# Non-asymptotic statistical test for the diffusion coefficient in
2-dimensional stochastic diffusions

Anna Melnykova111Laboratoire de Mathématiques d’Avignon, EA 2151, E-mail:
<EMAIL_ADDRESS>Patricia Reynaud-Bouret222Université Côte
d’Azur, CNRS, LJAD E-mail<EMAIL_ADDRESS>Adeline
Samson 333Université Grenoble Alpes, LJK UMR-CNRS 5224, E-mail:
<EMAIL_ADDRESS>
###### Abstract

We develop several statistical tests of the determinant of the diffusion
coefficient of a stochastic differential equation, based on discrete
observations on a time interval $[0,T]$ sampled with a time step $\Delta$. Our
main contribution is to control the test Type I and Type II errors in a non
asymptotic setting, i.e. when the number of observations and the time step are
fixed. The test statistics are calculated from the process increments. In
dimension 1, the density of the test statistic is explicit. In dimension 2,
the test statistic has no explicit density but upper and lower bounds are
proved. We also propose a multiple testing procedure in dimension greater than
2. Every test is proved to be of a given non-asymptotic level and separability
conditions to control their power are also provided. A numerical study
illustrates the properties of the tests for stochastic processes with known or
estimated drifts.

##  1 Introduction

Stochastic diffusion is a classical tool for modeling physical, biological or
ecological dynamics. An open question is how stochasticity should be
introduced into the stochastic dynamic process, on what coordinate and at what
scale. For example, diffusions have been widely used to model neuronal
activity, either of a single neuron (Ditlevsen and Samson, 2014, Höpfner et
al., 2016, León and Samson, 2018), or of a large neural network (Ditlevsen and
Löcherbach, 2017, Ableidinger et al., 2017). Although the intrinsic
stochasticity of neurons is well established, where and on what scale this
stochasticity should be introduced (on ion channels or membrane potential or
both) is still a matter of debate (Goldwyn and Shea-Brown, 2011). Examples
also exist in other applications, for example in modeling oscillatory systems
or movement behavior in ecology. From a statistical point of view, this
corresponds to testing the noise level of a multivariate diffusion process.
The aim of this paper is to answer this question. We propose to do this by
testing whether the determinant of the diffusion coefficient is equal or
larger than a certain value. If the test is rejected, the process is
considered as elliptic, since there is sufficient noise on all coordinates.

Let us formally introduce the stochastic process. Consider a filtered
probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathds{P})$. Let $X$ be a $d-$dimensional process solution of the
following Stochastic Differential Equation (SDE):

$dX_{t}=b_{t}dt+\Sigma dW_{t},\quad X_{0}=x_{0},\quad t>0,$ (1)

with a drift function $b_{t}$, a diffusion matrix
$\Sigma\in\mathds{R}^{d\times d}$, and $W$ a $d$-dimensional Brownian motion.
In this paper, for simplicity’s stake, we assume a diagonal $\Sigma$. We
consider discrete observations of $X$ on a time interval $[0,T]$ with a
regular time step $\Delta$, denoted
$\left\\{X_{i\Delta}\right\\}_{i=0,\ldots,n}$.

The objective is to construct a statistical test procedure to decide between
the two following hypotheses :

$\displaystyle H_{0}:\det\Sigma\Sigma^{T}=\det\Sigma_{0}\Sigma_{0}^{T}$
$\displaystyle H_{1}:\det\Sigma\Sigma^{T}>\det\Sigma_{0}\Sigma_{0}^{T}.$

Our test consists in rejecting the null hypothesis when an estimator of
$\det\Sigma\Sigma^{T}$, chosen as the testing statistic, is greater than a
certain critical value. The main issue in constructing the test procedure is
the choice of the critical value guaranteeing that the test is exactly at the
desired level $\alpha$. In addition, to understand the performance of the
constructed procedure, we want to find conditions leading to non-asymptotic
control of the Type II error.

When working with real data, observations are sampled with a fixed time
interval $[0,T]$ and a fixed time step $\Delta$. The framework is therefore
non-asymptotic in the sense that we have to control the Type I and Type II
errors of the test procedure for fixed $n$ and $\Delta$. Controlling the Type
I and Type II errors of a statistical test in a non asymptotic setting is
difficult. Here, it is all the more difficult because the non asymptotic
framework is also an problem for SDE inference. Indeed, estimators of drift
and diffusion coefficients have been shown to be consistent in different
asymptotic settings (either $T$ fixed and $n$ going to infinity or $T$ going
to infinity) but few results are available in a non-asymptotic setting. Here,
we face both difficulties. Especially, here, we will need non asymptotic
behavior of the estimator of the diffusion coefficient.

Several tests have been proposed on the matrix $\Sigma\Sigma^{T}$ of a
diffusion process, but only in the asymptotic setting $\Delta$ goes to zero
and $n$ goes to infinity (Dette and Podolskij, 2008, Jacod et al., 2008,
Podolskij and Rosenbaum, 2012). The test statistic therefore has an asymptotic
distribution from which one can construct a statistical test with a given
asymptotic level $\alpha$ through a rejection area. Among others, we can cite
Dette and Podolskij (2008) which propose to test the parametric form of the
volatility with the empirical processes of the integrated volatility.
Podolskij and Rosenbaum (2012) construct a test statistic and derive its
asymptotic behavior to test the local volatility hypothesis. Their test
statistic is a function of the increments of the stochastic process. Jacod et
al. (2008), Jacod and Podolskij (2013) test the rank of the matrix
$\Sigma\Sigma^{T}$. In Jacod et al. (2008), they consider continuous-time
observations of $X$ and construct a test statistic based on the process
perturbed by a random noise. Introducing a random perturbation of the
increment matrix enables to apply a ratio statistic based on the
multilinearity property of the determinant and ensures that the denominator of
the ratio never vanishes. The authors prove that the limit of the ratio
statistic identifies the rank of the volatility and they also study the
asymptotic distribution of this statistic. In Jacod and Podolskij (2013), they
extend their work to the case of discrete observations
$\left\\{X_{i\Delta}\right\\}_{i\in\mathds{N}}$. They also prove its
asymptotic distribution when $\Delta$ goes to zero. Fissler and Podolskij
(2017) consider testing the maximal rank of the volatility process for a
continuous diffusion observed with noise, using a pre-averaging approach with
weighted averages of process increments that eliminate the influence of noise.
Reiss and Winkelmann (2021) extend their work to time-varying covariance
matrices, again in an asymptotic setting.

In all these cases, the distribution of the test statistic is not explicit and
only asymptotic distributions have been obtained by applying asymptotic
convergence theorems when $\Delta$ goes to zero.

As already mentioned, our framework is different: we assume that the time step
$\Delta$ is fixed, which places us in a non-asymptotic setting. So we want to
construct a test procedure that guarantees a given level $\alpha$ in the non-
asymptotic setting with $\Delta$ and $n$ fixed. This is a major difference
with the works cited above. Although statistical tests reveal good properties
in the asymptotic setting, they are generally difficult to apply in a non-
asymptotic setting. In some examples, even if the rank of $\Sigma\Sigma^{T}$
is strictly less than $d$, the corresponding empirical covariance matrix may
be numerically full rank, i.e. in the non-asymptotic setting. This problem is
circumvented in the asymptotic setting in Jacod and Podolskij (2013) by adding
a random perturbation and studying the convergence of determinant ratio
statistics. But if we want to work in the non-asymptotic setting, we need to
use other estimators and probabilistic tools.

We have chosen to test the determinant of $\Sigma\Sigma^{t}$ rather than the
rank. The test statistic is therefore the determinant of the diffusion
increments matrix. In the asymptotic case, the influence of the drift is
negligible, since it is of order $O(\Delta)$. In the non-asymptotic case,
drift must be taken into account. We therefore propose to center the
statistics by removing the drift. We then study the distribution of the test
statistic. Under the assumption that the drift does not depend on $X_{t}$
itself (model (1)), the increments are independent. This makes possible to
derive the analytic distribution of the statistic in some simple cases, and in
other cases to prove lower and upper bounds of the distribution using
concentration inequalities. The case of a drift depending on $X_{t}$ is also
discussed. However in this case, the statistics is only approximately centered
and its distribution is much harder to access, even if we derive bounds for
the critical value.

Our first main contribution is to construct procedures for testing $H_{0}$
versus $H_{1}$ that satisfy non-asymptotic performance properties. In
particular, we propose a choice of critical values based either on the
explicit distribution of the test statistic (for one-dimensional SDE with
known drift) or on the lower bounds of the test statistic (for two-dimensional
SDE). In particular, for each $\alpha$ in $[0,1]$, these tests are of level
$\alpha$, i.e. they have a probability of Type I error at most equal to
$\alpha$. For particular models, they are even of size $\alpha$, the
probability of Type I error being exactly $\alpha$ since they are based on the
exact non-asymptotic distribution of the test statistic.

Our second main contribution consists in deriving non-asymptotic conditions on
the alternative hypothesis which guarantee that the probability of Type II
error is at most equal to a prescribed constant $\beta$. This can be done for
one-dimension SDE with necessary and sufficient conditions, when the drift is
fully known or even known up to a linear parameter. For two-dimension SDE, the
distribution is not exact and we use concentration inequalities to prove upper
bounds on the test statistic. The separability condition can then be deduced.
When the drift parameter is unknown, the test procedure is adapted. However,
power deteriorates slightly when the parameter is estimated on the first half
of the sample. For a dimension greater than 2, this is much more difficult,
and we are unable to prove the lower and upper bounds of the test statistics.
Instead, we propose an approach based on multiple one-dimensional tests and
prove that we control the level of the overall procedure. This procedure gives
very good results in practice.

This paper is organized as follows. First, we consider the case of a one-
dimensional diffusion process in Section 2. We calculate the exact
distribution for the non-centered and centered statistics, then deduce the
critical value and study conditions to control the Type II error. We show
that, from a non-asymptotic point of view, the centering of the test
statistics has a considerable influence on the test separation rates. We also
extend thus result to the case of unknown drift and discuss the case of a
drift depending on the process itself. In Section 3, we deal with a two-
dimensional process with known drift. We consider the center statistic and
prove the lower and upper bounds of its distribution. We then propose critical
values and conditions such that Type I and II errors are controlled. Section 4
presents the multiple testing approach. Next, Section 5 presents a numerical
study to illustrate the properties of the testing procedure on different SDEs.
We conclude with a discussion and perspective.

##  2 Test for a one-dimensional SDE

We start with a simple one-dimensional Brownian motion with drift depending on
time:

$dX_{t}=b_{t}dt+\sigma dW_{t},\quad X_{0}=x_{0},\quad t>0,$ (2)

where $b_{t}:\mathds{R}\to\mathds{R}$ is the drift function that depends only
on time $t$, $\sigma\in\mathds{R}$ is a constant diffusion coefficient and $W$
is a one-dimensional Brownian motion. The case of a drift $b$ depending on
$X_{t}$ is discussed at the end of Section 2.

We assume that the process $(X_{t})_{t\geq 0}$ is discretely observed on a
time interval $[0,T]$ at equidistant time step $\Delta$,
$t_{0}=0,t_{1}=\Delta,\ldots,t_{n}=n\Delta=T$. Our aim is to construct a
statistical test to decide between the two following hypotheses:

$\displaystyle H_{0}:\sigma^{2}=\sigma^{2}_{0}\quad\mbox{versus}\quad
H_{1}:\sigma^{2}>\sigma^{2}_{0},$

where $\sigma^{2}_{0}$ is a pre-chosen positive constant.

