# Efficient estimation for ergodic diffusion processes sampled at high
frequency

Michael Sørensen
Department of Mathematical Sciences
University of Copenhagen
Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark

###### Abstract

A general theory of efficient estimation for ergodic diffusion processes
sampled at high frequency with an infinite time horizon is presented. High
frequency sampling is common in many applications, with finance as a prominent
example. The theory is formulated in term of approximate martingale estimating
functions and covers a large class of estimators including most of the
previously proposed estimators for diffusion processes. Easily checked
conditions ensuring that an estimating function is an approximate martingale
are derived, and general conditions ensuring consistency and asymptotic
normality of estimators are given. Most importantly, simple conditions are
given that ensure rate optimality and efficiency. Rate optimal estimators of
parameters in the diffusion coefficient converge faster than estimators of
drift coefficient parameters because they take advantage of the information in
the quadratic variation. The conditions facilitate the choice among the
multitude of estimators that have been proposed for diffusion models. Optimal
martingale estimating functions in the sense of Godambe and Heyde and their
high frequency approximations are, under weak conditions, shown to satisfy the
conditions for rate optimality and efficiency. This provides a natural
feasible method of constructing explicit rate optimal and efficient estimating
functions by solving a linear equation.

Key words: Approximate martingale estimating functions, discrete time
observation of a diffusion process, efficiency, Euler approximation, explicit
estimating functions, generalized method of moments, optimal estimating
function, optimal rate, maximum likelihood estimation, stochastic differential
equations.

##  1 Introduction

Dynamic phenomena affected by random noise are often modelled in continuous
time by stochastic differential equations. Among the advantages of this
approach are interpretable model parameters and easy communication with other
scientists by using a common modelling tool, viz. differential equations. A
few examples are applications in the areas of animal movement (Michelot et al.
(2019)), climate research (Ditlevsen and Ditlevsen (2023)), finance (Chan et
al. (1992), Dipple et al. (2020)), protein structure evolution (Golden et al.
(2017)), neuro science (Bibbona et al. (2010), Jensen et al. (2012), Bachar et
al. (2013)), transmission of infectious diseases (Guy et al. (2015), Arnst et
al. (2022)) and physiology (Picchini et al. (2008)). While the dynamics is
formulated in continuous time, observations are made at discrete points in
time. This complicates statistical inference for these models, which is an
intensive area of research, where a profusion of estimators have been proposed
for parametric diffusion models, see e.g. Sørensen (2012) and Iacus and
Yoshida (2018). Many simulation studies have been performed to compare the
relative merits of estimators, but have not provided a clear general picture.
The simple and easily checked criteria for efficiency and rate optimality of
estimators obtained in this paper are useful for identifying the best
estimators and explain findings in simulation studies.

We consider a scalar diffusion given by the stochastic differential equation

$dX_{t}=b(X_{t};\alpha)dt+\sigma(X_{t};\beta)dW_{t},$ (1.1)

where $(\alpha,\beta)=\theta\in\Theta\subseteq{\rm
I\mkern-2.5mu\nonscript\mkern-0.5muR}^{2}$ are parameters to be estimated. The
restrictions to a scalar process and to two scalar parameters are made to
simplify the presentation. The results can be generalized to multivariate
diffusions as indicated in Section 4, and to multivariate parameters by
considering estimating functions of the same dimension as the parameter vector
and replacing partial derivatives by vectors or matrices of partial
derivatives. The process $X$ is assumed to be observed at equidistant time
points $i\Delta_{n}$, $i=0,\ldots,n$, and we consider the high
frequency/infinite time horizon asymptotic scenario, where

$n\rightarrow\infty,\hskip 17.07164pt\Delta_{n}\rightarrow 0,\hskip
17.07164ptn\Delta_{n}\rightarrow\infty.$

The length of the time interval in which observations are made goes to
infinity, which is necessary to ensure that the drift parameter $\alpha$ can
be estimated consistently. At the same time the sampling frequency goes to
infinity. This is particularly important for diffusion processes, because the
quadratic variation of a diffusion process contains information about the
parameter $\beta$ in the diffusion coefficient, which good estimators can use
when the sampling frequency is sufficiently high. For such estimators, $\beta$
is estimated at a higher rate than the $1/\sqrt{n\Delta_{n}}$-rate, which is
optimal for the drift parameter $\alpha$; see Gobet (2002) where it is shown
that the model (1.1) is locally asymptotically normal under the high
frequency/infinite time horizon asymptotic scenario. In the present paper we
give easily checked conditions for rate optimality and efficiency. If the
drift coefficient is known, consistent estimators can be found without the
infinite time horizon assumption. Results on rate optimality and efficiency
when the sampling interval is bounded are given in Jakobsen and Sørensen
(2017). High frequency asymptotics is often relevant in applications, because
the sampling frequency needs not be particularly high for the asymptotic
optimality results to work. It only needs to be high relative to the
characteristic time scale of the diffusion process. For some types of economic
data, even weekly observations can in be considered a high sampling frequency,
see e.g. Larsen and Sørensen (2007).

Our theory is phrased in terms of estimating functions of the general form

$G_{n}(\theta)=\sum_{i=1}^{n}g(\Delta_{n},X_{i\Delta_{n}},X_{(i-1)\Delta_{n}};\theta),$
(1.2)

where the function $g(\Delta,y,x;\theta)$, with values in ${\rm
I\mkern-2.5mu\nonscript\mkern-0.5muR}^{2}$, is such that $G_{n}$ is, exactly
or approximately, a martingale estimating function. Specifically,
$E_{\theta}(g(\Delta,X_{i\Delta_{n}},X_{(i-1)\Delta_{n}};\theta)\,|\,X_{(i-1)\Delta_{n}})$
is equal to zero or of order $\Delta^{\kappa}$ for some $\kappa\geq 2$.
Estimators are obtained as solutions to the estimating equation
$G_{n}(\theta)=0$. We call such an estimator a $G_{n}$-estimator. For
estimating functions that are not exact martingales, the extra condition that
$n\Delta^{2(\kappa-1)}\rightarrow 0$ is needed to ensure the asymptotic
results.

The theory developed here covers a large class of estimators for diffusion
processes including most of the previously proposed estimators. The few that
are not covered are likely to be less efficient, because non-martingale
estimating functions, in general, do not approximate the score function as
well as martingales. In particular, the theory covers the martingale
estimating functions proposed by Bibby and Sørensen (1995) and Kessler and
Sørensen (1999), GMM-estimators based on conditional moments, Hansen (1982,
1985, 1993), and the maximum likelihood estimator and Bayesian estimators; for
numerical methods to calculate these estimators, see Pedersen (1995), Roberts
and Stramer (2001), Aït-Sahalia (2002), Durham and Gallant (2002), Aït-Sahalia
and Mykland (2003), Beskos et al. (2009), Golightly and Wilkinson (2011),
Bladt and Sørensen (2014), van der Meulen and Schauer (2017) and Bladt et al.
(2021). The pseudo-likelihood function obtained from the Gaussian Euler
approximation to the transition density is covered too. Estimators closely
related to the Euler pseudo-likelihood were considered by Florens-Zmirou
(1989), Yoshida (1992) and Uchida (2010). These and pseudo-likelihood
functions based on more accurate Gaussian approximations to the likelihood
function, such as those considered by Kessler (1997), Uchida and Yoshida
(2012) and Kitagawa and Uchida (2014), are also covered. Sørensen and Uchida
(2003) and Gloter and Sørensen (2009) considered the Euler pseudo likelihood
under a combination of high frequency and small diffusion asymptotics, where
the diffusion coefficient goes to zero as $n\rightarrow\infty$. The latter
condition replaces the infinite time horizon condition.

The following condition on the function $g(\Delta,y,x;\theta)$ ensures rate
optimality of estimators.

###### Condition 1.1

$\partial_{y}g_{2}(0,x,x;\theta)=0$ (1.3)

for all $x$ in the state-space of the diffusion process and all
$\theta\in\Theta$.

By $\partial_{y}g_{2}(0,x,x;\theta)$ we mean $\partial_{y}g_{2}(0,y,x;\theta)$
evaluated at $y=x$. Here $\partial_{y}$ denotes the partial derivative w.r.t.
$y$, and $g_{i}$ is the $i$th coordinate of $g$. More precisely, it is
sufficient that a linear combination of the two coordinates of $g$ satisfies
(1.3), but without loss of generality it can be assumed to be $g_{2}$. This
will be explained in Section 2. We will refer to (1.3) as Jacobsen’s condition
because it equals one of the conditions for small $\Delta$-optimality in the
sense of Jacobsen (2001) of martingale estimating functions; see Jacobsen
(2002). Jacobsen considered an asymptotic scenario where the time between
observations $\Delta$ does not depend on $n$. In his approach the Condition
1.1 was introduced to avoid a singularity in the asymptotic variance of the
estimators at $\Delta=0$. In our high frequency approach, the condition
implies rate optimality for estimation of the diffusion coefficient parameter.

Our condition for efficiency is

###### Condition 1.2

$\partial_{y}g_{1}(0,x,x;\theta)=\partial_{\alpha}b(x;\alpha)/\sigma^{2}(x;\beta)$
(1.4)

and

$\partial^{2}_{y}g_{2}(0,x,x;\theta)=\partial_{\beta}\sigma^{2}(x;\beta)/\sigma^{4}(x;\beta),$
(1.5)

for all $x$ in the state space of the diffusion process and all
$\theta\in\Theta$.

The Conditions 1.1 and 1.2 are, under weak regularity conditions, shown to be
satisfied by martingale estimating functions that are optimal in the sense of
Godambe and Heyde (1987), which is both very useful and quite surprising.
Useful because it provides an easy method of constructing rate optimal and
efficient estimators, and surprising because Godambe-Heyde optimality is a
local property in the sense that it is a property of a particular class of
estimating functions. Therefore, there is no a priori reason to except this
property to imply global properties like rate optimality and efficiency.
Martingale estimating functions give consistent estimators at all sampling
frequencies, see Bibby et al. (2010), and Godambe-Heyde optimal martingale
estimating functions are known to often provide estimators with a high
efficiency, see e.g. the simulation studies in Overbeck and Rydén (1997) and
Larsen and Sørensen (2007). The results in this paper explain why this is the
case.

The paper is organized as follows. Section 2 sets up the model, the class of
approximate martingale estimating functions, and the assumptions and the
notation used throughout the paper. A number of well-known estimators are
shown to be covered by the theory, and a lemma of independent interest gives
fundamental identities and characterizes approximate martingale estimating
functions of order $\kappa$. Section 3 develops the high frequency asymptotic
theory for general estimating functions as well as for estimating functions
satisfying Condition 1.1. The conditions for efficiency are derived in Section
4, and it is proved that Godambe-Heyde optimal martingale estimating functions
and their high frequency approximations are rate optimal and efficient, which
provides a feasible method of constructing explicit rate optimal and efficient
estimating functions. Examples are considered, including the Euler pseudo-
likelihood and maximum likelihood estimation. Proofs and some lemmas are given
in Section 5, where also tools for studying high frequency asymptotic
properties of estimators are provided. Section 6 concludes.

##  2 Model, conditions and notation

We consider observations $X_{t_{0}^{n}},\ldots,X_{t_{n}^{n}}$ of the process
given by (1.1) at the time points $t_{i}^{n}=i\Delta_{n}$, $i=0,\ldots,n$. We
suppose that a solution of the stochastic differential equation (1.1) exists,
is unique in law, and is adapted to the filtration generated by the Wiener
process $W$ and the initial value $X_{0}$. The state-space of $X$ is denoted
by $(\ell,r)$, where $-\infty\leq\ell<r\leq\infty$, and we assume that
$v(x;\beta)=\sigma^{2}(x;\beta)>0$ for all $x\in(\ell,r)$. Furthermore, we
assume that $\theta=(\alpha,\beta)\in\Theta$, where $\Theta$ is a subset of
${\rm I\mkern-2.5mu\nonscript\mkern-0.5muR}^{2}$, and that the true parameter
value $\theta_{0}=(\alpha_{0},\beta_{0})\in\mbox{int}\,\Theta$, the interior
of $\Theta$. It is no serious restriction to assume that $\Theta$ is convex.

A function $f(y,x;\theta)$ is said to be of polynomial growth in $y$ and $x$
uniformly for $\theta$ in a compact set if, for any compact subset
$K\subseteq\Theta$, there exists a constant $C>0$ such that $\sup_{\theta\in
K}|f(y,x;\theta)|\leq C(1+|x|^{C}+|y|^{C})$ for all $x,y\in(\ell,r)$. The
assumptions of polynomial growth in this paper are made to simplify the
theory. These assumptions are satisfied for most models used in practice, but
could be relaxed.

Here and in the rest of the paper, $R(\Delta,y,x;\theta)$ denotes a (generic)
function such that $|R(\Delta,y,x;\theta)|$ $\leq F(y,x;\theta)$, for all
$\Delta$, where $F$ is some function of polynomial growth in $y$ and $x$
uniformly for $\theta$ in a compact set. Similarly for $R(\Delta,x;\theta)$.

