# Nonparametric Bayesian posterior contraction rates for scalar diffusions
with high-frequency data

Kweku<EMAIL_ADDRESS>[ Statistical Laboratory,
Department of Pure Mathematics and Mathematical Statistics, University of
Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK.  University of Cambridge

###### Abstract

We consider inference in the scalar diffusion model
$\mathop{}\\!\mathrm{d}X_{t}=b(X_{t})\mathop{}\\!\mathrm{d}t+\sigma(X_{t})\mathop{}\\!\mathrm{d}W_{t}$
with discrete data $(X_{j\Delta_{n}})_{0\leq j\leq n}$,
$n\to\infty,~{}\Delta_{n}\to 0$ and periodic coefficients. For $\sigma$ given,
we prove a general theorem detailing conditions under which Bayesian
posteriors will contract in $L^{2}$–distance around the true drift function
$b_{0}$ at the frequentist minimax rate (up to logarithmic factors) over Besov
smoothness classes. We exhibit natural nonparametric priors which satisfy our
conditions. Our results show that the Bayesian method adapts both to an
unknown sampling regime and to unknown smoothness.

adaptive estimation,

Bayesian nonparametrics,

concentration inequalities,

diffusion processes,

discrete time observations,

drift function,

###### keywords:

##  1 Introduction

Consider a scalar diffusion process $(X_{t})_{t\geq 0}$ starting at some
$X_{0}$ and evolving according to the stochastic differential equation

$\mathop{}\\!\mathrm{d}X_{t}=b(X_{t})\mathop{}\\!\mathrm{d}t+\sigma(X_{t})\mathop{}\\!\mathrm{d}W_{t},$

where $W_{t}$ is a standard Brownian motion. It is of considerable interest to
estimate the parameters $b$ and $\sigma$, which are arbitrary functions (until
we place further assumptions on their form), so that the model is naturally
_nonparametric_. As we will explain in Section 2, the problems of estimating
$\sigma$ and $b$ can essentially be decoupled in the setting to be considered
here, so in this paper we consider estimation of the drift function $b$ when
the diffusion coefficient $\sigma$ is assumed to be given.

It is realistic to assume that we do not observe the full trajectory
$(X_{t})_{t\leq T}$ but rather the process sampled at discrete time intervals
$(X_{k\Delta})_{k\leq n}$. The estimation problem for $b$ and $\sigma$ has
been studied extensively and minimax rates have been attained in two sampling
frameworks: _low-frequency_ , where $\Delta$ is fixed and asymptotics are
taken as $n\to\infty$ (see Gobet–Hoffmann–Reiss [16]), and _high-frequency_ ,
where asymptotics are taken as $n\to\infty$ and $\Delta=\Delta_{n}\to 0$,
typically assuming also that $n\Delta^{2}\to 0$ and $n\Delta\to\infty$ (see
Hoffmann [18], Comte et al. [8]). See also eg. [9], [17], [26], [32] for more
papers addressing nonparametric estimation for diffusions.

For typical frequentist methods, one must know which sampling regime the data
is drawn from. In particular, the low-frequency estimator from [16] is
consistent in the high-frequency setting but numerical simulations suggest it
does not attain the minimax rate (see the discussion in Chorowski [7]), while
the high-frequency estimators of [18] and [8] are not even consistent with
low-frequency data. The only previous result known to the author regarding
adaptation to the sampling regime in the nonparametric setting is found in
[7], where Chorowski is able to estimate the diffusion coefficient $\sigma$
but not the drift, and obtains the minimax rate when $\sigma$ has 1 derivative
but not for smoother diffusion coefficients.

For this paper we consider estimation of the parameters in a diffusion model
from a nonparametric Bayesian perspective. Bayesian methods for diffusion
estimation can be implemented in practice (eg. see Papaspiliopoulos et al.
[24]). For Bayesian estimation, the statistician need only specify a prior,
and for estimating diffusions from discrete samples the prior need not
reference the sampling regime, so Bayesian methodology provides a natural
candidate for a unified approach to the high- and low-frequency settings. Our
results imply that Bayesian methods can adapt both to the sampling regime and
also to unknown smoothness of the drift function (see the remarks after
Proposition 4 and Proposition 2 respectively for details). These results are
proved under the frequentist assumption of a fixed true parameter, so this
paper belongs to the field of _frequentist analysis of Bayesian procedures_.
See, for example, Ghosal & van der Vaart [12] for an introduction to this
field.

It has previously been shown that in the low-frequency setting we have a
_posterior contraction rate_ , guaranteeing that posteriors corresponding to
reasonable priors concentrate their mass on neighbourhoods of the true
parameter shrinking at the fastest possible rate (up to log factors) – see
Nickl & Söhl [23]. To complete a proof that such posteriors contract at a rate
adapting to the sampling regime, it remains to prove a corresponding
contraction rate in the high-frequency setting. This forms the key
contribution of the current paper: we prove that a large class of “reasonable”
priors will exhibit posterior contraction at the optimal rate (up to log
factors) in $L^{2}$–distance. This in turn guarantees that point estimators
based on the posterior will achieve the frequentist minimax optimal rate (see
the remark after Theorem 1) in both high- and low-frequency regimes.

The broad structure of the proof is inspired by that in [23]: we use the
testing approach of Ghosal–Ghosh–van der Vaart [10], coupled with the insight
of Giné and Nickl [14] that one may prove the existence of the required tests
by finding an estimator with good enough concentration around the true
parameter. The main ingredients here are:

  * •

A concentration inequality for a (frequentist) estimator, from which we
construct tests of the true $b_{0}$ against a set of suitable (sufficiently
separated) alternatives. See Section 4.

  * •

A small ball result, to relate the $L^{2}$–distance to the information-
theoretic Kullback–Leibler “distance”. See Section 5.

Though the structure reflects that of [23] the details are very different.
Estimators for the low-frequency setting are typically based on the mixing
properties of $(X_{k\Delta})$ viewed as a Markov chain and the spectral
structure of its transition matrix (see Gobet–Hoffmann–Reiss [16]) and fail to
take full advantage of the local information one sees when $\Delta\to 0$. Here
we instead use an estimator introduced in Comte et al. [8] which uses the
assumption $\Delta\to 0$ to view estimation of $b$ as a regression problem. To
prove this estimator concentrates depends on a key insight of this paper: the
Markov chain concentration results used in the low-frequency setting (which
give _worse_ bounds as $\Delta\to 0$) must be supplemented by Hölder type
continuity results, which crucially rely on the assumption $\Delta\to 0$. We
further supplement by martingale concentration results.

Similarly, the small ball result in the low-frequency setting depends on
Markov chain mixing. Here, we instead adapt the approach of van der Meulen &
van Zanten [33]. They demonstrate that the Kullback–Leibler divergence in the
discrete setting can be controlled by the corresponding divergence in the
continuous data model; a key new result of the current paper is that in the
high-frequency setting this control extends to give a bound on the variance of
the log likelihood ratio.

As described above, a key attraction of the Bayesian method is that it allows
the statistician to approach the low- and high-frequency regimes in a unified
way. Another attraction is that it naturally suggests uncertainty
quantification via posterior credible sets. The contraction rate theorems
proved in this paper and [23] are not by themselves enough to prove that
credible sets behave as advertised. For that one may aim for a nonparametric
Bernstein–von Mises result – see for example Castillo & Nickl [5, 6]. The
posterior contraction rate proved here constitutes a key first step towards a
proof of a Bernstein–von Mises result for the high-frequency sampled diffusion
model, since it allows one to localise the posterior around the true
parameter, as in the proofs in Nickl [22] for a non-linear inverse problem
comparable to the problem here.

##  2 Framework and assumptions

The notation introduced throughout the paper is gathered in Appendix C.

We work with a scalar diffusion process $(X_{t})_{t\geq 0}$ starting at some
$X_{0}$ and evolving according to the stochastic differential equation

$\mathop{}\\!\mathrm{d}X_{t}=b(X_{t})\mathop{}\\!\mathrm{d}t+\sigma(X_{t})\mathop{}\\!\mathrm{d}W_{t},$
(1)

for $W_{t}$ a standard Brownian motion. The parameters $b$ and $\sigma$ are
assumed to be 1–periodic and we also assume the following.

######  Assumption 1.

$\sigma\in C_{\text{per}}^{2}([0,1])$ is given. Continuity guarantees the
existence of an upper bound $\sigma_{U}<\infty$ and we further assume the
existence of a lower bound $\sigma_{L}>0$ so that
$\sigma_{L}\leq\sigma(x)\leq\sigma_{U}$ for all $x\in[0,1]$. Here
$C_{\text{per}}^{2}([0,1])$ denotes $C^{2}([0,1])$ functions with periodic
boundary conditions (i.e. $\sigma(0)=\sigma(1)$,
$\sigma^{\prime}(0)=\sigma^{\prime}(1)$ and
$\sigma^{\prime\prime}(0)=\sigma^{\prime\prime}(1)$).

######  Assumption 2.

$b$ is continuously differentiable with given norm bound. Precisely, we assume
$b\in\Theta$, where, for some arbitrary but known constant $K_{0},$

$\Theta=\Theta(K_{0})=\\{f\in C_{\text{per}}^{1}([0,1]):~{}\lVert
f\rVert_{C_{\text{per}}^{1}}=\lVert f\rVert_{\infty}+\lVert
f^{\prime}\rVert_{\infty}\leq K_{0}\\}.$

($\lVert\cdot\rVert_{\infty}$ denotes the supremum norm, $\lVert
f\rVert_{\infty}=\sup_{x\in[0,1]}\lvert f(x)\rvert$.) Note in particular that
$K_{0}$ upper bounds $\lVert b\rVert_{\infty}$ and that $b$ is Lipschitz
continuous with constant at most $K_{0}$.

$\Theta$ is the maximal set over which we prove contraction, and we will in
general make the stronger assumption that in fact $b\in\Theta_{s}(A_{0})$,
where

$\Theta_{s}(A_{0}):=\\{f\in\Theta:\lVert f\rVert_{B_{2,\infty}^{s}}\leq
A_{0}<\infty\\},\quad A_{0}>0,~{}s\geq 1$

with $B_{p,q}^{s}$ denoting a periodic Besov space and
$\lVert\cdot\rVert_{B_{p,q}^{s}}$ denoting the associated norm: see Section
2.1 for a definition of the periodic Besov spaces we use (readers unfamiliar
with Besov spaces may substitute the $L^{2}$–Sobolev space
$H^{s}=B_{2,2}^{s}\subseteq B_{2,\infty}^{s}$ for $B_{2,\infty}^{s}$ and only
mildly weaken the results). We generally assume the regularity index $s$ is
unknown. Our results will therefore aim to be _adaptive_ , at least in the
smoothness index (to be fully adaptive we would need to adapt to $K_{0}$
also).

Under Assumptions 2 and 1, there is a unique strong solution to 1 (see, for
example, Bass [2] Theorem 24.3). Moreover, this solution is also weakly unique
(= unique in law) and satisfies the Markov property (see [2] Proposition 25.2
and Theorem 39.2). We denote by $P_{b}^{(x)}$ the law (on the cylindrical
$\sigma$–algebra of $C([0,\infty])$) of the unique solution of 1 started from
$X_{0}=x$.

We consider “high-frequency data” $(X_{k\Delta_{n}})_{k=0}^{n}$ sampled from
this solution, where asymptotics are taken as $n\to\infty$, with
$\Delta_{n}\to 0$ and $n\Delta_{n}\to\infty$. We will suppress the subscript
and simply write $\Delta$ for $\Delta_{n}$. Throughout we will write
$X^{(n)}=(X_{0},\dots,X_{n\Delta})$ as shorthand for our data and similarly we
write $x^{(n)}=(x_{0},\dots,x_{n\Delta})$. We will denote by $\mathcal{I}$ the
set $\\{K_{0},\sigma_{L},\sigma_{U}\\}$ so that, for example, $C(\mathcal{I})$
will be a constant depending on these parameters.

Beyond guaranteeing existence and uniqueness of a solution, our assumptions
also guarantee the existence of transition densities for the discretely
sampled process (see Gihman & Skorohod [13] Theorem 13.2 for an explicit
formula for the transition densities). Morever, there also exists an invariant
distribution $\mu_{b}$, with density $\pi_{b}$, for the periodised process
$\dot{X}=X\mod 1$. Defining
$I_{b}(x)=\int_{0}^{x}\frac{2b}{\sigma^{2}}(y)\mathop{}\\!\mathrm{d}y$ for
$x\in[0,1],$ the density is

$\displaystyle\pi_{b}(x)=\frac{e^{I_{b}(x)}}{H_{b}\sigma^{2}(x)}\Big{(}e^{I_{b}(1)}\int_{x}^{1}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y+\int_{0}^{x}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y\Big{)},\qquad
x\in[0,1],$ $\displaystyle
H_{b}=\int_{0}^{1}\frac{e^{I_{b}(x)}}{\sigma^{2}(x)}\Big{(}e^{I_{b}(1)}\int_{x}^{1}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y+\int_{0}^{x}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y\Big{)}\mathop{}\\!\mathrm{d}x,$

(see Bhattacharya et al. [3], equations 2.15 to 2.17; note we have chosen a
different normalisation constant so the expressions appear slightly
different).

Observe that $\pi_{b}$ is bounded uniformly away from zero and infinity, i.e.
there exist constants $0<\pi_{L},\pi_{U}<\infty$ depending only on
$\mathcal{I}$ so that for any $b\in\Theta$ and any $x\in[0,1]$ we have
$\pi_{L}\leq\pi_{b}(x)\leq\pi_{U}$. Precisely, we see that
$\sigma_{U}^{-2}e^{-6K_{0}\sigma_{L}^{-2}}\leq
H_{b}\leq\sigma_{L}^{-2}e^{6K_{0}\sigma_{L}^{-2}},$ and we deduce we can take
$\pi_{L}=\pi_{U}^{-1}=\sigma_{L}^{2}\sigma_{U}^{-2}e^{-12K_{0}\sigma_{L}^{-2}}.$

We assume that $X_{0}\in[0,1)$ and that $X_{0}=\dot{X}_{0}$ follows this
invariant distribution.

######  Assumption 3.

$X_{0}\sim\mu_{b}$.

We will write $P_{b}$ for the law of the full process $X$ under Assumptions 2,
1 and 3, and we will write $E_{b}$ for expectation according to this law. Note
$\mu_{b}$ is not invariant for $P_{b}$, but nevertheless
$E_{b}(f(X_{t}))=E_{b}(f(X_{0}))$ for any 1–periodic function $f$ (eg. see the
proof of Theorem 6). Since we will be estimating the 1–periodic function $b$,
the assumption that $X_{0}\in[0,1)$ is unimportant.

Finally, we need to assume that $\Delta\to 0$ at a fast enough rate.

######  Assumption 4.

$n\Delta^{2}\log(1/\Delta)\leq L_{0}$ for some (unknown) constant $L_{0}$.
Since we already assume $n\Delta\to\infty$, this new assumption is equivalent
to $n\Delta^{2}\log(n)\leq L_{0}^{\prime}$ for some constant $L_{0}^{\prime}$.

Throughout we make the frequentist assumption that the data is generated
according to some fixed true parameter denoted $b_{0}$. We use $\mu_{0}$ as
shorthand for $\mu_{b_{0}}$, and similarly for $\pi_{0}$ and so on. Where
context allows, we write $\mu$ for $\mu_{b}$ with a generic drift $b$.

######  Remarks (Comments on assumptions).

_Periodicity assumption._ We assume $b$ and $\sigma$ are periodic so that we
need only estimate $b$ on $[0,1]$. One could alternatively assume $b$
satisfies some growth condition ensuring recurrence, then estimate the
restriction of $b$ to $[0,1]$, as in Comte et al. [8] and van der Meulen & van
Zanten [33]. The proofs in this paper work in this alternative framework with
minor technical changes, provided one assumes the behaviour of $b$ outside
$[0,1]$ can be exactly matched by a draw from the prior.

_Assuming that $\sigma\in C^{2}_{\text{per}}$ is given._ If we observe
continuous data $(X_{t})_{t\leq T}$ then $\sigma$ is known exactly (at least
at any point visited by the process) via the expression for the quadratic
variation $\langle
X\rangle_{t}=\int_{0}^{t}\sigma^{2}(X_{s})\mathop{}\\!\mathrm{d}s$. With high-
frequency data we cannot perfectly reconstruct the diffusion coefficient from
the data, but we can estimate it at a much faster rate than the drift. When
$b$ and $\sigma$ are both assumed unknown, if $b$ is $s$-smooth and $\sigma$
is $s^{\prime}$-smooth, the minimax errors for $b$ and $\sigma$ respectively
scale as $(n\Delta)^{-s/(1+2s)}$ and $n^{-s^{\prime}/(1+2s^{\prime})}$, as can
be shown by slightly adapting Theorems 5 and 6 from Hoffmann [18] so that they
apply in the periodic setting we use here. Since we assume that
$n\Delta^{2}\to 0$, it follows that $n\Delta\leq n^{1/2}$ for large $n$, hence
we can estimate $\sigma$ at a faster rate than $b$ regardless of their
relative smoothnesses.

Further, note that the problems of estimating $b$ and $\sigma$ in the high-
frequency setting are essentially independent. For example, the smoothness of
$\sigma$ does not affect the rate for estimating $b$, and vice-versa – see
[18]. We are therefore not substantially simplifying the problem of estimating
$b$ through the assumption that $\sigma$ is given.

The assumption that $\sigma^{2}$ is twice differentiable is a typical minimal
assumption to ensure transition densities exist.

_Assuming a known bound on $\lVert b\rVert_{C_{\text{per}}^{1}}$._ The
assumption that $b$ has one derivative is a typical minimal assumption to
ensure that the diffusion equation 1 has a strong solution and that this
solution has an invariant density. The assumption of a _known_ bound for the
$C_{\text{per}}^{1}$–norm of the function is undesirable, but needed for the
proofs, in particular to ensure the existence of a uniform lower bound
$\pi_{L}$ on the invariant densities. This lower bound is essential for the
Markov chain mixing results as its reciprocal controls the mixing time in
Theorem 6. It is plausible that needing this assumption is inherent to the
problem rather than an artefact of the proofs: possible methods to bypass the
Markov chain mixing arguments, such as the martingale approach of [8] Lemma 1,
also rely on such a uniform lower bound. One could nonetheless hope that our
results apply to an unbounded prior placing sufficient weight on
$\Theta(K_{n})$ for some slowly growing sequence $K_{n}$, but the lower bound
$\pi_{L}$ scales unfavourably as $e^{-K_{n}}$, which rules out this approach.

These boundedness assumptions in principle exclude Gaussian priors, which are
computationally attractive. In practice, one could choose a very large value
for $K_{0}$ and approximate Gaussian priors arbitrarily well using truncated
Gaussian priors.

_Assuming $X_{0}\sim\mu_{b}$._ It can be shown (see the proof of Theorem 6)
that the law of $\dot{X}_{t}$ converges to $\mu_{b}$ at exponential rate from
any starting distribution, so assuming $X_{0}\sim\mu_{b}$ is not restrictive
(as mentioned, our fixing $X_{0}\in[0,1)$ is arbitrary but unimportant).

_Assuming $n\Delta^{2}\log(1/\Delta)\leq L_{0}$._ It is typical in the high-
frequency setting to assume $n\Delta^{2}\to 0$ (indeed the minimax rates in
[18] are only proved under this assumption) but for technical reasons in the
concentration section (Section 4.2) we need the above.

###  Spaces of approximation

We will throughout depend on a family $\\{S_{m}:m\in\mathbb{N}\cup\\{0\\}\\}$
of function spaces. For our purposes we will take the $S_{m}$ to be periodised
Meyer-type wavelet spaces

$S_{m}=\operatorname{span}(\\{\psi_{lk}:0\leq k<2^{l},0\leq
l<m\\}\cup\\{1\\}).$

We will denote $\psi_{-1,0}\equiv 1$ for convenience. Denote by
$\langle\cdot,\cdot\rangle$ the $L^{2}([0,1])$ inner product and by
$\lVert\cdot\rVert_{2}$ the $L^{2}$–norm, i.e. $\langle
f,g\rangle=\int_{0}^{1}f(x)g(x)\mathop{}\\!\mathrm{d}x$ and $\lVert
f\rVert_{2}=\langle f,f\rangle^{1/2}$ for $f,g\in L^{2}([0,1]).$ One
definition of the (periodic) Besov norm $\lVert f\rVert_{B_{2,\infty}^{s}}$
is, for $f_{lk}:=\langle f,\psi_{lk}\rangle$,

$\lVert f\rVert_{B_{2,\infty}^{s}}=\lvert f_{-1,0}\rvert+\sup_{l\geq
0}2^{ls}\bigg{(}\sum_{k=0}^{2^{l}-1}f_{lk}^{2}\bigg{)}^{1/2},$ (2)

with $B_{2,\infty}^{s}$ defined as those periodic $f\in L^{2}([0,1])$ for
which this norm is finite. See Giné & Nickl [15] Sections 4.2.3 and 4.3.4 for
a construction of periodised Meyer-type wavelets and a proof that this wavelet
norm characterisation agrees with other possible definitions of the desired
Besov space.

Note that the orthonormality of the wavelet basis means $\lVert
f\rVert_{2}^{2}=\sum_{l,k}f_{lk}^{2}$. Thus it follows from the above
definition of the Besov norm that for any $b\in B_{2,\infty}^{s}([0,1])$ we
have

$\lVert\pi_{m}b-b\rVert_{2}\leq K\lVert b\rVert_{B_{2,\infty}^{s}}2^{-ms},$
(3)

for all $m$, for some constant $K=K(s)$, where $\pi_{m}$ is the
$L^{2}$–orthogonal projection onto $S_{m}$.

######  Remarks.

_Uniform sup-norm convergence of the wavelet series._ The wavelet projections
$\pi_{m}b$ converge to $b$ in supremum norm for any $b\in\Theta$, uniformly
across $b\in\Theta$. That is,

$\sup_{b\in\Theta}\lVert\pi_{m}b-b\rVert_{\infty}\to 0\quad\text{as}\quad
m\to\infty.$ (4)

This follows from Proposition 4.3.24 in [15] since $K_{0}$ uniformly bounds
$\lVert b\rVert_{C_{\text{per}}^{1}}$ for $b\in\Theta$.

_Boundary regularity._ Functions in the periodic Besov space here denoted
$B_{2,\infty}^{s}$ are $s$ regular at the boundary, in the sense that their
weak derivatives of order $s$ are 1–periodic.