In Section 2.1, we consider an exact testing procedure by calculating the
exact distribution of the test statistic. We then introduce a centered version
of the test statistic in Section 2.2. We deal with the case where the drift is
unknown and estimated in Section 2.3. Finally, Section 2.4 introduces an
approximation of the statistics for the case of a drift $b$ depending on the
process $X_{t}$ itself.

For each test, we present the test statistic and its exact distribution. We
then construct the test by calculating the critical value that control s the
Type I error. Finally, we study the Type II error of the test by deriving non-
asymptotic and optimal conditions on the alternative hypothesis. We will use
the notations $\mathds{P}_{\sigma_{0}^{2}}$ and $\mathds{P}_{\sigma^{2}}$ to
distinguish the probability under the null or the alternative hypothesis.

#### Notations

In the following, we denote $\mathcal{N}(\mu,\omega^{2})$ a normal
distribution with mean $\mu$ and variance $\omega^{2}$, $\chi^{2}_{n}(0)$ a
chi-squared random distribution with $n$ degrees of freedom,
$\chi^{2}_{n}(\lambda)$ a chi-squared random distribution with $n$ degrees of
freedom and a non-centrality parameter $\lambda$. Let us also denote the
quantiles $q_{\mathcal{N},\beta}$, $q_{\chi^{2}_{n},\beta}$ and
$q_{\chi^{2}_{n}(\lambda),\beta}$ of order $\beta$ of the distributions
$\mathcal{N}(0,1)$, $\chi^{2}_{n}(0)$ and $\chi^{2}_{n}(\lambda)$,
respectively. Further, the symbol ”$\sim$” is used throughout the paper as an
alias for ”follows a certain probability distribution”.

###  2.1 Non-centered statistics

We consider the normalized increments of process $X$ defined as:

$\xi_{i}:=\frac{X_{i\Delta}-X_{(i-1)\Delta}}{\sqrt{\Delta}},\quad
i=1,\dots,n.$ (3)

Let $\xi=(\xi_{1},\ldots,\xi_{n})$. Note that the $\\{\xi_{i}\\}$ are
independent in $i$, since the increments do not overlap. We then define the
test statistic:

$S=\frac{1}{n}\sum_{i=1}^{n}\xi^{2}_{i}=\frac{1}{n}\|\xi\|^{2}.$ (4)

We calculate the distribution of $\xi_{i}$, $\|\xi\|^{2}$ and $S$ in the next
lemma:

######  Lemma 1.

Let $\xi_{i}$ be the random variables defined by (3). We have

  1. 1.

$\xi_{i}\sim\mathcal{N}\left(\frac{\int_{(i-1)\Delta}^{i\Delta}b_{s}ds}{\sqrt{\Delta}},\sigma^{2}\right)$.

  2. 2.

$\left\|\xi\right\|^{2}\sim\sigma^{2}\chi^{2}_{n}(\lambda(\sigma))$, with a
non-centrality parameter $\lambda(\sigma)$ equal to:

$\lambda(\sigma)=\frac{\sum_{i=1}^{n}\left(\int_{(i-1)\Delta}^{i\Delta}b_{s}ds\right)^{2}}{\sigma^{2}\Delta}.$

  3. 3.

$S\sim\frac{\sigma^{2}}{n}\chi^{2}_{n}(\lambda(\sigma))$. Its cumulative
distribution function is $\forall t>0$

$\mathds{P}_{\sigma^{2}}\left(S\leq
t\right)=1-Q_{n/2}\left(\sqrt{\lambda(\sigma)},\sqrt{\frac{nt}{\sigma^{2}}}\right),$

where $Q_{m}(u,v)$ is a Markum Q-function, defined as:

$\displaystyle Q_{m}(u,v)$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{u^{2}+v^{2}}{2}\right)\sum_{k=1-m}^{\infty}\left(\frac{u}{v}\right)^{k}I_{k}(uv),$
(5)

where $I_{k}$ is a modified Bessel function of the first kind of order $k$.

######  Remark.

  1. 1.

If the function $b_{s}$ is constant, the non-centrality parameter
$\lambda(\sigma)=n\Delta b^{2}/\sigma^{2}$ is of order $O(n\Delta)$. In the
asymptotic setting $T$ fixed, it is a constant. In the asymptotic setting
$\Delta$ fixed and $n\rightarrow\infty$, it converges to $\infty$.

  2. 2.

Note that expression (5) is not explicit, even though several packages or
approximations exist (Gil et al., 2014). We will show in the next section that
centering the statistic gives results that are easier to use.

The following proposition directly follows Lemma 1:

######  Proposition 1.

[1d-Test with noncentered statistics] Let $\alpha\in]0;1[$ be a fixed
constant. Let $S$ be the test statistic defined by (4) and let us define the
test $\Upsilon$ which rejects $H_{0}$ if

$S\geq
z_{1-\alpha}=:\frac{\sigma^{2}_{0}}{n}q_{\chi^{2}_{n}(\lambda(\sigma_{0})),1-\alpha}.$

Then, the test $\Upsilon$ is of Type I error $\alpha$ and therefore it is of
level $\alpha$.

Further, let $\beta\in]0;1[$ be a constant such that $1-\beta\geq\alpha$. For
all $\sigma^{2}>0$ such that

$\sigma^{2}\geq\sigma_{0}^{2}\frac{q_{\chi^{2}_{n}(\lambda(\sigma_{0})),1-\alpha}}{q_{\chi^{2}_{n}(\lambda(\sigma)),\beta}},$
(6)

the test $\Upsilon$ satisfies

$\mathds{P}_{\sigma^{2}}\left(\Upsilon\text{ accepts }H_{0}\right)\leq\beta.$

Condition (6) is sufficient and necessary.

###### Proof.

Since $S$ is distributed according to a non-centered chi-squared distribution,
it is straightforward to obtain

$\mathds{P}_{\sigma_{0}^{2}}\left(S\geq\frac{\sigma_{0}^{2}}{n}q_{\chi^{2}_{n}(\lambda(\sigma_{0})),1-\alpha}\right)=\alpha.$

For the Type II error, we have

$\mathds{P}\left(S\leq
z_{1-\alpha}\right)=\mathds{P}\left(\chi_{n}^{2}(\lambda(\sigma))\leq\frac{\sigma_{0}^{2}}{\sigma^{2}}q_{\chi^{2}_{n}(\lambda(\sigma_{0})),1-\alpha}\right).$

It implies that $\mathds{P}_{\sigma^{2}}\left(S\leq
z_{1-\alpha}\right)\leq\beta$ as soon as
$\frac{\sigma_{0}^{2}}{\sigma^{2}}q_{\chi^{2}_{n}(\lambda(\sigma_{0})),1-\alpha}\leq
q_{\chi^{2}_{n}(\lambda(\sigma)),\beta}$. Type II error is bounded by a
$\beta$ when (6) holds. ∎

We want to understand the influence of $n$ and $\Delta$ on the threshold
$z_{1-\alpha}$ and the separability condition (6). However, they are
implicitly defined as they depend on $\sigma^{2}_{0}$ and $\sigma^{2}$ via the
non-centrality parameters $\lambda(\sigma_{0})$ and $\lambda(\sigma)$. In what
follows, we consider the simplified case of a constant drift $b$, provide a
quantile approximation to detail the effect of $n$, $\Delta$ and deduce more
intuitive conditions on $\sigma$.

In the following, $\square$ denotes a positive quantity that is upper and
lower bounded by positive constants. Its value can change from line to line
and even within the same equation. In the same spirit, $\square_{\beta}$
designates a quantity that is upper and lower bounded by positive functions of
$\beta$.

Thanks to Lemma 7 in the appendix, we have for $\alpha<1/\sqrt{2\pi}$,

$n-1+\lambda(\sigma_{0})+\log(1/\alpha)\leq
q_{\chi^{2}_{n}(\lambda(\sigma_{0})),1-\alpha}\leq
n+\square\sqrt{n\log(1/\alpha)}+\square{\log(1/\alpha)}+\square\lambda(\sigma_{0}).$

So the critical value satisfies

$\sigma_{0}^{2}\frac{n-1}{n}+\square\sigma_{0}^{2}\frac{\log(1/\alpha)}{n}+\square\Delta
b^{2}\leq
z_{1-\alpha}\leq\sigma_{0}^{2}+\square\sigma_{0}^{2}\frac{\sqrt{\log(1/\alpha)}}{\sqrt{n}}+\square\sigma_{0}^{2}\frac{\log(1/\alpha)}{n}+\square\Delta
b^{2}.$

Let us now describe the behavior for the two asymptotic settings:

  1. 1.

$T$ fixed, $n\rightarrow\infty$ and $\Delta=T/n\rightarrow 0$. With the
previous inequality, we know that the critical value
$z_{1-\alpha}\underset{{n\to\infty}}{\longrightarrow}{\sigma_{0}^{2}}$.

  2. 2.

$\Delta$ fixed, $n\rightarrow\infty$ and $T=\Delta n\rightarrow\infty$. Then
the critical value does not converge towards $\sigma_{0}^{2}$. There is a bias
of the order of $\Delta b^{2}$ up to a multiplicative constant.

#### Study on the separability condition (6)

We have shown that the Type II error is less than $\beta$ if and only if

$\sigma^{2}\geq\bar{\sigma}_{\alpha,\beta}^{2}=\sigma_{0}^{2}\frac{q_{\chi^{2}_{n}(\lambda(\sigma_{0})),1-\alpha}}{q_{\chi^{2}_{n}(\lambda(\sigma)),\beta}}.$

We can approximate this bound thanks to Lemma 7 for $\alpha<1/\sqrt{2\pi}$ and
$\beta<0.5$. On one hand

$\bar{\sigma}_{\alpha,\beta}^{2}\leq\sigma_{0}^{2}\frac{n+\square\sqrt{\left(n+\frac{n\Delta
b^{2}}{\sigma_{0}^{2}}\right)\log(1/\alpha)}+\square{\log(1/\alpha)}+\frac{n\Delta
b^{2}}{\sigma_{0}^{2}}}{n+\frac{n\Delta
b^{2}}{\sigma^{2}}-\square_{\beta}\sqrt{n}-\square_{\beta}\sqrt{\frac{n\Delta
b^{2}}{\sigma_{0}^{2}}}}.$

On the other hand

$\bar{\sigma}_{\alpha,\beta}^{2}\geq\sigma_{0}^{2}\frac{n-1+\log(1/\alpha)+\frac{n\Delta
b^{2}}{\sigma_{0}^{2}}}{n+\frac{n\Delta b^{2}}{\sigma^{2}}+\square\sqrt{n}}.$

By introducing $u=1+\frac{\Delta b^{2}}{\sigma^{2}}$ and $u_{0}=1+\frac{\Delta
b^{2}}{\sigma_{0}^{2}}$, we get that Equation (6) is therefore implied by

$\sigma^{2}u\geq\sigma_{0}^{2}u_{0}\frac{1+\square_{\alpha}(nu_{0})^{-1/2}}{1-\square_{\beta}n^{-1/2}(1+\sqrt{u_{0}-1})u^{-1}}.$

But $u\geq 1$ hence $u^{-1}\leq 1$ and since we are under
$H_{1}:\sigma^{2}\geq\sigma_{0}^{2}$, we have $u_{0}\geq u$. Hence (6) is
implied by

$\sigma^{2}u\geq\sigma_{0}^{2}u_{0}\frac{1+\square_{\alpha}(nu_{0})^{-1/2}}{1-\square_{\beta}n^{-1/2}(1+\sqrt{\frac{\Delta
b^{2}}{\sigma_{0}^{2}}})}.$

This is equivalent to

$\sigma^{2}+\Delta b^{2}\geq(\sigma_{0}^{2}+\Delta
b^{2})\left(1+\frac{\square_{\alpha,\beta}}{\sqrt{n}}\left[\frac{\sigma_{0}}{\sqrt{\sigma_{0}^{2}+\Delta
b^{2}}}+1+\sqrt{\frac{\Delta b^{2}}{\sigma_{0}^{2}}}\right]\right)$

or finally to

$\sigma^{2}\geq\sigma_{0}^{2}+\frac{\square_{\alpha,\beta}}{\sqrt{n}}\left[\sigma_{0}\sqrt{\sigma_{0}^{2}+\Delta
b^{2}}+(\Delta b^{2}+\sigma_{0}^{2})\left(1+\sqrt{\frac{\Delta
b^{2}}{\sigma_{0}^{2}}}\right)\right].$

This is a sufficient condition for having a Type II error of value $\beta$. We
are losing the necessary condition because of the lower bound on the chi-
squared quantile of the numerator, where the term in $\sqrt{n}$ disappears.
This is why we cannot prove that it is a sufficient and necessary condition up
to a constant. However we believe it is the true rate in the sense that the
Gaussian concentration inequality has been proved recently to be tight for
two-sided quantiles of convex Lipschitz functions (Valettas, 2019). We have
just not been able to pass from two-sided to one sided bounds. Let us now
describe the behavior for the two asymptotic settings:

  1. 1.