###### Definition 2.1

We let $C_{p,k_{1},k_{2},k_{3}}({\rm
I\mkern-2.5mu\nonscript\mkern-0.5muR}_{+}\times(\ell,r)^{2}\times\Theta)$
denote the class of real functions $f(t,y,x;\theta)$ satisfying that

  * (i)

$f(t,y,x;\theta)$ is $k_{1}$ times continuously differentiable with respect to
$t$, $k_{2}$ times continuously differentiable with respect to $y$, and
$k_{3}$ times continuously differentiable with respect to $\theta$

  * (ii)

$f$ and all partial derivatives
$\partial^{i_{1}}_{t}\,\partial^{i_{2}}_{y}\,\partial^{i_{3}}_{\alpha}\,\partial^{i_{4}}_{\beta}f$,
$i_{j}=0,\ldots k_{j}$, $j=1,2$, $i_{3}+i_{4}\leq k_{3}$ are continuously
differentiable with respect to $x$ and are of polynomial growth in $x$ and $y$
uniformly for $\theta$ in compact sets (for fixed $t\leq 1$)

  * (iii)

$f$ has an expansion

$f(\Delta,y,x;\theta)=\sum_{i=0}^{k_{1}}\frac{\Delta^{i}}{i!}f^{(i)}(y,x;\theta)+\Delta^{k_{1}+1}R(\Delta,y,x;\theta).$
(2.1)

The classes $C_{p,k_{2},k_{3}}((\ell,r)\times\Theta)$ and
$C_{p,k_{2},k_{3}}((\ell,r)^{2}\times\Theta)$ are defined similarly (with
property (iii) omitted) for functions of the form $f(y;\theta)$ and
$f(y,x;\theta)$, respectively.

We assume that the stochastic differential equation (1.1) satisfies the
following condition.

###### Condition 2.2

The following holds for all $\theta\in\Theta$:

  * (1)

$\int_{x^{\\#}}^{r}s(x;\theta)dx=\int^{x^{\\#}}_{\ell}s(x;\theta)dx=\infty$
(2.2)

and

$\int_{\ell}^{r}x^{k}\tilde{\mu}_{\theta}(x)dx<\infty$ (2.3)

for all $k\in{\rm I\mkern-2.5mu\nonscript\mkern-0.5muN}$, where $x^{\\#}$ is
an arbitrary point in $(\ell,r)$,

$s(x;\theta)=\exp\left(-2\int_{x^{\\#}}^{x}\frac{b(y;\alpha)}{v(y;\beta)}dy\right)$
(2.4)

and

$\tilde{\mu}_{\theta}(x)=[s(x;\theta)v(x;\beta)]^{-1}$ (2.5)

  * (2)

$\sup_{t}E_{\theta}(|X_{t}|^{k})<\infty$ for all $k\in{\rm
I\mkern-2.5mu\nonscript\mkern-0.5muN}$

  * (3)

$b,\sigma\in C_{p,4,1}((\ell,r)\times\Theta)$

  * (4)

There exists a constant $C_{\theta}$ such that for all $x,y\in(\ell,r)$

$|b(x;\alpha)-b(y;\alpha)|+|\sigma(x;\beta)-\sigma(y;\beta)|\leq
C_{\theta}|x-y|.$

The conditions (2.2) and (2.3) with $k=1$ ensure that the process $X$ is
ergodic with an invariant probability measure with Lebesgue density

$\mu_{\theta}(x)=\tilde{\mu}_{\theta}(x)/\int_{\ell}^{r}\tilde{\mu}_{\theta}(y)dy.$
(2.6)

If $X$ is stationary, Condition 2.2 (2) is obviously satisfied under (2.3).
Similarly, if $X$ is sufficiently mixing that all moments converge as
$t\rightarrow\infty$.

We consider estimating functions of the general form (1.2) where the function
$g(\Delta,y,x;\theta)$ has values in ${\rm
I\mkern-2.5mu\nonscript\mkern-0.5muR}^{2}$ and satisfies the following
condition.

###### Condition 2.3

  * (1)

There exists a $\kappa\geq 2$ such that

$E_{\theta}(g_{i}(\Delta_{n},X_{t_{j}^{n}},X_{t_{j-1}^{n}};\theta)\,|\,X_{t_{j-1}^{n}})=\Delta_{n}^{\kappa}\,R(\Delta_{n},X_{t_{j-1}^{n}};\theta)\
\mbox{ for }i=1,2\mbox{ and all }\theta\in\Theta$ (2.7)

  * (2)

$\displaystyle g_{i}(\Delta,y,x;\theta)$ $\displaystyle\in$ $\displaystyle
C_{p,2,6,2}({\rm
I\mkern-2.5mu\nonscript\mkern-0.5muR}_{+}\times(\ell,r)^{2}\times\Theta),\
i=1,2$ $\displaystyle g_{i}^{(j)}(y,x;\theta)$ $\displaystyle\in$
$\displaystyle C_{p,2(3-i),2}((\ell,r)^{2}\times\Theta),\ i=1,2,\ j=0,1,2,$

where the $g^{(j)}$s are the functions appearing in the expansion (2.1).

We call an estimating function satisfying Condition 2.3 (1) an approximate
martingale estimating function of order $\kappa$. For any non-singular
$2\times 2$ matrix, $M(\Delta,n,\theta)$, the estimating functions
$M(\Delta_{n},n,\theta)G_{n}(\theta)$ and $G_{n}(\theta)$ give identical
estimators. We call them versions of the same estimating function. Since the
matrix $M$ may depend on $\Delta_{n}$, not all versions satisfy Condition 2.3
and other conditions in the paper, in particular Conditions 1.1 and 1.2. We
say that (1.2) is an approximate martingale estimating function of order
$\kappa$, if there exists a version which satisfies (2.7), and for which the
limit $g(0,y,x;\theta)$ is finite with, for at least one value of
$(x,y,\theta)$, all coordinates different from zero. We use this version in
the proofs. It is, typically, obtained by multiplying one or both of the
coordinates by a power of $\Delta_{n}$; examples are given in Section 4.

It could be argued, that an estimating functions that satisfies (2.7) with
$\kappa=1$ could equally well be called an approximate martingale. However,
the asymptotic theory in this case is entirely different from the case
$\kappa\geq 2$ and requires a separate study. Some particular examples are
studied in Jørgensen and Sørensen (2021).

The generator of the solution to (1.1) is the differential operator

$L_{\theta}=b(x;\alpha)\frac{d}{dx}+\dfrac{1}{2}v(x;\beta)\frac{d^{2}}{dx^{2}}$
(2.8)

Here we take the domain of $L_{\theta}$ to be the set of all twice
continuously differentiable functions defined on the state space. For $f\in
C_{p,2(k+1)}((\ell,r))$ and $b,\sigma\in C_{p,2k,0}((\ell,r)\times\Theta)$,

$\mbox{E}_{\theta}(f(X_{t+\Delta})\,|\,X_{t})=\sum_{i=0}^{k}\frac{\Delta^{i}}{i!}L_{\theta}^{i}f(X_{t})+\Delta^{k+1}R(\Delta,X_{t};\theta),$
(2.9)

where

$\Delta^{k+1}R(\Delta,X_{t};\theta)=\int_{0}^{\Delta}\int_{0}^{u_{1}}\cdots\int_{0}^{u_{k}}E_{\theta}(L_{\theta}^{k+1}f(X_{t+u_{k+1}})\,|\,X_{t})du_{k+1}\cdots
du_{1},$

see e.g. Sørensen (2012). The properties of the remainder term follow from
Lemma 5.1 in Section 5. When we apply the generator to a function $h(y,x)$ of
two variables, we mean

$L_{\theta}(h)(y,x)=b(y;\alpha)\partial_{y}h(y,x)+\mbox{
\small$\frac{1}{2}$}v(y;\beta)\partial^{2}_{y}h(y,x),$ (2.10)

and for a function $h(\Delta,y,x;\theta)$ that depends also on $\Delta$ and
$\theta$, we use the notation

$L_{\theta}(h(\Delta;\tilde{\theta}))(y,x)=b(y;\alpha)\partial_{y}h(\Delta,y,x;\tilde{\theta})+\mbox{
\small$\frac{1}{2}$}v(y;\beta)\partial^{2}_{y}h(\Delta,y,x;\tilde{\theta}).$

The following lemma provides identities that play an essential role in the
proofs of the asymptotic theory in the next section. Note that $L_{\theta}$ is
applied coordinate-wise to a vector valued function, and that, depending on
the context, $0$ can also denote a $0$-vector.

###### Lemma 2.4

Let $G_{n}$ be an estimating function of the form (1.2), where $g_{i}\in
C_{p,\kappa-1,2(\kappa-1),0}({\rm
I\mkern-2.5mu\nonscript\mkern-0.5muR}\times(\ell,r)^{2}\times\Theta)$, i=1,2,
for a $\kappa\geq 2$, and assume Condition 2.2.

Then $G_{n}$ is an approximate martingale estimating function of order
$\kappa\geq 2$ (i.e. it satisfies (2.7)) if and only if

$\sum_{i=0}^{k}{k\choose i}L_{\theta}^{k-i}(g^{(i)}(\theta))(x,x)=0,\ \ \
k=0,\ldots,\kappa-1,$

for all $x\in(\ell,r)$ and $\theta\in\Theta$ (the $g^{(i)}$s are the functions
in the expansion $(\ref{gdeltaexp})$).

In particular, if $G$ is an approximate martingale estimating function, then

$\displaystyle g^{(0)}(x,x;\theta)$ $\displaystyle=$ $\displaystyle 0$ (2.11)
$\displaystyle g^{(1)}(x,x;\theta)$ $\displaystyle=$ $\displaystyle-
L_{\theta}(g^{(0)}(\theta))(x,x)$ (2.12)

for all $x\in(\ell,r)$ and $\theta\in\Theta$.

###  2.1 Examples

The prototype of estimating functions satisfying the condition (2.7) are the
martingale estimating functions for which

$E_{\theta}(g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\,|\,X_{t_{i-1}^{n}})=0.$

They often have the form

$g(\Delta,y,x;\theta)=A(x,\Delta;\theta)\left[f(y;\theta)-\pi_{\theta}^{1,\Delta}f(x;\theta)\right],$
(2.13)

where $f(y;\theta)=(f_{1}(y;\theta),\ldots,f_{N}(y;\theta))^{T}$, with $f_{i}$
real-valued, $A(x,\Delta;\theta)$ a $2\times N$-matrix of weights, and
$\pi_{\theta}^{1,\Delta}$ denotes the transition operator given by

$\pi_{\theta}^{1,\Delta}h(x)=E_{\theta}(h(X_{\Delta})\,|\,X_{0}=x)$ (2.14)

for a real-valued function $h$. Here and later $x^{T}$ denotes the transpose
of a vector or matrix $x$. The weight matrix $A$ can be chosen suitably, for
instance to obtain rate optimality and efficiency. Examples are polynomial
estimating functions, where the real functions $f_{j}$ are power functions or
more general polynomials. The quadratic martingale estimating function,
obtained for $N=2$, $f_{1}(x)=x$ and $f_{2}(x)=x^{2}$, is as a useful simple
example, see Section 4. Polynomial estimating functions are particularly
useful for the class of Pearson diffusions, for which all (finite) moments
(conditional as well as unconditional) can be found explicitly, see Forman and
Sørensen (2008). Other instances are the estimating functions based on
eigenfunctions of the generator (2.8) proposed by Kessler and Sørensen (1999).

The econometric generalized method of moments (GMM) based on conditional
moments is covered by our theory. This method is in practice often implemented
as follows; see Campbell et al. (1997). The starting point is an
$N$-dimensional function $h(\Delta,y,x;\theta)$ for which each coordinate
satisfies that
$E_{\theta}(h_{j}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\,|\,X_{t_{i-1}^{n}})=0.$
Let $A_{n}$ be an $N\times N$-matrix such that
$m_{n}(\theta)=A_{n}\sum_{i=1}^{n}h(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)$
converges in probability. For the usual low frequency asymptotics, where
$\Delta_{n}$ does not depend on $n$, $A_{n}=n^{-1}I_{N}$ ($I_{N}$ denotes the
identity matrix), but for the high frequency asymptotics considered in this
paper, a different choice of $A_{n}$ is usually necessary, as will become
clear in the next section. The GMM-estimator is obtained by minimizing
$Q_{n}(\theta)=m_{n}(\theta)^{T}W_{n}\,m_{n}(\theta)$, where $W_{n}$ is an
$N\times N$-matrix such that $W_{n}\rightarrow W$ in probability. It is
typically the (suitably normalized) inverse of a consistent estimator of the
covariance matrix of $m_{n}(\theta)$. Under weak regularity conditions, the
GMM-estimator solves the estimating equation
$\partial_{\theta}Q_{n}(\theta)=\partial_{\theta}m_{n}(\theta)^{T}W_{n}\,m_{n}(\theta)=0$,
so if $\partial_{\theta}m_{n}(\theta)\rightarrow D(\theta)$ in probability (a
necessary condition for asymptotic results about the GMM-estimator), then the
GMM-estimator has the same asymptotic behavior as the estimator obtained from
$D(\theta)^{T}WA_{n}\sum_{i=1}^{n}h(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)$,
which is a martingale estimating function. The close relationship between
martingale estimating functions and the type of GMM-estimators described here
is discussed in detail in Christensen and Sørensen (2008). More general GMM-
estimators of the martingale estimating function type were considered in
Hansen (1985, 1993), and a discussion of links between the literature on
estimating functions and that on GMM-estimators can be found in Hansen (2001).

Approximate martingale estimating functions can be obtained by replacing the
exact conditional expectation in (2.13) by the approximation given by (2.9)
such that the function $g$ has the form

$g(\Delta,y,x;\theta)=A(x,\Delta;\theta)\left[f(y;\theta)-\pi_{\theta}^{\kappa,\Delta}f(x;\theta)\right],$
(2.15)

where

$\pi_{\theta}^{\kappa,\Delta}f(x;\theta)=\sum_{i=0}^{\kappa-1}\frac{\Delta^{i}}{i!}\,L_{\theta}^{i}f(x;\theta),\hskip
17.07164pt\kappa=2,3,\ldots,$ (2.16)

with the generator $L_{\theta}$ applied coordinate-wise. This estimating
functions satisfies (2.7). A simple example is
$g(\Delta,y,x;\theta)=a(x,\Delta;\theta)(y-x-b(x;\alpha)\Delta)$ with
$\kappa=2$, considered by Prakasa Rao (1988) and Florens-Zmirou (1989). Other
instances are the estimators proposed by Chan et al. (1992) and Kelly et al.
(2004). For all $\kappa\in{\rm I\mkern-2.5mu\nonscript\mkern-0.5muN}$,
($\kappa\geq 2$), Kessler (1997) proposed a Gaussian approximation to the
likelihood function, for which the corresponding pseudo-score function is an
approximate martingale estimating function that satisfies (2.7).