_Alternative approximation spaces._ The key property we need for our
approximation spaces is that 3 and 4 hold. Of these, only the first is needed
of our spaces for our main contraction result Theorem 1. A corresponding
inequality holds for many other function spaces if we replace $2^{m}$ by
$D_{m}=\dim(S_{m})$; for example, for $S_{m}$ the set of trigonometric
polynomials of degree at most $m$, or (provided $s\leq s_{\max}$ for some
given $s_{\max}\in\mathbb{R}$) for $S_{m}$ generated by periodised Daubechies
wavelets. Priors built using these other spaces will achieve the same
posterior contraction rate.

##  3 Main contraction theorem

Let $\Pi$ be a (prior) probability distribution on some $\sigma$–algebra
$\mathcal{S}$ of subsets of $\Theta$. Given $b\sim\Pi$ assume that
$(X_{t}:t\geq 0)$ follows the law $P_{b}$ as described in Section 2. Write
$p_{b}(\Delta,x,y)$ for the transition densities

$p_{b}(\Delta,x,y)\mathop{}\\!\mathrm{d}y=P_{b}(X_{\Delta}\in\mathop{}\\!\mathrm{d}y\mid
X_{0}=x),$

and recall we use $p_{0}$ as shorthand for $p_{b_{0}}$. Assume that the
mapping $(b,\Delta,x,y)\mapsto p_{b}(\Delta,x,y)$ is jointly measurable with
respect to the $\sigma$–algebras $\mathcal{S}$ and $\mathcal{B}_{\mathbb{R}}$,
where $\mathcal{B}_{\mathbb{R}}$ is the Borel $\sigma$–algebra on
$\mathbb{R}$. Then it can be shown by standard arguments that the Bayesian
posterior distribution given the data is

$b\mid
X^{(n)}\sim\frac{\pi_{b}(X_{0})\prod_{i=1}^{n}p_{b}(\Delta,X_{(i-1)\Delta},X_{i\Delta})\mathop{}\\!\mathrm{d}\Pi(b)}{\int_{\Theta}\pi_{b}(X_{0})\prod_{i=1}^{n}p_{b}(\Delta,X_{(i-1)\Delta},X_{i\Delta})\mathop{}\\!\mathrm{d}\Pi(b)}\equiv\frac{p_{b}^{(n)}(X^{(n)})\mathop{}\\!\mathrm{d}\Pi(b)}{\int_{\Theta}p_{b}^{(n)}(X^{(n)})\mathop{}\\!\mathrm{d}\Pi(b)},$

where we introduce the shorthand
$p_{b}^{(n)}(x^{(n)})=\pi_{b}(x_{0})\prod_{i=1}^{n}p_{b}(\Delta,x_{(i-1)\Delta},x_{i\Delta})$
for the joint probability density of the data $(X_{0},\dots,X_{n\Delta})$.

A main result of this paper is the following. 1A is designed to apply to
adaptive sieve priors, while 1B is designed for use when the smoothness of the
parameter $b$ is known. See Section 3.1 for explicit examples of these results
in use and see Section 6 for the proof.

######  Theorem 1.

Consider data $X^{(n)}=(X_{k\Delta})_{0\leq k\leq n}$ sampled from a solution
$X$ to 1 under Assumptions 2, 1, 4 and 3. Let the true parameter be $b_{0}$.
Assume the appropriate sets below are measurable with respect to the
$\sigma$–algebra $\mathcal{S}$.

  1. A.

Let $\Pi$ be a sieve prior on $\Theta$, i.e. let
$\Pi=\sum_{m=1}^{\infty}h(m)\Pi_{m}$, where $\Pi_{m}(S_{m}\cap\Theta)=1$, for
$S_{m}$ a periodic Meyer-type wavelet space of resolution $m$ as described in
Section 2.1, and $h$ some probability mass function on $\mathbb{N}$. Suppose
we have, for all $\varepsilon>0$ and $m\in\mathbb{N}$, and for some constants
$\zeta,\beta_{1},\beta_{2},B_{1},B_{2}>0,$

    1. (i)

$B_{1}e^{-\beta_{1}D_{m}}\leq h(m)\leq B_{2}e^{-\beta_{2}D_{m}}$,

    2. (ii)

$\Pi_{m}(\\{b\in S_{m}:\lVert
b-\pi_{m}b_{0}\rVert_{2}\leq\varepsilon\\})\geq(\varepsilon\zeta)^{D_{m}}$,

where $\pi_{m}$ is the $L^{2}$–orthogonal projection onto $S_{m}$ and
$D_{m}=\dim(S_{m})=2^{m}$. Then for some constant
$M=M(A_{0},s,\mathcal{I},L_{0},\beta_{1},\beta_{2},B_{1},B_{2},\zeta)$ we
have, for any $b_{0}\in\Theta_{s}(A_{0})$,

$\Pi\Big{(}big\\{b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq
M(n\Delta)^{-s/(1+2s)}\log(n\Delta)^{1/2}big\\}\mid X^{(n)}\Big{)}\to 1$

in probability under the law $P_{b_{0}}$ of $X$.

  2. B.

Suppose now $b_{0}\in\Theta_{s}(A_{0})$ where $s\geq 1$ and $A_{0}>0$ are both
known. Let $j_{n}\in\mathbb{N}$ be such that
$D_{j_{n}}\sim(n\Delta)^{1/(1+2s)},$ i.e. for some positive constants
$L_{1},L_{2}$ and all $n\in\mathbb{N}$ let $L_{1}(n\Delta)^{1/(1+2s)}\leq
D_{j_{n}}\leq L_{2}(n\Delta)^{1/(1+2s)}$. Let $(\Pi^{(n)})_{n\in\mathbb{N}}$
be a sequence of priors satisfying, for some constant $\zeta>0$ and for
$\varepsilon_{n}=(n\Delta)^{-s/(1+2s)}\log(n\Delta)^{1/2}$,

    1. (I)

$\Pi^{(n)}(\Theta_{s}(A_{0})\cap\Theta)=1$ for all $n$,

    2. (II)

$\Pi^{(n)}(\\{b\in\Theta:\lVert\pi_{j_{n}}b-\pi_{j_{n}}b_{0}\rVert_{2}\leq\varepsilon_{n}\\})\geq(\varepsilon_{n}\zeta)^{D_{j_{n}}}$.

Then we achieve the same rate of contraction; i.e. for some
$M=M(A_{0},s,\mathcal{I},L_{0},\zeta)$,

$\Pi^{(n)}\Big{(}\big{\\{}b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq
M(n\Delta)^{-s/(1+2s)}\log(n\Delta)^{1/2}\big{\\}}\mid X^{(n)}\Big{)}\to 1$

in probability under the law $P_{b_{0}}$ of $X$.

######  Remark.

_Optimality._ The minimax lower bounds of Hoffmann [18] do not strictly apply
because we have assumed $\sigma$ is given. Nevertheless, the minimax rate in
this model should be $(n\Delta)^{-s/(1+2s)}$. This follows by adapting
arguments for the continuous data case from Kutoyants [20] Section 4.5 to
apply to the periodic model and observing that with high-frequency data we
cannot outperform continuous data.

Since a contraction rate of $\varepsilon_{n}$ guarantees the existence of an
estimator converging to the true parameter at rate $\varepsilon_{n}$ (for
example, the centre of the smallest posterior ball of mass at least 1/2 – see
Theorem 8.7 in Ghosal & van der Vaart [12]) the rates attained in Theorem 1
are optimal, up to the log factors.

###  Explicit examples of priors

Our results guarantee that the following priors will exhibit posterior
contraction. Throughout this section we continue to adopt Assumptions 2, 1, 4
and 3, and for technical convenience, we add an extra assumption on $b_{0}$.
Precisely, recalling that $\\{\psi_{lk}\\}$ form a family of Meyer-type
wavelets as in Section 2.1 and $\psi_{-1,0}$ denotes the constant function 1,
we assume the following.

######  Assumption 5.

For a sequence $(\tau_{l})_{l\geq-1}$ to be specified and a constant $B$, we
assume

$b_{0}=\sum_{\begin{subarray}{c}l\geq-1\\\ 0\leq
k<2^{l}\end{subarray}}\tau_{l}\beta_{lk}\psi_{lk},\quad\text{with
}\lvert\beta_{lk}\rvert\leq B\text{ for all }l\geq-1\text{ and all }0\leq
k<2^{l}.$ (5)

The explicit priors for which we prove contraction will be random wavelet
series priors. Let $u_{lk}\overset{iid}{\sim}q$, where $q$ is a density on
$\mathbb{R}$ satisfying

$q(x)\geq\zeta\text{ for }\lvert x\rvert\leq B,\quad\text{and}\quad
q(x)=0\text{ for }\lvert x\rvert>B+1,$

where $\zeta>0$ is a constant and $B>0$ is the constant from Assumption 5. For
example one might choose $q$ to be the density of a $\operatorname{Unif}[0,B]$
random variable or a truncated Gaussian density.

We define a prior $\Pi_{m}$ on $S_{m}$ as the law associated to a random
wavelet series

$b(x)=\sum_{\begin{subarray}{c}-1\leq l<m\\\ 0\leq
k<2^{l}\end{subarray}}\tau_{l}u_{lk}\psi_{lk}(x),\qquad x\in[0,1],$ (6)

for $\tau_{l}$ as in Assumption 5. We give three examples of priors built from
these $\Pi_{m}$.

######  Example 1 (Basic sieve prior).

Let $\tau_{-1}=\tau_{0}=1$ and $\tau_{l}=2^{-3l/2}l^{-2}$ for $l\geq 1$. Let
$h$ be a probability distribution on $\mathbb{N}$ as described in 1A, for
example, $h(m)=\gamma e^{-2^{m}},$ where $\gamma$ is a normalising constant.
Let $\Pi=\sum_{m=1}^{\infty}h(m)\Pi_{m}$ where $\Pi_{m}$ is as above.

######  Proposition 2.

The preceding prior meets the conditions of 1A for any $b_{0}$ satisfying
Assumption 5 with the same $\tau_{l}$ used to define the prior, and for an
appropriate constant $K_{0}$. Thus, if also $b_{0}\in\Theta_{s}(A_{0})$ for
some constant $A_{0}$, $\Pi\left(\\{b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq
M(n\Delta)^{-s/(1+2s)}\log(n\Delta)^{1/2}\\}\mid X^{(n)}\right)\to 1$ in
$P_{b_{0}}$-probability, for some constant $M$.

The proof can be found in Section 6.1.

######  Remark.

_Adaptive estimation._ If we assume $b_{0}\in\Theta_{s_{\text{min}}}(A_{0})$,
for some $s_{\text{min}}>3/2,$ Assumption 5 automatically holds with
$\tau_{l}$ as in Example 1 for some constant $B=B(s_{\text{min}},A_{0})$, as
can be seen from the wavelet characterisation 2. Thus, in contrast to the low-
frequency results of [23], the above prior adapts to unknown $s$ in the range
$s_{\text{min}}\leq s<\infty$.

When $s>1$ is known, we fix the rate of decay of wavelet coefficients to
ensure a draw from the prior lies in $\Theta_{s}(A_{0})$ by hand, rather than
relying on the hyperparameter to choose the right resolution of wavelet space.
We demonstrate with the following example. The proofs of Propositions 3 and 4,
also given in Section 6.1, mimic that of Proposition 2 but rely on 1B in place
of 1A.

######  Example 2 (Known smoothness prior).

Let $\tau_{-1}=1$ and $\tau_{l}=2^{-l(s+1/2)}$ for $l\geq 0$. Let
$\bar{L}_{n}\in\mathbb{N}\cup\\{\infty\\}$. Define a sequence of priors
$\Pi^{(n)}=\Pi_{\bar{L}_{n}}$ for $b$ (we can take $\bar{L}_{n}=\infty$ to
have a genuine prior, but a sequence of priors will also work provided
$\bar{L}_{n}\to\infty$ at a fast enough rate).

######  Proposition 3.

Assume $\bar{L}_{n}/(n\Delta)^{1/(1+2s)}$ is bounded away from zero. Then for
any $s>1$, the preceding sequence of priors meets the conditions of 1B for any
$b_{0}$ satisfying Assumption 5 with the same $\tau_{l}$ used to define the
prior, and for an appropriate constant $K_{0}$. Thus, for some constant $M$,
$\Pi^{(n)}\left(\\{b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq
M(n\Delta)^{-s/(1+2s)}\log(n\Delta)^{1/2}\\}\mid X^{(n)}\right)\to 1$ in
$P_{b_{0}}$-probability.

######  Remark.

Assumption 5 with $\tau_{l}=2^{-l(s+1/2)}$ in fact forces $b_{0}\in
B^{s}_{\infty,\infty}\subsetneq B^{s}_{2,\infty}$ with fixed norm bound.
Restricting to this smaller set does not change the minimax rate, as can be
seen from the fact that the functions by which Hoffmann perturbs in the lower
bound proofs in [18] lie in the smaller class addressed here. In principle,
one could remove this assumption by taking $\tau_{l}=2^{-ls}$ and taking the
prior $\Pi^{(n)}$ to be the law of $b\sim\Pi_{\bar{L}_{n}}$ conditional on
$b\in\Theta_{s}(A_{0})$.

######  Example 3 (Prior on the invariant density).

In some applications it may be more natural to place a prior on the invariant
density and only implicitly on the drift function. With minor adjustments, 1B
can still be applied to such priors. We outline the necessary adjustments.

  1. (i)

$b$ is not identifiable from $\pi_{b}$ and $\sigma^{2}$. We therefore
introduce the identifiability constraint $I_{b}(1)=0$. We could fix $I_{b}(1)$
as any positive constant and reduce to the case $I_{b}(1)=0$ by a translation,
so we choose $I_{b}(1)=0$ for simplicity (this assumption is standard in the
periodic model, for example see van Waaij & van Zanten [34]). With this
restriction, we have $\pi_{b}(x)=\frac{e^{I_{b}(x)}}{G_{b}\sigma^{2}(x)}$ for
a normalising constant $G_{b}$, so that
$b=((\sigma^{2})^{\prime}+\sigma^{2}(\log\pi_{b})^{\prime})/2$.

  2. (ii)

In place of Assumption 5, we need a similar assumption but for
$H_{0}:=\log\pi_{b_{0}}$. Precisely, we assume

$H_{0}=\sum_{\begin{subarray}{c}l\geq-1\\\ 0\leq
k<2^{l}\end{subarray}}\tau_{l}h_{lk}\psi_{lk},\quad\text{with }\lvert
h_{lk}\rvert\leq B\text{ for all }l\geq-1\text{ and all }0\leq k<2^{l},$ (7)

for $\tau_{-1}=\tau_{0}=1$ and $\tau_{l}=2^{-l(s+3/2)}l^{-2}$ for $l\geq 1$,
for some known constant $B$, and where $s\geq 1$ is assumed known.

  3. (iii)

Induce a prior on $b=((\sigma^{2})^{\prime}+\sigma^{2}H^{\prime})/2$ by
putting the prior $\Pi^{(n)}=\Pi_{\bar{L}_{n}}$ on $H$, where $\bar{L}_{n}$ is
as in Proposition 3.

  4. (iv)

To ensure $b\in\Theta_{s}(A_{0})$ we place further restrictions on $\sigma$;
for example, we could assume $\sigma^{2}$ is smooth. More tightly, it is
sufficient to assume (in addition to Assumption 1) that
$\sigma^{2}\in\Theta_{s+1}(A_{1})$ and
$\lVert\sigma^{2}\rVert_{C^{s}_{\text{per}}}\leq A_{1}$, where
$C^{s}_{\text{per}}$ is the Hölder norm, for some $A_{1}>0$. These conditions
on $\sigma$ can be bypassed with a more careful statement of 1B and a more
careful treatment of the bias.

######  Proposition 4.

Make changes i, ii, iii and iv as listed. Then, the obtained sequence of
priors meets the conditions of 1B for an appropriate constant $K_{0}$, hence
for some constant $M$ we have $\Pi^{(n)}\left(\\{b\in\Theta:\lVert
b-b_{0}\rVert_{2}\leq M(n\Delta)^{-s/(1+2s)}\log(n\Delta)^{1/2}\\}\mid
X^{(n)}\right)\to 1$ in $P_{b_{0}}$-probability.

######  Remarks.

_Minimax rates._ The assumption 7 restricts $b_{0}$ beyond simply lying in
$\Theta_{s}(A_{0})$. As with Nickl & Söhl [23] Remark 5, this further
restriction does not change the minimax rates, except for a log factor induced
by the weights $l^{-2}$.

_Adaptation to sampling regime._ The prior of Proposition 4 is the same as the
prior on $b$ in [23]. However, since here we assume $\sigma$ is given while in
[23] it is an unknown parameter, the results of [23] do not immediately yield
contraction of this prior at a near-minimax rate in the low-frequency setting.
In particular, when $\sigma$ is known the minimax rate for estimating $b$ with
low-frequency data is $n^{-s/(2s+3)}$ (for example see Söhl & Trabs [30]),
rather than the slower rate $n^{-s/(2s+5)}$ attained in Gobet–Hoffmann–Reiss
[16] when $\sigma$ is unknown (this improvement is possible because one
bypasses the delicate interweaving of the problems of estimating $b$ and
$\sigma$ with low-frequency data). Nevertheless, the prior of Proposition 4
will indeed exhibit near-minimax contraction also in the low-frequency
setting. An outline of the proof is as follows. The small ball results of [23]
still apply, with minor changes to the periodic model used here in place of
their reflected diffusion, so it is enough to exhibit tests of the true
parameter against suitably separated alternatives. The identification
$b=((\sigma^{2})^{\prime}+\sigma^{2}(\log\pi_{b})^{\prime})/2$ means one can
work with the invariant density rather than directly with the drift. Finally
one shows the estimator from [30] exhibits sufficiently good concentration
properties (alternatively, one could use general results for Markov chains
from Ghosal & van der Vaart [11]).

It remains an interesting open problem to simultaneously estimate $b$ and
$\sigma$ with a method which adapts to the sampling regime. Extending the
proofs of this paper to the case where $\sigma$ is unknown would show that the
Bayesian method fulfils this goal. The key difficulty in making this extension
arises in the small ball section (Section 5), because Girsanov’s Theorem does
not apply to diffusions with different diffusion coefficients.

_Intermediate sampling regime._ Strictly speaking, we only demonstrate
robustness to the sampling regime in the extreme cases where $\Delta>0$ is
fixed or where $n\Delta^{2}\to 0$. The author is not aware of any papers
addressing the intermediate regime (where $\Delta$ tends to $0$ at a slower
rate than $n^{-1/2}$) for a nonparametric model: the minimax rates do not even
appear in the literature. Since the Bayesian method adapts to the extreme
regimes, one expects that it attains the correct rates in this intermediate
regime (up to log factors). However, the proof would require substantial extra
work, primarily in exhibiting an estimator with good concentration properties
in this regime. Kessler’s work on the intermediate regime in the parametric
case [19] would be a natural starting point for exploring this regime in the
nonparametric setting.

##  4 Construction of tests

In this section we construct the tests needed to apply the general contraction
rate theory from Ghosal–Ghosh–van der Vaart [10]. The main result of this
section is the following. Recall that $S_{m}$ is a periodic Meyer-type wavelet
space of resolution $m$ as described in Section 2.1, $\pi_{m}$ is the
$L^{2}$–orthogonal projection onto $S_{m}$ and $D_{m}=\dim(S_{m})=2^{m}$.

######  Lemma 5.

Consider data $X^{(n)}=(X_{k\Delta})_{0\leq k\leq n}$ sampled from a solution
$X$ to 1 under Assumptions 2, 1, 4 and 3. Let $\varepsilon_{n}\to 0$ be a
sequence of positive numbers and let $l_{n}\to\infty$ be a sequence of
positive integers such that
$n\Delta\varepsilon_{n}^{2}/\log(n\Delta)\to\infty$ and, for some constant $L$
and all $n$, $D_{l_{n}}\leq Ln\Delta\varepsilon_{n}^{2}$. Let
$\Theta_{n}\subseteq\\{b\in\Theta:\lVert\pi_{l_{n}}b-b\rVert_{2}\leq\varepsilon_{n}\\}$
contain $b_{0}$.

Then for any $D>0$, there is an $M=M(\mathcal{I},L_{0},D,L)>0$ for which there
exist tests $\psi_{n}$ (i.e. $\\{0,1\\}$–valued functions of the data) such
that, for all $n$ sufficiently large,

$\max\Big{(}E_{b_{0}}\psi_{n}(X^{(n)}),\sup\big{\\{}E_{b}[1-\psi_{n}(X^{(n)})]:b\in\Theta_{n},\lVert
b-b_{0}\rVert_{2}>M\varepsilon_{n}\big{\\}}\Big{)}\leq
e^{-Dn\Delta\varepsilon_{n}^{2}}.$

The proof is given in Section 4.2 and is a straightforward consequence of our
constructing an estimator with appropriate concentration properties. First, we
introduce some general concentration results we will need.

###  General concentration results

We will use three forms of concentration results as building blocks for our
theorems. The first comes from viewing the data $(X_{j\Delta})_{0\leq j\leq
n}$ as a Markov chain and applying Markov chain concentration results; these
results are similar to those used in Nickl & Söhl [23] for the low-frequency
case, but here we need to track the dependence of constants on $\Delta$. The
second form are useful only in the high-frequency case because they use a
quantitative form of Hölder continuity for diffusion processes. An inequality
of the third form, based on martingale properties, is introduced only where
needed (in Lemma 13).

####  Markov chain concentration results applied to diffusions

Our main concentration result arising from the Markov structure is the
following. We denote by $\lVert\cdot\rVert_{\mu}$ the
$L^{2}_{\mu}([0,1])$–norm, $\lVert
f\rVert_{\mu}^{2}=E_{\mu}[f^{2}]=\int_{0}^{1}f(x)^{2}\mathop{}\\!\mathrm{d}\mu(x)$.

######  Theorem 6.

There exists a constant $\kappa=\kappa(\mathcal{I})$ such that, for all $n$
sufficiently large and all bounded 1–periodic functions
$f:\mathbb{R}\to\mathbb{R}$,

$P_{b}\left(\Big{\lvert}\sum_{k=1}^{n}f(X_{k\Delta})-E_{\mu}[f]\Big{\rvert}\geq
t\right)\leq 2\exp\left(-\frac{1}{\kappa}\Delta\min\left(\frac{t^{2}}{n\lVert
f\rVert_{\mu}^{2}},\frac{t}{\lVert f\rVert_{\infty}}\right)\right),$ (8)

or equivalently

$P_{b}\left(\Big{\lvert}\sum_{j=1}^{n}f(X_{j\Delta})-E_{\mu}[f]\Big{\rvert}\geq\max(\sqrt{\kappa
v^{2}x},\kappa ux)\right)\leq 2e^{-x},$ (9)

where $v^{2}=n\Delta^{-1}\lVert f\rVert_{\mu}^{2}$ and $u=\Delta^{-1}\lVert
f\rVert_{\infty}$.

