# Filtering of SPDEs:
The Ensemble Kalman Filter and related methods

Sebastian Ertel, TU Berlin

###### Abstract

This paper is concerned with the derivation and mathematical analysis of
continuous time Ensemble Kalman Filters (EnKBFs) and related data assimilation
methods for Stochastic Partial Differential Equations (SPDEs) with finite
dimensional observations. The signal SPDE is allowed to be nonlinear and is
posed in the standard abstract variational setting. Its coefficients are
assumed to satisfy global one-sided Lipschitz conditions.
We first review classical filtering algorithms in this setting, namely the
Kushner–Stratonovich and the Kalman–Bucy filter, proving a law of total
variance.
Then we consider mean-field filtering equations, deriving both a Feedback
Particle Filter and a mean-field EnKBF for nonlinear signal SPDEs.
The second part of the paper is devoted to the elementary mathematical
analysis of the EnKBF in this infinite dimensional setting, showing the well
posedness of both the mean-field EnKBF and its interacting particle
approximation. Finally we prove the convergence of the particle approximation.
Under the additional assumption that the observation function is bounded, we
even recover explicit and (nearly) optimal rates.

###### Contents

  1. 1 Introduction
  2. 2 Problem setting, assumptions and notations
  3. 3 The Kushner-Stratonovich equation and the law of total variance
  4. 4 Linear and Gaussian Filtering
    1. 4.1 The classical Kalman–Bucy filter
    2. 4.2 The consistent mean field EnKBF
  5. 5 The Feedback Particle Filter
  6. 6 The mean field EnKBF for nonlinear signals
  7. 7 The EnKBF as an interacting particle system
    1. 7.1 Analysis of the particle approximations
    2. 7.2 Quantitative propagation of chaos

##  1 Introduction

In recent years the field of data assimilation, that is the (optimal)
integration of real world data into mathematical models, has become an
important tool for practitioners in various scientific fields. As it shares
similar objectives to stochastic filtering, which is essentially the
discipline of Bayesian estimation of dynamic processes from noisy, potentially
incomplete data, many algorithms from filtering are used for data assimilation
tasks. Vice versa algorithms for data assimilation of dynamical processes can
be viewed through the lens of filtering, the mathematical model is then
referred to as the signal and the available data as the observations. To
combine these two objects in an optimal manner one aims to compute/approximate
the posterior distribution, that is the conditional distribution of the signal
given all past observations.

One such algorithm is the Ensemble Kalman Filter (EnKF), which was introduced
by Geir Evensen in 94 [22] and employs an ensemble of interacting particles to
estimate the state of a dynamical system. Since its inception many different
variants of the EnKF have been introduced. For an overview and historical
context we refer to [13], [23] or [50]. The EnKF has by now become one of the
most widely used techniques for data assimilation in high dimensional
settings, particularly popular amongst practitioners in the geosciences and
numerical weather forecasting. Besides its usage for state estimation in
dynamical systems, the EnKF and related algorithms have also been applied to
parameter estimation in inverse problems [36],[46].

While the original EnKF is a discrete time recursion, continuous time
counterparts, referred to as Ensemble Kalman–Bucy Filters (EnKBFs) have by now
been firmly established in the literature. In many cases they can also be
shown to be the limit of their discrete time counterparts for vanishing step
size, see e.g. [31] and the references found therein. In this paper we will
only consider the basic continuous time framework. In this case the signal $u$
is given by a stochastic (partial) differential equation

$\displaystyle\mathrm{d}u_{t}=\mathcal{A}(u_{t})\mathrm{d}t+\mathcal{B}(u_{t})\mathrm{d}W_{t}$
(S)

for a given initial datum $u_{0}$. We allow for $u$ to be an element of a
possibly infinite dimensional Hilbert space, $\mathcal{A}$ to be a general
coercive operator and the signal noise $W$ to be a Wiener process with a
covariance of finite trace. The observations $Y$ shall be given by the
stochastic differential

$\displaystyle\mathrm{d}Y_{t}=H(u_{t})\mathrm{d}t+\Gamma_{t}\mathrm{d}V_{t},\leavevmode\nobreak\
Y_{0}=0.$ (O)

We only consider finite dimensional observations taking values in
$\mathbb{R}^{d_{y}}$ for some $d_{y}\in\mathbb{N}$. This is the more
practically relevant case and also avoids discussions of the
regularity/degeneracy of the observation noise $V$, which in this work is
assumed to be white, i.e. some finite dimensional standard Brownian motion. A
more thorough discussion of the setting we consider, and the assumptions we
have to make, is found in section 2.

In our setting the EnKBF we consider in this paper takes the form

$\displaystyle\begin{split}\mathrm{d}u^{i}_{t}&=\mathcal{A}(u^{i}_{t})\mathrm{d}t+\mathcal{B}(u^{i}_{t})\mathrm{d}\bar{W}^{i}_{t}\\\
&\phantom{=}+\frac{1}{N}\sum_{j=1}^{N}u^{j}_{t}\left(H(u^{j}_{t})-\frac{1}{N}\sum_{k=1}^{N}H(u^{k}_{t})\right)^{\prime}R^{-1}_{t}\left(\mathrm{d}Y_{t}-\frac{H(u^{i}_{t})+\frac{1}{N}\sum_{k=1}^{N}H(u^{k}_{t})}{2}\mathrm{d}t\right)\end{split}$
(1)

for $i=1,\cdots,N$, where $(\bar{W}^{i})_{i=1,\cdots,N}$ are independent
copies of the Wiener process $\bar{W}$. The system (1) is often referred to as
the deterministic EnKBF [6], which is the continuous time counterpart of the
Ensemble Square Root Filter [32]. Our main results can be generalized to other
types of EnKBFs, in particular the more classical version which involves
randomness in the innovation term.

As already mentioned the task of filtering is to compute the posterior
distribution $(\eta)_{t\geq 0}$, which in our setting is given by

$\displaystyle\eta_{t}:=\mathbb{P}\left(\leavevmode\nobreak\
u_{t}\in\cdot\leavevmode\nobreak\ |\leavevmode\nobreak\
Y_{s},\leavevmode\nobreak\ s\leq t\leavevmode\nobreak\
\right),\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ t\geq 0.$ (P)

Even though the EnK(B)F has seen wide success in applications. Its
mathematical foundations and connections to the filtering problem only started
to emerge in last decade and still contain many important gaps. The algorithm
is so far best understood in the case of linear signal and observation
dynamics under Gaussian noise perturbations and Gaussian initial conditions,
henceforth called the linear Gaussian setting (see Assumption 4 in section 4).
In this particular case the mean-field limit111That is the limiting process
one obtains by taking the ensemble size $N\to+\infty$ in (1). of (1) is a
McKean–Vlasov process with the property, that its mean and covariance evolve
according to the Kalman–Bucy equations. Thus the time marginals of this mean-
field limit coincide with the posterior for all times, which was first shown
in [26] and [35]. Therefore quantitative propagation of chaos results are
important in the Gaussian setting, as they provide error bounds for the
EnK(B)F as an approximation of the true posterior (P). In the continuous time
setting a quantitative convergence result that is even uniform in time was
first obtained by [20]. By now there exists an extensive body of work on the
mean-field theory of EnKBFs in the linear Gaussian setting, including very
strong convergence and stability results. For more details we refer the reader
to the survey paper [6] (see also Remark 43). Interestingly infinite
dimensional signals do not seem to have been considered in this continuous
time, linear Gaussian setting so far.