$T$ fixed, $n\rightarrow\infty$ and $\Delta=T/n\rightarrow 0$. We have
$\Delta=T/n$. We recover a rate (at least for the upper bound) equal to
$\sigma_{0}^{2}(1+\frac{\square_{\alpha,\beta}}{\sqrt{n}})$.

  2. 2.

$\Delta$ fixed, $n\rightarrow\infty$ and $T=\Delta n\rightarrow\infty$. In
this case the limit deteriorates and converges at the same $\sqrt{n}$ rate but
with a multiplicative constant that worsens for large $\Delta$. If $\Delta
b^{2}/\sigma_{0}^{2}\geq 1$, the upper bound is at least in
$\sigma_{0}^{2}(1+\frac{\square_{\alpha,\beta}}{\sqrt{n}}\frac{\Delta
b^{2}}{\sigma_{0}^{2}})$ and up to
$\sigma_{0}^{2}\left(1+\frac{\square_{\alpha,\beta}}{\sqrt{n}}\left(\frac{\Delta
b^{2}}{\sigma_{0}^{2}}\right)^{3/2}\right)$, depending on whether one is
looking only at the numerator or if we take into account the concentration of
the denominator in Equation (6). In both cases, it means that the
multiplicative factor is increasing with $\Delta$ and we loose the $\sqrt{n}$
rate of separability of the two conditions when $\Delta$ is ”large”.

###  2.2 Centered statistics with known drift

In this section, we propose a new statistic to remove the dependency on the
drift and avoid the rate lost in the separability condition. To do so, we
introduce a centered test statistic. For $i=1,\ldots,n$, let us denote

$\dot{\xi}_{i}=\xi_{i}-\frac{1}{\sqrt{\Delta}}\int_{(i-1)\Delta}^{i\Delta}b_{s}ds=\frac{X_{i\Delta}-X_{(i-1)\Delta}-\int_{(i-1)\Delta}^{i\Delta}b_{s}ds}{\sqrt{\Delta}},$
(7)

such that $\dot{\xi}_{i}\sim\mathcal{N}(0,\sigma^{2})$. Then, we define the
statistics $\dot{S}$ as follows:

$\dot{S}=\frac{1}{n}\sum_{i=1}^{n}\dot{\xi}^{2}_{i}.$ (8)

Note that $\dot{S}$ follows a rescaled centered chi-squared distribution with
$n$ degrees of freedom

$\dot{S}\sim\frac{\sigma^{2}}{n}\chi^{2}_{n}(0).$

######  Proposition 2 (1d-Test with centered statistics and known drift).

Let $\alpha\in]0;1[$ be a fixed constant. Let $\dot{S}$ be the statistic
defined in (8) and let us define the test $\dot{\Upsilon}$ which rejects
$H_{0}$ if

$\dot{S}\geq\dot{z}_{1-\alpha}=:\frac{\sigma_{0}^{2}}{n}q_{\chi^{2}_{n},1-\alpha}.$
(9)

Then, the test $\dot{\Upsilon}$ is of Type I error $\alpha$ and therefore it
is of level $\alpha$.

Let $\beta\in]0;1[$ be a constant such that $1-\beta\geq\alpha$. For all
$\sigma^{2}$ such that

$\sigma^{2}\geq\frac{q_{\chi^{2}_{n},1-\alpha}}{q_{\chi^{2}_{n},\beta}}\sigma_{0}^{2},$
(10)

the test $\dot{\Upsilon}$ satisfies

$\mathds{P}_{\sigma^{2}}\left(\dot{\Upsilon}\text{ accepts
}H_{0}\right)\leq\beta.$

It is again a necessary and sufficient condition.

###### Proof.

Since $\dot{S}$ is distributed according to the centered chi-squared law with
$n$ degrees of freedom, it is straightforward to show that

$\mathds{P}_{\sigma^{2}_{0}}\left(\dot{S}\geq\dot{z}_{1-\alpha}\right)=\alpha.$
(11)

For the power of the test, we first note that

$\mathds{P}_{\sigma^{2}}\left(\dot{S}\leq\dot{z}_{1-\alpha}\right)=\mathds{P}_{\sigma^{2}}\left(\chi^{2}_{n}(0)\leq\frac{n}{\sigma^{2}}\frac{\sigma_{0}^{2}}{n}q_{\chi^{2}_{n},1-\alpha}\right),$

It implies that
$\mathds{P}_{\sigma^{2}}\left(\dot{S}\leq\dot{z}_{1-\alpha}\right)\leq\beta$
as soon as $\frac{\sigma_{0}^{2}}{\sigma^{2}}q_{\chi^{2}_{n},1-\alpha}\leq
q_{\chi^{2}_{n},\beta}$. Thus Type II error is bounded by a fixed risk level
$\beta\in]0;1[$ when (10) holds. ∎

#### Study of the threshold
$\dot{z}_{1-\alpha}=\frac{\sigma_{0}^{2}}{n}q_{\chi^{2}_{n},1-\alpha}$

We use again Lemma 7 to prove that

$\sigma_{0}^{2}\left(1+\frac{\square_{\alpha}}{n}\right)\leq\dot{z}_{1-\alpha}\leq\sigma_{0}^{2}\left(1+\frac{\square_{\alpha}}{\sqrt{n}}\right).$

This approximation does not depend on $\Delta$, only on the sample size $n$.
The order is thus the same for the setting $T$ fixed,
$n\rightarrow\infty,\Delta=T/n\rightarrow 0$, and the setting $\Delta$ fixed,
$n\rightarrow\infty,T=\Delta n\rightarrow\infty$.

#### Study of condition (10)

We have

$\sigma^{2}\geq\frac{q_{\chi^{2}_{n},1-\alpha}}{q_{\chi^{2}_{n},\beta}}\sigma^{2}_{0}.$

The same study as Section 2.2 leads again to a discrepancy between the upper
and lower bound. However if we look only at the upper bound, a necessary
condition for (10) to hold is

$\sigma^{2}\geq\sigma^{2}_{0}\left(1+\frac{\square_{\alpha,\beta}}{\sqrt{n}}\right).$

Therefore we see here that whatever the asymptotic regime, the multiplicative
constant in front of the separability rate does not explode when $\Delta$
increases since it does not depend on it. Of course our reasoning to compare
the centered and non centered procedure is purely on the upper bound. But as
mentionned earlier, because of recent results in Gaussian concentration
(Valettas, 2019), we believe that the upper bounds are more tight than the
lower bounds, even if we have not been able to prove it. This difference
between the behavior of the centered and non centered procedures for large
$\Delta$ has been confirmed on simulations, see Section 5.

###  2.3 Centered statistics with unknown drift

The drift is rarely known and has to be estimated from the discrete
observations $\left\\{X_{i\Delta}\right\\}_{i=0,\ldots,n}$. We present in this
section an adaptation of the previous test to the specific case of a
parametric drift depending on a linear parameter:

$dX_{t}=\theta f_{t}dt+\sigma dW_{t},\quad X_{0}=x_{0},\quad t>0,$ (12)

where $\theta\in\mathbb{R}$ is an unknown scalar parameter and
$f_{t}:\mathbb{R}\rightarrow\mathbb{R}$ is a known function. A standard
estimator of $\theta$ is the mean square estimator:

$\hat{\theta}=\arg\min_{\theta}\sum_{i=1}^{n}\left(X_{i\Delta}-X_{(i-1)\Delta}-\theta\int_{(i-1)\Delta}^{i\Delta}f_{s}ds\right)^{2}.$
(13)

This estimator has an explicit form and is normally distributed even when
$\Delta$ is fixed.

######  Lemma 2.

Let $\hat{\theta}$ be defined by (13). Then, the following holds:

  * (i)

$\hat{\theta}=\frac{\sum_{i=1}^{n}\left(X_{i\Delta}-X_{(i-1)\Delta}\right)\int_{(i-1)\Delta}^{i\Delta}f_{s}ds}{\sum_{i=1}^{n}\left(\int_{(i-1)\Delta}^{i\Delta}f_{s}ds\right)^{2}}.$

  * (ii)

$\hat{\theta}\sim\mathcal{N}(\theta,\sigma^{2}_{\theta})$ with
$\sigma^{2}_{\theta}=\frac{\Delta\sigma^{2}}{\sum_{i=1}^{n}\left(\int_{(i-1)\Delta}^{i\Delta}f_{s}ds\right)^{2}}$.

Proof is given in Appendix.

Now, let $\hat{\xi}_{i}$ be the increments centered around the estimated
drift:

$\hat{\xi}_{i}=\xi_{i}-\frac{\hat{\theta}}{\sqrt{\Delta}}\int_{(i-1)\Delta}^{i\Delta}f_{s}ds=\frac{X_{i\Delta}-X_{(i-1)\Delta}-\hat{\theta}\int_{(i-1)\Delta}^{i\Delta}f_{s}ds}{\sqrt{\Delta}}.$
(14)

We study the distribution of the vector
$\hat{\xi}=(\hat{\xi}_{1},\ldots,\hat{\xi}_{n})$.

######  Lemma 3.

Let us introduce $i=1,\ldots,n$:

$Z_{i}=\frac{1}{\sqrt{\Delta}}\int_{(i-1)\Delta}^{i\Delta}f_{s}ds,$

and $Z=(Z_{1},\ldots,Z_{n})^{t}$. Let $H$ be the projection matrix:

$H:=Z(Z^{t}Z)^{-1}Z^{t}.$

Let $C$ be a matrix such that $(C^{t}C)^{+}=(I-H)$, where $A^{+}$ denotes a
Moore-Penrose inverse of a matrix $A$. Then

  * •

$\hat{\xi}\sim\mathcal{N}(0,\sigma^{2}(I-H)),$

  * •

$\frac{1}{\sigma^{2}}\|C^{t}\hat{\xi}\|^{2}\sim\chi^{2}_{n-1}(0).$

Proof is given in Appendix. In practice, as the matrix $I-H$ has rank $n-1$,
we use the singular value decomposition (SVD) of $I-H$. SVD produces two
unitary matrices $U$ and $V$, and a diagonal matrix $D$ with $n-1$ non zero
values such that $I-H=UDV^{t}$. Then we take $C=UD^{-1/2}$.

We also define a new statistic:

$\tilde{S}=\frac{1}{n-1}\|C^{t}\hat{\xi}\|^{2},$ (15)

such that

$\frac{n-1}{\sigma^{2}}\tilde{S}\sim\chi^{2}_{n-1}(0).$

We can now define the test procedure.