Further, if $\mathcal{F}$ is a space of such functions indexed by some (subset
of a) $d$–dimensional vector space, then for
$V^{2}=\sup_{f\in\mathcal{F}}v^{2}$ and $U=\sup_{f\in\mathcal{F}}u$, we also
have

$P_{b}\left(\sup_{f\in\mathcal{F}}\Big{\lvert}\sum_{j=1}^{n}f(X_{j\Delta})-E_{\mu}[f]\Big{\rvert}\geq\tilde{\kappa}\max\left\\{\sqrt{V^{2}(d+x)},U(d+x)\right\\}\right)\leq
4e^{-x}.$ (10)

for some constant $\tilde{\kappa}=\tilde{\kappa}(\mathcal{I})$.

The proof is an application of the following abstract result for Markov
chains.

######  Theorem 7 (Paulin [25], Proposition 3.4 and Theorem 3.4).

Let $M_{1},\dots,M_{n}$ be a time-homogeneous Markov chain taking values in
$S$ with transition kernel $P(x,\mathop{}\\!\mathrm{d}y)$ and invariant
density $\pi$. Suppose $M$ is uniformly ergodic, i.e. $\sup_{x\in S}\lVert
P^{n}(x,\cdot)-\pi\rVert_{TV}\leq K\rho^{n}$ for some constants $K<\infty$,
$\rho<1$, where $P^{n}(x,\cdot)$ is the $n-$step transition kernel and
$\lVert\cdot\rVert_{TV}$ is the total variation norm for signed measures.
Write $t_{\text{mix}}=\min\\{n\geq 0:\sup_{x\in S}\lVert
P^{n}(x,\cdot)-\pi\rVert_{TV}<1/4\\}.$ Suppose $M_{1}\sim\pi$ and
$f:S\to\mathbb{R}$ is bounded. Let $V_{f}=\operatorname{Var}[f(M_{1})]$, let
$C=\lVert f-E[f(M_{1})]\rVert_{\infty}$. Then

$P\left(\lvert\sum_{i=1}^{n}f(M_{i})-E[f(M_{i})]\rvert\geq t\right)\leq
2\exp\left(\frac{-t^{2}}{2t_{\text{mix}}(8(n+2t_{\text{mix}})V_{f}+20tC)}\right).$

###### Proof of Theorem 6.

Since $f$ is assumed periodic we see that
$f(X_{k\Delta})=f(\dot{X}_{k\Delta}),$ where we recall $\dot{X}=X\mod 1$.
Denote by $\dot{p}_{b}(t,x,y)$ the transition densities of $\dot{X}$, i.e.
$\dot{p}_{b}(t,x,y)=\sum_{j\in\mathbb{Z}}p_{b}(t,x,y+j)$ (see the proof of
Proposition 9 in Nickl & Söhl [23] for an argument that the sum converges).
Theorem 2.6 in Bhattacharya et al. [3] tells us that if $\dot{X}_{0}$ has a
density $\eta_{0}$ on $[0,1]$, then $\dot{X}_{t}$ has a density $\eta_{t}$
satisfying

$\lVert\eta_{t}-\pi_{b}\rVert_{TV}\leq\frac{1}{2}\lVert\eta_{0}/\pi_{b}-1\rVert_{TV}\exp\Big{(}-\frac{1}{2M_{b}}t\Big{)},$

where
$M_{b}:=\sup_{z\in[0,1]}\Big{\\{}(\sigma^{2}(z)\pi_{b}(z))^{-1}\int_{0}^{z}\pi_{b}(x)\mathop{}\\!\mathrm{d}x\int_{z}^{1}\pi_{b}(y)\mathop{}\\!\mathrm{d}y\Big{\\}}.$
We can regularise to extend the result so that it also applies when the
initial distribution of $\dot{X}$ is a point mass: if $\dot{X}_{0}=x$ then
$\dot{X}_{1}$ has density $\dot{p}_{b}(1,x,\cdot),$ hence the result applies
to show

$\lVert\dot{p}_{b}(t,x,\cdot)-\pi_{b}\rVert_{TV}\leq\frac{1}{2}\lVert\dot{p}_{b}(1,x,\cdot)/\pi_{b}-1\rVert_{TV}\exp\Big{(}-\frac{1}{2M_{b}}(t-1)\Big{)}.$

Moreover, note
$\lVert\dot{p}_{b}(1,x,\cdot)/\pi_{b}-1\rVert_{TV}\leq\pi_{L}^{-1}\lVert\dot{p}_{b}(1,x,\cdot)-\pi_{b}\rVert_{TV}\leq\pi_{L}^{-1}.$
Also note we can upper bound $M_{b}$ by a constant $M=M(\mathcal{I})$:
precisely, we can take $M=\sigma_{L}^{-2}\pi_{L}^{-1}\pi_{U}^{2}$.

Thus, we see that for $t\geq 1$, we have

$\lVert\dot{p}_{b}(t,x,\cdot)-\pi_{b}\rVert_{TV}\leq
K\exp\Big{(}-\frac{1}{2M}t\Big{)}$

for some constant $K=K(\mathcal{I})$, uniformly across $x\in[0,1]$. It follows
that, for each fixed $\Delta$, the discrete time Markov chain
$(\dot{X}_{k\Delta})_{k\geq 0}$ is uniformly ergodic with mixing time
$t_{\text{mix}}\leq 1+2M\log(4K)\Delta^{-1}\leq K^{\prime}\Delta^{-1}$ for
some constant $K^{\prime}$. Theorem 7 applies to tell us

$P\left(\lvert\sum_{i=1}^{n}f(X_{k\Delta})-E_{\mu}[f]\rvert\geq t\right)\leq
2\exp\left(-\frac{t^{2}}{2K^{\prime}\Delta^{-1}(8(n+2K^{\prime}\Delta^{-1})V_{f}+20tC)}\right).$

Since $n\Delta\to\infty$ by assumption, we see
$8(n+2K^{\prime}\Delta^{-1})\leq K^{\prime\prime}n$ for some constant
$K^{\prime\prime}$. Using the bound $2/(a+b)\geq\min(1/a,1/b)$ for $a,b>0$ and
upper bounding the centred moments $V_{f}$ and $C$ by the uncentred moments
$\lVert f\rVert_{\mu}^{2}$ and $\lVert f\rVert_{\infty}$, we deduce (8).

The result 9 is obtained by a change of variables. For the supremum result 10,
we use a standard chaining argument, eg. as in Baraud [1] Theorem 2.1, where
we use 9 in place of Baraud’s Assumption 2.1, noting that Baraud only uses
Assumption 2.1 to prove an expression mirroring 9, and the rest of the proof
follows through exactly. Precisely, following the proof, we can take
$\tilde{\kappa}=36\kappa$. ∎

######  Remark.

The proof simplifies if we restrict $\Theta$ to only those $b$ satisfying
$I_{b}(1)=0$. In this case, the invariant density (upon changing normalising
constant to some $G_{b}$) reduces to the more familiar form
$\pi_{b}(x)=(G_{b}\sigma^{2}(x))^{-1}e^{I_{b}(x)}$. The diffusion is
reversible in this case, and we can use Theorem 3.3 from [25] instead of
Theorem 3.4 to attain the same results but with better constants.

####  Hölder continuity properties of diffusions

Define

$w_{m}(\delta)=\delta^{1/2}((\log\delta^{-1})^{1/2}+\log(m)^{1/2}),\qquad\delta\in(0,1]$

for $m\geq 1$, and write $w_{m}(\delta):=w_{1}(\delta)$ for $m<1$. The key
result of this section is the following.

######  Lemma 8.

Let $X$ solve the scalar diffusion equation 1, and grant Assumptions 2 and 1.
Then there exist positive constants $\lambda$, $C$ and $\tau$, all depending
on $\mathcal{I}$ only, such that for any $u>C\max(\log(m),1)^{1/2}$ and for
any initial value $x$,

$P_{b}^{(x)}\left(\sup_{\begin{subarray}{c}s,t\in[0,m],\\\ t\not=s,\lvert
t-s\rvert\leq\tau\end{subarray}}\left(\frac{\lvert
X_{t}-X_{s}\rvert}{w_{m}(\lvert t-s\rvert)}\right)>u\right)\leq 2e^{-\lambda
u^{2}}.$

######  Remarks.

  1. i.

We will need to control all increments $X_{(j+1)\Delta}-X_{j\Delta}$
simultaneously, hence we include the parameter $m$, which we will take to be
the time horizon $n\Delta$ when applying this result. Simply controlling over
$[0,1]$ and using a union bound does not give sharp enough results.

  2. ii.

The lemma applies for any distribution of $X_{0}$, not only point masses, by
an application of the tower law.

The modulus of continuity $w_{m}$ matches that of Brownian motion, and indeed
the proof, given in Appendix B, is to reduce to the corresponding result for
Brownian motion. First, by applying the scale function one transforms $X$ into
a local martingale, reducing Lemma 8 to the following result, also useful in
its own right.

######  Lemma 9.

Let $Y$ be a local martingale with quadratic variation satisfying
$\lvert\langle Y\rangle_{t}-\langle Y\rangle_{s}\rvert\leq A\lvert t-s\rvert$
for a constant $A\geq 1$. Then there exist positive constants
$\lambda=\lambda(A)$ and $C=C(A)$ such that for any
$u>C\max(\log(m),1)^{1/2}$,

$\Pr\left(\sup_{\begin{subarray}{c}s,t\in[0,m],s\not=t,\\\ \lvert
t-s\rvert\leq A^{-1}e^{-2}\end{subarray}}\left(\frac{\lvert
Y_{t}-Y_{s}\rvert}{w_{m}(\lvert t-s\rvert)}\right)>u\right)\leq 2e^{-\lambda
u^{2}}.$

In particular the result applies when $Y$ is a solution to
$\mathop{}\\!\mathrm{d}Y_{t}=\tilde{\sigma}(Y_{t})\mathop{}\\!\mathrm{d}W_{t},$
provided $\lVert\tilde{\sigma}^{2}\rVert_{\infty}\leq A.$

Lemma 9 follows from the corresponding result for Brownian motion by a time
change (i.e. the (Dambis–)Dubins-Schwarz Theorem). It is well known that
Brownian motion has modulus of continuity
$\delta^{1/2}(\log\delta^{-1})^{1/2}$ in the sense that there almost surely
exists a constant $C>0$ such that $\lvert B_{t}-B_{s}\rvert\leq C\lvert
t-s\rvert^{1/2}(\log(\lvert t-s\rvert^{-1}))^{1/2},$ for all $t,s\in[0,1]$
sufficiently close, but Lemmas 8 and 9 depend on the following quantitative
version of this statement, proved using Gaussian process techniques. The
proofs of Lemmas 9 and 10 are given in Appendix B.

######  Lemma 10.

Let $B$ be a standard Brownian motion on $[0,m]$. There are postive
(universal) constants $\lambda$ and $C$ such that for
$u>C\max(\log(m),1)^{1/2}$,

$\Pr\left(\sup_{\begin{subarray}{c}s,t\in[0,m],\\\ s\not=t,\lvert
t-s\rvert\leq e^{-2}\end{subarray}}\left(\frac{\lvert
B_{t}-B_{s}\rvert}{w_{m}(\lvert t-s\rvert)}\right)>u\right)\leq 2e^{-\lambda
u^{2}}.$

###  Concentration of a drift estimator

####  Defining the estimator

We adapt an estimator introduced in Comte et al. [8]. The estimator is
constructed by considering drift estimation as a regression-type problem.
Specifically, defining

$Z_{k\Delta}=\frac{1}{\Delta}\int_{k\Delta}^{(k+1)\Delta}\sigma(X_{s})\mathop{}\\!\mathrm{d}W_{s},\qquad
R_{k\Delta}=\frac{1}{\Delta}\int_{k\Delta}^{(k+1)\Delta}(b(X_{s})-b(X_{k\Delta}))\mathop{}\\!\mathrm{d}s,$

we can write

$\frac{X_{(k+1)\Delta}-X_{k\Delta}}{\Delta}=b(X_{k\Delta})+Z_{k\Delta}+R_{k\Delta}.$

Note $R_{k\Delta}$ is a discretization error which vanishes as $\Delta\to 0$
and $Z_{k\Delta}$ takes on the role of noise. We define the _empirical norm_
and the related _empirical loss function_

$\lVert
u\rVert_{n}=\frac{1}{n}\sum_{k=1}^{n}u(X_{k\Delta})^{2},\quad\gamma_{n}(u)=\frac{1}{n}\sum_{k=1}^{n}[\Delta^{-1}(X_{(k+1)\Delta}-X_{k\Delta})-u(X_{k\Delta})]^{2},\quad
u:[0,1]\to\mathbb{R}.$

In both we leave out the $k=0$ term for notational convenience.

Recalling that $S_{m}$ is a Meyer-type wavelet space as described in Section
2.1 and $K_{0}$ is an upper bound for the $C_{\text{per}}^{1}$–norm of any
$b\in\Theta$, for $l_{n}$ to be chosen we define $\tilde{b}_{n}$ as a solution
to the minimisation problem

$\tilde{b}_{n}\in\operatorname*{argmin}_{u\in\tilde{S}_{l_{n}}}\gamma_{n}(u),\qquad\tilde{S}_{m}:=\\{u\in
S_{m}:\lVert u\rVert_{\infty}\leq K_{0}+1\\},$

where we choose arbitrarily among minimisers if there is no unique
minimiser.111It is typical that we do not have uniqueness, since if $u$ is a
minimiser of $\gamma_{n}$, then so is any $\tilde{u}\in\tilde{S}_{l_{n}}$ such
that $\tilde{u}(X_{k\Delta})=u(X_{k\Delta})$ for $1\leq k\leq n$.

####  Main concentration result

For the estimator defined above we will prove the following concentration
inequality.

######  Theorem 11.

Consider data $X^{(n)}=(X_{k\Delta})_{0\leq k\leq n}$ sampled from a solution
$X$ to 1 under Assumptions 2, 1, 4 and 3. Let $\varepsilon_{n}\to 0$ be a
sequence of positive numbers and let $l_{n}\to\infty$ be a sequence of
positive integers such that
$n\Delta\varepsilon_{n}^{2}/\log(n\Delta)\to\infty$ and, for some constant $L$
and all $n$, $D_{l_{n}}\leq Ln\Delta\varepsilon_{n}^{2}$. For these $l_{n}$,
let $\tilde{b}_{n}$ be defined as above and let
$\Theta_{n}\subseteq\\{b\in\Theta:\lVert\pi_{l_{n}}b-b\rVert_{2}\leq\varepsilon_{n}\\}$
contain $b_{0}$, where $\pi_{l_{n}}$ is the $L^{2}-$orthogonal projection onto
$S_{l_{n}}$.

Then for any $D>0$ there is a $C=C(\mathcal{I},L_{0},D,L)>0$ such that,
uniformly across $b\in\Theta_{n}$,

$P_{b}\left(\lVert\tilde{b}_{n}-b\rVert_{2}>C\varepsilon_{n}\right)\leq
e^{-Dn\Delta\varepsilon_{n}^{2}},$

for all $n$ sufficiently large.

######  Remark.

Previous proofs of Bayesian contraction rates using the concentration of
estimators approach (see [14],[23],[28]) have used duality arguments, i.e. the
fact that $\lVert f\rVert_{2}=\sup_{v:\lVert v\rVert_{2}=1}\langle
f,v\rangle$, to demonstrate that the linear estimators considered satisfy a
concentration inequality of the desired form. A key insight of this paper is
that for the model we consider we can achieve the required concentration using
the above _minimum contrast_ estimator (see Birgé & Massart [4]), for which we
need techniques which differ substantially from duality arguments.

Before proceeding to the proof, we demonstrate how this can be used to prove
the existence of tests of $b_{0}$ against suitably separated alternatives.

###### Proof of Lemma 5.

Let $\tilde{b}_{n}$ be the estimator outlined above and let $D>0$. Let
$C=C(\mathcal{I},L_{0},D,L)$ be as in Theorem 11 and let $M=2C$. It’s not hard
to see that
$\psi_{n}=\mathbbm{1}\\{\lVert\tilde{b}_{n}-b\rVert_{2}>C\varepsilon_{N}\\}$
is a test with the desired properties. ∎

###### Proof of Theorem 11.

It is enough to show that, uniformly across $b\in\Theta_{n}$, for any $D>0$
there is a $C>0$ such
$P_{b}\left(\lVert\tilde{b}_{n}-b\rVert_{2}>C\varepsilon_{n}\right)\leq
14e^{-Dn\Delta\varepsilon_{n}^{2}},$ because by initially considering a
$D^{\prime}>D$ and finding the corresponding $C^{\prime}$, we can eliminate
the factor of $14$ in front of the exponential.

The proof is structured as follows. Our assumptions ensure that the $L^{2}$–
and $L^{2}(\mu)$–norms are equivalent. We further show that the
$L^{2}(\mu)$–norm is equivalent to the empirical norm $\lVert\cdot\rVert_{n}$
on an event of sufficiently high probability. Finally, the definition of the
estimator will allow us to control the empirical distance
$\lVert\tilde{b}_{n}-b\rVert_{n}$.

To this end, write
$\tilde{t}_{n}=(\tilde{b}_{n}-\pi_{l_{n}}b)\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{\mu}^{-1}$
(defining $\tilde{t}_{n}=0$ if $\tilde{b}_{n}=\pi_{l_{n}}b$) and introduce the
following set and events:

$\displaystyle I_{n}$ $\displaystyle=\left\\{t\in S_{l_{n}}:\lVert
t\rVert_{\mu}=1,\lVert t\rVert_{\infty}\leq
C_{1}\varepsilon_{n}^{-1}\right\\},$ $\displaystyle\mathcal{A}_{n}$
$\displaystyle=\left\\{\tilde{t}_{n}\in
I_{n}\right\\}\cup\\{\tilde{t}_{n}=0\\},$ $\displaystyle\Omega_{n}$
$\displaystyle=\left\\{\left\lvert\lVert
t\rVert_{n}^{2}-1\right\rvert\leq\frac{1}{2},\>\forall t\in I_{n}\right\\},$

where the constant $C_{1}$ is to be chosen. Then we can decompose

$P_{b}\big{(}\lVert\tilde{b}_{n}-b\rVert_{2}>C\varepsilon_{n}\big{)}\leq
P_{b}\big{(}\lVert\tilde{b}_{n}-b\rVert_{2}\mathbbm{1}_{\mathcal{A}_{n}^{c}}>C\varepsilon_{n}\big{)}+P_{b}\big{(}\Omega_{n}^{c}\big{)}+P_{b}(\big{(}\lVert\tilde{b}_{n}-b\rVert_{2}\mathbbm{1}_{\mathcal{A}_{n}\cap\Omega_{n}}>C\varepsilon_{n}\big{)}.$

Thus, we will have proved the theorem once we have completed the following:

  1. 1.

Show the theorem holds (deterministically) on $\mathcal{A}_{n}^{c}$, for a
large enough constant $C$.

  2. 2.

Show that $P_{b}(\Omega_{n}^{c})\leq 4e^{-Dn\Delta\varepsilon_{n}^{2}}$ for a
suitable choice of $C_{1}$.

  3. 3.

Show that, for any $D$, we can choose a $C$ such that
$P_{b}\big{(}\lVert\tilde{b}_{n}-b\rVert_{2}\mathbbm{1}_{\mathcal{A}_{n}\cap\Omega_{n}}>C\varepsilon_{n}\big{)}\leq
10e^{-Dn\Delta\varepsilon_{n}^{2}}$.

##### Step 1:

Intuitively we reason thus. The event $\mathcal{A}_{n}^{c}$ can only occur if
the $L^{2}(\mu)$–norm of $\tilde{b}_{n}-\pi_{l_{n}}b$ is small compared to the
$L^{\infty}$–norm. Since we have assumed a uniform supremum bound on functions
$b\in\Theta$, in fact $\mathcal{A}_{n}$ holds unless the $L^{2}(\mu)$–norm is
small in absolute terms. But if $\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{\mu}$
is small, then so is $\lVert\tilde{b}_{n}-b\rVert_{2}$. We formalise this
reasoning now.

For a constant $C_{2}$ to be chosen, define

$\mathcal{A}_{n}^{\prime}=\\{\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{\mu}>C_{2}\varepsilon_{n}\\}.$

On $\mathcal{A}_{n}^{\prime}$ we have
$\lVert\tilde{t}_{n}\rVert_{\infty}\leq(\lVert\tilde{b}_{n}\rVert_{\infty}+\lVert\pi_{l_{n}}b\rVert_{\infty})C_{2}^{-1}\varepsilon_{n}^{-1}.$
Note $\lVert\tilde{b}_{n}\rVert_{\infty}\leq K_{0}+1$ by definition. Since,
for $n$ large enough, $\lVert\pi_{l_{n}}b-b\rVert_{\infty}\leq 1$ uniformly
across $b\in\Theta_{n}\subseteq\Theta$ by 4 so that
$\lVert\pi_{l_{n}}b\rVert_{\infty}\leq\lVert b\rVert_{\infty}+1\leq K_{0}+1$,
we deduce that on $\mathcal{A}_{n}^{\prime}$,
$\lVert\tilde{t}_{n}\rVert_{\infty}\leq(2K_{0}+2)C_{2}^{-1}\varepsilon_{n}^{-1}$.
Since also $\lVert\tilde{t}_{n}\rVert_{\mu}=1$ (or $\tilde{t}_{n}=0$) by
construction, we deduce $\mathcal{A}_{n}^{\prime}\subseteq\mathcal{A}_{n}$ if
$C_{2}\geq C_{1}^{-1}(2K_{0}+2)$.