For nonlinear dynamics or non-Gaussian initial conditions the true posterior
can not be expected to be given by the mean-field limit of the EnK(B)F.
Therefore the mathematical foundations of the EnK(B)F and rigorous
justifications for its usage in nonlinear regimes are still sparse and an
ongoing research topic. Numerous works so far have focussed on investigating
stability and accuracy. In particular we mention here the seminal paper [29],
which investigated well posedness and accuracy of the EnK(B)F for finite
ensemble size. The signals considered were allowed to be infinite dimensional
(but deterministic) and included the (2D) Navier–Stokes equations. The
perspective that was put forward in this paper was to view the EnK(B)F as a
state estimator, rather than as an approximation of the true posterior.
Nevertheless the mean-field theory for EnK(B)Fs in finite dimensional settings
has lately received increasing interest [13] as it allows to connect the
EnK(B)F to the true posterior (P). Indeed one can obtain the mean-field limit
of the EnK(B)F from a McKean–Vlasov representation of the posterior, called
the Feedback Particle Filter [51][52] (also [49] for a survey), either by
Gaussian approximation or by what is usually referred to as the constant gain
approximation [48]. Even though these connections are well known in the
literature, only recently [14] were able to derive error bounds for the
inconsistency of the mean-field EnKF (as an approximation to the true
posterior) under restricting conditions that ensure a near Gaussian setting.
This indicates the relevance of the mean-field theory to providing firm
mathematical foundations for EnKBFs and their relation to the optimal filter.
So far the mean-field theory of the (continuous time) EnKBF seems to only have
been studied in a finite dimensional setting. [32] proved propagation of chaos
with implicit rates for the EnK(B)F in discrete and continuous time for linear
observations requiring the assumption that the mean-field limit actually
exists. [15] investigated the EnKBF from a rough paths perspective. Treating
the observation data as a continuous rough path they proved well posedness of
the mean-field equation under the assumption of bounded observation functions.
Under the additional assumption that the observation data is of bounded
variation they were also able to derive propagation of chaos with logarithmic
rates. [21] showed well posedness of the mean-field equation and propagation
of chaos with implicit rates in a correlated noise setting. A good summary of
the mean-field picture of EnK(B)Fs can be found in the recent survey paper
[13].

In this paper we derive and analyse the EnKBF and its mean-field limit for
signals that are given by SPDEs under finite dimensional observations. For the
signal SPDEs we consider a variational setting (see [43]), which is in
particular interesting for pracitical applications due to its relation to
popular numerical approximation methods like Finite Elements or other Galerkin
schemes. Nevertheless, to the best knowledge of the author, continuous time
Ensemble Kalman methods for SPDEs do not seem covered by the existing
literature. The problem setting and our basic assumptions are discussed in
section 2. As we analyse the EnKBF from a Bayesian point of view, we first
discuss the Kushner–Stratonovich equation describing the posterior
distribution in section 3. In particular we prove a priori bounds for the
variance of the Kushner–Stratonovich equation that in the Bayesian literature
are often referred to as the law of total variance. Similar bounds also hold
for the EnKBF and are the main tool for its analysis. In section 4 we restrict
our investigation to the linear Gaussian setting. After first reviewing a
classical result by Bensoussan [5] on the Kalman–Bucy filter in infinite
dimensions, we then introduce the mean-field EnKBF as McKean–Vlasov
representation of the Kalman–Bucy filter and discuss its well posedness. In
section 5 we then generalize this principle and derive the Feedback Particle
Filter (FPF) as a McKean–Vlasov representation of the posterior in nonlinear
filtering problems. We also show that the mean-field EnKBF for nonlinear
signals is the constant gain approximation of the FPF. Section 6 is devoted to
proving the well posedness of the mean-field EnKBF. Finally in section 7 we
investigate the EnKBF as a particle approximation of the mean-field version,
proving well posedness and a propagation of chaos result with implicit rates.
Under the additional assumption of a bounded observation function we are able
to improve on this convergence results, deriving (almost) optimal rates. While
optimal rates are well established in the linear Gaussian setting [20], to our
best knowledge this is the first result that shows such rates for general
nonlinear signal dynamics even in finite dimensions.

##  2 Problem setting, assumptions and notations

For the signal we consider SPDEs in a variational setting as they are found in
e.g. [39],[43]. To fix notation and for the convenience of the reader we
repeat some key concepts and results of this field in this section, for a more
detailed introduction to this topic we refer the reader to [43].

Let $\mathscr{H}$ be a Hilbert space and $\mathscr{V}$ be a Banach space that
form a Gelfand triple [43, Section 4.1]
$\mathscr{V}\hookrightarrow\mathscr{H}\hookrightarrow\mathscr{V}^{\prime}$. We
assume that that there exists an Orthonormal basis of $\mathscr{H}$, denoted
by $\left(\nu_{k}\right)_{k\in\mathbb{N}}\subset\mathscr{H}$ such that
$\nu_{k}\in\mathscr{V}$ for all $k\in\mathbb{N}$.

For a given separable real Hilbert space $\mathscr{U}$, we consider the
$\mathscr{U}$-valued $\mathcal{Q}$-Wiener process $(W_{t})_{t\geq 0}$ with
finite trace. To this end assume that $\mathcal{Q}$ is a symmetric, positive
semidefinite linear operator on $\mathscr{U}$ with finite trace
$\mathrm{tr}_{\mathscr{U}}\mathcal{Q}<+\infty$ and Eigenvalues
$\left(q_{k}\right)_{k\in\mathbb{N}}$ with corresponding orthonormal
Eigenbasis $\left(e_{k}\right)_{k\in\mathbb{N}}$. Then there are exist
independent Brownian motions $(w^{k})_{k\in\mathbb{N}}$ such that

$\displaystyle
W_{t}=\sum_{k\in\mathbb{N}}\sqrt{q_{k}}e_{k}w^{k}_{t}\leavevmode\nobreak\
\text{for all times}\leavevmode\nobreak\ t\geq 0.$

######  Definition 1.

We will always identify the Hilbert spaces $\mathscr{H},\mathscr{U}$ with
their duals $\mathscr{H}^{\prime},\mathscr{U}^{\prime}$. For any operator $B$
acting on Hilbert spaces we denote its adjoint by $B^{\prime}$. The adjoint of
an element $u\in\mathscr{H}$ is just it’s image under the Riesz embedding,
i.e. $u^{\prime}:=\left\langle u,\cdot\right\rangle_{\mathscr{H}}$. This
notation is consistent with finite dimensional settings. We note that
therefore $uu^{\prime}\in\mathrm{L}\left(\mathscr{H};\mathscr{H}\right)$
defines a bounded linear operator on $\mathscr{H}$.

To rigorously formulate the signal (S) as a variational SPDE on the Gelfand
triple $(\mathscr{V},\mathscr{H},\mathscr{V}^{\prime})$, we make the following
standard assumptions [43, page 56] that shall hold throughout this paper.

######  Assumption 2 (Signal assumptions).

Denote by $\mathrm{L}_{2}\left(\mathscr{U};\mathscr{H}\right)$ the space of
Hilbert–Schmidt operators, that is the space of all linear operators
$B:\mathscr{U}\to\mathscr{H}$ such that their Hilbert–Schmidt norm
$\left\|B\right\|_{\mathrm{L}_{2}\left(\mathscr{U};\mathscr{H}\right)}^{2}:=\sum_{k\in\mathbb{N}}\left\|Be_{k}\right\|_{\mathscr{H}}^{2}$
is finite.
We assume that $\mathcal{A}:\mathscr{V}\to\mathscr{V}^{\prime}$ and
$\mathcal{B}:\mathscr{V}\to\mathrm{L}_{2}\left(\mathscr{U};\mathscr{H}\right)$
satisfy the following conditions

  1. 1.

Hemicontinuity: For all $u,v,w\in\mathscr{V}$ the mapping

$\displaystyle
r\to\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(v+ru),w\right\rangle_{\mathscr{V}}}\leavevmode\nobreak\
\text{is continuous.}$

  2. 2.

Weak monotonicity/one-sided Lipschitz: There exists $\lambda>0$ such that for
all $u,v\in\mathscr{V}$

$\displaystyle
2\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(u)-\mathcal{A}(v),u-v\right\rangle_{\mathscr{V}}}+\left\|\left(\mathcal{B}(u)-\mathcal{B}(v)\right)\circ\sqrt{\mathcal{Q}}\right\|_{\mathrm{L}_{2}\left(\mathscr{U};\mathscr{H}\right)}^{2}\leq\lambda\left\|u-v\right\|_{\mathscr{H}}^{2}.$
(2)

  3. 3.

