}^{N}\int_{0}^{T}\nu_{j}Q_{j}(s)(dZ_{j}(s)+\rho_{j}Q_{j}(s)dw_{H}(s))}{\sum_{j=1}^{N}\int_{0}^{T}\nu_{j}^{2}Q_{j}^{2}(s)dw_{H}(s)}.$

Theorem 4.2 : Suppose the Hurst index $H>\frac{1}{2}.$ Suppose that

(4.20) $\sum_{j=1}^{\infty}\frac{\nu_{j}^{2}}{\mu_{j}(\theta)}=\infty.$

Then

$\lim_{N\rightarrow\infty}\hat{\theta}_{N}=\theta\;\;\mbox{a.s.}$

where $\theta$ is the true parameter. Furthermore, if the condition (i) holds,
then

(4.21)
$\lim_{N\rightarrow\infty}(\sum_{j=J}^{N}\frac{\nu_{j}^{2}}{\mu_{j}(\theta)})^{1/2}(\hat{\theta}_{N}-\theta)\stackrel{{\scriptstyle{\cal
L}}}{{\rightarrow}}\;\;N(0,1)\mbox{as}\;\;N\rightarrow\infty$

where $J=\min\\{j:\mu_{i}(\theta)=0\;\;\mbox{for all}\;\;i\geq j\\}.$ Proof :
Observe that

(4.22)
$\hat{\theta}_{N}-\theta=-\frac{\sum_{j=1}^{N}\int_{0}^{T}\nu_{j}Q_{j}(s)dM_{j}^{H}(s)}{\sum_{j=1}^{N}\int_{0}^{T}\nu_{j}^{2}Q_{j}^{2}(s)dw^{H}(s)}.$

The numerator of the expression on the right side (4.22) is a local martingale
and the denominator is its quadratic variation. It is known that

(4.23)
$\lim_{j\rightarrow\infty}\mu_{j}(\theta)\;\;E[\int_{0}^{T}Q_{j}^{2}(s)dw_{H}(s)]=\frac{T}{2}>0$

from the equation (2.1) in Chigansky and Kleptsyna (2019) (cf. Maruskeevych
(2016)) which implies that

$\sum_{j=1}^{N}\int_{0}^{T}\nu_{j}^{2}Q_{j}^{2}(s)dw^{H}(s)\rightarrow\infty\;\;\;\mbox{a.s.
as}\;\;N\rightarrow\infty$

under the condition (4.20). Hence, an application of the Strong law of large
numbers for local martingales (cf. Prakasa Rao (1999), Liptser and Shiryayev
(1989)) will imply that the left side of the equation (4.22) converges to zero
almost surely and hence

$\hat{\theta}_{N}-\theta\rightarrow 0\;\;\mbox{almost surely}$

as $N\rightarrow\infty.$ Theorem 4.3: Suppose there exists a norming function
$I_{N}$ such that

(4.24)
$I_{N}^{2}\sum_{j=1}^{N}\int_{0}^{T}\nu_{j}^{2}Q_{j}^{2}(s)dw^{H}(s)\rightarrow\eta^{2}\;\;\mbox{in
probability as}\;\;N\rightarrow\infty$

where $\eta$ is a random variable such that $P(\eta>0)=1.$ Let

(4.25) $R_{N}=\sum_{j=1}^{N}\int_{0}^{T}\nu_{j}Q_{j}(s)dM_{j}^{H}(s).$

Then

(4.26) $(I_{N}R_{n},I_{N}^{2}<R_{N}>)\rightarrow(\eta Z,\eta^{2})\;\;\mbox{in
law as }\;\;N\rightarrow\infty$

where the random variable $Z$ has the standard normal distribution and the
random vatiables $Z$ and $\eta$ are independent.Here $<R_{N}>,N\geq 1$ is the
quadratic variation of the process $\\{R_{N},N\geq 1\\}.$ Proof: This theorem
follows as a consequence of the central limit theorem for local martingales
(cf. Theorem 1.49 and remark 1.47, Prakasa Rao (1999b)). Observe that

(4.27)
$I_{N}^{-1}(\hat{\theta}_{N}-\theta)=\frac{I_{N}R_{N}}{I_{N}^{2}<R>_{N}},N\geq
1.$

Applying Theorem 4.3, we obtain the following result . Theorem 4.4: Suppose
the conditions stated in Theorem 4.2 hold. Then

(4.28)
$I_{N}^{-1}(\hat{\theta}_{N}-\theta)\rightarrow\frac{Z}{\eta}\;\;\mbox{in law
as}\;\;N\rightarrow\infty$

where the random variable $Z$ has the standard normal distribution and the
random variable $Z$ and $\eta$ are independent. Remarks : If the random
variable $\eta$ is a constant almost surely, then the limiting distribution of
the maximum likelihood estimator $\hat{\theta}_{N}$ is Gaussian with mean zero
and variance $\eta^{-2}.$ Otherwise it is a mixture of the normal distribution
with mean zero and variance $\eta^{-2}$ with the mixing distribution as that
the random variable $\eta.$ Applying Berry-Esseen bound for sums of
independent random variables, it is possible to obtain the limiting
distribution and rate of convergence of the distribution of the MLE if a bound
on the variance of the random variable

$\int_{0}^{T}Q_{j}^{2}(s)\;dw^{H}(s)$

can be obtained as $j\rightarrow\infty$ after suitable norming as in Lemma A.2
in Cialenco et al. (2009) in the fractional Brownian case. It has not been
possible to obtain a similar result in the case mixed fractional Brownian
motion as the function $g_{H}(s,t)$ defining the martingale $M_{j}^{H}$
defined by (3.8) is not explicitly known. The problem of obtaining a closed
form for the function $g_{H}(s,t)$ remains open. Acknowledgment This work was
supported under the scheme “INSA Senior Scientist” by the Indian National
Science Academy (INSA) at the CR Rao Advanced Institute of Mathematics,
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