Then on $(\mathcal{A}_{n}^{\prime})^{c}\supseteq\mathcal{A}_{n}^{c}$ we find,
using that $b\in\Theta_{n}$ and using
$\lVert\cdot\rVert_{2}\leq\pi_{L}^{-1/2}\lVert\cdot\rVert_{\mu}$,

$\lVert\tilde{b}_{n}-b\rVert_{2}\leq\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{2}+\lVert\pi_{l_{n}}b-b\rVert_{2}\leq(C_{2}\pi_{L}^{-1/2}+1)\varepsilon_{n}.$

So on $\mathcal{A}_{n}^{c}$, we have $\lVert\tilde{b}_{n}-b\rVert_{2}\leq
C\varepsilon_{n}$ deterministically for any $C\geq C_{2}\pi_{L}^{-1/2}+1$.
That is, for $C$ large enough (depending on $C_{1}$ and $\mathcal{I}$),
$P_{b}\big{(}\lVert\tilde{b}_{n}-b\rVert_{2}\mathbbm{1}_{\mathcal{A}_{n}^{c}}>C\varepsilon_{n}\big{)}=0$.

##### Step 2:

We show that for $n$ sufficiently large, and $C_{1}=C_{1}(\mathcal{I},D,L)$
sufficiently small, $P_{b}(\Omega_{n}^{c})\leq
4e^{-Dn\Delta\varepsilon_{n}^{2}}.$

For $t\in I_{n}$ we have $\Big{\lvert}\lVert
t\rVert_{n}^{2}-1\Big{\rvert}=n^{-1}\Big{\lvert}\sum_{k=1}^{n}t^{2}(X_{k\Delta})-E_{\mu}[t^{2}]\Big{\rvert}.$
Thus Theorem 6 can be applied to $\Omega_{n}^{c}=\left\\{\sup_{t\in
I_{n}}n^{-1}\Big{\lvert}\sum_{k=1}^{n}t^{2}(X_{k\Delta})-E_{\mu}[t^{2}]\Big{\rvert}>1/2\right\\}.$
Each $t\in I_{n}$ has $\lVert t^{2}\rVert_{\infty}\leq
C_{1}^{2}\varepsilon_{n}^{-2}$ and $\lVert
t^{2}\rVert_{\mu}^{2}=E_{\mu}[t^{4}]\leq\lVert t^{2}\rVert_{\infty}\lVert
t\rVert_{\mu}^{2}\leq C_{1}^{2}\varepsilon_{n}^{-2}.$ Since the indexing set
$I_{n}$ lies in a vector space of dimension $D_{l_{n}}$, we apply the theorem
with $x=Dn\Delta\varepsilon_{n}^{2}$ to see

$P_{b}\left(\sup_{t\in
I_{n}}\left\lvert\sum_{k=1}^{n}t^{2}(X_{k\Delta})-E_{\mu}[t^{2}]\right\rvert\geq
36\max\\{A,B\\}\right)\leq 4e^{-Dn\Delta\varepsilon_{n}^{2}}.$

where
$A=\sqrt{\tilde{\kappa}C_{1}^{2}n\Delta^{-1}\varepsilon_{n}^{-2}(Dn\Delta\varepsilon_{n}^{2}+D_{l_{n}})}$
and
$B=\tilde{\kappa}C_{1}^{2}\Delta^{-1}\varepsilon_{n}^{-2}(Dn\Delta\varepsilon_{n}^{2}+D_{l_{n}})$,
for some constant $\tilde{\kappa}=\tilde{\kappa}(\mathcal{I})$. Provided we
can choose $C_{1}$ so that $36\max\\{A/n,B/n\\}\leq 1/2$ the result is proved.
Such a choice for $C_{1}$ can be made as we have assumed $D_{l_{n}}\leq
Ln\Delta\varepsilon_{n}^{2}$.

##### Step 3:

Since $b\in\Theta_{n}$ and $\pi_{l_{n}}$ is $L^{2}$-orthogonal projection, we
have
$\lVert\tilde{b}_{n}-b\rVert_{2}^{2}\leq\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{2}^{2}+\varepsilon_{n}^{2}$.
Recall that $\lVert\cdot\rVert_{2}\leq\pi_{L}^{-1/2}\lVert\cdot\rVert_{\mu}$
and note that on $\mathcal{A}_{n}\cap\Omega_{n}$, we further have
$\frac{1}{2}\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{\mu}^{2}\leq\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{n}^{2}.$

Since also $\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{n}^{2}\leq
2(\lVert\pi_{l_{n}}b-b\rVert_{n}^{2}+\lVert\tilde{b}_{n}-b\rVert_{n}^{2})$ we
deduce that

$\lVert\tilde{b}_{n}-b\rVert_{2}^{2}\mathbbm{1}_{\mathcal{A}_{n}\cap\Omega_{n}}\leq\frac{1}{\pi_{L}}\left(4\lVert\pi_{l_{n}}b-b\rVert_{n}^{2}+4\lVert\tilde{b}_{n}-b\rVert_{n}^{2}\mathbbm{1}_{\mathcal{A}_{n}\cap\Omega_{n}}\right)+\varepsilon_{n}^{2},$

where we have dropped indicator functions from terms on the right except where
we will need them later. Thus, using a union bound,

$P_{b}(\lVert\tilde{b}_{n}-b\rVert_{2}\mathbbm{1}_{\mathcal{A}_{n}\cap\Omega_{n}}>C\varepsilon_{n})\leq
P_{b}\big{(}\lVert\pi_{l_{n}}b-b\rVert_{n}^{2}>C^{\prime}\varepsilon_{n}^{2}\big{)}+P_{b}\big{(}\lVert\tilde{b}_{n}-b\rVert_{n}^{2}\mathbbm{1}_{\mathcal{A}_{n}\cap\Omega_{n}}>C^{\prime}\varepsilon_{n}^{2}\big{)},$

for some constant $C^{\prime}$ (precisely we can take
$C^{\prime}=\pi_{L}(C^{2}-1)/8$). It remains to show that both probabilities
on the right are exponentially small.

##### Bounding
$P_{b}\left(\lVert\pi_{l_{n}}b-b\rVert_{n}>C\varepsilon_{n}\right)$:

We show that for any $D>0$ there is a constant $C$ such that
$P_{b}\left(\lVert\pi_{l_{n}}b-b\rVert_{n}>C\varepsilon_{n}\right)\leq
2e^{-Dn\Delta\varepsilon_{n}^{2}},$ for all $n$ sufficiently large. Since
$E_{b}\lVert g\rVert_{n}^{2}=\lVert g\rVert_{\mu}^{2}$ for any 1–periodic
deterministic function $g$ and
$\lVert\pi_{l_{n}}b-b\rVert_{\mu}^{2}\leq\pi_{U}\lVert\pi_{l_{n}}b-b\rVert_{2}^{2}\leq\pi_{U}\varepsilon_{n}^{2}$
for $b\in\Theta_{n}$, it is enough to show that

$P_{b}\left(\big{\lvert}\lVert\pi_{l_{n}}b-b\rVert_{n}^{2}-E_{b}\lVert\pi_{l_{n}}b-b\rVert_{n}^{2}\big{\rvert}>C\varepsilon_{n}^{2}\right)\leq
2e^{-Dn\Delta\varepsilon_{n}^{2}}$ (11)

for some different $C$. As in Step 2, we apply Theorem 6, but now working with
the single function $(\pi_{l_{n}}b-\nobreak b)^{2}$. For large enough $n$ we
have the bounds $\lVert\pi_{l_{n}}b-b\rVert_{\infty}\leq 1$ (derived from 4),
and $\lVert(\pi_{l_{n}}b-\nobreak b)^{2}\rVert_{\mu}\leq$
$\lVert\pi_{l_{n}}b-b\rVert_{\infty}\lVert\pi_{l_{n}}b-b\rVert_{\mu}\leq\pi_{U}^{1/2}\varepsilon_{n}$
(because $b\in\Theta_{n}$) and so applying the theorem with
$x=Dn\Delta\varepsilon_{n}^{2}$ gives

$P_{b}\left(\left\lvert\sum_{k=1}^{n}\left[(\pi_{l_{n}}b-b)^{2}(X_{k\Delta})-\lVert\pi_{l_{n}}b-b\rVert_{\mu}^{2}\right]\right\rvert\geq\max\\{a,b\\}\right)\leq
2e^{-Dn\Delta\varepsilon_{n}^{2}},$

for $a=\sqrt{\kappa
n\Delta^{-1}\pi_{U}\varepsilon_{n}^{2}Dn\Delta\varepsilon_{n}^{2}}=n\varepsilon_{n}^{2}\sqrt{\kappa\pi_{U}D}$
and $b=\kappa\Delta^{-1}Dn\Delta\varepsilon_{n}^{2}=n\varepsilon_{n}^{2}\kappa
D$, for some constant $\kappa=\kappa(\mathcal{I})$. We see that $a/n$ and
$b/n$ are both upper bounded by a constant multiple of $\varepsilon_{n}^{2}$,
hence, by choosing $C$ large enough, 11 holds.

##### Bounding
$P_{b}\big{(}\lVert\tilde{b}_{n}-b\rVert_{n}^{2}\mathbbm{1}_{\mathcal{A}_{n}\cap\Omega_{n}}>C\varepsilon_{n}^{2}\big{)}$:

We show that
$P_{b}\big{(}\lVert\tilde{b}_{n}-b\rVert_{n}^{2}\mathbbm{1}_{\mathcal{A}_{n}\cap\Omega_{n}}>C\varepsilon_{n}^{2}\big{)}\leq
8e^{-Dn\Delta\varepsilon_{n}^{2}}$ for some constant $C$.

Recall an application of 4 showed us that
$\lVert\pi_{l_{n}}b\rVert_{\infty}\leq K_{0}+1$ for sufficiently large $n$,
hence we see that $\pi_{l_{n}}b$ lies in $\tilde{S}_{l_{n}}$, so by definition
$\gamma_{n}(\tilde{b}_{n})\leq\gamma_{n}(\pi_{l_{n}}b)$. We now use this to
show that

$\frac{1}{4}\lVert\tilde{b}_{n}-b\rVert_{n}^{2}\mathbbm{1}_{\mathcal{A}_{n}\cap\Omega_{n}}\leq\frac{7}{4}\lVert\pi_{l_{n}}b-b\rVert_{n}^{2}+8\nu_{n}(\tilde{t}_{n})^{2}\mathbbm{1}_{\mathcal{A}_{n}}+\frac{8}{n}\sum_{k=1}^{n}R_{k\Delta}^{2},$
(12)

where $\nu_{n}(t)=\frac{1}{n}\sum_{k=1}^{n}t(X_{k\Delta})Z_{k\Delta}$ and we
recall that
$\tilde{t}_{n}=(\tilde{b}_{n}-\pi_{l_{n}}b)\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{\mu}^{-1}$.
The argument, copied from [8] Sections 3.2 and 6.1, is as follows. Using
$\Delta^{-1}(X_{(k+1)\Delta}-X_{k\Delta})=b(X_{k\Delta})+Z_{k\Delta}+R_{k\Delta}$
and
$\gamma_{n}(\tilde{b}_{n})-\gamma_{n}(b)\leq\gamma_{n}(\pi_{l_{n}}b)-\gamma_{n}(b)$,
one shows that

$\lVert\tilde{b}_{n}-b\rVert_{n}^{2}\leq\lVert\pi_{l_{n}}b-b\rVert_{n}^{2}+2\nu(\tilde{b}_{n}-\pi_{l_{n}}b)+\frac{2}{n}\sum_{k=1}^{n}R_{k\Delta}(\tilde{b}_{n}-\pi_{l_{n}}b)(X_{k\Delta}).$
(13)

Repeatedly applying the AM-GM–derived inequality $2ab\leq 8a^{2}+b^{2}/8$
yields

$\displaystyle\frac{2}{n}\sum_{k=1}^{n}R_{k\Delta}(\tilde{b}_{n}-\pi_{l_{n}}b)(X_{k\Delta})$
$\displaystyle\leq\frac{8}{n}\sum_{k=1}^{n}R_{k\Delta}^{2}+\frac{1}{8}\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{n}^{2},$
$\displaystyle
2\nu(\tilde{b}_{n}-\pi_{l_{n}}b)=2\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{\mu}\nu(\tilde{t}_{n})$
$\displaystyle\leq
8\nu_{n}(\tilde{t}_{n})^{2}+\frac{1}{8}\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{\mu}^{2}.$

Next recall that on $\mathcal{A}_{n}\cap\Omega_{n}$, we have
$\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{\mu}^{2}\leq
2\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{n}^{2},$ and further recall
$\lVert\tilde{b}_{n}-\pi_{l_{n}}b\rVert_{n}^{2}\leq
2\lVert\tilde{b}_{n}-b\rVert_{n}^{2}+2\lVert\pi_{l_{n}}b-b\rVert_{n}^{2}$.
Putting all these bounds into 13 yields 12, where on the right hand side we
have only included indicator functions where they will help us in future
steps. Next, by a union bound, we deduce

$P_{b}(\lVert\tilde{b}_{n}-b\rVert_{n}^{2}\mathbbm{1}_{\mathcal{A}_{n}\cap\Omega_{n}}>C\varepsilon_{n}^{2})\\\
\leq
P_{b}(\lVert\pi_{l_{n}}b-b\rVert_{n}^{2}>C^{\prime}\varepsilon_{n}^{2})+P_{b}(\nu_{n}(\tilde{t}_{n})^{2}\mathbbm{1}_{\mathcal{A}_{n}}>C^{\prime}\varepsilon_{n}^{2})+P_{b}\Big{(}\frac{1}{n}\sum_{k=1}^{n}R_{k\Delta}^{2}>C^{\prime}\varepsilon_{n}^{2}\Big{)},$

for some constant $C^{\prime}$ (we can take $C^{\prime}=C/96$). We have
already shown that $P_{b}(\lVert\pi_{l_{n}}b-b\rVert_{n}>C\varepsilon_{n})\leq
2e^{-Dn\Delta\varepsilon_{n}^{2}}$ for a large enough constant $C$, thus the
following two lemmas conclude the proof. ∎

######  Lemma 12.

Under the conditions of Theorem 11, for each $D>0$ there exists a constant
$C=C(\mathcal{I},L_{0},D)>0$ for which, for $n$ sufficiently large,
$P_{b}\left(\frac{1}{n}\sum_{k=1}^{n}R_{k\Delta}^{2}>C\varepsilon_{n}^{2}\right)\leq
2e^{-Dn\Delta\varepsilon_{n}^{2}}.$

######  Lemma 13.

Under the conditions of Theorem 11, for each $D>0$ there exists a constant
$C=C(\mathcal{I},L,D)>0$ for which, for $n$ sufficiently large,
$P_{b}(\nu_{n}(\tilde{t}_{n})\mathbbm{1}_{\mathcal{A}_{n}}>C\varepsilon_{n})\leq
4e^{-Dn\Delta\varepsilon_{n}^{2}}.$

###### Proof of Lemma 12.

Recall
$R_{k\Delta}=\frac{1}{\Delta}\int_{k\Delta}^{(k+1)\Delta}(b(X_{s})-b(X_{k\Delta}))\mathop{}\\!\mathrm{d}s,$
and recall any $b\in\Theta$ is Lipschitz, with Lipschitz constant at most
$K_{0}$, so $\lvert R_{k\Delta}\rvert\leq K_{0}\max_{s\leq\Delta}\lvert
X_{k\Delta+s}-X_{k\Delta}\rvert.$ It is therefore enough to bound
$\sup\\{\lvert X_{t}-X_{s}\rvert:\>s,t\in[0,n\Delta],~{}\lvert
t-s\rvert\leq\Delta\\}$.

We apply the Hölder continuity result (Lemma 8) with
$u=D^{1/2}\lambda^{-1/2}(n\Delta\varepsilon_{n}^{2})^{1/2}$ for
$\lambda=\lambda(\mathcal{I})$ the constant of the lemma, noting that the
assumption $n\Delta\varepsilon_{n}^{2}/\log(n\Delta)\to\infty$ ensures that
$u$ is large enough compared to $m=n\Delta$ that the conditions for the lemma
are met, at least when $n$ is large. We see that

$\sup_{\begin{subarray}{c}s,t\in[0,n\Delta]\\\ \lvert
t-s\rvert\leq\Delta\end{subarray}}\lvert
X_{t}-X_{s}\rvert\leq\Delta^{1/2}\left(\log(n\Delta)^{1/2}+\log(\Delta^{-1})^{1/2}\right)D^{1/2}\lambda^{-1/2}(n\Delta\varepsilon_{n}^{2})^{1/2},$

on an event $\mathcal{D}$ of probability at least
$1-2e^{-Dn\Delta\varepsilon_{n}^{2}}$, (we have used that, for $n$ large
enough, $\Delta\leq\min(\tau,e^{-1})$ in order to take the supremum over
$\lvert t-s\rvert\leq\Delta$ and to see
$\sup_{\delta\leq\Delta}w_{m}(\delta)=w_{m}(\Delta)$).

Now observe that $\log(n\Delta)^{1/2}\leq(\log(\Delta^{-1})^{1/2})$ for large
enough $n$ because $n\Delta^{2}\to 0$ (so $n\Delta\leq\Delta^{-1}$
eventually). Further, from the assumption $n\Delta^{2}\log(\Delta^{-1})\leq
L_{0}$ we are able to deduce that
$\Delta^{1/2}\log(\Delta^{-1})^{1/2}(n\Delta\varepsilon_{n}^{2})^{1/2}\leq
L_{0}^{1/2}\varepsilon_{n}$. It follows that on $\mathcal{D}$, we have
$R_{k\Delta}\leq C\varepsilon_{n}$ for a suitably chosen constant $C$
(independent of $k$ and $n$), which implies the desired concentration. ∎

###### Proof of Lemma 13.

Recall for
$Z_{k\Delta}=\frac{1}{\Delta}\int_{k\Delta}^{(k+1)\Delta}\sigma(X_{s})\mathop{}\\!\mathrm{d}W_{s}$
we set $\nu_{n}(t)=\frac{1}{n}\sum_{k=1}^{n}t(X_{k\Delta})Z_{k\Delta}.$ The
martingale-derived concentration result Lemma 2 in Comte et al. [8] (the model
assumptions in [8] are slightly different to those made here, but the proof of
the lemma equally applies in our setting) tells us
$P_{b}(\nu_{n}(t)\geq\xi,\lVert t\rVert_{n}^{2}\leq
u^{2})\leq\exp\left(-\frac{n\Delta\xi^{2}}{2\sigma_{U}^{2}u^{2}}\right),$ for
any $t,u$, and for any drift function $b\in\Theta$, so that

$P_{b}(\nu_{n}(t)\geq\xi)\leq\exp\left(-\frac{n\Delta\xi^{2}}{2\sigma_{U}^{2}u^{2}}\right)+P_{b}(\lVert
t\rVert_{n}^{2}>u^{2}).$ ($\star$)

We can apply Theorem 6 to see that, for some constant
$\kappa=\kappa(\mathcal{I})$,

$\displaystyle P_{b}(\lVert t\rVert_{n}^{2}>u^{2})$
$\displaystyle=P_{b}\left(\frac{1}{n}\left(\sum_{k=1}^{n}t(X_{k\Delta})^{2}-\lVert
t\rVert_{\mu}^{2}\right)>u^{2}-\lVert t\rVert_{\mu}^{2}\right)$
$\displaystyle\leq\exp\left(-\frac{1}{\kappa}\Delta\min\left\\{\frac{n^{2}(u^{2}-\lVert
t\rVert_{\mu}^{2})^{2}}{n\lVert t^{2}\rVert_{\mu}^{2}},\frac{n(u^{2}-\lVert
t\rVert_{\mu}^{2})}{\lVert t^{2}\rVert_{\infty}}\right\\}\right)$
$\displaystyle\leq\exp\left(-\frac{1}{\kappa}n\Delta(u^{2}-\lVert
t\rVert_{\mu}^{2})\lVert t\rVert_{\infty}^{-2}\min(u^{2}\lVert
t\rVert_{\mu}^{-2}-1,1)\right),$

where to obtain the last line we have used that $\lVert
t^{2}\rVert_{\mu}^{2}\leq\lVert t\rVert_{\infty}^{2}\lVert t\rVert_{\mu}^{2}$.

Now choose $u^{2}=\lVert t\rVert_{\mu}^{2}+\xi\lVert t\rVert_{\infty}$. Then
$\xi^{2}/u^{2}\geq\frac{1}{2}\min(\xi^{2}/\lVert t\rVert_{\mu}^{2},\xi/\lVert
t\rVert_{\infty})$ so that, returning to $\star$ ‣ Section 4.2.2, we find

$\displaystyle P_{b}(\nu_{n}(t)\geq\xi)$
$\displaystyle\leq\exp\left(-\frac{n\Delta}{4\sigma_{U}^{2}}\min(\xi^{2}\lVert
t\rVert_{\mu}^{-2},\xi\lVert
t\rVert_{\infty}^{-1})\right)+\exp\Big{(}-\frac{1}{\kappa}n\Delta\xi\min(\xi\lVert
t\rVert_{\mu}^{-2},\lVert t\rVert_{\infty}^{-1})\Big{)}$ $\displaystyle\leq
2\exp\left(-\frac{1}{\kappa^{\prime}}n\Delta\min(\xi^{2}\lVert
t\rVert_{\mu}^{-2},\xi\lVert t\rVert_{\infty}^{-1})\right),$

for some constant $\kappa^{\prime}=\kappa^{\prime}(\mathcal{I})$.