Coercivity: There exist constants
$\alpha_{V}>0,\alpha_{H},\alpha_{0}\in\mathbb{R}$ and
$\alpha_{\mathrm{p}}\in(1,+\infty)$, such that $u\in\mathscr{V}$

$\displaystyle
2\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(u),u\right\rangle_{\mathscr{V}}}+\left\|\mathcal{B}(u)\circ\sqrt{\mathcal{Q}}\right\|_{\mathrm{L}_{2}\left(\mathscr{U};\mathscr{H}\right)}^{2}\leq-\alpha_{V}\left\|u\right\|_{\mathscr{V}}^{\alpha_{\mathrm{p}}}+\alpha_{H}\left\|u\right\|_{\mathscr{H}}^{2}+\alpha_{0}.$
(3)

  4. 4.

Boundedness: There exists a constant $c_{\mathcal{A}}>0$ such that

$\displaystyle\left\|\mathcal{A}(u)\right\|_{\mathscr{V}^{\prime}}\leq
c_{\mathcal{A}}\left(1+\left\|u\right\|_{\mathscr{V}}\right)\leavevmode\nobreak\
\forall u\in\mathscr{V}.$

Next let us briefly discuss which (S)PDEs can be treated in this variational
framework.

######  Remark 3 (Possible Signals).

A classical example of a differential operator that satisfies our Assumptions
2 is the $p$-Laplacian

$\displaystyle\mathcal{A}(v):=\Delta\left(v|v|^{p-2}\right)$ (4)

for any $p\in[2,+\infty)$ on a suitable domain $\Lambda$ with Dirichlet
boundary conditions. In this case
$\mathscr{V}:=\mathcal{W}^{1,p}_{0}(\Lambda)$ is the classical $p$-integrable,
first order Sobolev space of functions that vanish on the boundary. The
Hilbert space is then set to $\mathscr{H}:=L(\Lambda)$. Neumann, or mixed
boundary conditions can be treated as well. Thus we can allow for signals that
are given by a (noisy) heat or porous media equation. Another differential
operator satisfying our assumptions is given by

$\displaystyle\mathcal{A}(v):=-\Delta
v-\mathfrak{a}v^{3}+\mathfrak{b}v+\mathfrak{c},$

where $\mathfrak{a}\geq 0,\leavevmode\nobreak\
\mathfrak{b},\mathfrak{c}\in\mathbb{R}$, for Dirichlet, Neumann or mixed
boundary conditions on suitable domains. Therefore we can treat signals
evolving by a stochastic reaction diffusion equation with a double well
potential. In particular Allen–Cahn and FitzHugh–Nagumo equations can be
treated. For a more detailed discussion we refer to [43, Section 4.1].

Under Assumption 2 it can be shown [43] that for a given initial condition
$u_{0}\in\mathscr{H}$ the signal SPDE (S) has a unique solution, which will
henceforth be referred to as the signal.

An important tool to our analysis is Itô’s Lemma for variational SPDEs [39,
page 136] first derived in [40]. For later reference let us specify in the
following definition for which functions one can use Itô’s Lemma.

######  Definition 4.

Any function $\phi:\mathscr{H}\to\mathbb{R}$ is said to be an Itô function, if

  1. 1.

$\phi$ is twice Fréchet differentiable, with the first two derivatives denoted
by $\mathrm{D}^{1}_{\mathscr{H}}\phi$ and $\mathrm{D}^{2}_{\mathscr{H}}\phi$.

  2. 2.

All of $\phi$, $\mathrm{D}^{1}_{\mathscr{H}}\phi$ and
$\mathrm{D}^{2}_{\mathscr{H}}\phi$ are locally bounded.

  3. 3.

For any operator $\mathcal{Q}:\mathscr{H}\to\mathscr{H}$ that is of trace
class, the functional
$v\mapsto\mathrm{tr}_{\mathscr{H}}\left[\mathcal{Q}\mathrm{D}^{2}_{\mathscr{H}}\phi(v)\right]$
is continuous on $\mathscr{H}$.

  4. 4.

For any $v\in\mathscr{V}$ it holds that
$\mathrm{D}^{1}_{\mathscr{H}}\phi(v)\in\mathscr{V}$ and the map
$\left.\mathrm{D}^{1}_{\mathscr{H}}\phi\right|_{\mathscr{V}}:\mathscr{V}\to\mathscr{V}$
is continuous when the domain is equipped with the strong and the image is
equipped with the weak topology.

  5. 5.

There exists a constant $C>0$ such that
$\left\|\mathrm{D}^{1}_{\mathscr{H}}v\right\|_{\mathscr{V}}\leq
C\left(1+\left\|v\right\|_{\mathscr{V}}\right)$ for all $v\in\mathscr{V}$.

If an Itô function $\phi$ is twice continuously Fréchet differentiable with
compact support, we refer to it as an Itô testfunction.

One important example of an Itô function is the squared norm
$\left\|.\right\|_{\mathscr{H}}^{2}$. With the Itô Lemma it is easy to show
the following basic identities for the signal distribution.

######  Lemma 5.

The signal mean $m_{t}:=\mathbb{E}\left[u_{t}\right]$ satisfies

$\displaystyle\partial_{t}m_{t}=\mathbb{E}\left[\mathcal{A}(u_{t})\right],$
(5)

and the covariance
$P_{t}:=\mathbb{Cov}\left[u_{t}\right]:=\mathbb{E}\left[\left(u_{t}-m_{t}\right)\left(u_{t}-m_{t}\right)^{\prime}\right]$
satisfies

$\displaystyle\begin{split}\partial_{t}\left\langle
v,\mathbb{Cov}\left[u_{t}\right]w\right\rangle_{\mathscr{H}}&=\mathbb{E}\left[\left\langle
v,u_{t}-m_{t}\right\rangle_{\mathscr{H}}\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(u_{t})-\mathcal{A}(m_{t}),w\right\rangle_{\mathscr{V}}}\right]\\\
&\phantom{=}+\mathbb{E}\left[\left\langle
w,u_{t}-m_{t}\right\rangle_{\mathscr{H}}\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(u_{t})-\mathcal{A}(m_{t}),v\right\rangle_{\mathscr{V}}}\right]\\\
&\phantom{=}+\left\langle
v,\mathbb{E}\left[\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}\left(\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}\right)^{\prime}\right]w\right\rangle_{\mathscr{H}}\end{split}$
(6)

for all $v,w\in\mathscr{V}$. The generator of the signal, denoted by
$\mathcal{L}$, is given by

$\displaystyle\mathcal{L}\phi=\prescript{}{\mathscr{V}^{\prime}}{{\left\langle\mathcal{A}(\cdot),\mathrm{D}^{1}_{\mathscr{H}}\phi\right\rangle_{\mathscr{V}}}}+\frac{1}{2}\mathrm{tr}_{\mathscr{H}}\left[\left(\mathrm{D}^{2}_{\mathscr{H}}\phi\right)\leavevmode\nobreak\
\mathcal{B}(\cdot)\sqrt{\mathcal{Q}}\left(\mathcal{B}(\cdot)\sqrt{\mathcal{Q}}\right)^{\prime}\right]$
(7)

for every Itô testfunction $\phi$ as per Definition 4.

###### Proof.

The equation of the mean (5) is a simple consequence of the centeredness of
the Wiener process $W$.

Using the fact that for any $v\in\mathscr{H}$ we have

$\displaystyle\left\langle
v,\mathcal{B}(\bar{u}_{t})\mathrm{d}W_{t}\right\rangle_{\mathscr{H}}=\sum_{k=1}^{+\infty}\sqrt{q_{k}}\left\langle
v,\mathcal{B}(\bar{u}_{t})e_{k}\right\rangle_{\mathscr{H}}\mathrm{d}\beta_{k}(t).$