##  3 Optimal rate

In this section we present asymptotic results for approximate martingale
estimating functions. We begin with a general approximate martingale
estimating function. Then we will see how Condition 1.1 implies rate
optimality, so that the estimator of the parameter in the diffusion
coefficient converges faster than the estimator of the parameter in the drift
coefficient. As previously, $x^{T}$ denotes the transpose of a vector or
matrix $x$ and $\theta_{0}=(\alpha_{0},\beta_{0})$ denotes the true parameter
value.

###### Theorem 3.1

Assume that the Conditions 2.2 and 2.3 hold. Suppose, moreover, the
identifiability condition that

$\displaystyle\gamma(\theta,\theta_{0})=\int_{\ell}^{r}[b(x,\alpha_{0})-b(x,\alpha)]\partial_{y}g(0,x,x;\theta)\mu_{\theta_{0}}(x)dx$
$\displaystyle\hskip
51.21495pt\mbox{}+\mbox{\small$\frac{1}{2}$}\int_{\ell}^{r}[v(x,\beta_{0})-v(x,\beta)]\partial^{2}_{y}g(0,x,x;\theta)\mu_{\theta_{0}}(x)dx\neq
0$

for all $\theta\neq\theta_{0}$, and that the matrix

$S=\int_{\ell}^{r}J_{\theta_{0}}(x)\mu_{\theta_{0}}(x)dx$ (3.2)

is invertible, where

$J_{\theta}(x)=\left(\begin{array}[]{cc}\partial_{\alpha}b(x;\alpha)\partial_{y}g_{1}(0,x,x;\theta)&\mbox{\small$\frac{1}{2}$}\partial_{\beta}v(x;\beta)\partial^{2}_{y}g_{1}(0,x,x;\theta)\\\
&\\\
\partial_{\alpha}b(x;\alpha)\partial_{y}g_{2}(0,x,x;\theta)&\mbox{\small$\frac{1}{2}$}\partial_{\beta}v(x;\beta)\partial^{2}_{y}g_{2}(0,x,x;\theta)\end{array}\right).$
(3.3)

Then with a probability that goes to one as $n\rightarrow\infty$, a consistent
$G_{n}$-estimator $\hat{\theta}_{n}=(\hat{\alpha}_{n},\hat{\beta}_{n})$ exists
and is unique in any compact subset $K$ of $\Theta$ with $\theta_{0}\in$
int$\,K$. If $n\Delta_{n}^{2\kappa-1}\rightarrow 0$, then

$\sqrt{n\Delta_{n}}(\hat{\theta}_{n}-\theta_{0})\stackrel{{\scriptstyle\cal
D}}{{\longrightarrow}}N_{2}\left(0,S^{-1}V_{0}(S^{T})^{-1}\right)$ (3.4)

under $P_{\theta_{0}}$, where $V_{0}=V(\theta_{0})$ with

$V(\theta)=\int_{\ell}^{r}v(x,\beta_{0})\partial_{y}g(0,x,x;\theta)\partial_{y}g(0,x,x;\theta)^{T}\mu_{\theta_{0}}(x)dx.$
(3.5)

For a martingale estimating function (3.4) holds without the extra condition
on the rate of convergence of $\Delta_{n}$.

A consistent estimator of the asymptotic variance of $\hat{\theta}_{n}$ can be
obtained from

$\frac{1}{n\Delta_{n}}\sum_{i=1}^{n}\partial_{\theta^{T}}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\hat{\theta}_{n})\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}-S$ (3.6)
$\frac{1}{n\Delta_{n}}\sum_{i=1}^{n}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\hat{\theta}_{n})g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\hat{\theta}_{n})^{T}\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}V_{0}$ (3.7)

The theorem follows from general asymptotic statistical results for stochastic
processes, see e.g. Jacod and Sørensen (2018). The proof is given in Section
5. The precise meaning of the uniqueness statement is that for any
$G_{n}$-estimator $\tilde{\theta}_{n}$ with
$P_{\theta_{0}}(\tilde{\theta}_{n}\in K)\rightarrow 1$ as
$n\rightarrow\infty$, it holds that
$P_{\theta_{0}}(\hat{\theta}_{n}\neq\tilde{\theta}_{n})\rightarrow 0$ as
$n\rightarrow\infty$.

We see from (3.4) that the rate of convergence of both $\hat{\alpha}$ and
$\hat{\beta}$ is $1/\sqrt{n\Delta_{n}}$, provided that the matrix $V_{0}$ is
regular. Here $n\Delta_{n}$ is the length of the interval in which the
diffusion is observed. Gobet (2002) showed that under weak regularity
conditions a discretely sampled diffusion model is local asymptotically normal
in the high frequency/infinite time horizon asymptotic scenario considered
here, and that the optimal rate of convergence for estimators of parameters in
the drift coefficient is $1/\sqrt{n\Delta_{n}}$, whereas the optimal rate for
estimators of parameters in the diffusion coefficient is $1/\sqrt{n}$.

The next theorem shows what happens when Jacobsen’s condition, Condition 1.1,
is satisfied, or more precisely, when a version of the estimating function
satisfies the condition.

###### Theorem 3.2

Suppose the Conditions 1.1, 2.2 and 2.3 hold. Assume, moreover, that the
following identifiability condition is satisfied

$\displaystyle\int_{\ell}^{r}[b(x,\alpha_{0})-b(x,\alpha)]\partial_{y}g_{1}(0,x,x;\theta)\mu_{\theta_{0}}(x)dx$
$\displaystyle\neq$ $\displaystyle 0\hskip 17.07164pt\mbox{when
}\alpha\neq\alpha_{0}$
$\displaystyle\int_{\ell}^{r}[v(x,\beta_{0})-v(x,\beta)]\partial^{2}_{y}g_{2}(0,x,x;\theta)\mu_{\theta_{0}}(x)dx$
$\displaystyle\neq$ $\displaystyle 0\hskip 17.07164pt\mbox{when
}\beta\neq\beta_{0},$

and that $S_{11}\neq 0$ and $S_{22}\neq 0$ with $S$ given by (3.2). Then with
a probability that goes to one as $n\rightarrow\infty$, a consistent
$G_{n}$-estimator $\hat{\theta}_{n}=(\hat{\alpha}_{n},\hat{\beta}_{n})$ exists
and is unique in any compact subset K of $\Theta$ with $\theta_{0}\in$
int$\,K$.

If, moreover,

$\partial_{\alpha}\partial^{2}_{y}g_{2}(0,x,x;\theta)=0$ (3.8)

and $n\Delta_{n}^{2(\kappa-1)}\rightarrow 0$, then

$\left(\begin{array}[]{c}\sqrt{n\Delta_{n}}(\hat{\alpha}_{n}-\alpha_{0})\\\
\sqrt{n}(\hat{\beta}_{n}-\beta_{0})\rule{0.0pt}{17.07164pt}\end{array}\right)\stackrel{{\scriptstyle\cal
D}}{{\longrightarrow}}N_{2}\left(\left(\begin{array}[]{c}0\\\
0\rule{0.0pt}{11.38109pt}\end{array}\right),\left(\begin{array}[]{cc}W_{1}(\theta_{0})/S^{2}_{11}&0\\\
0&W_{2}(\theta_{0})/S^{2}_{22}\rule{0.0pt}{17.07164pt}\end{array}\right)\right)$
(3.9)

where

$\displaystyle W_{1}(\theta)$ $\displaystyle=$
$\displaystyle\int_{\ell}^{r}v(x;\beta_{0})[\partial_{y}g_{1}(0,x,x;\theta)]^{2}\mu_{\theta_{0}}(x)dx=V(\theta)_{11}$
$\displaystyle W_{2}(\theta)$ $\displaystyle=$
$\displaystyle\mbox{\small$\frac{1}{2}$}\int_{\ell}^{r}[v(x;\beta_{0})^{2}+\frac{1}{2}(v(x;\beta_{0})-v(x;\beta))^{2}][\partial^{2}_{y}g_{2}(0,x,x;\theta)]^{2}\mu_{\theta_{0}}(x)dx$

with $V(\theta)$ given by (3.5). For a martingale estimating function (3.9)
holds without the extra condition on the rate of convergence of $\Delta_{n}$.

A consistent estimator of the asymptotic variance of $\hat{\theta}_{n}$ can be
obtained from (3.6) and

$D_{n}\sum_{i=1}^{n}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\hat{\theta}_{n})g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\hat{\theta}_{n})^{T}D_{n}\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}\left(\begin{array}[]{cc}W_{1}(\theta_{0})&0\\\
0&W_{2}(\theta_{0})\rule{0.0pt}{17.07164pt}\end{array}\right),$ (3.10)

where

$D_{n}=\left(\begin{array}[]{cc}\frac{1}{\sqrt{n\Delta_{n}}}&0\\\
0&\frac{1}{\Delta_{n}\sqrt{n}}\rule{0.0pt}{17.07164pt}\end{array}\right).$
(3.11)

Thus Jacobsen’s condition (1.3) and the additional condition (3.8) imply rate
optimal estimators and that the estimators of the drift parameter and of the
diffusion coefficient parameter are asymptotically independent. In the next
section we shall see that (3.8) is automatically satisfied for efficient
estimating functions. Note that for non-martingale estimating functions
$\Delta_{n}$ must go a bit faster to zero than was required in Theorem 3.1.
Note also that if the first coordinate of $g$ satisfies Jacobsen’s condition
too, then the first part of the identifiability condition in Theorem 3.2 does
not hold, and the parameter $\alpha$ cannot be consistently estimated by the
estimating function (1.2). The proof of Theorem 3.2 is given in Section 5.

###### Example 3.3

Consider a quadratic martingale estimating function of the form

$g(\Delta,y,x;\theta)=\left(\begin{array}[]{c}a_{1}(x,\Delta;\theta)[y-F(\Delta,x;\theta)]\\\
a_{2}(x,\Delta;\theta)\left[(y-F(\Delta,x;\theta))^{2}-\phi(\Delta,x;\theta)\right]\rule{0.0pt}{17.07164pt}\end{array}\right),$
(3.12)

where $F(\Delta,x;\theta)=E_{\theta}(X_{\Delta}|X_{0}=x)$ and
$\phi(\Delta,x;\theta)=\mbox{Var}_{\theta}(X_{\Delta}|X_{0}=x)$. Since, by
(2.9), $F(\Delta,x;\theta)=x+O(\Delta)$ and $\phi(\Delta,x;\theta)=O(\Delta)$,
we find that

$g(0,y,x;\theta)=\left(\begin{array}[]{c}a_{1}(x,0;\theta)(y-x)\\\
a_{2}(x,0;\theta)(y-x)^{2}\rule{0.0pt}{17.07164pt}\end{array}\right).$ (3.13)

Jacobsen’s condition (1.3) is satisfied because
$\partial_{y}g_{2}(0,y,x;\theta)=2a_{2}(x,\Delta;\theta)(y-x)$. Thus
estimators obtained from (3.12) are rate optimal, provided that (3.8) is
satisfied, for instance if $a_{2}$ does not depend on $\alpha$.

Clearly (3.13) holds if $F$ and $\phi$ in (3.12) are replaced by expansions of
order $O(\Delta^{\kappa-1})$ with $\kappa\geq 2$, using again (2.9). Thus rate
optimal estimators are also obtained in this more easily calculated case,
provided again that (3.8) holds. The simplest example ($\kappa=2$) is

$g(\Delta,y,x;\theta)=\left(\begin{array}[]{c}a_{1}(x,\Delta;\theta)[y-x-b(x;\alpha)\Delta]\\\
a_{2}(x,\Delta;\theta)\left[(y-x-b(x;\alpha)\Delta)^{2}-v(\Delta,x;\beta)\Delta\right]\rule{0.0pt}{17.07164pt}\end{array}\right).$
(3.14)

It is instructive to consider an example of an estimating function for which
estimators are not rate optimal. The martingale estimating function

$g(\Delta,y,x;\theta)=\left(\begin{array}[]{c}a_{1}(x,\Delta;\theta)[y-F(\Delta,x;\theta)]\\\
a_{2}(x,\Delta;\theta)\left[y^{2}-(\phi(\Delta,x;\theta)+F(\Delta,x;\theta)^{2})\right]\rule{0.0pt}{17.07164pt}\end{array}\right)$
(3.15)

does not satisfy (1.3). It is easy to check that a version of $g$ satisfying
(1.3) exists if and only if $a_{1}(x,0;\theta)=c_{\theta}a_{2}(x,0;\theta)x$
for some real constant $c_{\theta}$. In all other cases, the estimating
function given by (3.15) is not rate optimal. We can obtain a particular case
of (3.12) from (3.15) by choosing
$a_{1}(x,\Delta;\theta)=a_{2}(x,\Delta;\theta)F(\Delta,x;\theta)$ and using
the version $\tilde{g}_{1}=g_{1},\ \tilde{g}_{2}=g_{2}-2g_{1}$. This
particular case of (3.15), obviously satisfies that
$a_{1}(x,0;\theta)=a_{2}(x,0;\theta)x$.

$\hfill\Box$

##  4 Efficient estimating functions

In this section we study the conditions under which an approximate martingale
estimating function, $G_{n}(\theta)$, yields an efficient estimator. In
particular, we show that Condition 1.2 ensures efficiency, and that Godambe-
Heyde optimal estimating functions yield rate optimal and efficient
estimators.