By changing variables we attain the bound
$P_{b}(\nu_{n}(t)\geq\max(\sqrt{v^{2}x},ux))\leq 2\exp\left(-x\right),$ where
$v^{2}=\kappa^{\prime}(n\Delta)^{-1}\lVert t\rVert_{\mu}^{2}$ and
$u=\kappa^{\prime}(n\Delta)^{-1}\lVert t\rVert_{\infty}$. Then, as in Theorem
6, a standard chaining argument allows us to deduce that

$P_{b}\left(\sup_{t\in
I_{n}}\nu_{n}(t)\geq\tilde{\kappa}\Big{(}\sqrt{V^{2}(D_{l_{n}}+x)}+U(D_{l_{n}}+x)\Big{)}\right)\leq
4e^{-x},$

for $V^{2}=\sup_{t\in I_{n}}\lVert
t\rVert_{\mu}^{2}(n\Delta)^{-1}=(n\Delta)^{-1}$, $U=\sup_{t\in I_{n}}\lVert
t\rVert_{\infty}(n\Delta)^{-1}=C_{1}\varepsilon_{n}^{-1}(n\Delta)^{-1}$, and
for a constant $\tilde{\kappa}=\tilde{\kappa}(\mathcal{I})$. Taking
$x=Dn\Delta\varepsilon_{n}^{2}$ and recalling the assumption $D_{l_{n}}\leq
Ln\Delta\varepsilon_{n}^{2}$ we obtain the desired result (conditional on
$\tilde{t}_{n}\in I_{n}$, which is the case on the event $\mathcal{A}_{n}$). ∎

##  5 Small ball probabilities

Now we show that the Kullback–Leibler divergence between the laws
corresponding to different parameters $b_{0},b$ can be controlled in terms of
the $L^{2}$–distance between the parameters. Denote by $K(p,q)$ the
Kullback–Leibler divergence between probability distributions with densities
$p$ and $q$, i.e.
$K(p,q)=E_{p}\log(\frac{p}{q})=\int\log(\frac{p(x)}{q(x)}){\mathop{}\\!\mathrm{d}p}(x).$
Also write

$\operatorname{KL}(b_{0},b)=~{}~{}E_{b_{0}}\left[\log\left(\frac{p_{0}(\Delta,X_{0},X_{\Delta})}{p_{b}(\Delta,X_{0},X_{\Delta})}\right)\right].$

Recalling that
$p_{b}^{(n)}(x^{(n)})=\pi_{b}(x_{0})\prod_{i=1}^{n}p_{b}(\Delta,x_{(i-1)\Delta},x_{i\Delta})$
is the density on $\mathbb{R}^{n+1}$ of $X^{(n)}$ under $P_{b}$, we introduce
the following Kullback–Leibler type neighbourhoods: for $\varepsilon>0$,
define

$\displaystyle
B_{KL}^{(n)}(\varepsilon)=\left\\{b\in\Theta:K(p_{0}^{(n)},p_{b}^{(n)})\leq(n\Delta+1)\varepsilon^{2},~{}\operatorname{Var}_{b_{0}}\Big{(}\log\frac{p_{0}^{(n)}}{p_{b}^{(n)}}\Big{)}\leq(n\Delta+1)\varepsilon^{2}\right\\},$
$\displaystyle
B_{\varepsilon}=\left\\{b\in\Theta:K(\pi_{0},\pi_{b})\leq\varepsilon^{2},~{}\operatorname{Var}_{b_{0}}\Big{(}\log\frac{\pi_{0}}{\pi_{b}}\Big{)}\leq\varepsilon^{2},~{}\operatorname{KL}(b_{0},b)\leq\Delta\varepsilon^{2},~{}\operatorname{Var}_{b_{0}}\Big{(}\log\frac{p_{0}}{p_{b}}\Big{)}\leq\Delta\varepsilon^{2}\right\\}.$

Note that $\operatorname{KL}(b_{0},b)$ and $B_{\varepsilon}$ implicitly depend
on $n$ via $\Delta$.

The main result of this section is the following.

######  Theorem 14.

Consider data $X^{(n)}=(X_{k\Delta})_{0\leq k\leq n}$ sampled from a solution
$X$ to 1 under Assumptions 2, 1, 4 and 3. Let $\varepsilon_{n}\to 0$ be a
sequence of positive numbers such that $n\Delta\varepsilon_{n}^{2}\to\infty$.
Then there is a constant $A=A(\mathcal{I})$ such that, for all $n$
sufficiently large, $\\{b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq
A\varepsilon_{n}\\}\subseteq B_{KL}^{(n)}(\varepsilon_{n})$.

###### Proof.

Applying Lemma 23 in the appendix where it is shown that

$\operatorname{Var}_{b_{0}}\log\left(\frac{p_{0}^{(n)}(X^{(n)})}{p_{b}^{(n)}(X^{(n)})}\right)\leq
3\operatorname{Var}_{b_{0}}\left(\log\frac{\pi_{0}(X_{0})}{\pi_{b}(X_{0})}\right)+3n\operatorname{Var}_{b_{0}}\left(\log\frac{p_{0}(X_{0},X_{\Delta})}{p_{b}(X_{0},X_{\Delta})}\right),$

and noting also that
$K(p_{0}^{(n)},p_{b}^{(n)})=K(\pi_{0},\pi_{b})+n\operatorname{KL}(b_{0},b)$ by
linearity, we observe that $B_{\varepsilon_{n}/\sqrt{3}}\subseteq
B_{KL}^{(n)}(\varepsilon_{n})$. It is therefore enough to show that for some
$A=A(\mathcal{I})$ we have $\\{b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq
A\varepsilon_{n}\\}\subseteq B_{\varepsilon_{n}/\sqrt{3}}$. This follows
immediately by applying Lemma 15 below to $\xi_{n}=\varepsilon_{n}/\sqrt{3}$.
∎

######  Lemma 15.

Under the conditions of Theorem 14, there is an $A=A(\mathcal{I})$ such that,
for all $n$ sufficiently large, $\\{b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq
A\varepsilon_{n}\\}\subseteq B_{\varepsilon_{n}}$.

The key idea in proving Lemma 15 is to use the Kullback–Leibler divergence
between the laws $P_{b_{0}}^{(x)},P_{b}^{(x)}$ of the continuous-time paths to
control the Kullback–Leibler divergence between $p_{b}$ and $p_{0}$. This will
help us because we can calculate the Kullback–Leibler divergence between the
full paths using Girsanov’s Theorem, which gives us an explicit formula for
the likelihood ratios.

Let $P_{b,T}^{(x)}$ denote the law of $(X_{t})_{0\leq t\leq T}$ conditional on
$X_{0}=x$, i.e. the restriction of $P_{b}^{(x)}$ to $C([0,T])$. We write
$\mathbb{W}_{\sigma,T}^{(x)}$ for $P_{b,T}^{(x)}$ when $b=0$. Throughout this
section we will simply write $P_{b}^{(x)}$ for $P_{b,\Delta}^{(x)}$ and
similarly with $\mathbb{W}_{\sigma}^{(x)}$. We have the following.

######  Theorem 16 (Girsanov’s Theorem).

Assume $b_{0}$ and $b$ lie in $\Theta$, and $\sigma$ satisfies Assumption 1.
Then the laws $P_{b_{0},T}^{(x)}$ and $P_{b,T}^{(x)}$ are mutually absolutely
continuous with, for $X\sim P_{b,T}^{(x)}$, the almost sure identification

$\frac{\mathop{}\\!\mathrm{d}P_{b_{0},T}^{(x)}}{\mathop{}\\!\mathrm{d}P_{b,T}^{(x)}}((X_{t})_{t\leq
T})=\exp\left[\int_{0}^{T}\frac{b_{0}-b}{\sigma^{2}}(X_{t})\mathop{}\\!\mathrm{d}X_{t}-\frac{1}{2}\int_{0}^{T}\frac{b_{0}^{2}-b^{2}}{\sigma^{2}}(X_{t})\mathop{}\\!\mathrm{d}t\right].$

###### Proof.

See Liptser & Shiryaev [21], Theorem 7.19, noting that the assumptions are met
because $b,b_{0}$ and $\sigma$ are all Lipschitz and bounded, and $\sigma$ is
bounded away from 0. ∎

We write

$\tilde{p}_{0}^{(x)}=\frac{\mathop{}\\!\mathrm{d}P_{b_{0}}^{(x)}}{\mathop{}\\!\mathrm{d}\mathbb{W}^{(x)}_{\sigma}},\qquad\tilde{p}_{b}^{(x)}=\frac{\mathop{}\\!\mathrm{d}P_{b}^{(x)}}{\mathop{}\\!\mathrm{d}\mathbb{W}_{\sigma}^{(x)}}$
(14)

for the Radon-Nikodym derivatives (i.e. densities on $C([0,\Delta])$ with
respect to $\mathbb{W}_{\sigma}^{(x)}$) whose existence Girsanov’s Theorem
guarantees. We will simply write $X$ for $(X_{t})_{t\leq\Delta}$ where context
allows, and similarly with $U$. Since $\tilde{p}_{0}^{(x)}(X)=0$ for any path
$X$ with $X_{0}\not=x$, we will further omit the superscripts on our densities
in general, writing $\tilde{p}_{0}(X)$ for $\tilde{p}_{0}^{(X_{0})}(X)$, and
similarly for $\tilde{p}_{b}$.

###### Proof of Lemma 15.

We break the proof into a series of lemmas. We will upper bound the variances
in the definition of $B_{\varepsilon_{n}}$ by the corresponding uncentred
second moments. For some constant $A=A(\mathcal{I})$ we show the following.

  1. 1.

$A^{2}\operatorname{KL}(b_{0},b)\leq\Delta\lVert b-b_{0}\rVert_{2}^{2},$ which
shows that $\operatorname{KL}(b_{0},b)\leq\Delta\varepsilon_{n}^{2}$ whenever
$\lVert b-b_{0}\rVert_{2}\leq A\varepsilon_{n}$. This is the content of Lemma
17.

  2. 2.

If $\lVert b-b_{0}\rVert_{2}\leq A\varepsilon_{n}$ then we have
$E_{b_{0}}[\log(p_{0}/p_{b})^{2}]\leq\Delta\varepsilon_{n}^{2}.$ This is the
content of Lemma 18. Note that the other steps do not need any assumptions on
$\varepsilon_{n}$, but this step uses $n\Delta\varepsilon_{n}^{2}\to\infty$.

  3. 3.

$A^{2}\max\left\\{K(\pi_{0},\pi_{b}),E_{b_{0}}[\log(\pi_{0}/\pi_{b})^{2}]\right\\}\leq\lVert
b_{0}-b\rVert_{2}^{2}.$ From this it follows that
$K(\pi_{0},\pi_{b})\leq\varepsilon_{n}^{2}$ and
$E_{b_{0}}[\log(\pi_{0}/\pi_{b})^{2}]\leq\varepsilon_{n}^{2}$ whenever $\lVert
b-b_{0}\rVert_{2}\leq A\varepsilon_{n}$. This is the content of Lemma 19.

Together, then, the three lemmas below conclude the proof. ∎

######  Lemma 17.

Under the conditions of Theorem 14, there is a constant $A$ depending only on
$\mathcal{I}$ such that $A^{2}\operatorname{KL}(b_{0},b)\leq\Delta\lVert
b_{0}-b\rVert_{2}^{2}$.

The proof is essentially the same as that in van der Meulen & van Zanten [33]
Lemma 5.1, with minor adjustments to fit the periodic model and non-constant
$\sigma$ used here. Further, all the ideas needed are exhibited in the proof
of Lemma 18. Thus, we omit the proof.

######  Lemma 18.

Under the conditions of Theorem 14, there is a constant $A=A(\mathcal{I})$ so
that, for $n$ sufficiently large,
$E_{b_{0}}[\log(p_{0}/p)^{2}]\leq\Delta\varepsilon_{n}^{2}$ whenever $\lVert
b-b_{0}\rVert_{2}\leq A\varepsilon_{n}$.

###### Proof.

We first show that we can control the second moment of $\log(p_{0}/p_{b})$ by
the second moment of the corresponding expression
$\log(\tilde{p}_{0}/\tilde{p}_{b})$ for the full paths, up to an approximation
error which is small when $\Delta$ is small. Consider the smallest convex
function dominating $\log(x)^{2}$, given by

$h(x)=\begin{cases}\log(x)^{2}&x<e\\\ 2e^{-1}x-1&x\geq e\end{cases}$

(it is in fact more convenient, and equivalent, to think of $h$ as dominating
the function $x\mapsto(\log x^{-1})^{2}$). Let $X\sim P_{b_{0}}^{(x)}$ and let
$U\sim\mathbb{W}_{\sigma}^{(x)}$. Intuitively, the probability density of a
transition of $X$ from $x$ to $y$, with respect to the (Lebesgue) density
$p_{*}$ of transitions of $U$ from $x$ to $y$, can be calculated by
integrating the likelihood $\tilde{p}_{0}(U)$ over all paths of $U$ which
start at $x$ and end at $y$, and performing this integration will yield the
conditional expectation of $\tilde{p}_{0}^{(x)}(U)$ given $U_{\Delta}$. That
is to say,

$\frac{p_{0}(\Delta,x,y)}{p_{*}(\Delta,x,y)}=E_{\mathbb{W}_{\sigma}^{(x)}}\left[\tilde{p}_{0}(U)\mid
U_{\Delta}=y\right].$ (15)

The above argument is not rigorous because we condition on an event of
probability zero, but the formula 15 is true, and is carefully justified in
Lemma 24 in Appendix A. A corresponding expression holds for
$p_{b}(\Delta,x,y)$, so that

$E_{b_{0}}\left[\log\Big{(}\frac{p_{0}(\Delta,X_{0},X_{\Delta})}{p_{b}(\Delta,X_{0},X_{\Delta})}\Big{)}^{2}\right]\leq
E_{b_{0}}[h(p_{b}/p_{0})]=E_{b_{0}}\left[h\bigg{(}\frac{E_{\mathbb{W}_{\sigma}^{(X_{0})}}[\tilde{p}_{b}(U)\mid
U_{\Delta}=X_{\Delta}]}{E_{\mathbb{W}_{\sigma}^{(X_{0})}}[\tilde{p}_{0}(U)\mid
U_{\Delta}=X_{\Delta}]}\bigg{)}\right].$

Lemma 22 in Appendix A allows us to simplify the ratio of conditional
expectations. We apply with $\mathbb{P}=\mathbb{W}_{\sigma}^{(X_{0})}$,
$\mathbb{Q}=P_{b_{0}}^{(X_{0})}$ and
$g=\tilde{p}_{b}^{(X_{0})}/\tilde{p}_{0}^{(X_{0})}$, then further apply
conditional Jensen’s inequality and the tower law to find

$\displaystyle
E_{b_{0}}\left[\Big{(}\log\frac{p_{0}}{p_{b}}\Big{)}^{2}\right]$
$\displaystyle\leq
E_{b_{0}}\left[h\Big{(}E_{P_{b_{0}}^{(X_{0})}}\Big{[}\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)\mid
X_{\Delta}\Big{]}\Big{)}\right]\leq
E_{b_{0}}\Big{[}h\Big{(}\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)\Big{)}\Big{]}$
$\displaystyle\qquad\leq
E_{b_{0}}\left[\left(\log\frac{\tilde{p}_{0}}{\tilde{p}_{b}}(X)\right)^{2}\right]+E_{b_{0}}\left[(2e^{-1}\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)-1)\mathbbm{1}\Big{\\{}\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)\geq
e\Big{\\}}\right],$

which is the promised decomposition into a corresponding quantity for the
continuous case and an approximation error. We conclude by showing that each
of these two terms is bounded by $\frac{1}{2}\Delta\varepsilon_{n}^{2}$,
provided $\lVert b-b_{0}\rVert_{2}\leq A\varepsilon_{n}$ for some sufficiently
small constant $A=A(\mathcal{I})$.

##### Showing
$E_{b_{0}}\left[\left(\log\frac{\tilde{p}_{0}}{\tilde{p}_{b}}\right)^{2}\right]\leq\frac{1}{2}\Delta\varepsilon_{n}^{2}$:

Write $f=\frac{b_{0}-b}{\sigma}$. Then we apply Girsanov’s Theorem (Theorem
16) to find

$\displaystyle
E_{b_{0}}\left[\left(\log\frac{\tilde{p}_{0}}{\tilde{p}_{b}}(X)\right)^{2}\right]$
$\displaystyle=E_{b_{0}}\left[\Big{(}\int_{0}^{\Delta}f(X_{t})\mathop{}\\!\mathrm{d}W_{t}+\frac{1}{2}\int_{0}^{\Delta}f^{2}(X_{t})\mathop{}\\!\mathrm{d}t\Big{)}^{2}\right],$
$\displaystyle=E_{b_{0}}\Big{[}\Big{(}\int_{0}^{\Delta}f(X_{t})\mathop{}\\!\mathrm{d}W_{t}\Big{)}^{2}\Big{]}+\frac{1}{4}E_{b_{0}}\Big{[}\big{(}\int_{0}^{\Delta}f^{2}(X_{t})\mathop{}\\!\mathrm{d}t\big{)}^{2}\Big{]}$

The cross term has vanished in the final expression because
$\int_{0}^{\Delta}f(X_{t})\mathop{}\\!\mathrm{d}W_{t}$ is a martingale for
$X\sim P_{b_{0}}$ (since $f$ is bounded thanks to Assumptions 2 and 1 and a
bounded semimartingale integrated against a square integrable martingale
yields a martingale, as in [29] IV.27.4), while
$\int_{0}^{\Delta}f^{2}(X_{t})\mathop{}\\!\mathrm{d}t$ is a finite variation
process, and the expectation of a martingale against a finite variation
process is zero (eg. see [29] IV.32.12).

For the first term on the right, we use Itô’s isometry ([29] IV.27.5),
Fubini’s Theorem, periodicity of $f$ and stationarity of $\mu_{0}$ for the
periodised process $\dot{X}=X\mod 1$ to find

$E_{b_{0}}\Big{(}\int_{0}^{\Delta}f(X_{t})\mathop{}\\!\mathrm{d}W_{t}\Big{)}^{2}=E_{b_{0}}\int_{0}^{\Delta}f^{2}(X_{t})\mathop{}\\!\mathrm{d}t=\int_{0}^{\Delta}E_{b_{0}}f^{2}(\dot{X}_{t})\mathop{}\\!\mathrm{d}t=\Delta\lVert
f\rVert_{\mu_{0}}^{2}.$

The second term
$\frac{1}{4}E_{b_{0}}\Big{[}\big{(}\int_{0}^{\Delta}f^{2}(X_{t})\mathop{}\\!\mathrm{d}t\big{)}^{2}\Big{]}$
is upper bounded by $\frac{1}{4}\Delta^{2}\lVert f\rVert_{\infty}^{2}\lVert
f\rVert_{\mu_{0}}^{2}$ (this can be seen from the bound
$(\int_{0}^{\Delta}f^{2})^{2}\leq\Delta\lVert
f\rVert_{\infty}^{2}\int_{0}^{\Delta}f^{2}$), hence is dominated by
$\Delta\lVert f\rVert_{\mu_{0}}^{2}$ when $n$ is large. Thus, for some
constant $A=A(\mathcal{I})$ we find

$E_{b_{0}}\left[\left(\log\frac{\tilde{p}_{0}}{\tilde{p}_{b}}(X)\right)^{2}\right]\leq
2\Delta\lVert f\rVert_{\mu_{0}}^{2}\leq\frac{1}{2}A^{-2}\Delta\lVert
b_{0}-b\rVert_{2}^{2},$

where Assumptions 2 and 1 allow us to upper bound $\lVert f\rVert_{\mu_{0}}$
by $\lVert b_{0}-b\rVert_{2}$, up to a constant depending only on
$\mathcal{I}$. For $\lVert b_{0}-b\rVert_{2}\leq A\varepsilon_{n}$ we then
have
$E_{b_{0}}\big{[}\big{(}\log(\tilde{p}_{b}/\tilde{p}_{0})\big{)}^{2}\big{]}\leq\Delta\varepsilon_{n}^{2}/2.$

##### Showing
$E_{b_{0}}\left[(2e^{-1}\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)-1)\mathbbm{1}\\{\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)\geq
e\\}\right]\leq\frac{1}{2}\Delta\varepsilon_{n}^{2}$:

We have

$E_{b_{0}}\Big{[}\Big{(}2e^{-1}\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)-1\Big{)}\mathbbm{1}\Big{\\{}\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)\geq
e\Big{\\}}\Big{]}\leq
2e^{-1}P_{b}\Big{[}\frac{\tilde{p}_{b}}{\tilde{p}_{0}}\geq e\Big{]}\leq
P_{b}\Big{[}\log\Big{(}\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)\Big{)}\geq
1\Big{]}.$

By the tower law it suffices to show
$P_{b}^{(x)}\Big{[}\log\Big{(}\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)\Big{)}\geq
1\Big{]}\leq\frac{1}{2}\Delta\varepsilon_{n}^{2}$ for each $x\in[0,1]$.
Applying Girsanov’s Theorem (Theorem 16) we have, for $f=(b_{0}-b)/\sigma$,
and for $n$ large enough that $\Delta\lVert f\rVert_{\infty}^{2}\leq 1$,

$\displaystyle
P_{b}^{(x)}\Big{(}\log\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)>1\Big{)}$
$\displaystyle=P_{b}^{(x)}\Big{(}\int_{0}^{\Delta}-f(X_{t})\mathop{}\\!\mathrm{d}W_{t}+\frac{1}{2}\int_{0}^{\Delta}f(X_{t})^{2}\mathop{}\\!\mathrm{d}t>1\Big{)}$
$\displaystyle\leq
P_{b}^{(x)}\Big{(}\int_{0}^{\Delta}-f(X_{t})\mathop{}\\!\mathrm{d}W_{t}>1/2\Big{)}.$

Write $M_{t}=\int_{0}^{t}-f(X_{s})\mathop{}\\!\mathrm{d}W_{s}$. Then, for
$A=\max(1,(2K_{0}/\sigma_{L})^{2})$, since $A$ uniformly upper bounds $\lVert
f\rVert_{\infty}^{2}$ for $b\in\Theta$, we see that $M$ is a martingale whose
quadratic variation satisfies $\lvert\langle M\rangle_{t}-\langle
M\rangle_{s}\rvert\leq A\lvert t-s\rvert$. Recalling that
$w_{1}(\delta)=\delta^{1/2}\log(\delta^{-1})^{1/2}$, we apply Lemma 9 with
$u=w_{1}(\Delta)^{-1}/2$ to yield that, for $n$ large enough,

$\displaystyle
P_{b}^{(x)}\Big{(}\log\frac{\tilde{p}_{b}}{\tilde{p}_{0}}(X)>1\Big{)}$
$\displaystyle\leq P_{b}^{(x)}\Big{(}\sup_{s,t\leq\Delta,s\not=t}\frac{\lvert
M_{t}-M_{s}\rvert}{w_{1}(\lvert
t-s\rvert)}>\frac{1}{2}w_{1}(\Delta)^{-1}\Big{)}$ $\displaystyle\leq
2\exp\Big{(}-\lambda w_{1}(\Delta)^{-2}\Big{)},$

where $\lambda$ is a constant depending only on $\mathcal{I}$.