we derive by Itô’s formula on Hilbert spaces [39, page 136] the identity

$\displaystyle\mathrm{d}\left(\left\langle
v,u_{t}-m_{t}\right\rangle_{\mathscr{H}}\left\langle
w,u_{t}-_{t}\right\rangle_{\mathscr{H}}\right)$ $\displaystyle=\left\langle
v,u_{t}-m_{t}\right\rangle_{\mathscr{H}}\mathrm{d}\left\langle
w,u_{t}-m_{t}\right\rangle_{\mathscr{H}}+\left\langle
w,u_{t}-m_{t}\right\rangle_{\mathscr{H}}\mathrm{d}\left\langle
v,u_{t}-m_{t}\right\rangle_{\mathscr{H}}$
$\displaystyle\phantom{=}+\mathrm{d}\left[\left\langle
v,u-m\right\rangle_{\mathscr{H}},\left\langle
w,u-m\right\rangle_{\mathscr{H}}\right]_{t}$ $\displaystyle=\left(\left\langle
v,u_{t}-m_{t}\right\rangle_{\mathscr{H}}\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(u_{t})-\mathbb{E}\left[\mathcal{A}(u_{t})\right],w\right\rangle}_{\mathscr{V}}+\left\langle
w,u_{t}-m_{t}\right\rangle_{\mathscr{H}}\prescript{}{\mathscr{V}^{\prime}}{\left\langle
v,\mathcal{A}(u_{t})-\mathbb{E}\left[\mathcal{A}(u_{t})\right]\right\rangle}_{\mathscr{V}}\right)\mathrm{d}t$
$\displaystyle\phantom{=}+\sum_{k=1}^{+\infty}q_{k}\left\langle
v,\mathcal{B}(u_{t})e_{k}\right\rangle_{\mathscr{H}}\left\langle
w,\mathcal{B}(u_{t})e_{k}\right\rangle_{\mathscr{H}}$
$\displaystyle\phantom{=}+\left\langle
v,u_{t}-m_{t}\right\rangle_{\mathscr{H}}\left\langle
w,\mathcal{B}(u_{t})\mathrm{d}W_{t}\right\rangle_{\mathscr{H}}+\left\langle
w,u_{t}-m_{t}\right\rangle_{\mathscr{H}}\left\langle
v,\mathcal{B}(u_{t})\mathrm{d}W_{t}\right\rangle_{\mathscr{H}}.$

for all $v,w\in\mathscr{V}$. Now we note that we can also write

$\displaystyle\sum_{k=1}^{+\infty}q_{k}\left\langle
v,\mathcal{B}(u_{t})e_{k}\right\rangle_{\mathscr{H}}\left\langle
w,\mathcal{B}(u_{t})e_{k}\right\rangle_{\mathscr{H}}$
$\displaystyle=\left\langle
v,\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}\sum_{k=1}^{+\infty}\left\langle
w,\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}e_{k}\right\rangle_{\mathscr{H}}e_{k}\right\rangle_{\mathscr{H}}$
$\displaystyle=\left\langle
v,\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}\sum_{k=1}^{+\infty}\left\langle\left(\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}\right)^{\prime}w,e_{k}\right\rangle_{\mathscr{U}}e_{k}\right\rangle_{\mathscr{H}}.$

Due to Parseval, this gives us the identity

$\displaystyle\sum_{k=1}^{+\infty}q_{k}\left\langle
v,\mathcal{B}(u_{t})e_{k}\right\rangle_{\mathscr{H}}\left\langle
w,\mathcal{B}(u_{t})e_{k}\right\rangle_{\mathscr{H}}=\left\langle
v,\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}\left(\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}\right)^{\prime}w\right\rangle_{\mathscr{H}}.$
(8)

By taking the expectation in the previous evolution equation of the product
and using the projection properties of the expectation we then derive (6).

Finally we address the generator. Using Itô’s formula [39, page 136] and
taking the expectation, we see immediately that

$\displaystyle\partial_{t}\mathbb{E}\left[\phi(u_{t})\right]=\mathbb{E}\left[\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(u_{t}),\left(\mathrm{D}^{1}_{\mathscr{H}}\phi\right)(u_{t})\right\rangle_{\mathscr{V}}}\right]+\frac{1}{2}\mathrm{tr}_{\mathscr{H}}\left[\mathbb{E}\left[\left(\mathrm{D}^{2}_{\mathscr{H}}\phi\right)(u_{t})\sum_{k\in\mathbb{N}}q_{k}\left((\mathcal{B}(u_{t})e_{k})(\mathcal{B}(u_{t})e_{k})\right)\right]\right]$

By using (8) again, we thus derive (7). ∎

Let us address the observations next. As stated in the introduction we
consider continuous, $d_{y}$-dimensional observations given by the
differential equation (O). We make the following standard assumptions for the
coefficients $H$ and $\Gamma$.

######  Assumption 6 (Observation assumptions).

The observation function $H:\mathscr{H}\to\mathbb{R}^{d_{y}}$ is assumed to be
globally Lipschitz continuous, $\Gamma\in
C^{0}\left([0,+\infty),\mathbb{R}^{d_{y}\times d_{v}}\right)$ and $V$ is a
$d_{v}$-dimensional standard Brownian motion, independent of the signal $u$
and its driving noise $W$. As usual we set
$R_{t}:=\Gamma_{t}\Gamma^{\mathrm{T}}_{t}$ and assume that $R_{t}$ is
invertible for all times $t\geq 0$.

In the following variance bounds for both the posterior (P) and the law of the
EnKBF will play a crucial role in our analysis. That is why, besides the
standard assumptions for the signal 2 and the observations 6, we make the
following additional assumption, which will give us a priori bounds for the
signal variance.

######  Assumption 7 (Bounded signal diffusion).

There exists a constant $\beta>0$ such that

$\displaystyle\sup_{v\in\mathscr{V}}\mathrm{tr}_{\mathscr{H}}\left[\mathcal{B}(v)\sqrt{\mathcal{Q}}\left(\mathcal{B}(v)\sqrt{\mathcal{Q}}\right)^{\prime}\right]\leq\beta.$

##  3 The Kushner-Stratonovich equation and the law of total variance

In this section we investigate the Kushner–Stratonovich equation (KSE), which
describes the evloution of the posterior distribution (P). We focus on proving
bounds for the posterior covariance. Similar bounds will later on be key to
the investigation of the EnKBF, and remarkably seem to be one of the few areas
of consistency for general signals. By consistency we mean that it is a
property that is shared by both the EnKBF and the optimal filter, regardless
of whether the filter is (close to) a Gaussian. However first we note that in
our setting with finite dimensional observations, the KSE can be derived just
as for finite dimensional settings using an innovation process approach [4].
To this end we define the innovation Wiener process $\hat{I}$ by

$\displaystyle\mathrm{d}\hat{I}_{t}:=R^{-1/2}_{t}\mathrm{d}Y_{t}-R^{-1/2}_{t}\eta_{t}(H)\mathrm{d}t.$

By design $\hat{I}$ is an $\mathbb{R}^{d_{y}}$-valued diffusion process with
continuous sample paths. Its quadratic variation is given by

$\displaystyle\left[\hat{I}\right]_{t}$
$\displaystyle=\left[\int_{0}^{\cdot}R^{-1/2}_{s}\mathrm{d}Y_{s}-\int_{0}^{\cdot}R^{-1/2}_{s}\eta_{s}(H)\mathrm{d}s\right]_{t}=\left[\int_{0}^{\cdot}R^{-1/2}_{s}\mathrm{d}Y_{s}\right]_{t}$
$\displaystyle=\int_{0}^{t}R^{-1/2}_{s}\mathrm{d}\left[Y\right]_{s}R^{-1/2}_{s}=t\leavevmode\nobreak\
\mathrm{id}_{\mathbb{R}^{d_{y}}}.$

Next we show that $\hat{I}$ is a $(Y_{0:t})_{t\geq 0}$ martingale. Let
therefore $s<t$ and $\phi_{s}$ be a bounded and $Y_{0:s}$-measurable function,
then by the martingale property of $V$ and the projection property of the
posterior $\eta$, we have

$\displaystyle\mathbb{E}\left[\left(\hat{I}_{t}-\hat{I}_{s}\right)\phi_{s}\right]$
$\displaystyle=\mathbb{E}\left[\int_{s}^{t}R^{-1/2}_{r}\Gamma_{r}\mathrm{d}V_{r}\leavevmode\nobreak\
\phi_{s}\right]+\mathbb{E}\left[\int_{s}^{t}R^{-1/2}_{r}\left(H(u_{r})-\eta_{r}(H)\right)\mathrm{d}r\leavevmode\nobreak\
\phi_{s}\right]=0.$

Thus by Levys characterization of the Brownian motion, we know that $\hat{I}$
is indeed a standard $\mathbb{R}^{d_{y}}$-valued Brownian motion.