###### Theorem 4.1

Suppose Condition 1.2 and the conditions of Theorem 3.2 except (3.8) are
satisfied. Then the conclusions of Theorem 3.2 hold and the estimating
function (1.2) is efficient, i.e., the asymptotic covariance matrix of the
estimator $\hat{\theta}_{n}=(\hat{\alpha}_{n},\hat{\beta}_{n})$ equals

$\Sigma(\theta_{0})=\left(\begin{array}[]{cc}\left({\displaystyle\int_{\ell}^{r}}\frac{(\partial_{\alpha}b(x;\alpha_{0}))^{2}}{v(x;\beta_{0})}\mu_{\theta_{0}}(x)dx\right)^{-1}&0\\\
0&2\left({\displaystyle\int_{\ell}^{r}}\left[\frac{\partial_{\beta}v(x;\beta_{0})}{v(x;\beta_{0})}\right]^{2}\mu_{\theta_{0}}(x)dx\right)^{-1}\rule{0.0pt}{17.07164pt}\end{array}\right).$
(4.1)

Consistent estimators of the asymptotic variances can be obtained from

$\frac{1}{n\Delta_{n}}\sum_{i=1}^{n}g_{1}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\hat{\theta}_{n})^{2}\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}\int_{\ell}^{r}\frac{(\partial_{\alpha}b(x;\alpha_{0}))^{2}}{v(x;\beta_{0})}\mu_{\theta_{0}}(x)dx$

and

$\frac{1}{n\Delta_{n}^{2}}\sum_{i=1}^{n}g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\hat{\theta}_{n})^{2}\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}\int_{\ell}^{r}\left[\frac{\partial_{\beta}v(x;\beta_{0})}{v(x;\beta_{0})}\right]^{2}\mu_{\theta_{0}}(x)dx.$

Note that an efficient estimating function automatically satisfies (3.8). An
asymptotic martingale estimating function is efficient if and only if there
exists a version that satisfies the conditions of Theorem 4.1.

The covariance matrix (4.1) is equal to the leading term in the expansion of
the asymptotic variance of the maximum likelihood estimator in powers of
$\Delta$ found by Dacunha-Castelle and Florens-Zmirou (1986). The asymptotic
variance of $\hat{\alpha}_{n}$ equals that of the maximum likelihood estimator
based on continuous time observation, see e.g. Kutoyants (2004).

###### Example 4.2

Consider again the quadratic martingale estimating function (3.12). The
function $g(0,y,x;\theta)$, given by (3.13), satisfies the conditions for
efficiency (1.4) and (1.5) if we choose
$a_{1}(x,\Delta;\theta)=\partial_{\alpha}b(x;\alpha)/\sigma^{2}(x;\beta)$ and
$a_{2}(x,\Delta;\theta)=\partial_{\beta}\sigma^{2}(x;\beta)/\sigma^{4}(x;\beta)$,
as proposed by Bibby and Sørensen (1995, 1996). The same is true of weight
functions $a_{1}$ and $a_{2}$ that converge to $\partial_{\alpha}b/\sigma^{2}$
and $\partial_{\beta}\sigma^{2}/\sigma^{4}$ as $\Delta\rightarrow 0$. An
example is
$a_{1}(x,\Delta;\theta)=\partial_{\alpha}F(\Delta,x;\theta)/\phi(\Delta,x;\theta)$
and
$a_{2}(x,\Delta;\theta)=\Delta\partial_{\beta}\phi(\Delta,x;\theta)/\phi(\Delta,x;\theta)^{2}$.
This is the optimal quadratic martingale estimating function in the sense of
Godambe and Heyde (1987) (after multiplication of the second coordinate by
$\Delta$), see Bibby and Sørensen (1995, 1996).

Consider the pseudo-likelihood function obtained from the likelihood function
by replacing the transition density $p(\Delta,y,x;\theta)$ by the Gaussian
density with mean $F(\Delta,x;\theta)$ and variance $\phi(\Delta,x;\theta)$.
The exact conditional moments are used to ensure consistency of the estimator
also in the case of low frequency asymptotics, where $\Delta$ is not small.
The corresponding pseudo-score function is the quadratic estimating function
(3.12) with
$a_{1}(x,\Delta;\theta)=\partial_{\alpha}F(\Delta,x;\theta)/\phi(\Delta,x;\theta)$
and
$a_{2}(x,\Delta;\theta)=\partial_{\beta}\phi(\Delta,x;\theta)/\phi(\Delta,x;\theta)^{2}$,
which we have just seen is efficient.

A pseudo-likelihood function that works for data sampled at a high frequency
is the likelihood function obtained by replacing the original diffusion model
by its Euler approximation. It can be obtained from the original likelihood
function by replacing the transition density by the Gaussian density with mean
and variance given by the expansions $x-b(x;\alpha)\Delta$ and
$\sigma^{2}(x;\beta)\Delta$. The corresponding pseudo score is of the form
(3.14) with
$a_{1}(x,\Delta;\theta)=\partial_{\alpha}b(x;\alpha)/\sigma^{2}(x;\beta)$ and
(after multiplication by $\Delta$)
$a_{2}(x,\Delta;\theta)=\partial_{\beta}\sigma^{2}(x;\beta)/\sigma^{4}(x;\beta)$.
Since (3.13) holds, this Euler pseudo score function satisfies the conditions
for efficiency. This estimator has often been used in empirical work in
finance. Similarly, it follows that the estimators considered by Dorogovcev
(1976), Prakasa Rao (1988), Florens-Zmirou (1989), Yoshida (1992), Kessler
(1997), Kelly et al. (2004), and Uchida and Yoshida (2013) are efficient under
suitable conditions on the rate of convergence of $\Delta_{n}$.

$\hfill\Box$

###### Example 4.3

A final example is maximum likelihood estimation. In broad generality, the
score function is a martingale estimating function, see e.g. Barndorff-Nielsen
and Sørensen (1994). The transition density can, under weak regularity
conditions, be expanded in powers of $\Delta$

$p(\Delta,y,x;\theta)=r(\Delta,y,x;\theta)(1+O(\Delta)),$

where

$\displaystyle r(\Delta,y,x;\theta)=$
$\displaystyle\frac{1}{\sqrt{2\pi\sigma^{2}(y;\beta)\Delta}}\exp\left(-\frac{(k(y;\beta)-k(x;\beta))^{2}}{2\Delta}+m(y;\alpha,\beta)-m(x;\alpha,\beta)-\mbox{\small$\frac{1}{2}$}\log\left(\frac{\sigma(y;\beta)}{\sigma(x;\beta)}\right)\right),$

$k(x;\beta)=\int^{x}\sigma^{-1}(z;\beta)dz$ and
$m(x;\alpha,\beta)=\int^{x}b(z;\alpha)/\sigma^{2}(z;\beta)dz$, see e.g.
Dacunha-Castelle and Florens-Zmirou (1986) or Gihman and Skorohod (1972), Part
I, Chapter 13\. Therefore, under regularity conditions, the (suitably
normalized) score function $g_{1}(\Delta,y,x;\theta)=\partial_{\alpha}\log
p(\Delta,y,x;\theta)$ and $g_{2}(\Delta,y,x;\theta)=\Delta\partial_{\beta}\log
p(\Delta,y,x;\theta)$ satisfies that

$\displaystyle g_{1}(\Delta,y,x;\theta)$ $\displaystyle=$
$\displaystyle\int_{x}^{y}\frac{\partial_{\alpha}b(z;\alpha)}{\sigma^{2}(z;\beta)}dz+O(\Delta)$
$\displaystyle g_{2}(\Delta,y,x;\theta)$ $\displaystyle=$
$\displaystyle-[k(y;\beta)-k(x;\beta)][\partial_{\beta}k(y;\beta)-\partial_{\beta}k(x;\beta)]+O(\Delta).$

From these expansions it follows that the score functions (normalized as
above) satisfies the conditions (1.3), (1.4) and (1.5) for rate optimality and
efficiency. In particular,
$\partial^{2}_{y}g_{2}(0,x,x;\theta)=-2\partial_{x}k(x;\beta)\partial_{\beta}\partial_{x}k(x;\beta)=\partial_{\beta}\sigma^{2}(x;\beta)/\sigma^{4}(x;\beta)$.
Obviously, also the pseudo-likelihood function obtained by replacing the
transition density $p$ by $r$ has a pseudo-score function that satisfies the
conditions for rate optimality and efficiency.

$\hfill\Box$

Consider martingale estimating functions of the form (2.13) and the related
approximate martingale estimating functions (2.15), i.e.

$G_{n}(\theta)=\sum_{i=1}^{n}A(X_{t_{i-1}^{n}},\Delta;\theta)[f(X_{t_{i}^{n}};\theta)-\pi_{\theta}^{\kappa,\Delta}f(X_{t_{i-1}^{n}};\theta)],\hskip
14.22636pt\kappa=1,2,\ldots,$ (4.2)

with $\pi_{\theta}^{\kappa,\Delta}$ given by (2.14) for $\kappa=1$ (the
martingale case) and by (2.16) for $\kappa=2,3,\ldots$. Moreover, $A$ is a
$2\times N$-matrix of weights, and we assume that the coordinates of the
$N$-dimensional function $f(x,\theta)$ are twice continuously differentiable
w.r.t. $x$.

For estimating functions of the form (4.2), the condition for rate optimality
is

$\sum_{j=1}^{N}a_{2j}(x,0;\theta)\partial_{x}f_{j}(x;\theta)=0,$ (4.3)

where $a_{ij}$ denotes the $ij$th entry of $A$, and the condition for
efficiency is

$\displaystyle\sum_{j=1}^{N}a_{1j}(x,0;\theta)\partial_{x}f_{j}(x;\theta)$
$\displaystyle=$
$\displaystyle\partial_{\alpha}b(x;\alpha)/\sigma^{2}(x;\beta)$ (4.4)
$\displaystyle\sum_{j=1}^{N}a_{2j}(x,0;\theta)\partial^{2}_{x}f_{j}(x;\theta)$
$\displaystyle=$
$\displaystyle\partial_{\beta}\sigma^{2}(x;\beta)/\sigma^{4}(x;\beta)$ (4.5)

For a given function $f$, we want to find a weight-matrix $A$ such that these
equations are satisfied. Obviously, it is necessary that $N\geq 2$ in order
that all three equations are satisfies. If $N=1$, an efficient approximate
martingale estimating function can be obtained by solving (4.4), provided that
the diffusion coefficient is known, so that only the drift depends on a
parameter.

First consider $N=2$, and assume that the matrix

$M(x)=\left(\begin{array}[]{cc}\partial_{x}f_{1}(x;\theta)&\partial^{2}_{x}f_{1}(x;\theta)\\\
\partial_{x}f_{2}(x;\theta)&\partial^{2}_{x}f_{2}(x;\theta)\rule{0.0pt}{17.07164pt}\end{array}\right)$
(4.6)

is invertible for $\mu_{\theta}$-almost all $x$. Then the linear equations
(4.3) - (4.5) are satisfied for

$A(x,0;\theta)=\left(\begin{array}[]{cc}\partial_{\alpha}b(x;\alpha)/\sigma^{2}(x;\beta)&c(x;\theta)\\\
0&\partial_{\beta}\sigma^{2}(x;\beta)/\sigma^{4}(x;\beta)\rule{0.0pt}{19.91692pt}\end{array}\right)M(x)^{-1},$
(4.7)

where $c(x;\theta)$ is any (measurable) function. As a simple example, the
quadratic estimating function ($N=2$, $f_{1}(x)=x$ and
$f_{2}(x)=\frac{1}{2}x^{2}$) is rate optimal and efficient if the weights are
weights $a_{11}(x)=\partial_{\alpha}b(x;\alpha)/\sigma^{2}(x;\beta)$,
$a_{12}(x)=0$,
$a_{22}(x)=\partial_{\beta}\sigma^{2}(x;\beta)/\sigma^{4}(x;\beta)$ and
$a_{21}(x)=-2xa_{22}(x)$ (we have chosen $c=0$). Note that a simple choice for
the weight matrix $A(x,\Delta;\theta)$ for $\Delta>0$ is to let it be given by
(4.7) for all $\Delta$.

It is easily seen that, for any $N\geq 2$, there exist many solutions to (4.3)
- (4.5) provided that there are two coordinates of $f$ (without loss of
generality, we can assume these to be $f_{1}$ and $f_{2}$) such that $M(x)$ is
invertible. In the special case $\kappa=1$, this result follows from Theorem
2.2 of Jacobsen (2002). The conditions for Jacobsen’s concept small
$\Delta$-optimality for martingale estimating functions are identical to our
conditions for rate optimality and efficiency, so we can take advantage of his
thorough study of when martingale estimating functions of the type (4.2) with
$\kappa=1$ are small $\Delta$-optimal.

Here we will, however, go another way and give a natural and generally useful
way of finding a rate optimal and efficient weight matrix $A$ for all
$\kappa\geq 1$. A weight matrix $A^{*}$ in a martingale estimating function of
the type (4.2) (i.e. $\kappa=1$) is optimal in the sense of Godambe and Heyde
(1987), see also Heyde (1997), if it solves the linear equation

$\displaystyle A^{*}(x,\Delta;\theta)$ $\displaystyle
E_{\theta}\left([f(X_{\Delta};\theta)-\pi_{\theta}^{1,\Delta}f(x;\theta)][f(X_{\Delta};\theta)-\pi_{\theta}^{1,\Delta}f(x;\theta)]^{T}\,|\,X_{0}=x\right)$
$\displaystyle\hskip
170.71652pt\mbox{}=\partial_{\theta}\pi_{\theta}^{1,\Delta}f^{T}(x;\theta)-\pi_{\theta}^{1,\Delta}\partial_{\theta}f^{T}(x;\theta).$

It can be assumed that the functions $f_{1},\ldots,f_{N}$ are affinely
independent such that the conditional covariance matrix in (4) is invertible.
The Godambe-Heyde optimal martingale estimating function gives an estimator
that minimizes the asymptotic variance of estimators obtained from the class
of martingale estimating functions of the form (4.2) (with $\kappa=1$) with a
fixed function $f$ and for a fixed, possibly large, $\Delta$. The next theorem
shows that the Godambe-Heyde optimal estimators are rate optimal and efficient
in the high frequency asymptotic scenario considered in the present paper.
Moreover, the same is true of the approximate martingale estimating functions
obtained by expanding all conditional moments (including those in $A^{*}$) in
powers of $\Delta$, which gives a feasible general way of constructing
explicit estimating functions that are rate optimal and efficient.