Recall we assume $n\Delta\to\infty$ and $n\Delta^{2}\to 0$. It follows that
for large enough $n$ we have $\log(\Delta^{-1})\leq\log(n)$, and
$\Delta\leq\lambda\log(n)^{-2}$. Then observe

$\displaystyle\Delta\leq\lambda\log(n)^{-2}\implies\Delta\leq\lambda(\log\Delta^{-1})^{-1}\log(n)^{-1}\implies\log(n)\leq\lambda\Delta^{-1}(\log\Delta^{-1})^{-1},$

so that $\exp\big{(}-\lambda w_{1}(\Delta)^{-2}\big{)}\leq n^{-1}$ for $n$
large. Finally, since $n\Delta\varepsilon_{n}^{2}\to\infty$, we see
$2n^{-1}\leq\frac{1}{2}\Delta\varepsilon_{n}^{2}$ for $n$ large enough, as
required. ∎

######  Lemma 19.

Under the conditions of Theorem 14, there is a constant $A$ depending only on
$\mathcal{I}$ such that
$A^{2}\max\left\\{K(\pi_{0},\pi_{b}),E_{b_{0}}[\log(\pi_{0}/\pi_{b})^{2}]\right\\}\leq\lVert
b_{0}-b\rVert_{2}^{2}.$

###### Proof.

By the comment after Lemma 8.3 in [10], it suffices to prove that
$h^{2}(\pi_{0},\pi_{b})\lVert\pi_{0}/\pi_{b}\rVert_{\infty}\leq C\lVert
b-b_{0}\rVert_{2}^{2}$ for some $C=C(\mathcal{I})$, where $h$ is the Hellinger
distance between densities defined by
$h^{2}(p,q)=\int(\sqrt{p}-\sqrt{q})^{2}$. Since $\pi_{0},\pi_{b}$ are
uniformly bounded above and away from zero, we can absorb the term
$\lVert\pi_{0}/\pi_{b}\rVert_{\infty}$ into the constant.

We initially prove pointwise bounds on the difference between the densities
$\pi_{0},\pi_{b}$. Recall we saw in Section 2 that, for
$I_{b}(x)=\int_{0}^{x}\frac{2b}{\sigma^{2}}(y)\mathop{}\\!\mathrm{d}y$, we
have

$\displaystyle\pi_{b}(x)=\frac{e^{I_{b}(x)}}{H_{b}\sigma^{2}(x)}\Big{(}e^{I_{b}(1)}\int_{x}^{1}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y+\int_{0}^{x}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y\Big{)},\qquad
x\in[0,1],$ $\displaystyle
H_{b}=\int_{0}^{1}\frac{e^{I_{b}(x)}}{\sigma^{2}(x)}\Big{(}e^{I_{b}(1)}\int_{x}^{1}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y+\int_{0}^{x}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y\Big{)}\mathop{}\\!\mathrm{d}x.$

We can decompose: $\lvert\pi_{b}(x)-\pi_{0}(x)\rvert\leq
D_{1}+D_{2}+D_{3}+D_{4},$ where

$\displaystyle
D_{1}=\frac{e^{I_{b}(x)}}{\sigma^{2}(x)}\Big{\lvert}\frac{1}{H_{b}}-\frac{1}{H_{b_{0}}}\Big{\rvert}\Big{(}e^{I_{b}(1)}\int_{x}^{1}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y+\int_{0}^{x}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y\Big{)},$
$\displaystyle D_{2}=\frac{\lvert
e^{I_{b}(x)}-e^{I_{b_{0}}(x)}\rvert}{H_{b_{0}}\sigma^{2}(x)}\Big{(}e^{I_{b}(1)}\int_{x}^{1}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y+\int_{0}^{x}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y\Big{)},$
$\displaystyle
D_{3}=\frac{e^{I_{b_{0}}(x)}}{H_{b_{0}}\sigma^{2}(x)}\Big{\lvert}\big{(}e^{I_{b}(1)}-e^{I_{b_{0}}(1)}\big{)}\int_{x}^{1}e^{-I_{b}(y)}\mathop{}\\!\mathrm{d}y\Big{\rvert},$
$\displaystyle
D_{4}=\frac{e^{I_{b_{0}}(x)}}{H_{b_{0}}\sigma^{2}(x)}\Bigg{\lvert}e^{I_{b_{0}}(1)}\int_{x}^{1}(e^{-I_{b}(y)}-e^{-I_{b_{0}}(y)})\mathop{}\\!\mathrm{d}y+\int_{0}^{x}(e^{-I_{b}(y)}-e^{-I_{b_{0}}(y)})\mathop{}\\!\mathrm{d}y\Bigg{\rvert}.$

We have the bounds $\sigma_{U}^{-2}e^{-6K_{0}\sigma_{L}^{-2}}\leq
H_{b}\leq\sigma_{L}^{-2}e^{6K_{0}\sigma_{L}^{-2}},$ and
$e^{-2K_{0}\sigma_{L}^{-2}}\leq e^{I_{b}(x)}\leq e^{2K_{0}\sigma_{L}^{-2}}.$
An application of the mean value theorem then tells us

$\Big{\lvert}e^{I_{b}(x)}-e^{I_{b_{0}}(x)}\Big{\rvert}\leq
C(\mathcal{I})\int_{0}^{x}\frac{2\lvert
b_{0}-b\rvert}{\sigma^{2}}(y)\mathop{}\\!\mathrm{d}y\leq
C^{\prime}(\mathcal{I})\lVert b_{0}-b\rVert_{2},$

for some constants $C$, $C^{\prime}\\!$, and the same expression upper bounds
$\lvert e^{-I_{b}(x)}-e^{-I_{b_{0}}(x)}\rvert$.

It follows that, for some constant $C=C(\mathcal{I})$, we have $D_{i}\leq
C\lVert b-b_{0}\rVert_{2}$ for $i=2,3,4$. For $i=1$ the same bound holds since
$\lvert\frac{1}{H_{b}}-\frac{1}{H_{b_{0}}}\rvert\leq\frac{\lvert
H_{b}-H_{b_{0}}\rvert}{H_{b}H_{b_{0}}}$ and a similar decomposition to the
above yields $\lvert H_{b}-H_{b_{0}}\rvert\leq C(\mathcal{I})\lVert
b-b_{0}\rVert_{2}$.

Thus, we have shown that $\lvert\pi_{b}(x)-\pi_{0}(x)\rvert\leq
C(\mathcal{I})\lVert b-b_{0}\rVert_{2}$. Integrating this pointwise bound, we
find that $\lVert\pi_{0}-\pi_{b}\rVert_{2}\leq C(\mathcal{I})\lVert
b_{0}-b\rVert_{2}$. Finally, since
$h^{2}(\pi_{0},\pi_{b})\leq\frac{1}{4\pi_{L}}\lVert\pi_{0}-\pi_{b}\rVert_{2}^{2}\leq
C^{\prime}(\mathcal{I})\lVert b_{0}-b\rVert_{2}^{2},$ for some different
constant $C^{\prime}$, we are done. ∎

##  6 Main contraction results: proofs

We now have the tools we need to apply general theory in order to derive
contraction rates. Recall that $K(p,q)$ denotes the Kullback–Leibler
divergence between probability distributions with densities $p$ and $q$, and
recall the definition

$B_{KL}^{(n)}(\varepsilon)=\left\\{b\in\Theta:K(p_{0}^{(n)},p_{b}^{(n)})\leq(n\Delta+1)\varepsilon^{2},\operatorname{Var}_{b_{0}}\Big{(}\log\frac{p_{0}^{(n)}}{p_{b}^{(n)}}\Big{)}\leq(n\Delta+1)\varepsilon^{2}\right\\}.$

We have the following abstract contraction result, from which we deduce
Theorem 1.

######  Theorem 20.

Consider data $X^{(n)}=(X_{k\Delta})_{0\leq k\leq n}$ sampled from a solution
$X$ to 1 under Assumptions 2, 1, 4 and 3. Let the true parameter be $b_{0}$.
Let $\varepsilon_{n}\to 0$ be a sequence of positive numbers and let $l_{n}$
be a sequence of positive integers such that, for some constant $L$ we have,
for all $n$,

$D_{l_{n}}=2^{l_{n}}\leq Ln\Delta\varepsilon_{n}^{2},\quad\text{and}\quad
n\Delta\varepsilon_{n}^{2}/\log(n\Delta)\to\infty.$ (16)

For each $n$ let $\Theta_{n}$ be $\mathcal{S}$-measurable and assume

$b_{0}\in\Theta_{n}\subseteq\\{b\in\Theta:\lVert\pi_{l_{n}}b-b\rVert_{2}\leq\varepsilon_{n}\\},$
(17)

where $\pi_{l_{n}}$ is the $L^{2}$–orthogonal projection onto $S_{l_{n}}$ as
described in Section 2.1. Let $\Pi^{(n)}$ be a sequence of priors on $\Theta$
satisfying

  1. (a)

$\Pi^{(n)}(\Theta_{n}^{c})\leq e^{-(\omega+4)n\Delta\varepsilon_{n}^{2}}$,

  2. (b)

$\Pi^{(n)}(B_{KL}^{(n)}(\varepsilon_{n}))\geq e^{-\omega
n\Delta\varepsilon_{n}^{2}}$,

for some constant222In fact we can replace the exponent $\omega+4$ in a with
any $B>\omega+1$. We choose $\omega+4$ because it simplifies the exposition
and the exact value is unimportant. $\omega>0$. Then
$\Pi^{(n)}\left(\\{b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq
M\varepsilon_{n}\\}\mid X^{(n)}\right)\to 1$ in probability under the law
$P_{b_{0}}$ of $X$, for some constant $M=M(\mathcal{I},L_{0},\omega,L)$.

The proof, given the existence of tests, follows the standard format of
Ghosal–Ghosh–van der Vaart [10]. A main step in the proof of Theorem 20 is to
demonstrate an evidence lower bound.

######  Lemma 21 ((Evidence lower bound, ELBO)).

Recall we defined

$B_{KL}^{(n)}(\varepsilon)=\left\\{b\in\Theta:K(p_{0}^{(n)},p_{b}^{(n)})\leq(n\Delta+1)\varepsilon^{2},\operatorname{Var}_{b_{0}}\Big{(}\log\frac{p_{0}^{(n)}}{p_{b}^{(n)}}\Big{)}\leq(n\Delta+1)\varepsilon^{2}\right\\},$

where $p_{b}^{(n)}$ is the joint probability density of
$X_{0},\dots,X_{n\Delta}$ started from the invariant distribution when $b$ is
the true parameter and $p_{0}^{(n)}$ denotes $p_{b_{0}}^{(n)}$. Let
$n\Delta\varepsilon_{n}^{2}\to\infty$ and write $B_{KL}^{(n)}$ for
$B_{KL}^{(n)}(\varepsilon_{n})$. Define the event

$A_{n}=\Big{\\{}\int_{\Theta}(p_{b}^{(n)}/p_{0}^{(n)})\mathop{}\\!\mathrm{d}\Pi(b)\geq\Pi(B_{KL}^{(n)})e^{-2n\Delta\varepsilon_{n}^{2}}\Big{\\}}.$

Then as $n\to\infty$, $P_{b_{0}}\left(A_{n}^{c}\right)\to 0.$

###### Proof.

Write $\Pi^{\prime}=\Pi/\Pi(B_{KL}^{(n)})$ for the renormalised restriction of
$\Pi$ to $B_{KL}^{(n)}$. Then by Jensen’s inequality we have

$\int_{\Theta}(p_{b}^{(n)}/p_{0}^{(n)})(X^{(n)})\mathop{}\\!\mathrm{d}\Pi(b)\geq\Pi(B_{KL}^{(n)})\exp\left(\int_{B_{KL}^{(n)}}\log(p_{b}^{(n)}/p_{0}^{(n)})(X^{(n)}))\mathop{}\\!\mathrm{d}\Pi^{\prime}(b)\right).$

Write
$Z=\int_{B_{KL}^{(n)}}\log(p_{b}^{(n)}/p_{0}^{(n)})\mathop{}\\!\mathrm{d}\Pi^{\prime}(b)=-\int_{B_{KL}^{(n)}}\log(p_{0}^{(n)}/p_{b}^{(n)})\mathop{}\\!\mathrm{d}\Pi^{\prime}(b)$.
Applying Fubini’s Theorem and using the definition of $B_{KL}^{(n)}$, we see
that

$E_{b_{0}}Z\geq-\sup_{b\in
B_{KL}^{(n)}}E_{b_{0}}\log(p_{0}^{(n)}/p_{b}^{(n)})\geq-(n\Delta+1)\varepsilon_{n}^{2}.$

Further, applying Jensen’s inequality and twice applying Fubini’s Theorem, we
see

$\displaystyle\operatorname{Var}_{b_{0}}Z$
$\displaystyle=E_{b_{0}}\left(\int_{B_{KL}^{(n)}}\log(p_{b}^{(n)}/p_{0}^{(n)})\mathop{}\\!\mathrm{d}\Pi^{\prime}(b)-E_{b_{0}}Z\right)^{2}$
$\displaystyle=E_{b_{0}}\left(\int_{B_{KL}^{(n)}}\Big{[}\log(p_{b}^{(n)}/p_{0}^{(n)})-E_{b_{0}}\log(p_{b}^{(n)}/p_{0}^{(n)})\Big{]}\mathop{}\\!\mathrm{d}\Pi^{\prime}(b)\right)^{2}$
$\displaystyle\leq
E_{b_{0}}\int_{B_{KL}^{(n)}}\left(\log(p_{b}^{(n)}/p_{0}^{(n)})-E_{b_{0}}\log(p_{b}^{(n)}/p_{0}^{(n)})\right)^{2}\mathop{}\\!\mathrm{d}\Pi^{\prime}(b)$
$\displaystyle=\int_{B_{KL}^{(n)}}\operatorname{Var}_{b_{0}}\left(\log(p_{0}^{(n)}/p_{b}^{(n)})\right)\mathop{}\\!\mathrm{d}\Pi^{\prime}(b)\leq(n\Delta+1)\varepsilon_{n}^{2},$

where to obtain the inequality in the final line we have used the bound on the
variance of $\log(p_{0}^{(n)}/p_{b}^{(n)})$ for $b\in B_{KL}^{(n)}$.

Together, these bounds on the mean and variance of $Z$ tell us that

$P_{b_{0}}\left(\exp(Z)<\exp(-2n\Delta\varepsilon_{n}^{2})\right)\leq
P_{b_{0}}\left(\lvert
Z-EZ\rvert>(n\Delta-1)\varepsilon_{n}^{2}\right)\leq\frac{(n\Delta+1)\varepsilon_{n}^{2}}{(n\Delta-1)^{2}\varepsilon_{n}^{4}},$

where we have applied Chebyshev’s inequality to obtain the final inequality.
The rightmost expression tends to zero since
$n\Delta\varepsilon_{n}^{2}\to\infty$ by assumption, and the result follows. ∎

######  Remark.

The same is true, but with $P_{b_{0}}(A_{n}^{c})$ tending to zero at a
different rate, if we define $A_{n}$ instead by
$A_{n}=\\{\int_{\Theta}(p_{b}^{(n)}/p_{0}^{(n)}\mathop{}\\!\mathrm{d}\Pi(b)\geq\Pi(B_{KL}^{(n)})e^{-Bn\Delta\varepsilon_{n}^{2}}\\}$
for any $B>1$. That is to say, the exact value 2 in the exponent is not
important for the proof.

###### Proof of Theorem 20.

We write $\Pi$ for $\Pi^{(n)}$. Since $\Pi(\Theta)=1$ by assumption, it is
enough to show $E_{b_{0}}\Pi\left(\\{b\in\Theta:\lVert
b-b_{0}\rVert_{2}>M\varepsilon_{n}\\}\mid X^{(n)}\right)\to 0$.

Observe, for any measurable sets $S$ and $\Theta_{n}$, any event $A_{n}$ and
any $\\{0,1\\}$–valued function $\psi_{n}$ we can decompose

$\Pi(S\mid X^{(n)})\leq\mathbbm{1}_{A_{n}^{c}}+\psi_{n}+\Pi(\Theta_{n}^{c}\mid
X^{(n)})\mathbbm{1}_{A_{n}}+\Pi(S\cap\Theta_{n}\mid
X^{(n)})\mathbbm{1}_{A_{n}}(1-\psi_{n}).$

We apply the above to

$S=S_{M}^{(n)}=\\{b\in\Theta:\lVert
b-b_{0}\rVert_{2}>M\varepsilon_{n}\\},\quad
A_{n}=\Big{\\{}\int_{\Theta}(p_{b}^{(n)}/p_{0}^{(n)})(X^{(n)})\mathop{}\\!\mathrm{d}\Pi(b)\geq
e^{-(\omega+2)n\Delta\varepsilon_{n}^{2}}\Big{\\}},$

with $\Theta_{n}$ as given in the statement of the theorem and with $\psi_{n}$
the tests given by Lemma 5, noting that the assumptions for Theorem 20 include
those needed for Lemma 5. We take the expectation and bound each of the terms
separately.

##### Bounding $E_{b_{0}}\mathbbm{1}_{A_{n}^{c}}$:

We have $P_{b_{0}}(A_{n}^{c})\to 0$ from Lemma 21, since by assumption
$\Pi(B_{KL}^{(n)}(\varepsilon_{n}))\geq e^{-\omega
n\Delta\varepsilon_{n}^{2}}$.

##### Bounding $E_{b_{0}}\psi_{n}$:

This expectation tends to zero by Lemma 5.

##### Bounding $E_{b_{0}}[\Pi(\Theta_{n}^{c}\mid
X^{(n)})\mathbbm{1}_{A_{n}}]$:

We have

$\displaystyle\Pi(\Theta_{n}^{c}\mid X^{(n)})\mathbbm{1}_{A_{n}}$
$\displaystyle=\frac{\int_{\Theta_{n}^{c}}p_{b}^{(n)}(X^{(n)})\mathop{}\\!\mathrm{d}\Pi(b)}{\int_{\Theta}p_{b}^{(n)}(X^{(n)})\mathop{}\\!\mathrm{d}\Pi(b)}\mathbbm{1}_{A_{n}}$
$\displaystyle=\frac{\int_{\Theta_{n}^{c}}(p_{b}^{(n)}/p_{0}^{(n)})(X^{(n)})\mathop{}\\!\mathrm{d}\Pi(b)}{\int_{\Theta}(p_{b}^{(n)}/p_{0}^{(n)})(X^{(n)})\mathop{}\\!\mathrm{d}\Pi(b)}\mathbbm{1}_{A_{n}}$
$\displaystyle\leq
e^{(\omega+2)n\Delta\varepsilon_{n}^{2}}\int_{\Theta_{n}^{c}}(p_{b}^{(n)}/p_{0}^{(n)})(X^{(n)})\mathop{}\\!\mathrm{d}\Pi(b).$

Since $E_{b_{0}}[(p_{b}^{(n)}/p_{0}^{(n)})(X^{(n)})]=E_{b}[1]=1$, taking
expectations and applying Fubini’s Theorem yields
$E_{b_{0}}[\Pi(\Theta_{n}^{c}\mid X^{(n)})\mathbbm{1}_{A_{n}}]\leq
e^{(\omega+2)n\Delta\varepsilon_{n}^{2}}\Pi(\Theta_{n}^{c}).$ Since we assumed
$\Pi(\Theta_{n}^{c})\leq e^{-(\omega+4)n\Delta\varepsilon_{n}^{2}}$, we deduce
that

$E_{b_{0}}[\Pi(\Theta_{n}^{c}\mid
X^{(n)})\mathbbm{1}_{A_{n}}]\leq\exp\left((\omega+2)n\Delta\varepsilon_{n}^{2}-(\omega+4)n\Delta\varepsilon_{n}^{2}\right)\to
0.$

##### Bounding $E_{b_{0}}[\Pi(S\cap\Theta_{n}\mid
X^{(n)})\mathbbm{1}_{A_{n}}(1-\psi_{n})]$:

By a similar argument to the above, observe that

$E_{b_{0}}[\Pi(S\cap\Theta_{n}\mid
X^{(n)})\mathbbm{1}_{A_{n}}(1-\psi_{n})]\leq
e^{(\omega+2)n\Delta\varepsilon_{n}^{2}}\int_{b\in\Theta_{n}:\lVert
b-b_{0}\rVert_{2}>M\varepsilon_{n}}E_{b}[1-\psi_{n}(X^{(n)})]\mathop{}\\!\mathrm{d}\Pi(b).$

The integrand is bounded by $\sup_{b\in\Theta_{n}:\lVert
b-b_{0}\rVert_{2}>M\varepsilon_{n}}E_{b}[1-\psi_{n}(X^{(n)})]\leq
e^{-Dn\Delta\varepsilon_{n}^{2}}$ by construction of the tests $\psi_{n}$,
where by choosing $M$ large enough we could attain any fixed $D$ in the
exponential term. Choosing $M$ corresponding to some $D>\omega+2$ we see

$E_{b_{0}}[\Pi(S\cap\Theta_{n}\mid X^{(n)})\mathbbm{1}_{A_{n}}(1-\psi_{n})]\to
0.\qed$

###### Proof of Theorem 1.

  1. A.

We apply Theorem 20. The key idea which allows us to control the bias and
obtain this adaptive result with a sieve prior is _undersmoothing_.
Specifically, when we prove the small ball probabilities, we do so by
conditioning on the hyperprior choosing a resolution $j_{n}$ which corresponds
to the minimax rate $(n\Delta)^{-s/(1+2s)}$ rather than corresponding to the
slower rate $(n\Delta)^{-s/(1+2s)}\log(n\Delta)^{1/2}$ at which we prove
contraction. This logarithmic gap gives us the room we need to ensure we can
achieve the bias condition a and the small ball condition b for the _same_
constant $\omega$. The argument goes as follows.

Write $\bar{\varepsilon}_{n}^{2}=(n\Delta)^{-2s/(1+2s)}$ and let
$\varepsilon_{n}^{2}=(n\Delta)^{-2s/(1+2s)}\log(n\Delta)$. Choose $j_{n}$ and
$l_{n}$ natural numbers satisfying (at least for $n$ large enough)

$\frac{1}{2}n\Delta\bar{\varepsilon}_{n}^{2}\leq D_{j_{n}}=2^{j_{n}}\leq
n\Delta\bar{\varepsilon}_{n}^{2},\qquad\frac{1}{2}Ln\Delta\varepsilon_{n}^{2}\leq
D_{l_{n}}=2^{l_{n}}\leq Ln\Delta\varepsilon_{n}^{2},$

where $L$ is a constant to be chosen. Note that 16 holds by definition. Recall
now from our choice of approximation spaces in Section 2.1 that we have
$\lVert\pi_{m}b_{0}-b_{0}\rVert_{2}\leq K(s)\lVert
b_{0}\rVert_{B_{2,\infty}^{s}}2^{-ms}$. For any fixed $L$ we therefore find
that for $n$ large enough, writing $K=K(b_{0})=K(s)2^{s}\lVert
b_{0}\rVert_{B_{2,\infty}^{s}}$, we have

$\displaystyle\lVert\pi_{l_{n}}b_{0}-b_{0}\rVert_{2}\leq
K(b_{0})(Ln\Delta\varepsilon_{n}^{2})^{-s}=K(Ln\Delta\bar{\varepsilon}_{n}^{2}\log(n\Delta))^{-s}=KL^{-s}\bar{\varepsilon}_{n}\log(n\Delta)^{-s}\leq\varepsilon_{n}.$

Similarly, it can be shown that, with $A=A(\mathcal{I})$ the constant of the
small ball result (Theorem 14) and for $n$ large enough, we have $\lVert
b_{0}-\pi_{j_{n}}b_{0}\rVert_{2}\leq A\varepsilon_{n}/2.$

Set $\Theta_{n}=\\{b_{0}\\}\cup(S_{l_{n}}\cap\Theta)$ and observe that the
above calculations show that the bias condition 17 holds (since also for
$b\in\Theta_{n},$ if $b\not=b_{0}$ we have
$\lVert\pi_{l_{n}}b-b\rVert_{2}=0$).