Just as in the finite dimensional case [4], one can thus verify that the
posterior (P) satisfies the Kushner–Stratonovich equation (KSE) in its weak
form

$\displaystyle\mathrm{d}\eta_{t}(\phi)=\eta_{t}(\mathcal{L}\phi)\mathrm{d}t+\left(\eta_{t}(\phi
H)-\eta_{t}(\phi)\eta_{t}(H)\right)R^{-1}_{t}\left(\mathrm{d}Y_{t}-\eta_{t}(H)\mathrm{d}t\right),$
(9)

where $\mathcal{L}$ is the generator of $u$ defined in (7) and $\phi$ is an
arbitrary Itô testfunction (see Definition 4).

######  Remark 8.

As mentioned, the path we took to derive the KSE is rather standard in finite
dimensional settings [4]. There are various works establishing the nonlinear
filtering equations in infinite dimensional settings, even in the more
difficult case of correlated noise [2],[10]. In [53] an extension of the KSE
to infinite dimensional filtering problems with infinite dimensional
observations was derived.

A key tool in the analysis of the EnKBF is an inequality that bounds its
conditional variance by the (same upper bounds as the) variance of the signal,
irrespective of the observations. This is a feature that is actually shared by
the posterior distribution (P), as the projection properties of the
conditional expectation imply

$\displaystyle\mathbb{E}\left[\leavevmode\nobreak\
\eta_{t}(\left\|\cdot\right\|_{\mathscr{H}}^{2})-\left\|\eta_{t}(\mathrm{id}_{\mathscr{H}})\right\|_{\mathscr{H}}^{2}\leavevmode\nobreak\
\right]=\mathbb{E}\left[\leavevmode\nobreak\
\mathrm{tr}_{\mathscr{H}}\leavevmode\nobreak\
\mathbb{Cov}_{\eta_{t}}\left[\mathrm{id}_{\mathscr{H}}\right]\leavevmode\nobreak\
\right]\leq\mathrm{tr}_{\mathscr{H}}\leavevmode\nobreak\
\mathbb{Cov}\left[u_{t}\right].$

Now we note that by the covariance dynamics (6) and Parseval222One could have
also directly looked at the dynamics of
$\mathbb{E}\left[\left\|u_{t}-m_{t}\right\|_{\mathscr{H}}^{2}\right]$ and
proved this via the well known Itô formula for the norm. we obtain

$\displaystyle\partial_{t}\mathrm{tr}_{\mathscr{H}}\leavevmode\nobreak\
\mathbb{Cov}\left[u_{t}\right]$
$\displaystyle=2\sum_{k\in\mathbb{N}}\mathbb{E}\left[\left\langle\nu_{k},u_{t}-m_{t}\right\rangle_{\mathscr{H}}\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(u_{t})-\mathcal{A}(m_{t}),\nu_{k}\right\rangle_{\mathscr{V}}}\right]$
$\displaystyle\phantom{=}+\sum_{k\in\mathbb{N}}\left\langle\nu_{k},\mathbb{E}\left[\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}\left(\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}\right)^{\prime}\right]\nu_{k}\right\rangle_{\mathscr{H}}$
$\displaystyle=\mathbb{E}\left[2\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(u_{t})-\mathcal{A}(m_{t}),u_{t}-m_{t}\right\rangle_{\mathscr{V}}}\right]+\mathbb{E}\left[\mathrm{tr}_{\mathscr{H}}\left[\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}\left(\mathcal{B}(u_{t})\sqrt{\mathcal{Q}}\right)^{\prime}\right]\right]$

To estimate the first term we use the one-sided Lipschitz condition (2) and
for the second term we use Assumption 7. This gives us

$\displaystyle\partial_{t}\mathrm{tr}_{\mathscr{H}}\leavevmode\nobreak\
\mathbb{Cov}\left[u_{t}\right]\leq\lambda\leavevmode\nobreak\
\mathrm{tr}_{\mathscr{H}}\leavevmode\nobreak\
\mathbb{Cov}\left[u_{t}\right]+\beta.$

Which, by Grönwall, implies $\mathrm{tr}_{\mathscr{H}}\leavevmode\nobreak\
\mathbb{Cov}\left[u_{t}\right]\leq\beta e^{\lambda t}$. Together with the
variance bound for the posterior this gives us

$\displaystyle\mathbb{E}\left[\leavevmode\nobreak\
\eta_{t}(\left\|\cdot\right\|_{\mathscr{H}}^{2})-\left\|\eta_{t}(\mathrm{id}_{\mathscr{H}})\right\|_{\mathscr{H}}^{2}\leavevmode\nobreak\
\right]=\mathbb{E}\left[\leavevmode\nobreak\
\mathrm{tr}_{\mathscr{H}}\leavevmode\nobreak\
\mathbb{Cov}_{\eta_{t}}\left[\mathrm{id}_{\mathscr{H}}\right]\leavevmode\nobreak\
\right]\leq\beta e^{\lambda t}.$ (10)

######  Remark 9 (Law of total variance).

In probability theory the identity

$\displaystyle\mathbb{E}\left[\mathbb{Var}\left[X|Y\right]\right]+\mathbb{Var}\left[\mathbb{E}\left[X|Y\right]\right]=\mathbb{Var}\left[X\right]$

for any random variables $X,Y$ is often referred to as the law of total
variance. As we have seen, inequality (10) is a direct consequence of this
identity. Thus in the following we also refer to (10) as the law of total
variance. We will later see that it also holds for the variance of the EnKBF
and is key to its analysis.

As one would expect, one can also show the law of total variance (10) for any
sufficiently regular solution of the KSE without invoking its connection to a
Bayesian estimation problem. While this may seem trivial at first glance, it
is in fact a highly non trivial task to connect a given solution of the KSE or
Zakai equation back to the posterior distribution (P) without using uniqueness
arguments for these equations. We therefore formulate this fact in a separate
Lemma. The proof only uses the dynamics of the KSE and does not require the
connection to the conditional distribution, which makes the law of total
variance (10) an attractive a priori estimate for the analysis of the KSE or
related equations.

######  Lemma 10.

Let $\eta$ be a solution to (9) and assume that at all times it’s support lies
in $\mathscr{V}$. Assume furthermore that $\mathrm{id}_{\mathscr{H}}$ and
$\left\|.\right\|_{\mathscr{H}}^{2}$ are always integrable with respect to
$\eta_{t},\leavevmode\nobreak\ t\geq 0$. Then the law of total variance (10)
holds.

To prove this Lemma one would be tempted to use $\mathrm{id}_{\mathscr{H}}$
and $\left\|.\right\|_{\mathscr{H}}^{2}$ as testfunctions in (9). However this
is not allowed by the conditions of Itô’s formula. Instead we make a Fourier
argument, at the basis of which the following Lemma lies. This Lemma will also
become useful in the next section.

######  Lemma 11.

For every $i,j\in\mathbb{N}$ denote the $i$-th Fourier coefficient function by
$\phi_{i}(v):=\left\langle v,\nu_{i}\right\rangle_{\mathscr{H}}$ and define
the quadratic function $\chi_{ij}(v):=\left\langle
v,\nu_{i}\right\rangle_{\mathscr{H}}\left\langle
v,\nu_{j}\right\rangle_{\mathscr{H}}$. Then it holds that