###### Theorem 4.4

Suppose Condition 2.2 is satisfied, that $f_{j}\in
C_{p,6,1}((\ell,r)\times\Theta)$, $j=1,\ldots,N$, that $N\geq 2$ and that the
$2\times 2$ matrix $M(x)$ given by (4.6) is invertible for
$\mu_{\theta}$-almost all $x$. Let $A^{*}(x,\Delta;\theta)$ satisfy (4), and
define

$B(x,\Delta,,\theta)=\left(\begin{array}[]{cc}1&0\\\
0&2\Delta\rule{0.0pt}{14.22636pt}\end{array}\right)A^{*}(x,\Delta;\theta).$

Then the limit $B(x,0,\theta)$ exists, and

$g^{*}(\Delta,y,x;\theta)=B(x,\Delta,,\theta)[f(y;\theta)-\pi_{\theta}^{\kappa,\Delta}f(x;\theta)]$
(4.9)

satisfies the conditions for rate optimality (1.3) and efficiency (1.4) and
(1.5) for all $\kappa\in{\rm I\mkern-2.5mu\nonscript\mkern-0.5muN}$. The same
is true if $B$ is replaced by a matrix $\tilde{B}$ satisfying that
$\tilde{B}(x,0,\theta)=B(x,0,\theta)$.

The matrix $\tilde{B}$ can be obtained by replacing the conditional moments in
$A^{*}$ by expansions in powers of $\Delta$, or simply by defining
$\tilde{B}(x,\Delta,\theta)=B(x,0,\theta)$. It is surprising that a local
property like Godambe-Heyde optimality, which ensures optimality only within a
particular class of estimating functions, implies global optimality properties
like rate optimality and efficiency. Phrased in terms of the concept small
$\Delta$-optimality, this result was conjectured by Jacobsen (2002) for
martingale estimating functions ($\kappa=1$). The fact that one of the
conditions for efficiency is $N\geq 2$ explains the finding in Larsen and
Sørensen (2007) that an optimal martingale estimating function based on two
eigenfunctions seemed to be efficient for weekly observations of exchange
rates in a target zone.

Let us conclude this section by stating the results for a $d$-dimensional
diffusion. In this case $b(x;\alpha)$ is $d$-dimensional and
$v(x;\beta)=\sigma(x;\beta)\sigma(x;\beta)^{T}$ is a $d\times d$-matrix. The
conditions for efficiency are

$\partial_{y}g_{1}(0,x,x;\theta)=\partial_{\alpha}b(x;\alpha)^{T}v(x;\beta)^{-1}$

and

$\mbox{vec}\left(\partial^{2}_{y}g_{2}(0,x,x;\theta)\right)=\mbox{vec}\left(\partial_{\beta}v(x;\beta)\right)\left(v^{\otimes
2}(x;\beta)\right)^{-1}.$

In the latter equation, vec$(M)$ denotes for a $d\times d$ matrix $M$ the
$d^{2}$-dimensional row vector consisting of the rows of $M$ placed one after
the other, and $M^{\otimes 2}$ is the $d^{2}\times d^{2}$-matrix with
$(i^{\prime},j^{\prime}),(ij)$th entry equal to
$M_{i^{\prime}i}M_{j^{\prime}j}$. Thus if $M=\partial_{\beta}v(x;\beta)$ and
$M^{\bullet}=(v^{\otimes 2}(x;\beta))^{-1}$, then the $(i,j)$th coordinate of
vec$(M)\,M^{\bullet}$ is
$\sum_{i^{\prime}j^{\prime}}M_{i^{\prime}j^{\prime}}M^{\bullet}_{(i^{\prime}j^{\prime}),(i,j)}$.
These expressions are the conditions for small $\Delta$-optimality for
multivariate diffusions given by Jacobsen (2002).

For a $d$-dimensional diffusion process, the condition analogous to the one
discussed shortly before Theorem 4.4 ensuring the existence of a rate optimal
and efficient estimating function of the form (4.2) is that $N\geq d(d+3)/2$,
and that the $N\times(d+d^{2})$-matrix

$\left(\begin{array}[]{cc}\partial_{x}f(x;\theta)&\partial^{2}_{x}f(x;\theta)\end{array}\right)$

has full rank $d(d+3)/2$. For $\kappa=1$ this follows from Theorem 2.2 of
Jacobsen (2002), and it is clear from the proof of this theorem that it holds
for $\kappa\geq 2$ too. When $\alpha$ and $\beta$ are multivariate, we further
need that $\\{\partial_{\alpha_{i}}b(x;\alpha)\\}$ and
$\\{\partial_{\beta_{i}}v(x;\beta)\\}$ are two sets of linearly independent
functions of $x$. These conditions also ensure that Theorem 4.4 holds for a
$d$-dimensional diffusion process, i.e. that the Godambe-Heyde optimal
martingale estimating function is rate optimal and efficient for a
$d$-dimensional diffusion process.

##  5 Proofs and lemmas

The first of the following lemmas is a slight generalization of Lemma 6 in
Kessler (1997), while the second lemma is essentially Lemma 8 in the same
paper. The proofs are analogous to those in Kessler’s paper. The result (5.3)
follows from (5.2). The notation $R(\Delta,y,x;\theta)$ was defined in Section
2. We sometimes use the notation $a\leq_{C}b$, which means that there exists a
$C>0$ such that $a\leq Cb$.

###### Lemma 5.1

Assume Condition 2.2. Then a constant $C_{k}>0$ exists for $k=1,2,\ldots$ such
that

$E_{\theta_{0}}(|X_{t+\Delta}-X_{t}|^{k}\,|\,X_{t})\leq
C_{k}\Delta^{k/2}(1+|X_{t}|)^{C_{k}}$ (5.1)

for $\Delta>0$. Let $f(y,x,\theta)$ be a real function of polynomial growth in
$x$ and $y$ uniformly for $\theta$ in a compact set $K$. Then for any fixed
$\Delta_{0}>0$ there exists a constant $C>0$ such that

$E_{\theta_{0}}(|f(X_{t+\Delta},X_{t},\theta)|\,|\,X_{t})\leq
C(1+|X_{t}|)^{C}\ \ \ \mbox{ for }\Delta\in[0,\Delta_{0}]\mbox{ and }\theta\in
K.$ (5.2)

Suppose the function $f(y,x,\theta)$ is, moreover, $2k$ times differentiable
($k\leq 3$) with respect to $y$ with all derivatives of polynomial growth in
$x$ and $y$ uniformly for $\theta$ in compact sets. Then

$\int_{0}^{\Delta}\int_{0}^{u_{1}}\cdots\int_{0}^{u_{k-1}}E_{\theta_{0}}\left(L_{\theta_{0}}^{k}(f)(X_{t+u_{k}},X_{t};\theta)\,|\,X_{t}\right)du_{k}\cdots
du_{1}=\Delta^{k}R(\Delta,X_{t},\theta).$ (5.3)

The result (5.3) is used to ensure that the remainder term in expansions of
the type (2.9) have the expected order. The result can be proved for larger
values of $k$ if stronger differentiability conditions are imposed on the
coefficients $b$ and $\sigma$.

###### Lemma 5.2

Assume Condition 2.2, and let $f(x,\theta)$ be a real function that is
differentiable with respect to $x$ and $\theta$ with derivatives of polynomial
growth in $x$ uniformly for $\theta$ in a compact set. Then

$\frac{1}{n}\sum_{i=1}^{n}f(X_{t_{i}^{n}};\theta)\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}\int_{\ell}^{r}f(x;\theta)\mu_{\theta_{0}}(x)dx$

uniformly for $\theta$ in a compact set.

Lemma 9 in Genon-Catalot and Jacod (1993) is used frequently in the proofs of
Lemma 5.5 and Lemma 5.6 to establish pointwise convergence. The result is
therefore cited here for the convenience of the reader.

###### Lemma 5.3

Let $Z^{n}_{i}$ ($i=1,\ldots,n,n\in{\rm
I\mkern-2.5mu\nonscript\mkern-0.5muN}$) be a triangular array of random
variables such that $Z^{n}_{i}$ is ${\cal G}^{n}_{i}$-measurable, where ${\cal
G}^{n}_{i}=\sigma(W_{s}\,:\,s\leq t^{n}_{i})$. If

$\sum_{i=1}^{n}E_{\theta}(Z^{n}_{i}\,|\,{\cal
G}^{n}_{i-1})\stackrel{{\scriptstyle P_{\theta}}}{{\longrightarrow}}U$

and

$\sum_{i=1}^{n}E_{\theta}((Z^{n}_{i})^{2}\,|\,{\cal
G}^{n}_{i-1})\stackrel{{\scriptstyle P_{\theta}}}{{\longrightarrow}}0,$

where $U$ is a random variable, then

$\sum_{i=1}^{n}Z^{n}_{i}\stackrel{{\scriptstyle
P_{\theta}}}{{\longrightarrow}}U.$

Proof of Lemma 2.4. Combining (2.1) and (2.9), we find that

$E_{\theta}(g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\,|\,X_{t_{i-1}^{n}})\\\
=\sum_{\ell=0}^{\kappa-1}\frac{\Delta_{n}^{\ell}}{\ell!}\sum_{j=0}^{\ell}{\ell\choose
j}L_{\theta}^{\ell-j}(g^{(j)}(\theta))(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}})+\Delta_{n}^{\kappa}R(\Delta,X_{t_{i-1}^{n}},\theta),$

from which the “if” statement of the lemma follows immediately. The “only if”
statement follows from the same expansion because an approximate martingale
estimating function satisfies
$E_{\theta}(g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\,|\,X_{t_{i-1}^{n}})=O(\Delta_{n}^{\kappa})$.

$\hfill\Box$

Theorem 3.1 follows via asymptotic statistical results for stochastic
processes, see e.g. Jacod and Sørensen (2018). To prove the theorem we need
two technical lemmas. The first is used to establish uniform convergence in
the proofs of Lemma 5.5 and Lemma 5.6. The lemma is easier to formulate with
the following definitions.

Let ${\cal C}_{0}$ denote the subclass of $C_{p,1,2,1}({\rm
I\mkern-2.5mu\nonscript\mkern-0.5muR}_{+},(\ell,r)^{2},\Theta)$ of functions
$f(\Delta,y,x;\theta)$ satisfying that $f(0,x,x;\theta)=0$ for all
$x\in(\ell,r)$ and $\theta\in\Theta$, and define the operators

$\displaystyle{\cal L}_{1}f(s,y,x;\theta)$ $\displaystyle=$
$\displaystyle\partial_{s}f(s,y,x;\theta)+\partial_{y}f(s,y,x;\theta)b(y;\alpha_{0})+\mbox{\small$\frac{1}{2}$}\partial^{2}_{y}f(s,y,x;\theta)v(y,\beta_{0})$
$\displaystyle{\cal L}_{2}f(s,y,x;\theta)$ $\displaystyle=$
$\displaystyle\partial_{y}f(s,y,x;\theta)\sigma(y;\beta_{0}).$

###### Lemma 5.4

Assume Condition 2.2, and consider

$\zeta^{(j)}_{n}(\theta)=\frac{1}{n\Delta^{j/2}_{n}}\sum_{i=1}^{n}f(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta),\
\ j=2,3,4,$ (5.4)

for $f\in{\cal C}_{0}$. Then the following holds for $j=2$. For every
$m\in{\rm I\mkern-2.5mu\nonscript\mkern-0.5muN}$ and for every compact
$K\subseteq\Theta$, a constant $C_{m,K}>0$ exists such that

$E_{\theta_{0}}\left(|\zeta_{n}^{(j)}(\theta_{2})-\zeta^{(j)}_{n}(\theta_{1})|^{2m}\right)\leq
C_{m,K}|\theta_{2}-\theta_{1}|^{2m}$ (5.5)

for all $\theta_{1}$ and $\theta_{2}$ in $K$ and for all $n$.

Moreover, if $h_{i}={\cal L}_{i}f\in{\cal C}_{0}$ for $i=1,2$, then (5.5)
holds for $j=3$, and if ${\cal L}_{2}h_{i}\in{\cal C}_{0}$ for $i=1,2$, then
(5.5) holds for $j=4$.