######  Proposition 3 (1d-Test with centered statistics and unknown drift).

Let $\alpha\in]0;1[$ be a fixed constant. Let $\tilde{S}$ be the test
statistic defined by (15) and let us define the test $\tilde{\Upsilon}$ which
rejects $H_{0}$ if

$\tilde{S}\geq\hat{z}_{\alpha}=\frac{\sigma_{0}^{2}}{n-1}q_{\chi^{2}_{n-1},1-\alpha}.$

Then, the test $\tilde{\Upsilon}$ is of Type I error $\alpha$ and therefore it
is of level $\alpha$.

Let $\beta\in]0;1[$ be a constant such that $1-\beta\geq\alpha$. For all
$\sigma^{2}$ such that

$\sigma^{2}\geq\sigma^{2}_{0}\frac{q_{\chi^{2}_{n-1},1-\alpha}}{q_{\chi^{2}_{n-1},\beta}}.$
(16)

the test $\tilde{\Upsilon}$ satisfies

$\mathds{P}_{\sigma^{2}}\left(\tilde{\Upsilon}\text{ accepts
}H_{0}\right)\leq\beta.$

This is a necessary and sufficient condition.

The proof is analogous to Proposition 3. The condition for the Type II error
is essentially the same as the previous test and especially it does not depend
on $\Delta$ as well.

######  Remark.

  1. 1.

This procedure could be generalized to the case of a multidimensional vector
$\theta$ when for example the drift $b_{t}$ is defined as
$b_{t}=\sum_{1}^{p}\theta_{k}f_{kt}$ for a set of $p$ known functions
$(f_{kt})_{k=1,\ldots,p}$.

  2. 2.

A non-linear drift $b_{t}=f(t,\theta)$ could be considered with estimators
obtained through contrasts for example. But, we would loose the exact level of
the test.

To conclude the study of the one-dimensional case, we proved that centering
the statistics is important in a non-asymptotic setting, since it allows us to
find separation rates that are of parametric rate $1/\sqrt{n}$ in both
settings ($T$ fixed, $\Delta\to 0$ and $\Delta$ fixed, $T\to\infty$). This is
not the case if the centering is not done and if $\Delta$ is fixed and
$T\to\infty$.

###  2.4 Approximated statistics for a drift depending on $X$

Let us now consider an SDE where the drift $b$ depends on $X$, that is:

$dX_{t}=b(X_{t})dt+\sigma dW_{t}.$

First we introduce a theoretical and ideal statistic centered by the integral
of the drift, that would correspond to continuous observations of the process
$X$. Then we discretise this integral to fit with discrete observations of $X$
and bound the corresponding approximation. Let us be more precise.

For $i=1,\ldots,n$, let us denote the theoretical quantity

${\dot{\xi}_{i}=\frac{X_{i\Delta}-X_{(i-1)\Delta}-\int_{(i-1)\Delta}^{i\Delta}b(X_{s})ds}{\sqrt{\Delta}}}$
(17)

such that $\dot{\xi}_{i}\sim\mathcal{N}(0,\sigma^{2})$ and are independent.
These quantities are not calculable because the process $X$ is only observed
at discrete times and not continuously. However, it can be approximated with a
numerical scheme, for example the Euler-Maruyama scheme. More generally, we
consider the following assumption:

  * (A1)

There exists a function $A:\mathds{R}\times\mathds{R}\to\mathds{R}$ (possibly
depending on parameters of the process), such that $\forall\Delta>0$, the
following holds:

$\left\|\max_{t\in[0,T]}\left|\int_{t}^{t+\Delta}b(X_{s})ds-A(X_{t},X_{t+\Delta})\right|\right\|_{\infty}\leq\Delta
K,$

where the constant $K$ depends on the function $b$.

In the case of the Euler-Maruyama approximation, $A(X_{t},X_{t+\Delta})=\Delta
b(X_{t})$ and $K$ can be a Lipschitz constant of the drift multiplied by the
bound of finite moments of the process. It could also be a global bound of the
drift function.

Then, we can introduce the following approximation of the $\xi_{i}$:

${\dot{\xi}_{i,A}=\frac{X_{i\Delta}-X_{(i-1)\Delta}-A(X_{i\Delta},X_{(i-1)\Delta})}{\sqrt{\Delta}}},$
(18)

and a test statistic $\dot{S}_{A}$:

$\displaystyle\dot{S}_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\dot{\xi}_{i,A}^{2}.$ (19)

We can now define the test procedure.

######  Proposition 4.

Let $\alpha\in]0;1[$ and $\eta\in]0;1[$ be fixed constants. Let $\dot{S}_{A}$
be the test statistics defined by (19) and let us define the test
$\dot{\Upsilon}_{A}$ which rejects $H_{0}$ if

$\dot{S}_{A}\geq(1+\eta)\dot{z}_{1-\alpha}+\left(\frac{\eta+1}{\eta}\right)\Delta
K^{2},$

where $\dot{z}_{1-\alpha}$ is defined in Proposition 2 ad $K$ is the constant
from assumption (A1). Then the test $\dot{\Upsilon}_{A}$ is of Type I error
less than $\alpha$ and therefore it is of level $\alpha$.

Let $\beta\in]0;1[$ be a constant such that $1-\beta\geq\alpha$. For all
$\sigma^{2}$ such that

$\sigma^{2}\geq\left(\frac{1+\eta}{1-\eta}\right)\frac{q_{\chi^{2}_{n},1-\alpha}}{q_{\chi^{2}_{n},\beta}}\sigma_{0}^{2}+\frac{2}{\eta(1-\eta)}\frac{n\Delta
K^{2}}{q_{\chi^{2}_{n},\beta}},$

the test $\dot{\Upsilon}_{A}$ satisfies

$\mathds{P}_{\sigma^{2}}\left(\dot{\Upsilon}_{A}\text{ accepts
}H_{0}\right)\leq\beta.$

It is only a sufficient condition.

###### Proof.

First, note that $\dot{S}_{A}$ can be written as follows:

$\dot{S}_{A}=\frac{1}{n}\sum_{i=1}^{n}\left(\dot{\xi}_{i}+\frac{\int_{i\Delta}^{(i+1)\Delta}b(X_{s})ds-A(X_{i\Delta},X_{(i-1)\Delta})}{\sqrt{\Delta}}\right)^{2}.$

Then, for any $\eta>0$, since for all $a,b$, $(1-\eta)a^{2}-(1-\eta)/\eta
b^{2}\leq(a+b)^{2}\leq(1+\eta)a^{2}+(1+\eta)/\eta b^{2}$, the following bounds
hold on $\dot{S}_{A}$:

$(1-\eta)\dot{S}-\frac{1-\eta}{\eta}\Delta
K^{2}\leq\dot{S}_{A}\leq(1+\eta)\dot{S}+\frac{\eta+1}{\eta}\Delta K^{2}$

Combined with Proposition 2, it implies that

$\displaystyle\mathds{P}$
$\displaystyle\left(\dot{S}_{A}\geq(1+\eta)\dot{z}_{1-\alpha}+\frac{\eta+1}{\eta}\Delta
K^{2}\right)\leq$ $\displaystyle\mathds{P}$
$\displaystyle\left((1+\eta)\dot{S}+\frac{\eta+1}{\eta}\Delta
K^{2}\geq(1+\eta)\dot{z}_{1-\alpha}+\frac{\eta+1}{\eta}\Delta K^{2}\right)=$
$\displaystyle\mathds{P}$
$\displaystyle\left(\dot{S}\geq\dot{z}_{1-\alpha}\right)\leq\alpha.$

The same inequalities grant the control over the power of the test:

$\displaystyle\mathds{P}$
$\displaystyle\left(\dot{S}_{A}\leq(1+\eta)\dot{z}_{1-\alpha}+\left(\frac{\eta+1}{\eta}\right)\Delta
K^{2}\right)\leq$ $\displaystyle\mathds{P}$
$\displaystyle\left((1-\eta)\dot{S}-\frac{1-\eta}{\eta}\Delta
K^{2}\leq(1+\eta)\dot{z}_{1-\alpha}+\left(\frac{\eta+1}{\eta}\right)\Delta
K^{2}\right)=$ $\displaystyle\mathds{P}$
$\displaystyle\left(\dot{S}\leq\frac{1}{1-\eta}\left((1+\eta)\dot{z}_{1-\alpha}+\frac{2}{\eta}\Delta
K^{2}\right)\right).$

Similar arguments as the one in the proof of Proposition 2 give the result. ∎

Thus this test procedure generalises the previous tests to the case of a drift
depending on the process $X$.

From a separation rate point of view, the first term is still (up to a
multiplicative constant) in
$\sigma_{0}^{2}\left(1+\frac{\square_{\alpha,\beta}}{\sqrt{n}}\right)$.
However, since $q_{\chi^{2}_{n},\beta}\sim n$, there is a residual in $\Delta
K^{2}$, that tends to zero only if $\Delta$ tends to $0$. This rate is even
worse than in the non centered case, where the factors in $\Delta$ where
divided by $\sqrt{n}$. It means in particular that if $\Delta>>\sqrt{n}$, the
rate is not the parametric rate anymore but is governed only by the size of
the step. In this sense, the problem of testing with drift depending on the
process itself can be much more complex than the other cases and might lead to
separation rates that strongly depend on the approximation and observation
scheme. We did not try to calibrate this test in practice, because even the
choice of $\eta$ might lead to huge variations in practical performance and
its calibration is beyond the scope of the present article.

##  3 Test for a two-dimensional SDE

Now let us turn to a two-dimensional SDE $X=(X^{1}_{t},X^{2}_{t})$, defined
as:

$dX_{t}=b_{t}dt+\Sigma dW_{t},\quad X_{0}=x_{0},\quad t>0,$ (20)

where $b_{t}=(b_{t,1},b_{t,2})^{T}$ is a known drift and $\Sigma$ is a
diagonal diffusion matrix with constant coefficients $\sigma_{1}$ and
$\sigma_{2}$ on the main diagonal and $W$ is a 2-dimensional Brownian motion.
In this section, we only consider the case of a drift depending on time $t$.
The goal is to construct a statistical test of the following hypothesis:

$\displaystyle H_{0}:\det\Sigma\Sigma^{T}=\det\Sigma_{0}\Sigma_{0}^{T}$
$\displaystyle H_{1}:\det\Sigma\Sigma^{T}>\det\Sigma_{0}\Sigma_{0}^{T}.$

As we assume $\Sigma$ diagonal, it is equivalent to testing

$\displaystyle
H_{0}:\sigma_{1}^{2}\sigma_{2}^{2}=\sigma_{1,0}^{2}\sigma_{2,0}^{2},\quad\mbox{versus}\quad
H_{1}:\sigma_{1}^{2}\sigma_{2}^{2}>\sigma_{1,0}^{2}\sigma_{2,0}^{2}.$

We define the 2-dimensional centered increments with shifted indices to allow
independent variables for $j=1,2,i=1,\dots,n/2$:

$\dot{\xi}_{ij}:=\frac{X_{(2i+j-2)\Delta}-X_{(2i+j-3)\Delta}-\int_{(2i+j-3)\Delta}^{(2i+j-2)\Delta}b_{s}ds}{\sqrt{\Delta}}.$
(21)

######  Lemma 4.

The vectors $\dot{\xi}_{ij}$ are independent in $i$ and $j$. Moreover $\forall
j\in\\{1,2\\}$, $i\in\\{1,\dots,n/2\\}$:

$\dot{\xi}_{ij}\sim\mathcal{N}\left(0,\Sigma\Sigma^{T}\right).$

Note that the independence in $i$ and $j$ is not true when the drift depends
on the process $X$ itself. Let us define the determinant of the following 2x2
matrices
$\dot{s}_{i}=\det[(\dot{\xi}_{i1})^{2},(\dot{\xi}_{i2})^{2}]=\dot{\xi}_{i11}^{2}\dot{\xi}_{i22}^{2}-\dot{\xi}_{i12}^{2}\dot{\xi}_{i21}^{2}$.
The first terms are

$\displaystyle\dot{s}_{1}$ $\displaystyle=$
$\displaystyle\det\left[\left(\frac{X_{\Delta}-X_{0}-\int_{0}^{\Delta}b_{s}ds}{\sqrt{\Delta}}\right)^{2},\left(\frac{X_{2\Delta}-X_{\Delta}-\int_{\Delta}^{2\Delta}b_{s}ds}{\sqrt{\Delta}}\right)^{2}\right]$
$\displaystyle\dot{s}_{2}$ $\displaystyle=$
$\displaystyle\det\left[\left(\frac{X_{3\Delta}-X_{2\Delta}-\int_{2\Delta}^{3\Delta}b_{s}ds}{\sqrt{\Delta}}\right)^{2},\left(\frac{X_{4\Delta}-X_{3\Delta}-\int_{3\Delta}^{4\Delta}b_{s}ds}{\sqrt{\Delta}}\right)^{2}\right],$

and so on. The statistic is thus the sum of independent variables:

$\dot{S}=\frac{1}{n/2}\sum_{i=1}^{n/2}\dot{s}_{i}.$ (22)

We start by some preliminary results on $\dot{S}$ in Section 3.1. Then, we
study the Type I and Type II errors of the test in Section 3.2. The two
previous sections consider the drift known. The case of an unknown drift is
presented in Section 3.3.