$\displaystyle\mathrm{d}\eta_{t}(\phi_{i})$
$\displaystyle=\eta_{t}\left(\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(\cdot)),\nu_{i}\right\rangle_{\mathscr{V}}}\right)\mathrm{d}t+\mathbb{Cov}_{\eta_{t}}\left[\phi_{i},H\right]R^{-1}_{t}\left(\mathrm{d}Y_{t}-\eta_{t}(H)\mathrm{d}t\right)$
(11)

and

$\displaystyle\begin{split}&\mathrm{d}\left(\eta_{t}(\chi_{ij})-\eta_{t}(\phi_{i})\eta_{t}(\phi_{j})\right)\\\
&=\left(\mathbb{Cov}_{\eta_{t}}\left[\phi_{i},\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(\cdot),\nu_{j}\right\rangle_{\mathscr{V}}}\right]+\mathbb{Cov}_{\eta_{t}}\left[\phi_{j},\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(\cdot),\nu_{i}\right\rangle_{\mathscr{V}}}\right]\right)\mathrm{d}t\\\
&\phantom{=}+\eta_{t}\left(\left\langle\nu_{i},\left((\mathcal{B}(\cdot)\sqrt{\mathcal{Q}}\right)\left((\mathcal{B}(\cdot)\sqrt{\mathcal{Q}}\right)^{\prime}\nu_{j}\right\rangle_{\mathscr{H}}\right)\mathrm{d}t\\\
&\phantom{=}-\mathbb{Cov}_{\eta_{t}}\left[\phi_{i},H\right]R^{-1}_{t}\mathbb{Cov}_{\eta_{t}}\left[H,\phi_{j}\right]+\mathfrak{X}_{ij}R^{-1}_{t}\left(\mathrm{d}Y_{t}-\eta_{t}(H)\mathrm{d}t\right),\end{split}$
(12)

with

$\displaystyle\mathfrak{X}_{ij}:=\mathbb{Cov}_{\eta_{t}}\left[\chi_{ij},H\right]-\eta_{t}(\phi_{j})\mathbb{Cov}_{\eta_{t}}\left[\phi_{i},H\right]-\eta_{t}(\phi_{i})\mathbb{Cov}_{\eta_{t}}\left[\phi_{j},H\right]$
(13)

###### Proof.

We note that $\left(\mathrm{D}^{1}_{\mathscr{H}}\phi_{i}\right)(v)=\nu_{i}$
and $\left(\mathrm{D}^{2}_{\mathscr{H}}\phi_{i}\right)(v)=0$, which implies
$\left(\mathcal{L}\phi_{i}\right)(v)=\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(v)),\nu_{k}\right\rangle_{\mathscr{V}}}$.
This gives us (11).

Now we use (11) and Itô’s product rule to derive

$\displaystyle\mathrm{d}\left(\eta_{t}(\phi_{i})\eta_{t}(\phi_{j})\right)$
$\displaystyle=\left(\eta_{t}(\phi_{j})\eta_{t}\left(\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(\cdot)),\nu_{i}\right\rangle_{\mathscr{V}}}\right)+\eta_{t}(\phi_{i})\eta_{t}\left(\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(\cdot)),\nu_{j}\right\rangle_{\mathscr{V}}}\right)\right)\mathrm{d}t$
$\displaystyle\phantom{=}+\left(\eta_{t}(\phi_{j})\mathbb{Cov}_{\eta_{t}}\left[\phi_{i},H\right]+\eta_{t}(\phi_{i})\mathbb{Cov}_{\eta_{t}}\left[\phi_{j},H\right]\right)R^{-1}_{t}\left(\mathrm{d}Y_{t}-\eta_{t}(H)\mathrm{d}t\right)$
$\displaystyle\phantom{=}+\mathbb{Cov}_{\eta_{t}}\left[\phi_{j},H\right]R^{-1}_{t}\mathbb{Cov}_{\eta_{t}}\left[H,\phi_{j}\right].$

Next we note that
$\left(\mathrm{D}^{1}_{\mathscr{H}}\chi_{ij}\right)(v)=\left\langle
v,\nu_{j}\right\rangle_{\mathscr{H}}\nu_{i}+\left\langle
v,\nu_{i}\right\rangle_{\mathscr{H}}\nu_{j}$ and
$\left(\mathrm{D}^{2}_{\mathscr{H}}\chi_{ij}\right)(v)=\nu_{i}\left(\nu_{j}\right)^{\prime}+\nu_{j}\left(\nu_{i}\right)^{\prime}$.
Just as in (7), this implies

$\displaystyle\mathcal{L}\chi_{ij}(v)$
$\displaystyle=\left\langle\nu_{i},v\right\rangle_{\mathscr{H}}\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(v),\nu_{j}\right\rangle_{\mathscr{V}}}+\left\langle\nu_{i},v\right\rangle_{\mathscr{H}}\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(v),\nu_{j}\right\rangle_{\mathscr{V}}}$
$\displaystyle\phantom{=}+\left\langle\nu_{i},\sum_{k\in\mathbb{N}}q_{k}\left((\mathcal{B}(v)e_{k})(\mathcal{B}(v)e_{k})^{\prime}\right)\nu_{j}\right\rangle_{\mathscr{H}}.$

By (8), we thus derive

$\displaystyle\mathrm{d}\eta_{t}(\chi_{ij})$
$\displaystyle=\left(\eta_{t}\left(\left\langle\nu_{i},v\right\rangle_{\mathscr{H}}\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(v),\nu_{j}\right\rangle_{\mathscr{V}}}\right)+\eta_{t}\left(\left\langle\nu_{i},v\right\rangle_{\mathscr{H}}\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(v),\nu_{j}\right\rangle_{\mathscr{V}}}\right)\right)\mathrm{d}t$
$\displaystyle\phantom{=}+\eta_{t}\left(\left\langle\nu_{i},\left((\mathcal{B}(\cdot)\sqrt{\mathcal{Q}}\right)\left((\mathcal{B}(\cdot)\sqrt{\mathcal{Q}}\right)^{\prime}\nu_{j}\right\rangle_{\mathscr{H}}\right)$
$\displaystyle\phantom{=}+\mathbb{Cov}_{\eta_{t}}\left[\chi_{ij},H\right]R^{-1}_{t}\left(\mathrm{d}Y_{t}-\eta_{t}(H)\mathrm{d}t\right).$

Therefore by subtracting the equation for
$\eta_{t}(\phi_{i})\eta_{t}(\phi_{j})$ from the evolution equation of
$\eta_{t}(\chi_{ij})$, we derive (12).

∎

Now we are able to prove the law of total variance (10) for the KSE.

###### Proof of Lemma 10.

By Parseval we have

$\displaystyle\left(\eta_{t}(\left\|\cdot\right\|_{\mathscr{H}}^{2})-\left\|\eta_{t}(\mathrm{id}_{\mathscr{H}})\right\|_{\mathscr{H}}^{2}\right)=\sum_{k\in\mathbb{N}}\left(\eta_{t}(\chi_{k})-\eta_{t}(\phi_{k})\eta_{t}(\phi_{k})\right).$

Thus we can use (12). Now we note that in equation (12) the innovation term
$\mathrm{d}Y_{t}-\eta_{t}(H)\mathrm{d}t$ vanishes under the expectation and
the contribution of
$-\mathbb{Cov}_{\eta_{t}}\left[\phi_{k},H\right]R^{-1}_{t}\mathbb{Cov}_{\eta_{t}}\left[H,\phi_{k}\right]$
is indeed negative. Thus we have

$\displaystyle\partial_{t}\mathbb{E}\left[\eta_{t}(\left\|\cdot\right\|_{\mathscr{H}}^{2})-\left\|\eta_{t}(\mathrm{id}_{\mathscr{H}})\right\|_{\mathscr{H}}^{2}\right]$
$\displaystyle\leq\sum_{k\in\mathbb{N}}\mathbb{E}\left[\mathrm{Cov}_{\eta_{t}}\left[\left\langle\nu_{k},\cdot\right\rangle_{\mathscr{H}},\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(v),\nu_{k}\right\rangle_{\mathscr{V}}}\right]+\mathrm{Cov}_{\eta_{t}}\left[\left\langle\nu_{k},\cdot\right\rangle_{\mathscr{H}},\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(v),\nu_{k}\right\rangle_{\mathscr{V}}}\right]\right]$
$\displaystyle\phantom{=}+\sum_{k\in\mathbb{N}}\mathbb{E}\left[\eta_{t}\left(\left\langle\nu_{k},\left((\mathcal{B}(\cdot)\sqrt{\mathcal{Q}}\right)\left((\mathcal{B}(\cdot)\sqrt{\mathcal{Q}}\right)^{\prime}\nu_{k}\right\rangle_{\mathscr{H}}\right)\right].$

Then using Parseval in combination with the same bounds as before gives the
desired inequality (10).
∎

##  4 Linear and Gaussian Filtering

Only in this section we make the following assumption, which we refer to as
the linear Gaussian setting.

######  Assumption 12.

[Linear Gaussian Setting] We assume that

  * •

$\mathcal{A}:\mathscr{V}\to\mathscr{V}^{\prime}$ is linear

  * •

$\mathcal{B}\in\mathrm{L}\left(\mathscr{U};\mathscr{H}\right)$ is a constant
linear operator, i.e. it is independent of the state $u$.