Proof. By Ito’s formula

$\displaystyle f(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)=$
$\displaystyle\int_{t_{i-1}^{n}}^{t_{i}^{n}}h_{1}(s-t_{i-1}^{n},X_{s},X_{t_{i-1}^{n}};\theta)ds+\int_{t_{i-1}^{n}}^{t_{i}^{n}}h_{2}(s-t_{i-1}^{n},X_{s},X_{t_{i-1}^{n}};\theta)dW_{s}.$

We treat the two terms on the right hand side of (5) separately. For a
function $k(s,y,x;\theta)$, define
$Dk(\cdot;\theta_{2},\theta_{1})=k(\cdot;\theta_{2})-k(\cdot;\theta_{1})$.
Because $f\in{\cal C}_{0}$, the partial derivatives $\partial_{\theta}h_{i}$,
$i=1,2$, are of polynomial growth in $y$ and $x$ uniformly for $\theta$ in a
compact set. Therefore

$\displaystyle\frac{1}{\Delta^{2m}_{n}}E_{\theta_{0}}\left(\left|\frac{1}{n}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}Dh_{1}(s-t_{i-1}^{n},X_{s},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})ds\right|^{2m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{n\Delta^{2m}_{n}}\sum_{i=1}^{n}E_{\theta_{0}}\left(\left|\int_{t_{i-1}^{n}}^{t_{i}^{n}}Dh_{1}(s-t_{i-1}^{n},X_{s},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})ds\right|^{2m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{n\Delta_{n}}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}E_{\theta_{0}}\left(|Dh_{1}(s-t_{i-1}^{n},X_{s},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})|^{2m}\right)ds$
$\displaystyle\leq_{C}$
$\displaystyle\frac{1}{n\Delta_{n}}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}E_{\theta_{0}}\left(\left|\int_{0}^{1}\partial_{\theta}h_{1}(s-t_{i-1}^{n},X_{s},X_{t_{i-1}^{n}};\theta_{1}+u(\theta_{2}-\theta_{1}))du\right|^{2m}\right)ds|\theta_{2}-\theta_{1}|^{2m}$
$\displaystyle\leq_{C}$ $\displaystyle|\theta_{2}-\theta_{1}|^{2m},$

where we have used Condition 2.2 and Jensen’s inequality (twice). Using the
Burkholder-Davis-Gundy inequality and Jensen’s inequality we obtain

$\displaystyle\frac{1}{\Delta^{2m}_{n}}E_{\theta_{0}}\left(\left|\frac{1}{n}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}Dh_{2}(s-t_{i-1}^{n},X_{s},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})dW_{s}\right|^{2m}\right)$
$\displaystyle\leq_{C}$
$\displaystyle\frac{1}{\Delta^{2m}_{n}}E_{\theta_{0}}\left(\left|\frac{1}{n^{2}}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}Dh_{2}(s-t_{i-1}^{n},X_{s},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})^{2}ds\right|^{m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{n^{m+1}\Delta^{2m}_{n}}\sum_{i=1}^{n}E_{\theta_{0}}\left(\left|\int_{t_{i-1}^{n}}^{t_{i}^{n}}Dh_{2}(s-t_{i-1}^{n},X_{s},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})^{2}ds\right|^{m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{(n\Delta_{n})^{m+1}}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}E_{\theta_{0}}\left(|Dh_{2}(s-t_{i-1}^{n},X_{s},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})|^{2m}\right)ds$
$\displaystyle\leq_{C}$
$\displaystyle\frac{1}{(n\Delta_{n})^{m}}|\theta_{2}-\theta_{1}|^{2m},$

which implies (5.5) for $j=2$.

The result for $j=3,4$ can be proved in a similar way. When $h_{i}\in{\cal
C}_{0}$ for $i=1,2$,

$\displaystyle f(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)=$
$\displaystyle\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}h_{11}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta)duds+\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}h_{21}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta)dW_{u}ds$
$\displaystyle\mbox{}+\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}h_{12}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta)dudW_{s}+\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}h_{22}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta)dW_{u}dW_{s}$

where $h_{ij}={\cal L}_{i}{\cal L}_{j}f$, $i,j=1,2$. In the two cases $h_{11}$
and $h_{12}$, we can prove the result for $j=4$, which implies the result for
$j=3$.

$\displaystyle\frac{1}{\Delta^{4m}_{n}}E_{\theta_{0}}\left(\left|\frac{1}{n}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}Dh_{11}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})duds\right|^{2m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{n\Delta^{4m}_{n}}\sum_{i=1}^{n}E_{\theta_{0}}\left(\left|\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}Dh_{11}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})duds\right|^{2m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{n\Delta_{n}^{2}}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}E_{\theta_{0}}\left(|Dh_{11}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})|^{2m}\right)duds$
$\displaystyle\leq_{C}$ $\displaystyle|\theta_{2}-\theta_{1}|^{2m},$

$\displaystyle\frac{1}{\Delta^{3m}_{n}}E_{\theta_{0}}\left(\left|\frac{1}{n}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}Dh_{21}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})dW_{u}ds\right|^{2m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{\Delta^{3m}_{n}n}\sum_{i=1}^{n}E_{\theta_{0}}\left(\left|\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}Dh_{21}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})dW_{u}ds\right|^{2m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{\Delta^{m+1}_{n}n}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}E_{\theta_{0}}\left(\left|\int_{t_{i-1}^{n}}^{s}Dh_{21}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})dW_{u}\right|^{2m}\right)ds$
$\displaystyle\leq_{C}$
$\displaystyle\frac{1}{\Delta^{m+1}_{n}n}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}E_{\theta_{0}}\left(\left|\int_{t_{i-1}^{n}}^{s}Dh_{21}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})^{2}ds\right|^{m}\right)ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{\Delta_{n}^{2}n}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}E_{\theta_{0}}\left(|Dh_{21}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})|^{2m}duds\right)$
$\displaystyle\leq_{C}$ $\displaystyle|\theta_{2}-\theta_{1}|^{2m},$

$\displaystyle\frac{1}{\Delta^{4m}_{n}}E_{\theta_{0}}\left(\left|\frac{1}{n}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}Dh_{12}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})dudW_{s}\right|^{2m}\right)$
$\displaystyle\leq_{C}$
$\displaystyle\frac{1}{\Delta^{4m}_{n}}E_{\theta_{0}}\left(\left|\frac{1}{n^{2}}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}\left(\int_{t_{i-1}^{n}}^{s}Dh_{12}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})du\right)^{2}ds\right|^{m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{n^{m+1}\Delta^{4m}_{n}}\sum_{i=1}^{n}E_{\theta_{0}}\left(\left|\int_{t_{i-1}^{n}}^{t_{i}^{n}}\left(\int_{t_{i-1}^{n}}^{s}Dh_{12}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})du\right)^{2}ds\right|^{m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{n^{m+1}\Delta_{n}^{m+2}}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}E_{\theta_{0}}\left(|Dh_{12}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})|^{2m}duds\right)$
$\displaystyle\leq_{C}$
$\displaystyle\frac{1}{(n\Delta_{n})^{m}}|\theta_{2}-\theta_{1}|^{2m},$

and

$\displaystyle\frac{1}{\Delta^{3m}_{n}}E_{\theta_{0}}\left(\left|\frac{1}{n}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}Dh_{22}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})dW_{u}dW_{s}\right|^{2m}\right)$
$\displaystyle\leq_{C}$
$\displaystyle\frac{1}{\Delta^{3m}_{n}}E_{\theta_{0}}\left(\left|\frac{1}{n^{2}}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}\left(\int_{t_{i-1}^{n}}^{s}Dh_{22}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})dW_{u}\right)^{2}ds\right|^{m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{n^{m+1}\Delta^{3m}_{n}}\sum_{i=1}^{n}E_{\theta_{0}}\left(\left|\int_{t_{i-1}^{n}}^{t_{i}^{n}}\left(\int_{t_{i-1}^{n}}^{s}Dh_{22}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})dW_{u}\right)^{2}ds\right|^{m}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{n^{m+1}\Delta^{2m+1}_{n}}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}E_{\theta_{0}}\left(\left|\int_{t_{i-1}^{n}}^{s}Dh_{22}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})dW_{u}\right|^{2m}\right)ds$
$\displaystyle\leq_{C}$
$\displaystyle\frac{1}{n^{m+1}\Delta^{2m+1}_{n}}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}E_{\theta_{0}}\left(\left|\int_{t_{i-1}^{n}}^{s}Dh_{22}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})^{2}du\right|^{m}\right)ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{n^{m+1}\Delta_{n}^{m+2}}\sum_{i=1}^{n}\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}E_{\theta_{0}}\left(|Dh_{22}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta_{2},\theta_{1})|^{2m}duds\right)$
$\displaystyle\leq_{C}$
$\displaystyle\frac{1}{(n\Delta_{n})^{m}}|\theta_{2}-\theta_{1}|^{2m}.$

Finally, we prove (5.5) for $j=4$. We have already taken care of two of the
terms in (5), but the terms involving $h_{21}$ and $h_{22}$ require more work.
Since $h_{2i}\in{\cal C}_{0}$, $i=1,2$, we find that

$\displaystyle\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}h_{21}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta)dW_{u}ds$
$\displaystyle=$
$\displaystyle\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}\int_{t_{i-1}^{n}}^{u}{\cal
L}_{1}h_{21}(v-t_{i-1}^{n},X_{v},X_{t_{i-1}^{n}};\theta)dvdW_{u}ds$
$\displaystyle+$
$\displaystyle\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}\int_{t_{i-1}^{n}}^{u}{\cal
L}_{2}h_{21}(v-t_{i-1}^{n},X_{v},X_{t_{i-1}^{n}};\theta)dW_{v}dW_{u}ds$

and

$\displaystyle\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}h_{22}(u-t_{i-1}^{n},X_{u},X_{t_{i-1}^{n}};\theta)dW_{u}dW_{s}$
$\displaystyle=$
$\displaystyle\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}\int_{t_{i-1}^{n}}^{u}{\cal
L}_{1}h_{22}(v-t_{i-1}^{n},X_{v},X_{t_{i-1}^{n}};\theta)dvdW_{u}dW_{s}$
$\displaystyle+$
$\displaystyle\int_{t_{i-1}^{n}}^{t_{i}^{n}}\int_{t_{i-1}^{n}}^{s}\int_{t_{i-1}^{n}}^{u}{\cal
L}_{2}h_{22}(v-t_{i-1}^{n},X_{v},X_{t_{i-1}^{n}};\theta)dW_{v}dW_{u}dW_{s}.$

The result is now obtained by evaluating the triple integrals using the
Burkholder-Davis-Gundy inequality and Jensen’s inequality exactly as above.
$\hfill\Box$

###### Lemma 5.5

Under the Conditions 2.2 and 2.3

$\frac{1}{n\Delta_{n}}\sum_{i=1}^{n}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}\gamma(\theta,\theta_{0}),$ (5.8)
$\displaystyle\frac{1}{n\Delta_{n}}\sum_{i=1}^{n}\partial_{\theta^{T}}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)$
$\displaystyle\stackrel{{\scriptstyle P_{\theta_{0}}}}{{\longrightarrow}}$
$\displaystyle\int_{\ell}^{r}[L_{\theta_{0}}(\partial_{\theta^{T}}g(0;\theta))(x,x)-L_{\theta}(\partial_{\theta^{T}}g(0;\theta))(x,x)-J_{\theta}(x)]\mu_{\theta_{0}}(x)dx,$

and

$\frac{1}{n\Delta_{n}}\sum_{i=1}^{n}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)^{T}\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}V(\theta),$ (5.10)

uniformly for $\theta$ in a compact set. The function $\gamma$ given by (3.1)
is a continuous function of $\theta$. For a martingale estimating function or
more generally if $n\Delta_{n}^{2\kappa-1}\rightarrow 0$,

$\frac{1}{\sqrt{n\Delta_{n}}}\sum_{i=1}^{n}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta_{0})\stackrel{{\scriptstyle\cal
D}}{{\longrightarrow}}N_{2}\left(0,V_{0}\right).$ (5.11)

Proof. By (2.1), (2.9), (2.11) and Lemma 5.1,

$\displaystyle
E_{\theta_{0}}\left(g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\,|\,X_{t_{i-1}^{n}}\right)$
$\displaystyle=$
$\displaystyle\Delta_{n}\left[g^{(1)}(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}};\theta)+L_{\theta_{0}}(g(0;\theta))(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}})\right]+\Delta_{n}^{2}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)$
$\displaystyle=$
$\displaystyle\Delta_{n}\left[L_{\theta_{0}}(g(0;\theta))(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}})-L_{\theta}(g(0;\theta))(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}})\right]+\Delta_{n}^{2}R(\Delta_{n},X_{t_{i-1}^{n}},\theta).$

The last equality follows from (2.12). Thus

$\displaystyle\frac{1}{n\Delta_{n}}\sum_{i=1}^{n}E_{\theta_{0}}\left(g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\,|\,X_{t_{i-1}^{n}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left[L_{\theta_{0}}(g(0;\theta))(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}})-L_{\theta}(g(0;\theta))(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}})\right]+\Delta_{n}\frac{1}{n}\sum_{i=1}^{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)$
$\displaystyle\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}\gamma(\theta,\theta_{0})$

by Lemma 5.2. Moreover,
$E_{\theta_{0}}\left(g_{j}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)^{2}\,|\,X_{t_{i-1}^{n}}\right)=\Delta_{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)$,
so

$\frac{1}{(n\Delta_{n})^{2}}\sum_{i=1}^{n}E_{\theta_{0}}\left(g_{j}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)^{2}\,|\,X_{t_{i-1}^{n}}\right)=\frac{1}{n\Delta_{n}}\frac{1}{n}\sum_{i=1}^{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}0.$

Therefore pointwise convergence in (5.8) follows from Lemma 5.3. In order to
prove that the convergence is uniform for $\theta$ in a compact set $K$, we
show that the sequence
$\zeta_{n}(\cdot)=\frac{1}{n\Delta_{n}}\sum_{i=1}^{n}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}},\cdot)$
converges weakly to the limit $\gamma(\cdot,\theta_{0})$ in the space, $C(K)$,
of continuous functions on $K$ with the supremum norm. Since the limit is non-
random, this implies uniform convergence in probability for $\theta\in K$.
That $\gamma(\cdot,\theta_{0})$ is continuous follows from the dominated
convergence theorem because of the imposed uniform polynomial growth
assumptions. Since pointwise convergence has been established, weak
convergence follows because the family of distributions of $\zeta_{n}(\cdot)$
is tight. The tightness follows from Lemma 5.4 with $f=g_{i}$, $j=2$ and
$m=2$. That (5.5) and pointwise convergence implies tightness follows from
Corollary 14.9 in Kallenberg (1997), which is a generalization of Theorem 12.3
in Billingsley (1968) (see also Lemma 3.1 in Yoshida (1990) and Theorem 20 in
Appendix I of Ibragimov and Has’minskii (1981)).