Next, for the small ball condition b, recall Theorem 14 tells us that
$\\{b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq\nobreak
A\varepsilon_{n}\\}\subseteq B_{KL}^{(n)}(\varepsilon_{n})$ for all $n$ large
enough. Thus it suffices to show, for some $\omega>0$ for which we can also
achieve a, that $\Pi(\\{b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq
A\varepsilon_{n}\\})\geq e^{-\omega n\Delta\varepsilon_{n}^{2}}$. Using that
$\lVert b-b_{0}\rVert_{2}\leq\lVert
b-\pi_{j_{n}}b_{0}\rVert_{2}+\lVert\pi_{j_{n}}b_{0}-b_{0}\rVert_{2}\leq\lVert
b-\pi_{j_{n}}b_{0}\rVert_{2}+A\varepsilon_{n}/2$, and using our assumptions on
$h$ and $\Pi_{m}$, we see that

$\displaystyle\Pi(\\{b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq
A\varepsilon_{n}\\})$ $\displaystyle=\sum_{m}h(m)\Pi_{m}(\\{b\in S_{m}:\lVert
b-b_{0}\rVert_{2}\leq A\varepsilon_{n}\\}),$ $\displaystyle\geq
h(j_{n})\Pi_{j_{n}}\left(\\{b\in S_{j_{n}}:\lVert
b-\pi_{j_{n}}b_{0}\rVert_{2}\leq A\varepsilon_{n}/2\\}\right)$
$\displaystyle\geq h(j_{n})(\varepsilon_{n}A\zeta/2)^{D_{j_{n}}}$
$\displaystyle\geq
B_{1}\exp\left({-\beta_{1}D_{j_{n}}}+D_{j_{n}}[\log(\varepsilon_{n})+\log(A\zeta/2)]\right)$
$\displaystyle\geq
B_{1}\exp\left(-Cn\Delta\bar{\varepsilon}_{n}^{2}-Cn\Delta\bar{\varepsilon}_{n}^{2}\log(\varepsilon_{n}^{-1})\right)$

for some constant $C=C(\mathcal{I},\beta_{1},\zeta)$. Since
$\log(\varepsilon_{n}^{-1})=\frac{s}{1+2s}\log(n\Delta)-\frac{1}{2}\log\log(n\Delta)\leq\log(n\Delta),$
we deduce that $\Pi(\\{b\in\Theta:\lVert b-b_{0}\rVert_{2}\leq
A\varepsilon_{n}\\})\geq
B_{1}e^{-C^{\prime}n\Delta\bar{\varepsilon}_{n}^{2}\log(n\Delta)}=B_{1}e^{-C^{\prime}n\Delta\varepsilon_{n}^{2}},$
with a different constant $C^{\prime}$. Changing constant again to some
$\omega=\omega(\mathcal{I},\beta_{1},B_{1},\zeta)$, we absorb the $B_{1}$
factor into the exponential for large enough $n$.

For a, since $\Pi(\Theta^{c})=0$ by assumption, we have
$\Pi(\Theta_{n}^{c})\leq\Pi(S_{l_{n}}^{c})=\sum_{m=l_{n}+1}^{\infty}h(m).$ We
have assumed that $h(m)\leq B_{2}e^{-\beta_{2}D_{m}}$, which ensures that the
sum is at most a constant times $e^{-\beta_{2}D_{l_{n}}}\leq
e^{-\frac{1}{2}L\beta_{2}n\Delta\varepsilon_{n}^{2}}$. For the
$\omega=\omega(\mathcal{I},\beta_{1},B_{1},\zeta)$ for which we proved b
above, we can therefore choose $L$ large enough to guarantee
$\Pi(\Theta_{n}^{c})\leq e^{-(\omega+4)n\Delta\varepsilon_{n}^{2}}$.

  2. B.

Let $\varepsilon_{n}$ and $j_{n}$ be as in the statement of the theorem and
define $l_{n}$ as above (here we can take $L=1$). Similarly to before, we
apply results from Section 2.1 to see

$\begin{rcases*}\lVert\pi_{l_{n}}b-b\rVert_{2}\leq\varepsilon_{n}\\\
\lVert\pi_{j_{n}}b-b\rVert_{2}\leq\varepsilon_{n}\end{rcases*}\text{ for all
$n$ sufficiently large and all $b\in\Theta_{s}(A_{0})$},$

Set $\Theta_{n}=\Theta_{s}(A_{0})$ for all $n$. Our assumptions then guarantee
the bias condition a will hold for any $\omega$ (indeed,
$\Pi^{(n)}(\Theta_{n}^{c})=0$). Thus it suffices to prove that there exists an
$\omega$ such that $\Pi^{(n)}(\\{b\in\Theta_{s}(A_{0}):\lVert
b-b_{0}\rVert_{2}\leq 3\varepsilon_{n}\\})\geq e^{-\omega
n\Delta\varepsilon_{n}^{2}},$ since we can absorb the factor of 3 into the
constant $M$ by applying Theorem 20 to $\xi_{n}=3\varepsilon_{n}$.

The prior concentrates on $\Theta_{s}(A_{0})$, so that we have
$\Pi^{(n)}(\\{b:\lVert\pi_{j_{n}}b-b\rVert_{2}\leq\varepsilon_{n}\\})=1$, and
$b_{0}$ lies in $\Theta_{s}(A_{0})$, so that
$\lVert\pi_{j_{n}}b_{0}-b_{0}\rVert_{2}\leq\varepsilon_{n}$. Thus

$\Pi^{(n)}(\\{b\in\Theta_{s}(A_{0}):\lVert b-b_{0}\rVert_{2}\leq
3\varepsilon_{n}\\})\geq\Pi^{(n)}(\\{b\in\Theta_{s}(A_{0}):\lVert\pi_{j_{n}}b-\pi_{j_{n}}b_{0}\rVert_{2}\leq\varepsilon_{n}\\}).$

From here the argument is very similar to the previous part (indeed, it is
slightly simpler) so we omit the remaining details. ∎

###  Explicit priors: proofs

###### Proof of Proposition 2.

We verify that the conditions of 1A are satisfied. Condition i holds by
construction. The $B_{\infty,1}^{s}$–norm can be expressed as

$\lVert f\rVert_{B_{\infty,1}^{s}}=\lvert
f_{-1,0}\rvert+\sum_{l=0}^{\infty}2^{l(s+1/2)}\max_{0\leq k<2^{l}}{\lvert
f_{lk}\rvert},$ (18)

(see [15] Section 4.3) so that any $b$ drawn from our prior lies in
$B^{1}_{\infty,1}$ and satisfies the bound $\lVert
b\rVert_{B^{1}_{\infty,1}}\leq(B+1)(2+\sum_{l\geq 1}l^{-2})$. It follows from
standard Besov spaces results (eg. [15] Proposition 4.3.20, adapted to apply
to periodic Besov spaces) that $b\in C_{\text{per}}^{1}([0,1])$, with a
$C_{\text{per}}^{1}$–norm bounded in terms of $B$. Thus $\Pi(\Theta)=1$ for an
appropriate choice of $K_{0}$. We similarly see that $b_{0}\in\Theta$. It
remains to show that ii holds. We have

$\displaystyle\lVert
b-\pi_{m}b_{0}\rVert_{2}^{2}=\sum_{\begin{subarray}{c}-1\leq l<m\\\ 0\leq
k<2^{l}\end{subarray}}\tau_{l}^{2}(u_{lk}-\beta_{lk})^{2}$
$\displaystyle\leq\Big{(}1+\sum_{l=0}^{m-1}2^{-2l}\Big{)}\max_{\begin{subarray}{c}-1\leq
l<m,\\\ 0\leq k<2^{l}\end{subarray}}\lvert
u_{lk}-\beta_{lk}\rvert^{2}<4\max_{\begin{subarray}{c}-1\leq l<m,\\\ 0\leq
k<2^{l}\end{subarray}}\lvert u_{lk}-\beta_{lk}\rvert^{2},$

so that $\Pi(\\{b\in S_{m}:\lVert
b-\pi_{m}b_{0}\rVert_{2}\leq\varepsilon\\})\geq\Pi(\lvert
u_{lk}-\beta_{lk}\rvert\leq\varepsilon/2~{}~{}\forall l,k,-1\leq
l<m,k<2^{l}).$ Since we have assumed $\lvert\beta_{lk}\rvert\leq B\tau_{l}$
and $q(x)\geq\zeta$ for $\lvert x\rvert\leq B$, it follows from independence
of the $u_{lk}$ that the right-hand side of this last expression is lower
bounded by $(\varepsilon\zeta/2)^{D_{m}},$ so that ii holds with $\zeta/2$ in
place of $\zeta$. ∎

###### Proof of Proposition 3.

We verify the conditions of 1B. Since $s>1$ similarly to the proof of
Proposition 2 we see $\Pi^{(n)}(\Theta)=1$ and $b_{0}\in\Theta$ for an
appropriate choice of $K_{0}$. Observe also that for $A_{0}=2B+2$ we have
$\Pi^{(n)}(\Theta_{s}(A_{0}))=1$ by construction, and
$b_{0}\in\Theta_{s}(A_{0})$ by Assumption 5, using the wavelet
characterisation 2 of $\lVert\cdot\rVert_{B_{2,\infty}^{s}}$. Thus I holds and
it remains to check II.

Let $j_{n}\in\mathbb{N}$ be such that $j_{n}\leq\bar{L}_{n}$,
$2^{j_{n}}\sim(n\Delta)^{1/(1+2s)}.$ Similarly to the proof of Proposition 2
we have

$\Pi^{(n)}(\\{b\in\Theta:\lVert\pi_{j_{n}}b-\pi_{j_{n}}b_{0}\rVert_{2}\leq\varepsilon_{n}\\})\geq\Pi^{(n)}(\lvert
u_{lk}-\beta_{lk}\rvert\leq\varepsilon_{n}/2~{}~{}\forall l<j_{n},~{}\forall
k<2^{l})\geq(\varepsilon_{n}\zeta/2)^{D_{j_{n}}},$

so we’re done. ∎

###### Proof of Proposition 4.

We include only the key differences to the previous proofs.

Adapting slightly the proof of Proposition 2, we see that $H$ and $H_{0}$ both
have $B_{\infty,1}^{2}$–norm bounded by $(B+1)(2+\sum_{l\geq 1}l^{-2}).$ Since
$\lVert
b\rVert_{C^{1}_{\text{per}}}\leq\frac{1}{2}\lVert\sigma^{2}\rVert_{C^{1}_{\text{per}}}(1+\lVert
H\rVert_{C^{2}_{\text{per}}})$ and using [15] Proposition 4.3.20, adapted to
apply to periodic Besov spaces, to control $\lVert
H\rVert_{C^{2}_{\text{per}}}$ by $\lVert H\rVert_{B^{2}_{\infty,1}}$, we see
that for some constant $K_{0}=K_{0}(B)$ we have $b_{0}\in\Theta(K_{0})$ and
$\Pi^{(n)}(\Theta(K_{0}))=1$. From the wavelet characterisation

$\lVert f\rVert_{B_{2,2}^{s}}=\lvert
f_{-1,0}\rvert+\Big{(}\sum_{l=0}^{\infty}2^{2ls}\sum_{k=0}^{2^{l}-1}f_{lk}^{2}\Big{)}^{1/2}$

it can be seen that $H$ and $H_{0}$ have Sobolev norm
$\lVert\cdot\rVert_{B_{2,2}^{s+1}}$ bounded by some $A_{0}^{\prime}$, hence
for some constant $K=K(A_{0}^{\prime},s)$ we have $\lVert
H-\pi_{m}H\rVert_{B_{2,2}^{1}}\leq K2^{-ms}$ and similarly for $H_{0}$. Since
the $B_{2,2}^{s+1}$ norm controls the $B^{s+1}_{2,\infty}$ norm, and we have
assumed $\sigma^{2}\in\Theta_{s+1}$, we additionally see that
$b_{0}\in\Theta_{s}(A_{0})$ and $\Pi^{(n)}(\Theta_{s}(A_{0}))=1$ for an
appropriate constant $A_{0}$. Note that here we also depend on the assumption
$\sigma^{2}\in C^{s}$ to allow us to control $\lVert
b\rVert_{B^{s}_{2,\infty}}$: Remark 1 on page 143 of Triebel [31] and
Proposition 4.3.20 from [15] together tell us that
$\lVert\sigma^{2}H^{\prime}\rVert_{B^{s}_{2,\infty}}\leq
c\lVert\sigma^{2}\rVert_{C^{\alpha}}\lVert
H^{\prime}\rVert_{B^{s}_{2,\infty}}$ for some constant $c=c(s)$, and similarly
for $H_{0}$.

Observe, for $j_{n}\in\mathbb{N}$ such that $j_{n}\leq\bar{L}_{n}$ and
$2^{j_{n}}\sim(n\Delta)^{1/(1+2s)}$,

$\displaystyle\lVert\pi_{j_{n}}b-\pi_{j_{n}}b_{0}\rVert_{2}$
$\displaystyle\leq\lVert
b-b_{0}\rVert_{2}=\lVert\sigma^{2}(H^{\prime}-H_{0}^{\prime})/2\rVert_{2}\leq\frac{1}{2}\sigma_{U}^{2}\lVert
H-H_{0}\rVert_{B_{2,2}^{1}}$
$\displaystyle\qquad\leq\frac{\sigma_{U}^{2}}{2}\Big{(}\lVert
H-\pi_{j_{n}}H\rVert_{B_{2,2}^{1}}+\lVert
H_{0}-\pi_{j_{n}}H_{0}\rVert_{B_{2,2}^{1}}+\lVert\pi_{j_{n}}H-\pi_{j_{n}}H_{0}\rVert_{B_{2,2}^{1}}\Big{)}.$

Now $\sigma_{U}^{2}\lVert
H-\pi_{j_{n}}H\rVert_{B_{2,2}^{1}}\leq\sigma_{U}^{2}K2^{-j_{n}s}\leq
C(n\Delta)^{-s/(1+2s)}\leq\frac{1}{2}(n\Delta)^{-s/(1+2s)}\log(n\Delta)^{1/2}=\frac{1}{2}\varepsilon_{n}$
for large enough $n$, and similarly for $H_{0}$.

Thus,

$\displaystyle\Pi^{(n)}\Big{(}\big{\\{}b:\lVert\pi_{j_{n}}b-\pi_{j_{n}}b_{0}\rVert_{2}\leq\varepsilon_{n}\big{\\}}\Big{)}$
$\displaystyle\geq\Pi^{(n)}\Big{(}\big{\\{}b:\lVert\pi_{j_{n}}H-\pi_{j_{n}}H_{0}\rVert_{B_{2,2}^{1}}\leq\sigma_{U}^{-2}\varepsilon_{n}/2\big{\\}}\Big{)}$
$\displaystyle\geq\Pi^{(n)}(\lvert
u_{lk}-\beta_{lk}\rvert\leq\kappa\varepsilon_{n}~{}~{}\forall
l<j_{n},~{}\forall k<2^{l}),$

where the final inequality can be seen to hold from the wavelet representation
of $\lVert\cdot\rVert_{B_{2,2}^{1}}$ (the constant $\kappa$ can be taken to be
$\kappa=\frac{1}{2}\sigma_{U}^{-2}(1+(\sum_{k=0}^{\infty}2^{-2l})^{1/2})^{-1}>\sigma_{U}^{-2}/6)$.
The small ball condition II follows from our updated assumptions. ∎

## Acknowledgements

This work was supported by the UK Engineering and Physical Sciences Research
Council (EPSRC) grant EP/L016516/1 for the University of Cambridge Centre for
Doctoral Training, the Cambridge Centre for Analysis. I would like to thank
Richard Nickl for his valuable support throughout the process of writing this
paper. I would also like to thank two anonymous referees for their very
helpful suggestions.

##  Appendix A Technical lemmas

######  Lemma 22.

Let $\mathbb{Q},\mathbb{P}$ be mutually absolutely continuous probability
measures and write
$f=\frac{\mathop{}\\!\mathrm{d}\mathbb{Q}}{\mathop{}\\!\mathrm{d}\mathbb{P}}$.
Then, for any measurable $g$ and any sub–$\sigma$–algebra $\mathcal{G}$,
${E_{\mathbb{Q}}[g\mid\mathcal{G}]}=\frac{E_{\mathbb{P}}[fg\mid\mathcal{G}]}{E_{\mathbb{P}}[f\mid\mathcal{G}]}.$

###### Proof.

This follows straightforwardly using the characterisation of conditional
expectation in terms of expectations against $\mathcal{G}$–measurable
functions. Precisely, we recall that

$E_{\mathbb{P}}[c(X)v(X)]=E_{\mathbb{P}}[u(X)v(X)]$ ($\star$)

holds for any $\mathcal{G}$–measurable function $v$ if
$c(X)=E_{\mathbb{P}}[u(X)\mid\mathcal{G}]$ a.s., and conversely if $c(X)$ is
$\mathcal{G}$–measurable and $\star$ ‣ Appendix A holds for any
$\mathcal{G}$–measurable $v$ then $c(X)$ is a version of the conditional
expectation $E_{\mathbb{P}}[u(X)]$. For the converse statement it is in fact
enough for $\star$ ‣ Appendix A to hold for all indicator functions
$v=\mathbbm{1}_{A}$, $A\in\mathcal{G}$.

Applying $\star$ ‣ Appendix A repeatedly we find, for $A\in\mathcal{G}$,

$E_{\mathbb{P}}\left[E_{\mathbb{Q}}[g\mid\mathcal{G}]E_{\mathbb{P}}[f\mid\mathcal{G}]\mathbbm{1}_{A}\right]=E_{\mathbb{P}}\left[fE_{\mathbb{Q}}[g\mid\mathcal{G}]\mathbbm{1}_{A}\right]=E_{\mathbb{Q}}\left[E_{\mathbb{Q}}[g\mid\mathcal{G}]\mathbbm{1}_{A}\right]=E_{\mathbb{Q}}\left[g\mathbbm{1}_{A}\right]=E_{\mathbb{P}}\left[fg\mathbbm{1}_{A}\right],$

so that, since also
$E_{\mathbb{Q}}[g\mid\mathcal{G}]E_{\mathbb{P}}[f\mid\mathcal{G}]$ is
$\mathcal{G}$-measurable, it is (a version of)
$E_{\mathbb{P}}\left[fg\mid\mathcal{G}\right]$, as required. ∎

######  Lemma 23.

The variance of the log likelihood ratio tensorises in this model, up to a
constant. Precisely,
$\operatorname{Var}_{b_{0}}\log\left(\frac{p_{0}^{(n)}(X^{(n)})}{p_{b}^{(n)}(X^{(n)})}\right)\leq
3\operatorname{Var}_{b_{0}}\left(\log\frac{\pi_{0}(X_{0})}{\pi_{b}(X_{0})}\right)+3n\operatorname{Var}_{b_{0}}\left(\log\frac{p_{0}(X_{0},X_{\Delta})}{p_{b}(X_{0},X_{\Delta})}\right).$

###### Proof.

We write
$\log\Big{(}\frac{p_{0}^{(n)}(X^{(n)})}{p_{b}^{(n)}(X^{(n)})}\Big{)}=U+V+W,$
where $U=\log\frac{\pi_{0}(X_{0})}{\pi_{b}(X_{0})}$ and

$V=\sum_{\begin{subarray}{c}1\leq k\leq n\\\ k\text{
odd}\end{subarray}}\log\frac{p_{0}(\Delta,X_{(k-1)\Delta},X_{k\Delta})}{p_{b}(\Delta,X_{(k-1)\Delta},X_{k\Delta})},\qquad
W=\sum_{\begin{subarray}{c}1\leq k\leq n\\\ k\text{
even}\end{subarray}}\log\frac{p_{0}(\Delta,X_{(k-1)\Delta},X_{k\Delta})}{p_{b}(\Delta,X_{(k-1)\Delta},X_{k\Delta})}.$

Note now that $V$ and $W$ are both sums are of _independent_ terms since
$(X_{k\Delta})_{k\leq n}$ is a Markov chain. We thus have

$\operatorname{Var}_{b_{0}}(V)=\\#\\{1\leq k\leq n:k\text{
odd}\\}\operatorname{Var}_{b_{0}}\left(\log\frac{p_{0}(X_{0},X_{\Delta})}{p_{b}(X_{0},X_{\Delta})}\right),$

and a corresponding result for $W$. Using
$\operatorname{Var}(R+S+T)=\operatorname{Var}(R)+\operatorname{Var}(S)+\operatorname{Var}(T)+2\operatorname{Cov}(R,S)+2\operatorname{Cov}(S,T)+2\operatorname{Cov}(T,R)$
and $2\operatorname{Cov}(R,S)\leq\operatorname{Var}(R)+\operatorname{Var}(S)$,
one derives the elementary inequality $\operatorname{Var}(U+V+W)\leq
3(\operatorname{Var}(U)+\operatorname{Var}(V)+\operatorname{Var}(W)).$ The
result follows. ∎

######  Lemma 24.

Let $\tilde{p}_{0}$ be as in 14. Let $p^{*}(\Delta,x,y)$ be the density of
transitions from $x$ to $y$ in time $\Delta$ for a process
$U\sim\mathbb{W}_{\sigma}^{(x)}$. Then

$\frac{p_{0}(\Delta,x,y)}{p_{*}(\Delta,x,y)}=E_{\mathbb{W}_{\sigma}^{(x)}}\left[\tilde{p}_{0}(U)\mid
U_{\Delta}=y\right].$