  * •

$H\in\mathrm{L}\left(\mathscr{H};\mathbb{R}^{d_{y}}\right)$ is a linear
operator

  * •

the initial condition (of the signal and posterior) $u_{0}$ is Gaussian in
$\mathscr{H}$, i.e.

$\displaystyle
u_{0}\sim\mathcal{N}\left(m_{0},P_{0}\right)\leavevmode\nobreak\
\text{with}\leavevmode\nobreak\ m_{0}\in\mathscr{H}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\
P_{0}\in\mathrm{L}\left(\mathscr{H};\mathscr{H}\right)\leavevmode\nobreak\
\text{is symmetric positive semidefinite}.$

In this setting $u$ and $Y$ are jointly Gaussian and thus the posterior
distribution is also Gaussian
$\eta_{t}=\mathcal{N}\left(m_{t},P_{t}\right),\leavevmode\nobreak\ t\geq 0$,
meaning that it is completely described by its (conditional) mean $m$ and
covariance $P$. We refer to [38] for conditional Gaussian distributions on
Hilbert spaces.

###  4.1 The classical Kalman–Bucy filter

It is well known that in finite dimensions $m$ and $P$ satisfy the Kalman–Bucy
equations [4]

$\mathrm{d}m_{t}=\mathcal{A}m_{t}\mathrm{d}t+P_{t}H^{\prime}R^{-1}_{t}\left(\mathrm{d}Y_{t}-Hm_{t}\mathrm{d}t\right)$
(14a)
$\frac{\mathrm{d}P_{t}}{\mathrm{d}t}=\mathcal{A}P_{t}+P_{t}\mathcal{A}^{\prime}-P_{t}H^{\prime}R^{-1}_{t}HP_{t}+\left(\mathcal{B}\sqrt{\mathcal{Q}}\right)\left(\mathcal{B}\sqrt{\mathcal{Q}}\right)^{\prime}.$
(14b)

This is also true in infinite dimensional settings, which is a direct
consequence of the Kushner–Stratonovich equation, see [53, Section 3.3.1] (as
well as [4] for the finite dimensional case). For the sake of completeness and
since, for finite dimensional observations, it is a trivial consequence of
Lemma 11, we also show the derivation here.

######  Lemma 13.

Assume that $\eta$ is a Gaussian solution to the KSE , i.e. $\eta_{t}$ is a
Gaussian on $\mathscr{V}$ with mean $m_{t}$ and covariance $P_{t}$ satisfying
(9) for all times $t\geq 0$. Then $m$ and $P$ satisfy the Kalman–Bucy
equations (14).

###### Proof.

The Kalman–Bucy filter is a direct consequence of Lemma 11. To show this we
first note that for arbitrary $i\in\mathbb{N}$, the linearity of $\mathcal{A}$
and $H$ turns (11) into

$\displaystyle\mathrm{d}\left\langle\nu_{i},m_{t}\right\rangle_{\mathscr{H}}$
$\displaystyle=\mathrm{d}\eta_{t}(\phi_{i})$
$\displaystyle=\eta_{t}\left(\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(\cdot)),\nu_{i}\right\rangle_{\mathscr{V}}}\right)\mathrm{d}t+\mathbb{Cov}_{\eta_{t}}\left[\phi_{i},H\right]R^{-1}_{t}\left(\mathrm{d}Y_{t}-\eta_{t}(H)\mathrm{d}t\right)$
$\displaystyle=\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}m_{t},\nu_{i}\right\rangle_{\mathscr{V}}}\mathrm{d}t+\left\langle\nu_{i},P_{t}H^{\prime}R^{-1}_{t}\left(\mathrm{d}Y_{t}-\eta_{t}(H)\mathrm{d}t\right)\right\rangle_{\mathscr{H}},$

which is the equation of the $i$-th Fourier coefficient of (14a). Thus $m$
indeed satisfies (14a).

Similarly one verifies immediately that the linearity of $\mathcal{A}$ and $H$
turns (12) into

$\displaystyle\begin{split}\mathrm{d}\left\langle\nu_{i},P_{t}\nu_{j}\right\rangle_{\mathscr{H}}&=\left(\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}P_{t}\nu_{j},\nu_{i}\right\rangle_{\mathscr{V}}}+\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}P_{t}\nu_{i},\nu_{j}\right\rangle_{\mathscr{V}}}\right)\mathrm{d}t\\\
&\phantom{=}+\left\langle\nu_{i},\left((\mathcal{B}\sqrt{\mathcal{Q}}\right)\left((\mathcal{B}\sqrt{\mathcal{Q}}\right)^{\prime}\nu_{j}\right\rangle_{\mathscr{H}}\mathrm{d}t-\left\langle\nu_{i},P_{t}H^{\prime}R^{-1}_{t}HP_{t}\nu_{j}\right\rangle_{\mathscr{H}}\mathrm{d}t\\\
&\phantom{=}+\mathfrak{X}_{ij}R^{-1}_{t}\left(\mathrm{d}Y_{t}-\eta_{t}(H)\mathrm{d}t\right),\end{split}$

for every $i,j\in\mathbb{N}$. Thus in order to prove (14b), it remains to show
that for $\mathfrak{X}$, defined in (13), it holds that $\mathfrak{X}_{ij}=0$
for all $i,j\in\mathbb{N}$.

The main idea is that this question reduces to identities for finite
dimensional Gaussians. Let $i,j\in\mathbb{N}$ be arbitrary, but fixed. Define
the linear map $\Psi:\mathscr{H}\to\mathbb{R}^{d_{y}+2}$ by

$\displaystyle\Psi_{t}(v):=\left(\begin{array}[]{c}\left\langle
v,\nu_{i}\right\rangle_{\mathscr{H}}\\\ \left\langle
v,\nu_{j}\right\rangle_{\mathscr{H}}\\\ Hv\end{array}\right).$

Now we take an arbitrary, $\eta_{t}$-distributed random variable
$\bar{u}_{t}\sim\eta_{t}$, then

$\displaystyle Z:=\Psi_{t}\left(\bar{u}_{t}\right)$

is an $\mathbb{R}^{d_{y}+2}$ Gaussian vector, with

$\displaystyle\mu:=\mathbb{E}_{Y}\left[\Psi_{t}\left(\bar{u}_{t}\right)\right]=\left(\begin{array}[]{c}\left\langle
m_{t},\nu_{i}\right\rangle_{\mathscr{H}}\\\ \left\langle
m_{t},\nu_{j}\right\rangle_{\mathscr{H}}\\\ Hm_{t}\end{array}\right)$

and

$\displaystyle\Sigma:=\mathbb{Cov}_{Y}\left[\Psi_{t}(\bar{u}_{t})\right]=\left(\begin{array}[]{ccc}\left\langle\nu_{i},P_{t}\nu_{i}\right\rangle_{\mathscr{H}}&\left\langle\nu_{i},P_{t}\nu_{j}\right\rangle_{\mathscr{H}}&\left(HP_{t}\nu_{i}\right)^{\prime}\\\
\left\langle\nu_{j},P_{t}\nu_{i}\right\rangle_{\mathscr{H}}&\left\langle\nu_{j},P_{t}\nu_{j}\right\rangle_{\mathscr{H}}&\left(HP_{t}\nu_{j}\right)^{\prime}\\\
HP_{t}\nu_{i}&HP_{t}\nu_{j}&HP_{t}H^{\prime}\\\ \end{array}\right).$