In a similar way it follows from (2.1), (2.9), (2.11), (2.12) and Lemma 5.1
that

$\displaystyle
E_{\theta_{0}}\left(\partial_{\theta^{T}}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\,|\,X_{t_{i-1}^{n}}\right)$
$\displaystyle=$
$\displaystyle\Delta_{n}\left[\partial_{\theta^{T}}g^{(1)}(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}};\theta)+L_{\theta_{0}}(\partial_{\theta^{T}}g(0;\theta))(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}})\right]+\Delta_{n}^{2}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)$
$\displaystyle=$
$\displaystyle\Delta_{n}\left[L_{\theta_{0}}(\partial_{\theta^{T}}g(0;\theta))(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}})-L_{\theta}(\partial_{\theta^{T}}g(0;\theta))(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}})-J_{\theta}(X_{t_{i-1}^{n}})\right]$
$\displaystyle\hskip
227.62204pt\mbox{}+\Delta_{n}^{2}R(\Delta_{n},X_{t_{i-1}^{n}},\theta),$

and from (2.1),(2.9), (2.11), and Lemma 5.1 that

$\displaystyle
E_{\theta_{0}}\left(g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)^{T}\,|\,X_{t_{i-1}^{n}}\right)$
$\displaystyle=$
$\displaystyle\Delta_{n}v(X_{t_{i-1}^{n}},\beta_{0})\partial_{y}g(0,X_{t_{i-1}^{n}},X_{t_{i-1}^{n}};\theta)\partial_{y}g(0,X_{t_{i-1}^{n}},X_{t_{i-1}^{n}};\theta)^{T}+\Delta_{n}^{2}R(\Delta_{n},X_{t_{i-1}^{n}},\theta).$

Since by (2.1),(2.9), (2.11), and Lemma 5.1

$E_{\theta_{0}}\left([\partial_{\theta}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)]^{2}\,|\,X_{t_{i-1}^{n}}\right)=\Delta_{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)$

and

$E_{\theta_{0}}\left([g_{j}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)g_{k}(\Delta,X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)]^{2}\,|\,X_{t_{i-1}^{n}}\right)=\Delta_{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta),$
(5.13)

we can, as above, use Lemma 5.2 and Lemma 5.3 to prove (5.5) and (5.10). As
above, uniform convergence for $\theta$ in a compact set $K$ follows by using
Lemma 5.4 with $f=\partial_{\theta_{j}}g_{k}$ and $f=g_{j}g_{k}$ to prove the
tightness of (5.4) (with $j=2$) in $C(K)$.

Finally, (5.11) follows from the central limit theorem for square integrable
martingale arrays under conditions which, in the martingale case, we have
already verified in the proof of (5.10), see e.g. Corollary 3.1 in Hall and
Heyde (1980) with the conditional Lindeberg condition replaced by the stronger
conditional Liapounov condition that follows from (5.13) and Lemma 5.2, e.g.

$\frac{1}{(n\Delta_{n})^{2}}\sum_{i=1}^{n}E_{\theta_{0}}\left(g_{j}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta_{0})^{4}\,|\,X_{t_{i-1}^{n}}\right)=\frac{1}{n\Delta_{n}}\frac{1}{n}\sum_{i=1}^{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta_{0})\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}0.$

The nestedness condition in Hall and Heyde’s Corollary 3.1 is not needed here
because the limit of the quadratic variation is non-random.

In the case of non-martingale estimating functions, we consider the martingale
$\sum_{i=1}^{n}\tilde{g}(\Delta_{n},$
$X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta_{0})$, where
$\tilde{g}=g-E_{\theta_{0}}\left(g|\,X_{t_{i-1}^{n}}\right)$. This martingale
satisfies the conditions of the central limit theorem, which follows from the
expansions of conditional expectations given above and
$E_{\theta_{0}}\left(g_{j}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta_{0})^{3}\,|\,X_{t_{i-1}^{n}}\right)=\Delta_{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta_{0})$.
Now, (5.11) follows because by (2.7)

$\frac{1}{\sqrt{n\Delta_{n}}}\sum_{i=1}^{n}E_{\theta_{0}}\left(g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta_{0})\,|\,X_{t_{i-1}^{n}}\right)=\sqrt{n}\Delta_{n}^{\kappa-1/2}\frac{1}{n}\sum_{i=1}^{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta_{0})\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}0.$ (5.14)

$\hfill\Box$

Proof of Theorem 3.1. By Lemma 5.5, the estimating function

$G_{n}(\theta)=\frac{1}{n\Delta_{n}}\sum_{i=1}^{n}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)$
(5.15)

satisfies the conditions that $G_{n}(\theta_{0})\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\rightarrow}}0$,
$\partial_{\theta}G_{n}(\theta)\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\rightarrow}}U(\theta)$ uniformly for $\theta$ in a compact
set, and that $U(\theta_{0})=-S$ is invertible. Here $U(\theta)$ denotes the
right hand side of (5.5). This implies the eventual existence and the
consistency of $\hat{\theta}_{n}$ as well as the eventual uniqueness of
consistent $G_{n}$-estimators; see Theorems 2.5 and 2.6 in Jacod and Sørensen
(2018). Now consider any $G_{n}$-estimator $\tilde{\theta}_{n}$ for which
$P_{\theta_{0}}(\tilde{\theta}_{n}\in K)\rightarrow 1$ as
$n\rightarrow\infty$, where $K$ is a compact subset of $\Theta$ with
$\theta_{0}\in$ int$\,K$. By Theorem 2.7 in Jacod and Sørensen (2018) the
facts that $\gamma(\theta,\theta_{0})$ (the limit of $G_{n}(\theta)$)
satisfies that $\gamma(\theta,\theta_{0})\neq 0$ for $\theta\neq\theta_{0}$
and is continuous in $\theta$ imply that
$P_{\theta_{0}}(\hat{\theta}_{n}\neq\tilde{\theta}_{n})\rightarrow 0$ as
$n\rightarrow\infty$. The asymptotic normality follows by standard arguments.
Finally, (3.6) and (3.7) follow from (5.5) and (5.10) because the convergence
is uniform for $\theta$ in compact sets. $\hfill\Box$

The next lemma is needed in the proof of Theorem 3.2

###### Lemma 5.6

Under the Conditions 1.1, 2.2, and 2.3

$D_{n}\sum_{i=1}^{n}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)^{T}D_{n}\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}\left(\begin{array}[]{cc}W_{1}(\theta)&0\\\
0&W_{2}(\theta)\rule{0.0pt}{17.07164pt}\end{array}\right)$ (5.16)

uniformly for $\theta$ in a compact set, where $D_{n}$ is given by (3.11).

For a martingale estimating function or if more generally
$n\Delta^{2(\kappa-1)}\rightarrow 0$,

$\left(\begin{array}[]{c}\frac{1}{\sqrt{n\Delta_{n}}}\sum_{i=1}^{n}g_{1}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta_{0})\\\
\frac{1}{\Delta_{n}\sqrt{n}}\sum_{i=1}^{n}g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta_{0})\rule{0.0pt}{19.91692pt}\end{array}\right)\stackrel{{\scriptstyle\cal
D}}{{\longrightarrow}}N_{2}\left(\left(\begin{array}[]{c}0\\\
0\end{array}\right),\left(\begin{array}[]{cc}W_{1}(\theta_{0})&0\\\
0&W_{2}(\theta_{0})\rule{0.0pt}{17.07164pt}\end{array}\right)\right).$ (5.17)

If, in addition, condition (3.8) holds, then

$\frac{1}{n\Delta^{3/2}_{n}}\sum_{i=1}^{n}\partial_{\alpha}g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}0$ (5.18)

uniformly for $\theta$ in a compact set.

Proof. By (2.1), (2.9), (2.11), (1.3) and Lemma 5.1,

$\displaystyle\frac{1}{n\Delta_{n}^{3/2}}\sum_{i=1}^{n}E_{\theta_{0}}\left(g_{1}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\,|\,X_{t_{i-1}^{n}}\right)$
$\displaystyle\hskip
170.71652pt=\Delta_{n}^{1/2}\frac{1}{n}\sum_{i=1}^{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}0$

and

$\displaystyle\frac{1}{n^{2}\Delta_{n}^{3}}\sum_{i=1}^{n}E_{\theta_{0}}\left([g_{1}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)]^{2}\,|\,X_{t_{i-1}^{n}}\right)$
$\displaystyle\hskip
170.71652pt=\frac{1}{n\Delta_{n}}\frac{1}{n}\sum_{i=1}^{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}0,$

so the pointwise convergence of the two off-diagonal entries in (5.16) follows
from Lemma 5.3. Similarly to the proof of Lemma 5.5, uniform convergence for
$\theta$ in a compact set $K$ follows by using Lemma 5.4 with $f=g_{1}g_{2}$
to prove the tightness of (5.4) (with $j=3$) in $C(K)$.

The convergence of
$(n\Delta_{n})^{-1}\sum_{i=1}^{n}g_{1}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)^{2}$
was taken care of in Lemma 5.5. By (2.1), (2.9), (2.11), (2.12), (1.3) and
Lemma 5.1, we see that

$\displaystyle
E_{\theta_{0}}\left(g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)^{2}\,|\,X_{t_{i-1}^{n}}\right)$
$\displaystyle=$
$\displaystyle\Delta_{n}^{2}\left[\mbox{\small$\frac{1}{2}$}L^{2}_{\theta_{0}}(g_{2}(0;\theta)^{2})(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}})+2L_{\theta_{0}}(g_{2}(0;\theta)g_{2}^{(1)}(\theta))(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}})\right.$
$\displaystyle\hskip
170.71652pt\left.\mbox{}+g_{2}^{(1)}(X_{t_{i-1}^{n}},X_{t_{i-1}^{n}};\theta)^{2}\right]+\Delta_{n}^{3}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)$
$\displaystyle=$
$\displaystyle\mbox{\small$\frac{1}{2}$}\Delta_{n}^{2}\left[v(X_{t_{i-1}^{n}};\beta_{0})^{2}+\mbox{\small$\frac{1}{2}$}(v(X_{t_{i-1}^{n}};\beta_{0})-v(X_{t_{i-1}^{n}};\beta))^{2}\right](\partial_{y}^{2}g_{2}(0,X_{t_{i-1}^{n}},X_{t_{i-1}^{n}};\theta))^{2}$
$\displaystyle\hskip
284.52756pt\mbox{}+\Delta_{n}^{3}R(\Delta_{n},X_{t_{i-1}^{n}},\theta),$

Thus

$\displaystyle\frac{1}{n\Delta_{n}^{2}}\sum_{i=1}^{n}E_{\theta_{0}}\left(g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)^{2}\,|\,X_{t_{i-1}^{n}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\mbox{\small$\frac{1}{2}$}\left[v(X_{t_{i-1}^{n}};\beta_{0})+\mbox{\small$\frac{1}{2}$}(v(X_{t_{i-1}^{n}};\beta_{0})-v(X_{t_{i-1}^{n}};\beta))^{2}\right](\partial_{y}^{2}g_{2}(0,X_{t_{i-1}^{n}},X_{t_{i-1}^{n}};\theta))^{2}$
$\displaystyle\hskip
227.62204pt\mbox{}+\Delta_{n}\frac{1}{n}\sum_{i=1}^{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)$
$\displaystyle\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}W_{2}(\theta)$

by Lemma 5.2. We conclude that
$(n\Delta_{n}^{2})^{-1}\sum_{i=1}^{n}g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}}\theta)^{2}$
converges to $W_{2}(\theta)$ by Lemma 5.3 because

$\frac{1}{n^{2}\Delta_{n}^{4}}\sum_{i=1}^{n}E_{\theta_{0}}\left(g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)^{4}\,|\,X_{t_{i-1}^{n}}\right)=\frac{1}{n\Delta_{n}}\frac{1}{n}\sum_{i=1}^{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}0.$ (5.20)

This follows from (2.1), (2.9), (2.11), (1.3), and Lemmas 5.1 and 5.2. Uniform
convergence for $\theta$ in a compact set $K$ follows by using Lemma 5.4 with
$f=g_{2}^{2}$ to prove the tightness of (5.4) (with $j=4$) in $C(K)$.

As in the proof of Lemma 5.5, (5.17) follows from the central limit theorem
for square integrable martingale arrays (Corollary 3.1 in Hall and Heyde
(1980)) under conditions which, in the martingale case, we have already
verified in the proof of (5.16). In particular, the conditional Liapounov
condition follows from (5.13), (5.20) and (5).

In the case of non-martingale estimating functions, consider the martingale
$\sum_{i=1}^{n}\tilde{g}(\Delta_{n},$
$X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta_{0})$, where
$\tilde{g}=g-E_{\theta_{0}}\left(g|\,X_{t_{i-1}^{n}}\right)$, which satisfies
the conditions of the central limit theorem. This follows from the expansions
of conditional expectations given above and
$E_{\theta_{0}}\left(g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta_{0})^{3}\,|\,X_{t_{i-1}^{n}}\right)=\Delta_{n}^{2}R(\Delta_{n},X_{t_{i-1}^{n}},\theta_{0})$.
Now, (5.17) follows because $g_{1}$ satisfies (5.14), and because by (2.7)

$\frac{1}{\Delta_{n}\sqrt{n}}\sum_{i=1}^{n}E_{\theta_{0}}\left(g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta_{0})\,|\,X_{t_{i-1}^{n}}\right)=\sqrt{n}\Delta_{n}^{\kappa-1}\frac{1}{n}\sum_{i=1}^{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta_{0})\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}0.$

Finally, to prove (5.18) note that (5), (1.3) and (3.8) imply that

$E_{\theta_{0}}\left(\partial_{\alpha}g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\,|\,X_{t_{i-1}^{n}}\right)=\Delta^{2}R(\Delta_{n},X_{t_{i-1}^{n}},\theta),$

and that it follows from (2.1), (2.9), (2.11), (2.12), (1.3), (3.8) and Lemma
5.1 that

$E_{\theta_{0}}\left([\partial_{\alpha}g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)]^{2}\,|\,X_{t_{i-1}^{n}}\right)=\Delta_{n}^{3}R(\Delta_{n},X_{t_{i-1}^{n}},\theta).$

Therefore by Lemma 5.2

$\frac{1}{n\Delta_{n}^{3/2}}\sum_{i=1}^{n}E_{\theta_{0}}\left(\partial_{\alpha}g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)\,|\,X_{t_{i-1}^{n}}\right)=\sqrt{\Delta_{n}}\frac{1}{n}\sum_{i=1}^{n}R(\Delta_{n},X_{t_{i-1}^{n}};\theta)\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}0.$

and

$\frac{1}{n^{2}\Delta_{n}^{3}}\sum_{i=1}^{n}E_{\theta_{0}}\left([\partial_{\alpha}g_{2}(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta)]^{2}\,|\,X_{t_{i-1}^{n}}\right)=\frac{1}{n}\frac{1}{n}\sum_{i=1}^{n}R(\Delta_{n},X_{t_{i-1}^{n}},\theta)\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}0,$

so that (5.18) follows from Lemma 5.3. Uniform convergence for $\theta$ in a
compact set $K$ follows by using Lemma 5.4 with $f=\partial_{\alpha}g_{2}$ to
conclude tightness of (5.4) (with $j=3$) in $C(K)$. To see that
$\partial_{\alpha}g_{2}$ satisfies the conditions of the lemma, we use (2.12)
and (3.8) to conclude that
$\partial_{\Delta}\partial_{\alpha}g_{2}(0,x,x;\theta)=\partial_{\alpha}g_{2}^{(1)}(x,x;\theta)=-\partial_{\alpha}L_{\theta}(g_{2}(0;\theta))(x,x)=0$.