###  3.1 Preliminary results on the test statistic $\dot{S}$

First, the distribution of $\dot{s}_{i}$ is studied. Thanks to the centered
statistics, its cumulative distribution function is explicitly known, as
detailed in the following proposition (proof is given in Appendix).

######  Proposition 5.

  1. 1.

The density function of $\dot{s}_{i}$ is given by:

$g_{\dot{s}_{i}}(x)=\frac{1}{2\sqrt{\sigma^{2}_{1}\sigma^{2}_{2}}}\frac{e^{-\sqrt{\frac{x}{\sigma^{2}_{1}\sigma^{2}_{2}}}}}{\sqrt{x}}.$
(23)

  2. 2.

Its expectation and variance are defined by:

$\displaystyle\mathds{E}\left[\dot{s}_{i}\right]$ $\displaystyle=$
$\displaystyle 2\sigma^{2}_{1}\sigma^{2}_{2},$ (24) $\displaystyle
Var\left[\dot{s}_{i}\right]$ $\displaystyle=$ $\displaystyle
20\sigma^{4}_{1}\sigma^{4}_{2}.$ (25)

  3. 3.

The following holds for all $i$, $\forall x$:

$\mathds{P}\left(\dot{s}_{i}\leq
x\right)=1-e^{-\sqrt{\frac{x}{\sigma^{2}_{1}\sigma^{2}_{2}}}}.$

The following Theorem provides that the lower bound of $\dot{S}$ is sub-
gaussian due to the fact that $\dot{S}>0$ and the upper bound of $\dot{S}$ is
obtained using Chebyshev’s inequality. The proof is given in Appendix.

######  Theorem 1.

Let $\dot{S}$ be defined by (22).

  1. 1.

For any $t\in\mathbb{R}$, we have the lower bound

$\mathds{P}\left(\dot{S}-\mathds{E}\left[\dot{S}\right]\leq-t\right)\leq\exp\left(-\frac{nt^{2}}{{96}\sigma^{4}_{1}\sigma^{4}_{2}}\right).$
(26)

  2. 2.

For any $t\in\mathbb{R}$, we have the upper bound

$\mathds{P}\left(\dot{S}-\mathds{E}\left[\dot{S}\right]\geq
t\right)\leq\frac{1}{n/2}\frac{20\sigma^{4}_{1}\sigma^{4}_{2}}{t^{2}}.$ (27)

Note that the lower bound (26) is decaying exponentially fast as $t$ grows. In
comparison, the upper bound (27) is decaying at much slower rate.

###  3.2 Control of Type I and Type II errors

Using Theorem 1 we can define the rejection zone for the test statistic
$\dot{S}$.

######  Theorem 2 (2-dimensional test with centered statistics).

Let $\alpha\in]0,1[$ be a fixed constant and let $\dot{S}$ be the test
statistic defined in (22). Let us define a test $\dot{\Upsilon}$ which rejects
$H_{0}:\det\Sigma\Sigma^{T}=\det\Sigma_{0}\Sigma_{0}^{T}$ if

$\dot{S}\geq\dot{z}_{\alpha}=2\det\Sigma_{0}\Sigma_{0}^{T}\left(\sqrt{\frac{10}{n\alpha}}+1\right).$

Then $\dot{\Upsilon}$ is a test of Type I error $\alpha$ and therefore it is
of level $\alpha$. Let $\beta\in]0,1[$ such that $1-\beta\geq\alpha$. If
$n>24(-\log\beta)$ and if

$\det\Sigma\Sigma^{T}\geq\frac{\det\Sigma_{0}\Sigma_{0}^{T}\left(\sqrt{\frac{10}{n\alpha}}+1\right)}{1-{2}\sqrt{-\frac{{6}}{n}\log\beta}},$
(28)

then the test $\dot{\Upsilon}$ satisfies

$\mathds{P}_{\sigma}\left(\dot{\Upsilon}\quad accepts\quad
H_{0}\right)\leq\beta.$

###### Proof.

We start with the Type I error. We apply Theorem 1 to control the probability
to surpass some given threshold $\dot{z}_{\alpha}$:

$\mathds{P}_{\sigma_{0}}\left(\dot{S}\geq\dot{z}_{\alpha}\right)=\mathds{P}_{\sigma_{0}}\left(\dot{S}-\mathds{E}\left[\dot{S}\right]\geq\dot{z}_{\alpha}-\mathds{E}\left[\dot{S}\right]\right)\leq\frac{1}{n/2}\frac{20(\det\Sigma_{0}\Sigma_{0}^{T})^{2}}{\left(\dot{z}_{\alpha}-2\det\Sigma_{0}\Sigma_{0}^{T}\right)^{2}}.$

We want to limit the risk of the Type I error to $\alpha$. We have to solve
the following inequality:

$\displaystyle\frac{1}{n/2}\frac{20(\det\Sigma_{0}\Sigma_{0}^{T})^{2}}{\left(\dot{z}_{\alpha}-2\det\Sigma_{0}\Sigma_{0}^{T}\right)^{2}}\leq\alpha.$

Thus

$\displaystyle\dot{z}_{\alpha}\geq
2\det\Sigma_{0}\Sigma_{0}^{T}\left(\sqrt{\frac{10}{n\alpha}}+1\right).$

It remains to control the power of the test. Under $H_{1}$,
$\mathds{E}[\dot{S}]=2\det\Sigma\Sigma^{T}$.We are looking for conditions on
$\det\Sigma\Sigma^{T}$, such that
$\mathds{P}_{\sigma}\left(\dot{S}\leq\dot{z}_{\alpha}\right)\leq\beta$. Then,
by Theorem 1:

$\displaystyle\mathds{P}_{\sigma}\left(\dot{S}\leq\dot{z}_{\alpha}\right)$
$\displaystyle=$
$\displaystyle\mathds{P}_{\sigma}\left(\dot{S}-2\det\Sigma\Sigma^{T}\leq\dot{z}_{\alpha}-2\det\Sigma\Sigma^{T}\right)$
$\displaystyle\leq\exp\left(-\frac{n\left(\dot{z}_{\alpha}-2\det\Sigma\Sigma^{T}\right)^{2}}{{96}(\det\Sigma\Sigma^{T})^{2}}\right).$

Now, the right part of the expression is bounded by a fixed risk level $\beta$
if

$\displaystyle\det\Sigma\Sigma^{T}\geq\frac{\dot{z}_{\alpha}}{2-4\sqrt{-\frac{{6}}{n}\log\beta}}.$

Replacing $\dot{z}_{\alpha}$ by its definition, we obtain the result. For
certain values of $n$ and $\log\beta$ it is possible that the lower bound of
condition (28) takes negative values. It is not the case as soon as
$n>24(-\log\beta)$. ∎

######  Remark.

Theorem 2 is valid under condition $n>24(-\log\beta)$. For example, for
$\beta=0.05$, one needs at least $150$ observations.

#### Study of condition (28)

Let us approximate condition (28):

$\det\Sigma\Sigma^{T}\geq\det\Sigma_{0}\Sigma_{0}^{T}\left(1+\frac{1}{\sqrt{n}}\left(\sqrt{\frac{10}{\alpha}}+{2}\sqrt{-{6}\log\beta}\right)+\frac{4}{n}\sqrt{\frac{-{15}\log\beta}{\alpha}}\right)$

This does not depend on the setting $T$ fixed, $n\rightarrow\infty$ or
$\Delta$ fixed, $n\rightarrow\infty$. For both cases, the separation rate has
order $1/\sqrt{n}$.

###  3.3 Test with unknown drift

As it is not realistic to assume the drift fully known, we consider the case
of a drift depending on a linear vector $\theta=(\theta_{1},\theta_{2})^{t}$
and a vector of drift $f_{t}=(f_{t,1},f_{t,2})^{t}$:

$dX_{t}=\theta^{t}f_{t}dt+\Sigma dW_{t}.$

If the parameter $\theta$ is estimated on the same sample than the one used
for testing, the centered increments used to define the test statistics are
not independent. Instead, we propose to split the sample in two sub-samples
$(X_{1},\ldots,X_{n_{e}})$ and $(X_{n_{e}+1},\ldots,X_{n})$.

Standard estimators of $\theta_{k}$, $k=1,2$ are the mean square estimators
calculated on $(X_{1},\ldots,X_{n_{e}})$ and their distribution is known, by
following the same steps as in one-dimension (Lemma 2). Then we prove the next
lemma:

######  Lemma 5.

Let us define the estimators of $\theta_{l}$, for $l=1,2$

$\hat{\theta}_{l}=\arg\min_{{\theta_{l}}}\sum_{i=1}^{n_{e}}\left(X_{i\Delta,l}-X_{(i-1)\Delta,l}-\theta_{l}\int_{(i-1)\Delta}^{i\Delta}f_{s,l}ds\right)^{2}.$

Their distributions are

$\hat{\theta}_{l}\sim\mathcal{N}(\theta_{l},\sigma^{2}_{\theta,l})\quad\mbox{with}\quad\sigma^{2}_{\theta,l}=\frac{\Delta\sigma_{l}^{2}}{\sum_{k=1}^{n_{e}}\left(\int_{(k-1)\Delta}^{k\Delta}f_{s,l}ds\right)^{2}}.$

The estimators $\hat{\theta}_{1},\hat{\theta}_{2}$ are calculated from the
first sub-sample $(X_{1},\ldots,X_{n_{e}})$ and are thus independent of the
second sub-sample $(X_{n_{e}+1},\ldots,X_{n})$. This allows to define
independent increments centered around the estimated value of the drift, for
$l=1,2$, $j=1,2$ and $i=\frac{n_{e}+3}{2},\ldots,\frac{n}{2}$:

$\displaystyle\tilde{\xi}_{ij,l}$ $\displaystyle=$
$\displaystyle\frac{X_{(2i+j-2)\Delta,l}-X_{(2i+j-3)\Delta,l}}{\sqrt{\Delta}}-\frac{\hat{\theta}_{l}}{\sqrt{\Delta}}\int_{(2i+j-2)\Delta}^{(2i+j-3)\Delta}f_{s,l}ds.$
(29)

For $l=1,2$, we have
$\tilde{\xi}_{ij,l}=\xi_{ij,l}+\frac{\hat{\theta}_{l}-\theta_{l}}{\sqrt{\Delta}}\int_{(2i+j-2)\Delta}^{(2i+j-3)\Delta}f_{s,1}ds$
and we prove the following Lemma.