###### Proof.

Let $U\sim\mathbb{W}_{\sigma}^{(x)}$ and let $\mathbb{B}_{\sigma}^{(x,y)}$
denote the law on $C([0,\Delta])$ of $U$ conditional on $U_{\Delta}=y$. We
define the conditional law rigorously via disintegration (eg. see [27] Chapter
5, Theorem 9, applied to $\lambda=\mathbb{W}_{\sigma}^{(x)}$,
$\mathcal{X}=C([0,\Delta])$ with the sup norm,
$T((U_{t})_{t\leq\Delta})=U_{\Delta}$ and
$\mu(\mathop{}\\!\mathrm{d}y)=p^{*}(\Delta,x,y)\mathop{}\\!\mathrm{d}y$), so
that

$E_{\mathbb{W}_{\sigma}^{(x)}}[f(U)]=\int_{-\infty}^{\infty}p^{*}(\Delta,x,y)E_{\mathbb{B}_{\sigma}^{(x,y)}}[f(U)]\mathop{}\\!\mathrm{d}y,$

for all non-negative measurable functions $f$. Taking
$f(U)=\tilde{p}_{0}(U)\mathbbm{1}\\{U_{\Delta}\in A\\}$ for an arbitrary Borel
set $A\subseteq\mathbb{R}$, we see

$P_{b_{0}}^{(x)}(X_{\Delta}\in
A)=\int_{-\infty}^{\infty}p^{*}(\Delta,x,y)\mathbbm{1}\\{y\in
A\\}E_{B_{\sigma}^{(x,y)}}[\tilde{p}_{0}]\mathop{}\\!\mathrm{d}y.$

The result follows. ∎

##  Appendix B Proofs for Section 4.1.2

###### Proof of Lemma 8.

Set $Y_{t}=S(X_{t})$, where

$S(x)=\int_{0}^{x}\exp\Big{(}-\int_{0}^{y}\frac{2b}{\sigma^{2}}(z)\mathop{}\\!\mathrm{d}z\Big{)}\mathop{}\\!\mathrm{d}y$

is the scale function, and let $\psi$ be the inverse of $S$. Since
$S^{\prime\prime}$ exists and is continuous, Itô’s formula applies to yield

$\mathop{}\\!\mathrm{d}Y_{t}=\tilde{\sigma}(Y_{t})\mathop{}\\!\mathrm{d}W_{t},\quad\tilde{\sigma}(y):=S^{\prime}(\psi(y))\sigma(\psi(y)).$

Let $A=A(\mathcal{I})=\max(\sigma_{U}^{2}\exp(4K_{0}/\sigma_{L}^{2}),1)$ and
observe that $\lVert\tilde{\sigma}^{2}\rVert_{\infty}\leq A$. Thus, there are
constants $C=C(\mathcal{I})$ and $\lambda=\lambda(\mathcal{I})$ so that for
any $u>C\max(\log m,1)^{1/2}$, the event

$\mathcal{D}=\left\\{\sup\bigg{\\{}\frac{\lvert
Y_{t}-Y_{s}\rvert}{w_{m}(\lvert t-s\rvert)}:{s,t\in[0,m],~{}s\not=t,~{}\lvert
t-s\rvert\leq A^{-1}e^{-2}}\bigg{\\}}\leq u\right\\},$

occurs with probability at least $1-2e^{-\lambda u^{2}}$, by Lemma 9. Now
$X_{t}=\psi(Y_{t})$ and $\psi$ is Lipschitz with constant
$\lVert\psi^{\prime}\rVert_{\infty}=\lVert
1/(S^{\prime}\circ\psi)\rVert_{\infty}\leq\exp(2K_{0}\sigma_{L}^{-2})$. It
follows that on $\mathcal{D}$, writing $\tau=A^{-1}e^{-2}$, we have for any
$s,t\in[0,m]$, $s\not=t$, $\lvert t-s\rvert\leq\tau$,

$\lvert X_{t}-X_{s}\rvert\leq\exp(2K_{0}\sigma_{L}^{-2})\lvert
Y_{t}-Y_{s}\rvert\leq\exp(2K_{0}\sigma_{L}^{-2})w_{m}(\lvert t-s\rvert)u$

The result follows by relabelling $(\exp(2K_{0}/\sigma_{L}^{2})u)\mapsto u$,
$\lambda\mapsto\lambda\exp(-4K_{0}/\sigma_{L}^{2})$ and $C\mapsto
C\exp(2K_{0}/\sigma_{L}^{2})$. ∎

###### Proof of Lemma 9.

Recall $w_{m}(\delta):=\delta^{1/2}(\log(\delta^{-1})^{1/2}+\log(m)^{1/2})$
for $m\geq 1$ and $w_{m}(\delta):=w_{1}(\delta)$ for $m<1$. We see that we may
assume $m\geq 1$ and the result for $m<1$ will follow. By the (Dambis–)Dubins-
Schwarz Theorem (Rogers & Williams [29], (34.1)), we can write
$Y_{t}=Y_{0}+B_{\eta_{t}}$ for $B$ a standard Brownian motion and for
$\eta_{t}=\langle Y\rangle_{t}$ the quadratic variation of $Y$. Define the
event

$\mathcal{C}=\left\\{\sup\bigg{\\{}\frac{\lvert
B_{t^{\prime}}-B_{s^{\prime}}\rvert}{w_{Am}(\lvert
t^{\prime}-s^{\prime}\rvert)}:~{}s^{\prime},t^{\prime}\in[0,Am],~{}s^{\prime}\not=t^{\prime},~{}\lvert
t^{\prime}-s^{\prime}\rvert\leq e^{-2}\bigg{\\}}\leq u\right\\}.$

By Lemma 10, there are universal constants $C$ and $\lambda$ so that for
$u>C\max(\log(Am),1)^{1/2}$, $\mathcal{C}$ occurs with probability at least
$1-2e^{-\lambda u^{2}}$, and note that by allowing $C$ to depend on $A$ we can
replace $\max(\log(Am),1)$ with $\max(\log(m),1)$. On this event, for
$s,t\in[0,m]$ with $\lvert t-s\rvert\leq A^{-1}e^{-2}$ and $s\not=t$ we have

$\displaystyle\lvert Y_{t}-Y_{s}\rvert$ $\displaystyle=\lvert
B_{\eta_{t}}-B_{\eta_{s}}\rvert$ $\displaystyle\leq\sup\\{\lvert
B_{t^{\prime}}-B_{s^{\prime}}\rvert:~{}s^{\prime},t^{\prime}\in[0,Am],~{}s^{\prime}\not=t^{\prime},~{}\lvert
t^{\prime}-s^{\prime}\rvert\leq A\lvert t-s\rvert\\}$ $\displaystyle\leq
u\sup\\{w_{Am}(\lvert
t^{\prime}-s^{\prime}\rvert):~{}s^{\prime},t^{\prime}\in[0,Am],~{}s^{\prime}\not=t^{\prime},~{}\lvert
t^{\prime}-s^{\prime}\rvert\leq A\lvert t-s\rvert\\}$ $\displaystyle\leq
w_{Am}(A\lvert t-s\rvert)u,$

where we have used that $w_{Am}(\delta)$ is increasing in the range
$\delta\leq e^{-2}$ to attain the final inequality. Recalling we assume $A\geq
1$, one sees that $w_{Am}(A\delta)\leq A^{1/2}w_{Am}(\delta)$ provided
$\delta\leq A^{-1}$, which holds in the relevant range. Thus, on
$\mathcal{C}$, and for $s,$ $t$ and $u$ in the considered ranges,

$\displaystyle\lvert Y_{t}-Y_{s}\rvert$ $\displaystyle\leq A^{1/2}u\lvert
t-s\rvert^{1/2}\left((\log(Am))^{1/2}+(\log\lvert
t-s\rvert^{-1})^{1/2}\right)$ $\displaystyle\leq A^{\prime}u\lvert
t-s\rvert^{1/2}\left((\log(m))^{1/2}+(\log\lvert
t-s\rvert^{-1})^{1/2}\right),$

where $A^{\prime}$ is a constant depending on $A$ (note we have absorbed a
term depending on $\log(A)$ into the constant, using that $\log(\lvert
t-s\rvert^{-1})\geq 2$). The desired result follows upon relabelling
$A^{\prime}u\mapsto u$ since $C$ and $\lambda$ are here allowed to depend on
$A$.

For the particular case
$\mathop{}\\!\mathrm{d}Y_{t}=\tilde{\sigma}(Y_{t})\mathop{}\\!\mathrm{d}W_{t}$,
we simply observe that $\lvert\langle Y\rangle_{t}-\langle
Y\rangle_{s}\rvert=\lvert\int_{s}^{t}\tilde{\sigma}^{2}(Y_{s})\mathop{}\\!\mathrm{d}s\rvert\leq\lVert\tilde{\sigma}^{2}\rVert_{\infty}\lvert
t-s\rvert$. ∎

###### Proof of Lemma 10.

Assume $m\geq 1$; the result for $m<1$ follows. For a Gaussian process $B$,
indexed by $T$ and with intrinsic covariance (pseudo-)metric
$d(s,t)=(E[(B_{t}-B_{s})^{2}])^{1/2}$, Dudley’s Theorem ([15] Theorem 2.3.8)
says

$E\left[\sup_{s,t\in T,s\not=t}\frac{\lvert
B_{t}-B_{s}\rvert}{\int_{0}^{d(s,t)}\sqrt{\log
N(T,d,x)}\mathop{}\\!\mathrm{d}x}\right]<\infty,$

where $N(T,d,x)$ is the number of (closed) balls of $d-$radius $x$ needed to
cover $T$. Inspecting the proof, it is in fact shown that the process

$C_{u}=\frac{B_{u_{2}}-B_{u_{1}}}{\int_{0}^{d(u_{1},u_{2})}\sqrt{\log(N(T,d,x))}\mathop{}\\!\mathrm{d}x}\quad\text{on}\quad
U=\\{u=(u_{1},u_{2}):u_{1},u_{2}\in T,~{}d(u_{1},u_{2})\not=0\\},$

is a Gaussian process on with bounded and continuous sample paths. It follows
by [15] Theorem 2.1.20 that

$\Pr\left\\{\left\lvert\sup_{v\in V}\lvert C_{v}\rvert-E\sup_{v\in V}\lvert
C_{v}\rvert\right\rvert>u\right\\}\leq 2e^{-u^{2}/2\sigma^{2}},$

for any subset $V$ of $U$, where $\sigma^{2}=\sup_{v\in V}E[C_{v}^{2}].$ We
can upper bound $C_{v}$ by applying the trivial lower bound for the
denominator $\int_{0}^{a}\sqrt{\log N(T,d,x)}\geq\frac{a}{2}\sqrt{\log 2}$ for
any $a=d(u,v)$ with $u,v\in T$ (this follows from the fact that $N(T,d,x)\geq
2$ if $x$ is less than half the diameter of $T$). Using also that $d$ is the
intrinsic covariance metric, we deduce that $EC_{v}^{2}\leq 4/\log 2$, so we
can take $\sigma^{2}=4/\log 2$.

We will apply the result to $B$ a standard Brownian motion on $T=[0,m]$, which
has intrinsic covariance metric $d(s,t)=\lvert t-s\rvert^{1/2}$. For this $T$
and $d$, we have $N(T,d,x)\leq mx^{-2}$. Then, applying Jensen’s inequality,
we see

$\displaystyle\int_{0}^{d(s,t)}\sqrt{\log N(T,d,x)}\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq
d(s,t)^{1/2}\Big{(}\int_{0}^{d(s,t)}\log(N(T,d,x)\Big{)}^{1/2}$
$\displaystyle\leq 2^{1/2}d(s,t)\left[1+\log(d(s,t)^{-1})+\log
m\right]^{1/2}.$

Set $V=\\{u=(s,t)\in U:\lvert t-s\rvert\leq e^{-2}\\}$ and observe that for
$(s,t)\in V$ we have $1+\log(d(s,t)^{-1})=1+\frac{1}{2}{\log(\lvert
t-s\rvert^{-1})}\leq\log(\lvert t-s\rvert^{-1}).$ Noting further that
$(a+b)^{1/2}\leq a^{1/2}+b^{1/2}$ for $a,b\geq 0$ and recalling we defined
$w_{m}(\delta)=\delta^{1/2}((\log\delta^{-1})^{1/2}+\log(m)^{1/2}),$ we see

$\int_{0}^{d(s,t)}\sqrt{\log N(T,d,x)}\mathop{}\\!\mathrm{d}x\leq
2^{1/2}w_{m}(\lvert t-s\rvert).$

Thus, writing $M=E\left[\sup\Big{\\{}\frac{\lvert
B_{t}-B_{s}\rvert}{\int_{0}^{d(s,t)}\sqrt{\log
N(T,d,x)}\mathop{}\\!\mathrm{d}x}:s,t\in T,s\not=t,\lvert t-s\rvert\leq
e^{-2}\Big{\\}}\right]$ we see

$\Pr\Big{[}\sup\Big{\\{}\frac{\lvert B_{t}-B_{s}\rvert}{w_{m}(\lvert
t-s\rvert)}:s,t\in T,s\not=t,\\\ \lvert t-s\rvert\leq
e^{-2}\Big{\\}}>2^{1/2}(M+u)\Big{]}\leq 2e^{-(u^{2}(\log 2)/8)}.$

As $M$ is a fixed finite number, we can write $M+u=(1+\varepsilon)u$ with
$\varepsilon\to 0$ as $u\to\infty$. Then

$\Pr\left[{\sup_{\begin{subarray}{c}s,t\in T,s\not=t,\\\ \lvert t-s\rvert\leq
e^{-2}\end{subarray}}{\frac{\lvert B_{t}-B_{s}\rvert}{w_{m}(\lvert
t-s\rvert)}}}>u\right]\leq 2e^{-(u^{2}(\log 2)/16(1+\varepsilon)^{2})}.$

Thus provided $u$ is larger than $M$, we have the result with the constant
$\lambda=(\log 2)/64$.

Finally we track how $M$ grows with $m$ in order to know when $u$ is large
enough for this lemma to apply. Observe that we can write
$M=E\max_{k}{M_{k}},$ where

$M_{k}=\sup_{\begin{subarray}{c}s,t\in T_{k},s\not=t,\\\ \lvert t-s\rvert\leq
e^{-2}\end{subarray}}\frac{\lvert
B_{t}-B_{s}\rvert}{\int_{0}^{d(s,t)}\sqrt{\log
N(T,d,x)}\mathop{}\\!\mathrm{d}x},\qquad T_{k}=[ke^{-2},(k+2)e^{-2}].$

As $N(T,d,x)\geq N(T_{k},d,x)$, defining

$M_{k}^{\prime}=\sup_{s,t\in T_{k},s\not=t,\lvert t-s\rvert\leq
e^{-2}}\frac{\lvert B_{t}-B_{s}\rvert}{\int_{0}^{d(s,t)}\sqrt{\log
N(T_{k},d,x)}\mathop{}\\!\mathrm{d}x},$

we see $M_{k}\leq M_{k}^{\prime}$. As with the whole process $C$ we can apply
[15] Theorem 2.1.20 to each $M_{k}^{\prime}$ to see that $\Pr(\lvert
M_{k}^{\prime}-EM_{k}^{\prime}\rvert>v)\leq 2e^{-v^{2}/2\sigma^{2}},$ with
$\sigma^{2}=4/\log 2$ as before. That is, each
$(M_{k}^{\prime}-EM_{k}^{\prime})$ is subgaussian with parameter
$12/\sqrt{\log 2}$ (see [15] Lemma 2.3.1). They all have the same constant
(i.e. not depending on $m$) expectation, we can bound their maximum, by
standard results for subgaussian variables (eg. see [15] Lemma 2.3.4):

$EM=E\big{[}EM_{0}^{\prime}+\max_{k}\\{M_{k}^{\prime}-EM_{0}^{\prime}\\}\big{]}\leq
EM_{0}^{\prime}+12\sqrt{2\log N/\log 2},$

where $N$ is the number of $M_{k}^{\prime}$ over which we take the maximum and
scales linearly with $m$. It follows that $M$ is of order bounded by
$\sqrt{\log(m)}$ as $m\to\infty$. ∎

##  Appendix C Notation

We collect most of the notation used in the course of this paper.

$X$: A solution to
$\mathop{}\\!\mathrm{d}X_{t}=b(X_{t})\mathop{}\\!\mathrm{d}t+\sigma(X_{t})\mathop{}\\!\mathrm{d}W_{t}$.

$\dot{X}$: The periodised diffusion $\dot{X}=X\mod 1$.

$b,\sigma$: Drift function, diffusion coefficient.

$\mu=\mu_{b}$; $\pi_{b}$: Invariant distribution/density of $\dot{X}$.

$P_{b}^{(x)}$: Law of $X$ on $C([0,\infty])$ (on $C([0,\Delta])$ in Section 5)
for initial condition $X_{0}=x$.

$E_{b}$; $P_{b}$; $\operatorname{Var}_{b}$: Expectation/probablity/variance
according to the law of $X$ started from $\mu_{b}$.

$E_{\mu};\operatorname{Var}_{\mu}$, and similar: Expectation/variance
according to the subscripted measure.

$\mathbb{W}_{\sigma}^{(x)}$: Notation for $P_{b}^{(x)}$ when $b=0$.

$p_{b}(t,x,y),\dot{p}_{b}(t,x,y)$: Transition densities of $X,\dot{X}$ (with
respect to Lebesgue measure).

$\tilde{p}_{b}$: Density (with respect to $\mathbb{W}_{\sigma}^{(x)}$) of
$P_{b}^{(x)}$ on $C([0,\Delta])$.

$I_{b}(x)=\int_{0}^{x}(2b/\sigma^{2}(y))\mathop{}\\!\mathrm{d}y$.

$X^{(n)}=(X_{0},\dots,X_{n\Delta})$; $x^{(n)}=(x_{0},\dots,x_{n\Delta})$;
$p_{b}^{(n)}(x^{(n)})=\pi_{b}(x_{0})\prod_{i=1}^{n}p_{b}(\Delta,x_{(i-1)\Delta},x_{i\Delta}).$

$b_{0}$: The true parameter generating the data.

$\mu_{0},\pi_{0},p_{0}$ etc.: Shorthand for
$\mu_{b_{0}},\pi_{b_{0}},p_{b_{0}}$ etc.

$\sigma_{L}>0;$ $\sigma_{U}<\infty$: A lower and upper bound for $\sigma$.

$L_{0}$: A constant such that $n\Delta^{2}\log(1/\Delta)\leq L_{0}$ for all
$n$.

$\Theta=\Theta(K_{0})$: The maximal paramater space: $\Theta=\\{f\in
C_{\text{per}}^{1}([0,1]):~{}\lVert f\rVert_{C_{\text{per}}^{1}}\leq
K_{0}\\}$.

$\Theta_{s}(A_{0})=\\{f\in\Theta:\lVert f\rVert_{B_{2,\infty}^{s}}\leq
A_{0}\\}$, for $B_{2,\infty}^{s}$ a (periodic) Besov space.

$\mathcal{I}=\\{K_{0},\sigma_{L},\sigma_{U}\\}$.

$S_{m}$: Wavelet approximation space of resolution $m$, generated by
periodised Meyer-type wavelets: $S_{m}=\operatorname{span}\\{\psi_{lk}:-1\leq
l<m,0\leq k<2^{l}\\},$ where $\psi_{-1,0}$ is used as notation for the
constant function 1.

$D_{m}=\dim(S_{m})=2^{m}$; $\pi_{m}=$($L^{2}$–)orthogonal projection onto
$S_{m}$.

$w_{m}(\delta)=\delta^{1/2}(\log(\delta^{-1})^{1/2}+\log(m)^{1/2})$ if $m\geq
1$, $w_{m}:=w_{1}$ if $m<1$.

$\mathbbm{1}_{A}$: Indicator of the set (or event) $A$.

$K(p,q)$: Kullback–Leibler divergence between densities $p,q$:
$K(p,q)=E_{p}[\log(p/q)].$

$\operatorname{KL}(b_{0},b)=E_{b_{0}}\log(p_{0}/p_{b})$.

$B_{KL}^{(n)}(\varepsilon)=\left\\{b\in\Theta:K(p_{0}^{(n)},p_{b}^{(n)})\leq(n\Delta+1)\varepsilon^{2},\operatorname{Var}_{b_{0}}\Big{(}\log\big{(}{p_{0}^{(n)}}/{p_{b}^{(n)}}\big{)}\Big{)}\leq(n\Delta+1)\varepsilon^{2}\right\\}.$

$B_{\varepsilon}=\left\\{b\in\Theta:K(\pi_{0},\pi_{b})\leq\varepsilon^{2},~{}\operatorname{Var}_{b_{0}}\Big{(}\log\frac{\pi_{0}}{\pi_{b}}\Big{)}\leq\varepsilon^{2},~{}\operatorname{KL}(b_{0},b)\leq\Delta\varepsilon^{2},~{}\operatorname{Var}_{b_{0}}\Big{(}\log\frac{p_{0}}{p_{b}}\Big{)}\leq\Delta\varepsilon^{2}\right\\}.$

$\Pi$: The prior distribution.

$\Pi(\cdot\mid X^{(n)})$: The posterior distribution given data $X^{(n)}$.

$\langle\cdot,\cdot\rangle$: the $L^{2}([0,1])$ inner product, $\langle
f,g\rangle=\int_{0}^{1}f(x)g(x)\mathop{}\\!\mathrm{d}x$.

$\lVert\cdot\rVert_{2}$: The $L^{2}([0,1])$–norm, $\lVert
f\rVert_{2}^{2}=\int_{0}^{1}f(x)^{2}\mathop{}\\!\mathrm{d}x$.

$\lVert\cdot\rVert_{\mu}$: The $L^{2}(\mu)$–norm, $\lVert
f\rVert_{\mu}^{2}=\int_{0}^{1}f(x)^{2}\mu(\mathop{}\\!\mathrm{d}x)=\int_{0}^{1}f(x)^{2}\pi_{b}(x)\mathop{}\\!\mathrm{d}x$.

$\lVert\cdot\rVert_{\infty}$: The $L^{\infty}$– (supremum) norm,333All
functions we use will be continuous hence we can take the supremum rather than
needing the essential supremum. $\lVert
f\rVert_{\infty}=\sup_{x\in[0,1]}\lvert f(x)\rvert$.

$\lVert\rVert_{C_{\text{per}}^{1}}\\!:$ The $C_{\text{per}}^{1}$–norm, $\lVert
f\rVert_{C_{\text{per}}^{1}}=\lVert f\rVert_{\infty}+\lVert