$\hfill\Box$

Proof of Theorem 3.2. The results on eventual existence, uniqueness and
consistence of $\hat{\theta}_{n}$ follow from Theorem 3.1: Because (1.3)
implies $S_{21}=0$, the assumptions that $S_{11}\neq 0$ and $S_{22}\neq 0$
ensure that $S$ is invertible, and similarly, under Condition 1.1 the
identifiability condition imposed in Theorem 3.2 ensures that
$\gamma(\theta,\theta_{0})\neq 0$ for $\theta\neq\theta_{0}$, where $\gamma$
is the limit of $G_{n}(\theta)$ given by (3.1).

To prove the asymptotic normality (3.9) of the estimator $\hat{\theta}_{n}$ we
consider

$\tilde{G}_{n}(\theta)=D_{n}\sum_{i=1}^{n}g(\Delta_{n},X_{t_{i}^{n}},X_{t_{i-1}^{n}};\theta),$

where $D_{n}$ is given by (3.11). On the set
$\\{\tilde{G}_{n}(\hat{\theta}_{n})=0\\}$ (the probability of which goes to
one)

$-\partial_{\theta^{T}}\tilde{G}_{n}(\theta^{(1)}_{n},\theta^{(2)}_{n})A_{n}^{-1}A_{n}(\hat{\theta}_{n}-\theta_{0})=\tilde{G}_{n}(\theta_{0}),$

where

$A_{n}=\left(\begin{array}[]{cc}\sqrt{\Delta_{n}n}&0\\\
0&\sqrt{n}\end{array}\right),$

$\partial_{\theta^{T}}\tilde{G}_{n}(\theta^{(1)}_{n},\theta^{(2)}_{n})$ is the
$2\times 2$-matrix whose $jk$th entry is
$\partial_{\theta_{k}}\tilde{G}_{n}(\theta^{(j)}_{n})_{j}$, and
$\theta^{(j)}_{n}$ is a random convex combination of $\hat{\theta}_{n}$ and
$\theta_{0}$. Since by (5.5) and (5.18)

$-\partial_{\theta^{T}}\tilde{G}_{n}(\theta^{(1)}_{n},\theta^{(2)}_{n})A_{n}^{-1}\stackrel{{\scriptstyle
P_{\theta_{0}}}}{{\longrightarrow}}\left(\begin{array}[]{cc}S_{11}&0\\\
0&S_{22}\end{array}\right),$

(3.9) follows from (5.17).

Finally, (3.10) follows from (5.16) because the convergence is uniform for
$\theta$ in compact sets.

$\hfill\Box$

Proof of Theorem 4.1. That the conclusions of Theorem 3.2 holds is obvious
because (1.5) implies (3.8). The efficiency follows from Theorem 4.1 in Gobet
(2002), where it is proved that the diffusion model (1.1) is locally
asymptotically normal with Fisher information matrix ${\cal
I}(\theta_{0})=\Sigma(\theta_{0})^{-1}$, with $\Sigma(\theta_{0})$ given by
(4.1). Under Condition 1.2, $S_{11}=W_{1}(\theta_{0})={\cal
I}_{11}(\theta_{0})$ and $S_{22}=W_{2}(\theta_{0})={\cal I}_{22}(\theta_{0})$,
so it follows from (3.9) that the asymptotic covariance matrix of
$\hat{\theta}_{n}$ equals the inverse of the Fisher information matrix. The
estimators of the asymptotic variances follow from (3.10).

$\hfill\Box$

Proof of Theorem 4.4. By (2.9)
$\pi^{1,\Delta}_{\theta}f(x;\theta)=f(x;\theta)+\Delta
L_{\theta}f(x;\theta)+O(\Delta^{2})$, so after another application of (2.9),
we see that
$h(\Delta,y,x;\theta)=f(y;\theta)-\pi_{\theta}^{\kappa,\Delta}f(x;\theta)$,
$\kappa\in{\rm I\mkern-2.5mu\nonscript\mkern-0.5muN}$, satisfies

$\displaystyle
E_{\theta}\left(h(\Delta,X_{\Delta},x;\theta)h(\Delta,X_{\Delta},x;\theta)^{T}\,|\,X_{0}=x\right)=\Delta
L_{\theta}(h(0;\theta)h(0;\theta)^{T})(x,x)$ $\displaystyle\mbox{}\hskip
28.45274pt+\Delta^{2}\left(\mbox{\small$\frac{1}{2}$}L^{2}_{\theta}(h(0;\theta)h(0;\theta)^{T})(x,x)-L_{\theta}f(x;\theta)L_{\theta}f^{T}(x;\theta)\right)+O(\Delta^{3})$
$\displaystyle=$ $\displaystyle\Delta
v(x;\beta)\partial_{x}f(x;\theta)\partial_{x}f(x;\theta)^{T}+\Delta^{2}K(x)+O(\Delta^{3}),$

where

$\displaystyle K(x)$ $\displaystyle=$ $\displaystyle
q_{1}(x;\theta)\partial_{x}f(x;\theta)\partial_{x}f(x;\theta)^{T}+q_{2}(x;\theta)\left(\partial^{2}_{x}f(x;\theta)\partial_{x}f(x;\theta)^{T}+\partial_{x}f(x;\theta)\partial^{2}_{x}f(x;\theta)^{T}\right)$
$\displaystyle\mbox{}+v(x;\beta)^{2}\left(\partial^{2}_{x}f(x;\theta)\partial^{2}_{x}f(x;\theta)^{T}+\mbox{\small$\frac{1}{2}$}(\partial^{3}_{x}f(x;\theta)\partial_{x}f(x;\theta)^{T}+\partial_{x}f(x;\theta)\partial^{3}_{x}f(x;\theta)^{T})\right)$

with

$\displaystyle q_{1}(x;\theta)$ $\displaystyle=$
$\displaystyle\mbox{\small$\frac{1}{2}$}[b(x;\alpha)(2+\partial_{x}v(x;\beta))-2b(x;\alpha)+\mbox{\small$\frac{1}{2}$}v(x;\beta)(4\partial_{x}b(x;\alpha)+\partial_{x}^{2}v(x;\beta))]$
$\displaystyle q_{2}(x,\theta)$ $\displaystyle=$
$\displaystyle\mbox{\small$\frac{3}{4}$}v(x;\beta)(1+\mbox{\small$\frac{1}{3}$}b(x;\alpha)+\partial_{x}v(x;\beta)).$

Since

$\displaystyle\partial_{\alpha}L_{\theta}f(x;\theta)-L_{\theta}\partial_{\alpha}f(x;\theta)$
$\displaystyle=$
$\displaystyle\partial_{\alpha}b(x;\alpha)\partial_{x}f(x;\theta)$
$\displaystyle\partial_{\beta}L_{\theta}f(x;\theta)-L_{\theta}\partial_{\beta}f(x;\theta)$
$\displaystyle=$
$\displaystyle\mbox{\small$\frac{1}{2}$}\partial_{\beta}v(x;\beta)\partial^{2}_{x}f(x;\theta),$

it also follows from (2.9) that

$\partial_{\theta^{T}}\pi^{\Delta}_{\theta}f(x)-\pi^{\Delta}_{\theta}\partial_{\theta^{T}}f(x)=\Delta
F(x)\left(\begin{array}[]{cc}\partial_{\alpha}b(x;\alpha)&0\\\
0&\frac{1}{2}\partial_{\beta}v(x;\beta)\rule{0.0pt}{17.07164pt}\end{array}\right)+O(\Delta^{2}),$

where $F(x)$ denotes the $N\times 2$-matrix
$F(x)=(\partial_{x}f(x),\partial^{2}_{x}f(x))$.

If $A^{*}(x,\Delta;\theta)$ satisfies (4), then the $2\times N$-matrix

$B(x,\Delta;\theta)=\left(\begin{array}[]{cc}1&0\\\
0&2\Delta\rule{0.0pt}{14.22636pt}\end{array}\right)A^{*}(x,\Delta;\theta).$

satisfies that

$\displaystyle
B(x,\Delta;\theta)\left[v(x;\beta)\partial_{x}f(x;\theta)\partial_{x}f(x;\theta)^{T}+\Delta
K(x;\theta)+O(\Delta^{2})\right]$ (5.26) $\displaystyle\hskip
85.35826pt=\left(\begin{array}[]{cc}\partial_{\alpha}b(x;\alpha)&0\\\
0&\Delta\partial_{\beta}v(x;\beta)\rule{0.0pt}{17.07164pt}\end{array}\right)F(x)^{T}+\left(\begin{array}[]{c}O(\Delta)\\\
O(\Delta^{2})\rule{0.0pt}{17.07164pt}\end{array}\right).$

Let $B(x,\Delta;\theta)_{i}$ denote the $i$th row of $B(x,\Delta;\theta)$
($i=1,2$). Then it follows by letting $\Delta$ tend to zero that

$v(x;\beta)B(x,0;\theta)_{2}\partial_{x}f(x;\theta)\partial_{x}f(x;\theta)^{T}=0.$
(5.27)

The condition that $M(x)$ is invertible implies that we can find a coordinate
of $\partial_{x}f(x;\theta)$ which is not equal to zero, so we conclude that

$\partial_{y}g_{2}^{*}(0,x,x;\theta)=B(x,0;\theta)_{2}\partial_{x}f(x;\theta)=0.$

Similarly we find that

$[v(x;\beta)B(x,0;\theta)_{1}\partial_{x}f(x;\theta)-\partial_{\alpha}b(x;\alpha)]\partial_{x}f(x;\theta)^{T}=0,$

which implies

$\partial_{y}g_{1}^{*}(0,x,x;\theta)=B(x,0;\theta)_{1}\partial_{x}f(x;\theta)=\partial_{\alpha}b(x;\alpha)/v(x;\beta).$

Finally, (5.26) and (5.27) imply that $B(x,\Delta;\theta)_{2}\left(\Delta
K(x;\theta)+O(\Delta^{2})\right)=\Delta\partial_{\beta}v(x;\beta)\partial^{2}_{x}f(x;\theta)^{T}+O(\Delta^{2})$,
so that

$B(x,0;\theta)_{2}K(x;\theta)=\partial_{\beta}v(x;\beta)\partial^{2}_{x}f(x;\theta)^{T}.$

Since we have shown that $B(x,0;\theta)_{2}\partial_{x}f(x;\theta)=0$, this
expression can be rewritten as

$c_{1}(x;\theta)\partial_{x}f(x;\theta)=c_{2}(x;\theta)\partial^{2}_{x}f(x;\theta)$

where

$\displaystyle c_{1}(x;\theta)$ $\displaystyle=$ $\displaystyle
q_{2}(x;\theta)B(x,0;\theta)_{2}\partial^{2}_{x}f(x;\theta)+\mbox{\small$\frac{1}{2}$}v(x;\beta)^{2}B(x,0;\theta)_{2}\partial^{3}_{x}f(x;\theta)$
$\displaystyle c_{2}(x;\theta)$ $\displaystyle=$
$\displaystyle\partial_{\beta}v(x;\beta)-v(x;\beta)^{2}B(x,0;\theta)_{2}\partial^{2}_{x}f(x;\theta).$

If $c_{2}(x;\theta)\neq 0$, then
$\partial^{2}_{x}f(x)=c_{1}(x;\theta)/c_{2}(x;\theta)\partial_{x}f(x;\theta)$,
which implies that det$(M(x))=0$. This contradicts the assumption that $M(x)$
is invertible, so we conclude that
$\partial_{\beta}v(x;\beta)-v(x;\beta)^{2}B(x,0;\theta)_{2}\partial^{2}_{x}f(x;\theta)=0$
or equivalently

$\partial^{2}_{y}g_{2}^{*}(0,x,x;\theta)=B(x,0;\theta)_{2}\partial^{2}_{x}f(x;\theta)=\partial_{\beta}v(x;\beta)/v(x;\beta)^{2}.$

That the results hold for $\tilde{B}$ is obvious. Note that it also follows
that neither $B(x,0;\theta)_{1}$ nor $B(x,0;\theta)_{2}$ is the zero vector,
so there exist $x$ and $y$ such that $g_{i}^{*}(0,y,x;\theta)\neq 0$, $i=1,2$.

$\hfill\Box$

## Acknowledgements

I am grateful to Arnaud Gloter, Nina Munkholt Jakobsen, Mathieu Kessler,
Masayuki Uchida and Nakahiro Yoshida for helpful discussions.

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