######  Lemma 6.

The distributions of the increments are, for $l=1,2$, $j=1,2$ and
$i=\frac{n_{e}+3}{2},\ldots,\frac{n}{2}$,

$\tilde{\xi}_{ij,l}\sim\mathcal{N}(0,\sigma_{l}^{2}(1+h_{ij,l}))\quad\mbox{with}\quad
h_{ij,l}=\frac{\left(\int_{(2i+j-2)\Delta}^{(2i+j-3)\Delta}f_{s,l}ds\right)^{2}}{\sum_{k=1}^{n_{e}}\left(\int_{(k-1)\Delta}^{k\Delta}f_{s,l}ds\right)^{2}}.$

We assume that there exists a constant $C$, smaller than $n_{e}$ such that

$\forall i,j,l,h_{ij,l}\leq\frac{C}{n_{e}}\leq 1.$

Let us define the determinant of the following 2x2 matrices
$\tilde{s}_{i}=\det[(\tilde{\xi}_{i1})^{2},(\tilde{\xi}_{i2})^{2}]$
$=\tilde{\xi}_{i1,1}^{2}\tilde{\xi}_{i2,2}^{2}-\tilde{\xi}_{i1,2}^{2}\tilde{\xi}_{i2,1}^{2}$.
Conditionally on $\hat{\theta}$, its expectation and variance are approximated
by:

$\displaystyle\mathds{E}\left[\tilde{s}_{i}\right]$ $\displaystyle=$
$\displaystyle 2\sigma^{2}_{1}\sigma^{2}_{2}(1-h_{ii,1})(1-h_{ii,2}){\leq
2\sigma^{2}_{1}\sigma^{2}_{2}\frac{(n_{e}-C)^{2}}{n_{e}^{2}}\leq
2\sigma^{2}_{1}\sigma^{2}_{2}},$ $\displaystyle Var\left[\tilde{s}_{i}\right]$
$\displaystyle=$ $\displaystyle
20\sigma^{4}_{1}\sigma^{4}_{2}(1-h_{ii,1})^{2}(1-h_{ii,2})^{2}{\leq
20\sigma^{4}_{1}\sigma^{4}_{2}\frac{(n_{e}-C)^{4}}{n_{e}^{4}}\leq
20\sigma^{4}_{1}\sigma^{4}_{2}}.$

We can then apply the same methodology developed for the known drift case. Let
us define the statistic

$\tilde{S}=\frac{2}{n-n_{e}-1}\sum_{i=\frac{n_{e}+3}{2}}^{\frac{n}{2}}\tilde{s_{i}}.$
(30)

Proposition 5 and Theorem 1 can be easily extended to this case. We can then
define the rejection zone for the test statistic $\tilde{S}$.

######  Theorem 3 (2-dimensional test with centered statistics and unknown
drift).

Let $\alpha\in]0,1[$ be a fixed constant and let $\tilde{S}$ be the test
statistic defined in (30). Let us define a test $\tilde{\Upsilon}$ which
rejects $H_{0}:\det\Sigma\Sigma^{T}=\det\Sigma_{0}\Sigma_{0}^{T}$ if

$\tilde{S}\geq\tilde{z}_{\alpha}=2\det\Sigma_{0}\Sigma_{0}^{T}\left(\sqrt{\frac{10}{{n_{t}}\alpha}}+1\right),$

where $n_{t}=n-n_{e}$ is the size of the second sample. Then
$\tilde{\Upsilon}$ is a test of Type I error $\alpha$ and therefore it is of
level $\alpha$. Let $\beta\in]0,1[$ such that $1-\beta\geq\alpha$. If
${n_{t}}>48(-\log\beta)$ and if

$\det\Sigma\Sigma^{T}\geq\frac{\det\Sigma_{0}\Sigma_{0}^{T}\left(\sqrt{\frac{10}{{n_{t}}\alpha}}+1\right)}{1-4\sqrt{-\frac{3}{{n_{t}}}\log\beta}},$
(31)

then the test $\tilde{\Upsilon}$ satisfies

$\mathds{P}_{\sigma}\left(\tilde{\Upsilon}\quad accepts\quad
H_{0}\right)\leq\beta.$

It is difficult to deduce a natural choice for $n_{e}$ from this result. But
we recommend using $n_{e}=n/2$.

As in dimension 1, this could also be extended to the case of a drift defined
as a linear combination of known functions
($b_{t}=\sum_{k=1}^{p}\theta_{k}^{t}f_{kt}$).

##  4 Test in dimension $d\geq 2$ with a multiple testing approach

The previous tests are difficult to adapt to the case $d>2$ because we lose
the equivalent of Proposition 5. An alternative is to consider several tests
$\delta_{j,\alpha}$, one for each component $j=1,...,d$ and then correct them
for multiplicity. This multiple procedure is not equivalent to the test of
$H_{0}="\det(\Sigma)=\sigma^{2}_{0,1}...\sigma^{2}_{0,d}"$ versus
$H_{1}="\det(\Sigma)>\sigma_{0,1}^{2}...\sigma_{0,d}^{2}"$. However, it is of
main interest when the primary objective is to identify on which SDE
coordinate the noise acts (for example in neurosciences).

More precisely, let us consider the test $\delta_{j,\alpha}$ testing
$H_{0,j}="\sigma_{j}^{2}=\sigma_{0,j}^{2}"$ versus
$H_{1,j}="\sigma_{j}^{2}>\sigma_{0,j}^{2}"$ at level $\alpha$. In particular,
we can use any of the tests developed in Section 2, coordinate per coordinate,
like the ones with centered statistics, that have been proved to have better
performance.

Note that if the hypotheses are considered as a set of probabilities where the
hypotheses hold and if the model consists in saying that
$\sigma_{j}\geq\sigma_{0,j}$ for all $j$, we have that

$H_{0}=\bigcap_{j=1,...,d}H_{0,j}\mbox{ and
}\bigcup_{j=1,...,d}H_{1,j}=H_{1}.$

So we can build a test of $H_{0}$ versus $H_{1}$ by saying that we reject
$H_{0}$ if there exists a test $\delta_{j,\alpha/d}$ that rejects. Note that
we use the level $\alpha/d$. This comes from the Bonferroni bound (Roquain,
2011):

$\displaystyle\mathbb{P}_{H_{0}}(\exists
j=1,...d,\quad\delta_{j,\alpha/d}\mbox{ rejects }H_{0,j})$ $\displaystyle\leq$
$\displaystyle\sum_{j=1..d}\mathbb{P}_{H_{0}}(\delta_{j,\alpha/d}\mbox{
rejects }H_{0,j})$ $\displaystyle\leq$ $\displaystyle d\alpha/d=\alpha.$

Thus, this multiple testing approach controls the first type error.

In addition to being a test of the same level, this aggregation of individual
tests gives us extra information: the indices $j$ for which the test
$\delta_{j,\alpha/d}$ rejects, that is the coordinates $j$ for which the noise
is large.

##  5 Numerical experiments

In this section, we illustrate the numerical properties of the test in
dimension 1 or 2. We focus on studying their power and the impact of designs
by letting $n$ and $\Delta$ varying. In dimension 1, we consider three test
statistics: the non-centered drift statistics $S$ (Section 2.1), the centered
statistics $\dot{S}$ with the drift being explicitly known (Section 2.2) and,
the centered statistics $\tilde{S}$ with the drift being estimated from the
discrete observations (Section 2.3). In dimension 2, we consider the test
statistic with the drift known (Section 3.2) or estimated (Section 3.3) and
the multiple testing approach (Section 4).

###  5.1 One dimensional process with known drift

Let us consider the following toy SDE, a randomly perturbed sinusoidal
function, defined as follows:

$dX_{t}=\theta\sin(t)dt+\sigma dW_{t},\quad X_{0}=0,$ (32)

where $\theta\in\mathds{R}$ and $\sigma\in\mathds{R}$. The parameter $\theta$
is fixed to $1$ in all simulations.

To study the power of the test procedures, processes are simulated under
$H_{1}$ for a given value of $\sigma^{2}$ and the test is applied to each
process. Different values of $\sigma^{2}$ are considered, varying from $0$ to
$0.36$ with a step $0.001$. For each value of $\sigma^{2}$, $N=5000$ processes
are simulated with Euler-Maruyama scheme with a time step 0.01, for different
values of time horizon $T$ and subsampled with different discretization step
$\Delta$. These processes are denoted $X_{\sigma}$. The power of a test
procedure $\Psi$ is then estimated as the proportion of processes for which
the test is rejected and is denoted $\Pi(\Psi)$:

$\Pi(\Psi)=\frac{\\#\text{ processes for which }H_{0}\text{ is rejected
according to test }\Psi}{N}.$ (33)

The power functions $\Pi(\Upsilon)$, $\Pi(\dot{\Upsilon})$ and
$\Pi(\tilde{\Upsilon})$ are computed in three settings: $T=1,\Delta=0.1$ and
$n=10$; $T=1,\Delta=0.01$ and $n=100$; and $T=10,\Delta=0.1$ and $n=100$. Note
that the decision rules are given in Propositions 1, 2 and 3, respectively.

Figure 1: Power functions of the test of $H_{0}:\sigma^{2}=0.1^{2}$ against
$H_{1}:\sigma^{2}>0.1^{2}$ as a function of $\sigma_{20}^{2}$. Processes
$X_{\sigma}$ are simulated for $\sigma_{20}^{2}$ varying between $0$ and
$0.36$. Three tests are considered: the one-dimensional non-centered test $S$
with known drift (Section 2.1) in dashed blue line, with centered statistic
$\dot{S}$ (Section 2.2) in dotted green line, with centered statistics and
estimated drift $\tilde{S}$ (Section 2.3) in plain red line. Three designs are
considered: $\Delta=0.01,T=1$ (left), $\Delta=0.1,T=1$ (middle) and
$\Delta=0.1,T=10$ (right).

All three power functions are plotted on Figure 1. The performance of the
centered statistics in tests $\dot{\Upsilon}$ and $\tilde{\Upsilon}$ are
almost identical and depend mostly on the number of available observations.
The performance of the non-centered test $\Upsilon$ is sensitive to the step
size $\Delta$.

Especially the power functions $\Pi(\dot{\Upsilon})$ and
$\Pi(\tilde{\Upsilon})$ are identical when $T=10,\Delta=0.1$ and
$T=1,\Delta=0.01$. This is in accordance with the concluding remark in Section
2.2: the performance of the test does not depend on the time horizon nor on
the step size, only on the number of observations. For the non-centered
statistics $\Pi(\Upsilon)$, however, it is not the case: as the law of the
statistics depends on the drift, the performance of the test depends both on
the number of observations, and on the discretization step.

###  5.2 2-dimensional process with known drift

To illustrate how the method works in dimension two, we use a randomly
perturbed sinusoïd $X_{t}=(X_{t,1},X_{t,2})$:

$\begin{gathered}dX_{t,1}=\theta_{1}\sin(t)dt+\sigma_{1}dW_{t,1},\\\
dX_{t,2}=\theta_{2}\cos(t)dt+\sigma_{2}dW_{t,2}.\end{gathered}$ (34)

Parameters used for simulations are
$X_{0,1}=X_{0,2}=0,\theta_{1}=\theta_{2}=1$, $\sigma_{2}=1$. We generate
$N=5000$ processes under $H_{1}$ with $\sigma^{2}_{1}$ varying between $0$ and
$0.36$ (with a step $0.001$) in order to study the power of the test. We use 3
different scenarios: with $T=1,\Delta=0.01$; $T=1,\Delta=0.1$ and
$T=10,\Delta=0.1$.

We define the power function as in (33) for the 2-dimensional tests with known
drift (Section 3.2) or estimated (Section 3.3), and for the multiple testing
procedure (Section 4) with either known drift, or estimated.

Figure 2: Power functions of the test of
$H_{0}:\sigma_{1}^{2}\sigma_{2}^{2}=\sigma_{20}^{2}$ against
$H_{1}:\sigma_{1}^{2}\sigma_{2}^{2}>\sigma_{20}^{2}$ as a function of
$\sigma_{20}^{2}$. Processes $X_{\sigma}$ are simulated for $\sigma_{20}^{2}$
varying between $0$ and $0.36$. Four tests are considered: the 2-dimensional
test with known drift (Section 3.2) in plain red line, with estimated drift
(Section 3.3) with dotted green line, the multiple testing procedure (Section
4) with known drift in blue dot-dashed line and estimated drift in magenta
dashed line. Three designs are considered: $\Delta=0.01,T=1$ (left),
$\Delta=0.1,T=1$ (middle) and $\Delta=0.1,T=10$ (right).

Results are presented in Figure 2. For 2-dimensional tests, the power is
influenced by the number of observations $n$. When $\Delta=0.1,T=10$, the
powers are almost identical to the case $\Delta=0.01,T=1$. This is in
accordance with the remark following Theorem 2, the separation rate of the two
hypotheses depends only on the number of observations. Unsurprisingly, in the
scenario with very few observations ($\Delta=0.1,T=1$), the hypotheses fail to
separate even when $\sigma^{2}>>\sigma^{2}_{0}$. When the parameters of the
drift are estimated, the power of the test is slightly smaller. This is
expected as the test statistic is build only on half of the sample (the first
half sample being used to estimate the parameters).

The multiple testing gives better results. For both known and estimated drift,
the multiple test gives a perfect separation already at $\sigma=0.14$ (for
settings $\Delta=0.01,T=1$ and $\Delta=0.1,T=10$), while for the two-
dimensional test a separation occurs closer to $\sigma=0.25$.

##  6 Conclusions

We develop various tests of the diffusion coefficients of SDEs. In dimension
one, we propose a test statistic that has an explicit distribution, even when
the (linear) drift parameter is unknown. The tests are of exact level
$\alpha$. We also prove separability conditions to achieve a given power. The
test with an unknown parameter can be applied to a non-parametric drift
estimated by a projection on a functional basis, e.g. on a spline basis. It
can therefore be used to test the diffusion coefficient of a one-dimensional
SDE even when the drift is unknown.

In dimension 2, we propose a test statistic, with a non-explicit distribution.
However, thanks to concentration inequalities, we prove a test procedure with
a non-asymptotic level. When the drift parameter is unknown, the test
procedure is adapted by estimating the parameters on the first half of the
sample and then applying the test statistic using data from the second half of
the sample. We therefore loose power when the parameters are estimated, as the
simulations also illustrate.

We therefore propose an alternative, which is also suitable for a dimension
$d$ greater than 2. This alternative uses a one-dimensional test on each
coordinate and corrects the procedure by a multiple testing approach. This
allows to control the Type I error of the global test. Since the one-
dimensional tests have the exact level even when the linear drift parameters
are estimated, the multiple testing procedure detects the diffusion
coefficient with the exact level, even when the drift is estimated on a
functional basis (spline basis, for example).

Further work will continue considering SDE whose drift depends on the process
itself, as introduced in Section 2.4. A strategy is to approximate the drift
by introducing an extra parameter, as done in this paper. The calibration of
this hyperparameter could be done by cross-validation or test aggregation.
Another strategy would require the development of further concentration
inequalities to prove the upper and lower bounds of the test statistics. Also
the separation rate of the test we provided in this case is strongly
deteriorated when $\Delta>>n^{-1/2}$. We do not know if this is minimax,
meaning that this deterioration is there whatever the test or if it is just
due to our method. This could be a very interesting aspect to search for
future work.

## Acknowledgments

A.S. was supported by MIAI@Grenoble Alpes, (ANR-19-P3IA-0003) and by the LabEx
PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement
d’avenir.

P.R.-B. was supported by the French government, through UCAJedi and 3IA Côte
d’Azur Investissements d’Avenir managed by the National Research Agency
(ANR-15 IDEX-01 and ANR-19-P3IA-0002), directly by the ANR project ChaMaNe
(ANR-19-CE40-0024-02), and by the interdisciplinary Institute for Modeling in
Neuroscience and Cognition (NeuroMod).

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##  Appendix A Appendix

#### Proofs for 1-D SDE

###### Proof of Lemma 2.

We only prove (ii) as (i) is trivial. Note that

$\left({X_{i\Delta}-X_{(i-1)\Delta}}\right)\sim\mathcal{N}\left(\theta\int_{(i-1)\Delta}^{i\Delta}f_{s}ds,\Delta\sigma^{2}\right)$
and the increments are independent. As $\hat{\theta}$ is normally distributed
as a linear combination of normal variables, it is easy to see that
$\mathbb{E}(\hat{\theta})=\theta$ and

$\displaystyle Var\left[\hat{\theta}\right]$ $\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{n}Var\left[{X_{i\Delta}-X_{(i-1)\Delta}}\right]\left({\int_{(i-1)\Delta}^{i\Delta}f_{s}ds}\right)^{2}}{\left(\sum_{i=1}^{n}\left(\int_{(i-1)\Delta}^{i\Delta}f_{s}ds\right)^{2}\right)^{2}}$
$\displaystyle=$
$\displaystyle\frac{\Delta\sigma^{2}}{\sum_{i=1}^{n}\left(\int_{(i-1)\Delta}^{i\Delta}f_{s}ds\right)^{2}}.$

∎

###### Proof of Lemma 3.

Let us introduce for $i=1,\ldots,n$,
$Y_{i}=\frac{X_{i\Delta}-X_{(i-1)\Delta}}{\sqrt{\Delta}}$ and the
corresponding vector $Y=(Y_{1},\ldots,Y_{n})^{t}$, such that
$Y=Z\theta+\varepsilon$, with $\varepsilon\sim\mathcal{N}(0,\sigma^{2}I)$.
Thus $\hat{\theta}=(Z^{t}Z)^{-1}Z^{t}Y$ and
$\hat{\xi}_{i}=Y_{i}-Z_{i}\hat{\theta}$. Then we have $\hat{\xi}=(I-H)Y$ and
$\hat{\xi}$ follows a normal distribution with variance $\sigma^{2}(I-H)$
(because $I-H$ is a projection matrix). Note that $\sum_{i=1}^{n}H_{ii}=1$ and
$rank(H)=1$. Thus $rank(C^{t}C)=n-1$ and we can deduce the last point. ∎

###### Proof of Proposition 5.

First, note that by Theorem 4.1.1. in Girko (1990)
$\dot{s}_{i}\sim\sigma^{2}_{1}\sigma^{2}_{2}\chi^{2}_{i,1}\chi^{2}_{i,2}$,
where $\chi^{2}_{i,k}$ denotes a variable distributed according to a chi-
squared distribution with $k$ degrees of freedom, all variables being
independent in $i$. Here we use the advantage that the covariance matrix of
each vector-column is the same. The distribution of
$\chi^{2}_{i,1}\chi^{2}_{i,2}$ is deduced from Wells et al. (1962) . The PDF
of a product $\chi^{2}_{i,1}\chi^{2}_{i,2}$ is written as follows:

$f(\omega)=\frac{\omega^{-1/4}K_{1/2}(\omega^{1/2})}{\sqrt{2}\Gamma(1)\Gamma(1/2)},$
(35)

where $K_{v}(x)$ is the modified Bessel function of the second kind. Further,
in our specific case,
$K_{1/2}(\omega^{1/2})=\frac{1}{2}\sqrt{2\pi}e^{-\sqrt{\omega}}\omega^{-1/4}$,
and simplify (35), obtaining:

$f(\omega)=\frac{1}{2}\frac{\omega^{-1/4}\sqrt{2\pi}e^{-\sqrt{\omega}}\omega^{-1/4}}{\sqrt{2\pi}}=\frac{1}{2}\omega^{-\frac{1}{2}}e^{-\sqrt{\omega}}.$

We can deduce the expectation:

$\mathds{E}\left[\dot{s}_{i}\right]=\frac{1}{2}\int_{0}^{\infty}\sqrt{\frac{x}{\sigma^{2}_{1}\sigma^{2}_{2}}}e^{-\sqrt{\frac{x}{\sigma^{2}_{1}\sigma^{2}_{2}}}}dx=2\sigma^{2}_{1}\sigma^{2}_{2}.$

For the second moment the computation is similar:

$\mathds{E}\left[\dot{s}^{2}_{i}\right]=\frac{1}{2}\int_{0}^{\infty}\frac{x^{3/2}}{\sqrt{\sigma^{2}_{1}\sigma^{2}_{2}}}e^{-\sqrt{\frac{x}{\sigma^{2}_{1}\sigma^{2}_{2}}}}dx=24\sigma^{4}_{1}\sigma^{4}_{2}.$

Finally, note that

$\mathds{P}\left(\dot{s}_{i}\leq
x\right)=\mathds{P}\left(\sigma^{2}_{1}\sigma^{2}_{2}\chi^{2}_{i,1}\chi^{2}_{i,2}\leq
x\right)=\mathds{P}\left(\chi^{2}_{i,1}\chi^{2}_{i,2}\leq\frac{x}{\sigma^{2}_{1}\sigma^{2}_{2}}\right)=\frac{1}{2}\int_{0}^{\frac{x}{\sigma^{2}_{1}\sigma^{2}_{2}}}\frac{e^{-\sqrt{\omega}}}{\sqrt{\omega}}d\omega.$

Computing the integral, we obtain the result. ∎

#### Proofs for 2-D SDE

Before we proceed, let us establish a state an auxillary result from analysis:

######  Proposition 6.

For any $u>0$, the following holds:

$0\leq\frac{e^{-u}+u-1}{u^{2}}\leq\frac{1}{2}.$

###### Proof.

First, we recall that $e^{-u}\geq 1-u,\forall u$. It proves the first part of
inequality automatically. For the second part, let us consider function
$f(u)=e^{-u}+u-1-\frac{u^{2}}{2}$. Note that $f(0)=0$. Then, it is enough to
prove that $f(u)$ is descending in order for the second part of inequality to
hold. For that, it is enough to look at the sign of $f(u)$:

$f^{\prime}(u)=-e^{-u}+1-u\leq 0,$

by using again that $e^{-u}\geq 1-u,\forall u$. ∎

###### Proof of Theorem 1.

Proof of 1. First, note that

$\mathds{P}\left(\dot{S}-\mathds{E}\left[\dot{S}\right]\leq-t\right)=\mathds{P}\left(\sum_{i=1}^{n/2}\left(\dot{s}_{i}-\mathds{E}\left[\dot{s}_{i}\right]\right)\leq-
nt/2\right).$

For all $\lambda>0$, we have:

$\mathds{P}\left(\sum_{i=1}^{n/2}\left(\dot{s}_{i}-\mathds{E}\left[\dot{s}_{i}\right]\right)\leq-
nt/2\right)=\mathds{P}\left(e^{-\lambda(\sum_{i=1}^{n/2}\dot{s}_{i}-\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}_{i}\right])}\geq
e^{\lambda nt/2}\right).$

Then, by Markov’s inequality, we have:

$\mathds{P}\left(e^{-\lambda(\sum_{i=1}^{n/2}\dot{s}_{i}-\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}_{i}\right])}\geq
e^{\lambda
nt/2}\right)\leq\mathds{E}\left[e^{-\lambda(\sum_{i=1}^{n/2}\dot{s}_{i}-\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}_{i}\right])}\right]e^{-\lambda
nt/2}.$

Since all the $\dot{s}_{i}$ are independent in $i$, we note that

$\mathds{E}\left[e^{-\lambda(\sum_{i=1}^{n/2}\dot{s}_{i}-\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}_{i}\right])}\right]e^{-\lambda
nt/2}=e^{-\lambda
nt/2}\prod_{i=1}^{n/2}\mathds{E}\left[e^{-\lambda(\dot{s}_{i}-\mathds{E}\left[\dot{s}_{i}\right])}\right]\\\
=e^{-\lambda
nt/2+\lambda\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}_{i}\right]}\prod_{i=1}^{n/2}\mathds{E}\left[e^{-\lambda\dot{s}_{i}}\right].$

Now, let us rewrite:

$\mathds{E}\left[e^{-\lambda\dot{s}_{i}}\right]=1-\lambda\mathds{E}[\dot{s}_{i}]+\lambda^{2}\mathds{E}\left[\dot{s}^{2}_{i}\frac{e^{-\lambda\dot{s}_{i}}+\lambda\dot{s}_{i}-1}{\left(\lambda\dot{s}_{i}\right)^{2}}\right].$

Now, let us define the following function:

$h(u):=\frac{e^{-u}+u-1}{u^{2}},$

which is decreasing for any $u>0$ (see Proposition 6). For $\lambda>0$ and as
$\dot{s}_{i}\geq 0$, we have:

$h\left(\lambda\dot{s}_{i}\right)\leq{\lim_{u\rightarrow
0}h(u)\leq\frac{1}{2}}.$

Then,
$\mathds{E}\left[\dot{s}^{2}_{i}h\left(\lambda\dot{s}_{i}\right)\right]\leq\mathds{E}\left[\dot{s}^{2}_{i}\right]$
and $\mathds{E}\left[e^{-\lambda\dot{s}_{i}}\right]\leq
1-\lambda\mathds{E}[\dot{s}_{i}]+{\frac{\lambda^{2}}{2}}\mathds{E}\left[\dot{s}^{2}_{i}\right]$.
Finally, we obtain:

$\mathds{P}\left(e^{-\lambda(\dot{S}-\mathds{E}\left[\dot{S}\right])}\geq
e^{\lambda t}\right)\leq e^{-\lambda
nt/2+\lambda\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}_{i}\right]}\prod_{i=1}^{n/2}\left(1-\lambda\mathds{E}[\dot{s}_{i}]+{\frac{\lambda^{2}}{2}}\mathds{E}\left[\dot{s}^{2}_{i}\right]\right)\\\
\leq\exp\left(-\lambda
nt/2+\lambda\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}_{i}\right]-\lambda\sum_{i=1}^{n/2}\mathds{E}[\dot{s}_{i}]+{\frac{\lambda^{2}}{2}}\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}^{2}_{i}\right]\right)\\\
=\exp\left(-\lambda
nt/2+{\frac{\lambda^{2}}{2}}\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}^{2}_{i}\right]\right).$
(36)

Then we maximize the last expression with respect to $\lambda$. The maximum is
obtained for
$\hat{\lambda}=\frac{nt}{{2}\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}^{2}_{i}\right]}.$
Using $\mathds{E}\left[\dot{s}^{2}_{i}\right]=24\sigma^{4}_{1}\sigma^{4}_{2}$
(Proposition 5), we obtain the bound

$\exp\left(-\hat{\lambda}nt/2+{\frac{\hat{\lambda}^{2}}{2}}\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}^{2}_{i}\right]\right)=\exp\left(-\frac{(nt/2)^{2}}{{2}\sum_{i=1}^{n/2}\mathds{E}\left[\dot{s}^{2}_{i}\right]}\right)=\exp\left(-\frac{(nt/2)^{2}}{n{24}\sigma^{4}_{1}\sigma^{4}_{2}}\right).$

It gives the result.

Proof of 2. By Chebyshev’s inequality we have

$\mathds{P}\left(\left|\sum_{i=1}^{n/2}\dot{s}_{i}-\mathds{E}\left[\sum_{i=1}^{n/2}\dot{s}_{i}\right]\right|\geq
nt/2\right)\leq\frac{Var\left[\sum_{i=1}^{n/2}\dot{s}_{i}\right]}{\left(nt/2\right)^{2}}.$

It implies:

$\mathds{P}\left(\sum_{i=1}^{n/2}\dot{s}_{i}-\mathds{E}\left[\sum_{i=1}^{n/2}\dot{s}_{i}\right]\geq
nt/2\right)\\\ \\\
\leq\frac{Var\left[\sum_{i=1}^{n/2}\dot{s}_{i}\right]}{\left(nt/2\right)^{2}}-\mathds{P}\left(\sum_{i=1}^{n/2}\dot{s}_{i}-\mathds{E}\left[\sum_{i=1}^{n/2}\dot{s}_{i}\right]\leq-
nt/2\right).$

Note that
$\mathds{P}\left(\sum_{i=1}^{n/2}\dot{s}_{i}-\mathds{E}\left[\sum_{i=1}^{n/2}\dot{s}_{i}\right]\leq-
nt/2\right)$ is evaluated in (26) and is decaying exponentially fast as $t$
grows. In comparison, the first term is decaying at a much slower rate. Thus,
the principal term which controls the upper bound is given by
$\frac{Var\left[\sum_{i=1}^{n/2}\dot{s}_{i}\right]}{\left(nt/2\right)^{2}}$.
Finally, the following result is obtained:

$\mathds{P}\left(\dot{S}-\mathds{E}\left[\dot{S}\right]\geq
t\right)\leq\frac{10n\sigma^{4}_{1}\sigma^{4}_{2}}{\left(nt/2\right)^{2}}=\frac{40}{nt^{2}}\sigma^{4}_{1}\sigma^{4}_{2}.$

∎

#### Sharp estimate on classical quantiles

######  Lemma 7.

For any $\alpha\in(0,0.5]$, we always have that

$q_{\mathcal{N},1-\alpha}\leq\sqrt{2\log(1/\alpha)}$

and that

$q_{\chi_{n}^{2}(\lambda),1-\alpha}\leq(\sqrt{n}+\sqrt{2\log(1/\alpha)})^{2}+2\sqrt{2\log(1/\alpha)}\lambda^{1/2}+\lambda.$

With $\alpha=0.5$, we get the slightly better bound

$q_{\chi_{n}^{2}(\lambda),0.5}\leq(\sqrt{n}+\sqrt{2\log(2)})^{2}+\lambda.$

Moreover if $\alpha\leq 1/\sqrt{2\pi}\simeq 0.39,$ we also have that

$q_{\mathcal{N},1-\alpha}\geq\sqrt{\log(1/\alpha)}$

and

$q_{\chi_{n}^{2}(\lambda),1-\alpha}\geq
n-1+2\sqrt{\log(1/\alpha)}\lambda^{1/2}+\lambda+\log(1/\alpha).$

For $\beta<0.5$, we have the following bound

$q_{\chi_{n}^{2}(\lambda),\beta}\geq
n+\lambda-\sqrt{\frac{2(n+2\lambda)}{\beta}}.$

###### Proof.

Gordon (1941) establishes that if $\Phi(x)$ is the c.d.f. of
$\mathcal{N}(0,1)$ then for all positive $x$,

$\frac{e^{-x^{2}/2}}{\sqrt{2\pi}\left(x+x^{-1}\right)}\leq\Phi(x)\leq\frac{e^{-x^{2}/2}}{\sqrt{2\pi}x}.$

Since the function $f(x)=\frac{e^{-x^{2}/2}}{x}$ is strictly decreasing, we
can obtain an upper bound on the quantile by finding a $z_{\alpha}$ such that
$f(z_{\alpha})\leq\sqrt{2\pi}\alpha$ for $\alpha<0.5$. Choosing
$x_{\alpha}=\sqrt{2\log(1/\alpha)}$, we get
$f(x_{\alpha})=\frac{\alpha}{\sqrt{2\log(1/\alpha)}}\leq\sqrt{2\pi}\alpha$
since $\alpha<0.5$.

For the lower bound of the Gaussian quantile, note that for all $v>0$,
$v-\log(v)\geq 1$, so $x_{t}=\sqrt{2\log(1/t)-2\log\log(1/t)}$ is well defined
for all $t\leq 1$ and satisfies $x_{t}\geq\sqrt{2}$, for all $t\leq 1$. So
Gordon’s lower bound implies that

$\Phi(x_{t})\geq\frac{1}{2\sqrt{2\pi}}f(x_{t})=\frac{t\log(1/t)}{2\sqrt{2\pi}\sqrt{2\log(1/t)-2\log\log(1/t)}}.$

But $v^{2}/4\geq 2v-2\log(v)$ holds for all positive $v$, so

$\log(1/t)^{2}/4\geq 2\log(1/t)-2\log\log(1/t)$

and $\Phi(x_{t})\geq\frac{t}{\sqrt{2\pi}}.$

By taking $t=\sqrt{2\pi}\alpha$, we get therefore that lower bound

$q_{\mathcal{N},1-\alpha}\geq\sqrt{2\log(\sqrt{2\pi}/\alpha)-2\log\log(\sqrt{2\pi}/\alpha)}.$

It holds for all $\alpha\leq 1/\sqrt{2\pi}$.

But by studying the function we can see that

$2\log(\sqrt{2\pi}/\alpha)-2\log\log(\sqrt{2\pi}/\alpha)\geq\log(1/\alpha).$

For the chi-square distribution with central parameter $\lambda$, we use the
results by Robert (1990), which states that

$q_{\chi_{n}^{2}(\lambda),1-\alpha}\leq
q_{\chi_{n}^{2}(0),1-\alpha}+2q_{\mathcal{N},1-\alpha}\lambda^{1/2}+\lambda.$

We use Gaussian concentration of measure (Boucheron et al., 2013) to derive
that, if $X$ is a standard Gaussian vector of dimension $n$ and $\|X\|$
designs its euclidean norm, then for all positive $x$

$\mathbb{P}(\|X\|\geq\mathbb{E}(\|X\|)+x)\leq e^{-x^{2}/2}.$

Since $\mathbb{E}(\|X\|)\leq\sqrt{n}$, we have that

$\mathbb{P}\left(\|X\|^{2}\geq(\sqrt{n}+\sqrt{2\log(1/\alpha)})^{2}\right)\leq\alpha.$

Combined with the bound on the Gaussian quantiles, it give us the upper bound.
For $\alpha=0.5,$ note that $q_{\mathcal{N},1-\alpha}=0$.

For the lower bound, Robert (1990) also proves that

$q_{\chi_{n}^{2}(\lambda),1-\alpha}\geq
n-1+q_{\mathcal{N},1-\alpha}^{2}+2q_{\mathcal{N},1-\alpha}\lambda^{1}/2+\lambda.$

Combined with the lower bound on Gaussian quantiles, we get the corresponding
lower bound.

For the rougher bound for small $\beta$, note that if
$Z\sim\chi^{2}_{n}(\lambda)$ then

$\mathbb{E}(Z)=n+\lambda\quad\mbox{and}\quad Var(Z)=2(n+2\lambda).$

So by Bienaymé Tchebicheff inequality, we obtain for all positive $x$

$\mathbb{P}(Z\leq\mathbb{E}(Z)-x)\leq\frac{Var(Z)}{x^{2}}.$

So by choosing $x=\sqrt{\frac{2(n+2\lambda)}{\beta}}$, we obtain the desired
lower bound. ∎