With these definitions we see that

$\displaystyle\mathfrak{X}_{ij}$
$\displaystyle=\mathbb{Cov}_{Y}\left[Z_{1}Z_{2},Z_{3}\right]-\mathbb{E}_{Y}\left[Z_{2}\right]\mathbb{Cov}_{Y}\left[Z_{1},Z_{3}\right]-\mathbb{E}_{Y}\left[Z_{1}\right]\mathbb{Cov}_{Y}\left[Z_{2},Z_{3}\right],$
(15)

and since

$\displaystyle\mathbb{Cov}_{Y}\left[Z_{1}Z_{2},Z_{3}\right]$
$\displaystyle=\mathbb{E}_{Y}\left[\left(Z_{1}Z_{2}-\mu_{1}\mu_{2}\right)\left(Z_{3}-\mu_{3}\right)^{\prime}\right]$
$\displaystyle=\mathbb{E}_{Y}\left[Z_{1}Z_{2}Z_{3}^{\prime}\right]-\mathbb{Cov}_{Y}\left[Z_{1},Z_{2}\right]\mathbb{E}_{Y}\left[Z_{3}^{\prime}\right]-\mathbb{E}_{Y}\left[Z_{1}\right]\mathbb{E}_{Y}\left[Z_{2}\right]\mathbb{E}_{Y}\left[Z_{3}^{\prime}\right]$

we can write (15) as

$\displaystyle\mathfrak{X}_{ij}=\mathbb{E}_{Y}\left[Z_{1}Z_{2}Z_{3}^{\prime}\right]-\Sigma_{12}\mu_{3}^{\prime}-\mu_{2}\Sigma_{13}-\mu_{1}\Sigma_{23}-\mu_{1}\mu_{2}\mu_{3}^{\prime}.$
(16)

Showing that $\mathfrak{X}_{ij}=0$ is equivalent to a simple identity for
third moments of finite dimensional Gaussians, which we prove here for the
sake of completeness and the convenience of the reader. We use the moment
generating function

$\displaystyle\mathcal{M}(r):=\mathbb{E}_{Y}\left[\exp\left(r_{1}Z_{1}+r_{2}Z_{2}+r_{3}\cdot
Z_{3}\right)\right]=\exp\left(r\cdot\mu+\frac{r^{\prime}\Sigma r}{2}\right),$

where $r=(r_{1},r_{2},r_{3})^{\mathrm{T}}$,
$r_{1},r_{2}\in\mathbb{R},\leavevmode\nobreak\ r_{3}\in\mathbb{R}^{d_{y}}$.
Then clearly we have

$\displaystyle\mathbb{E}_{Y}\left[Z_{1}Z_{2}Z_{3}\right]=\partial_{r_{1}}\partial_{r_{2}}\nabla_{r_{3}}\mathcal{M}(0)=\mu_{1}\Sigma_{32}+\mu_{2}\Sigma_{31}+\Sigma_{12}\mu_{3}+\mu_{1}\mu_{2}\mu_{3},$

which in turn implies $\mathfrak{X}_{ij}=0$ for all $i,j\in\mathbb{N}$.

∎

######  Remark 14 (Literature).

The first (mathematically rigorous) work treating the Kalman–Bucy Filter (14)
for infinite dimensional SDEs seems to be by Falb [24] in 1967, which required
$\mathcal{A}$ to be a bounded operator on a Hilbert space, thus not allowing
for differential operators. In the 70s, several works, which also allowed for
$\mathcal{A}$ to be a differential operator, expanded on this. For a good
overview on the progress that has been made in this time period we refer the
interested reader to an excellent survey paper by Curtain [17], who, in
various papers treated semigroup approaches to (14). Very recent results for
far more general infinite dimensional signals using a semigroup approach can
be found in [30]. A variational approach, which is the one we use, was already
developed in the 70s by Bensoussan [5]333The book is written in French, an
English text that also recites some of its results is [37].. In this book
Bensoussan studied two approaches for Kalman–Bucy filters, one introduced so
called random linear functionals to mathematically describe model errors, the
other one uses SPDEs. Many of the papers recited here, including the book by
Bensoussan, actually allow for more general observations, taking values in
Hilbert spaces.

Note that in the infinite dimensional setting the Riccati equation (14b) is
operator valued. Well posedness of such equations is a classical subject of
infinite dimensional control theory. Given the well posedness of the Riccati
equation (14b) the well posedness of the equation for the mean (14a) can be
followed from such results for linear SPDEs. Indeed the existence and
uniqueness of solutions to (14b) was shown by Bensoussan [5, Théorème
3.1],[37, Theorem 7.3.2]. For the convenience of the reader we recite this
result in the following.

######  Theorem 15 (Bensoussan 1971,[5]).

Let $T$ be an arbitrary timeframe, we denote the $\mathscr{V}$-valued Sobolev
space on $[0,T]$ by

$\displaystyle\mathcal{W}([0,T]):=\left\\{\leavevmode\nobreak\ y\in
L^{2}\left([0,T];\mathscr{V}\right)\leavevmode\nobreak\ :\leavevmode\nobreak\
\partial_{t}y\in
L^{2}\left([0,T];\mathscr{V}^{\prime}\right)\leavevmode\nobreak\
\right\\}\subseteq C^{0}\left([0,T];\mathscr{H}\right),$

where the last inclusion is a consequence of Sobolev embedding. We assume that
$A$ is coercive, i.e. that (3) holds for $\alpha_{0}=0$ and that both $P_{0}$
and $\mathcal{Q}$ are invertible.
Then there exists a unique family of operators $(P_{t})_{t\in[0,T]}$ such that
for all testfunctions $\phi\in\mathcal{W}([0,T])$ that satisfy

$\displaystyle\partial_{t}\phi+A^{\prime}\phi\in L^{2}([0,T];\mathscr{H}),$

it holds that

$\displaystyle P\phi\in\mathcal{W}([0,T])$ (17)

and that

$\displaystyle\partial_{t}(P\phi)=P\left(\partial_{t}\phi+\mathcal{A}^{\prime}\phi\right)+\mathcal{A}P\phi+(\mathcal{B}\mathcal{Q}\mathcal{B}^{\prime})\phi-(PH^{\prime}R^{-1}HP)\phi.$
(18)

######  Remark 16.

We remark that besides an appropriate well posedness of the Riccati equation
(14b), [5, Théorème 3.1] also derives the optimal filter in a random linear
functional framework. We did not recite this in Theorem 15 and refer the
interested reader to [5, Théorème 3.1] or [37, Theorem 7.3.2]. We also remark
that the invertibility conditions in Theorem 15 can be relaxed [17].

######  Remark 17.

Condition (17) is a relaxed regularity condition, as for constant
testfunctions $\phi\in\mathscr{V}$ the image of the adjoint
$\mathcal{A}^{\prime}\phi$ is not necessarily contained in $\mathscr{H}$ and
thus in (17) not all elements of $\mathscr{V}$ are assumed to be mapped into
the Sobolev space.

######  Remark 18 (Extended Kalman–Bucy).

While the Kalman–Bucy filter provides a consistent representation of the
posterior in the linear Gaussian setting, adaptations for nonlinear signals
called Extended Kalman–Bucy filters (EKF) are often used in practice, in
particular in engineering. Hereby the signal is (a priori) linearized by a
Taylor expansion and the standard Kalman–Bucy algorithm (14) is then applied
to these altered signal dynamics. There is a recent paper [1] that discusses
the derivation of an EKF for semilinear SPDEs and the elementary mathematical
analysis via a mild-solutions/semigroup approach.

###  4.2 The consistent mean field EnKBF

In the linear Gaussian setting, a representation of the posterior $\eta$ by a
stochastic process $\bar{u}$ is given by the mean-field EnKBF. In the finite
dimensional setting numerical approximations based on this equation have
proven very successful in many practical applications, particular for high
dimensional signals. One can also easily generalize this representation to our
SPDE setting, resulting in the McKean–Vlasov SPDE

$\displaystyle\mathrm{d}\bar{u}_{t}=\mathcal{A}\bar{u}_{t}\mathrm{d}t+\mathcal{B}\mathrm{d}\bar{W}_{t}+\bar{P}_{t}H^{\prime}R^{-1}_{t}\left(\mathrm{d}Y_{t}-H\frac{\bar{u}_{t}+\bar{m}_{t}}{2}\mathrm{d}t\right)$
(19)

with

$\displaystyle\begin{split}\bar{m}_{t}&:=\mathbb{E}\left[\bar{u}_{t}\right]\in\mathscr{V}\\\
\bar{P}_{t}&:=\mathbb{Cov}\left[\bar{u}_{t}\right]=\mathbb{E}\left[\left(\bar{u}_{t}-\bar{m}_{t}\right)\left(\bar{u}_{t}-\bar{m}_{t}\right)^{\prime}\right]\in\mathrm{L}\left(\mathscr{V}^{\prime};\mathscr{V}\right).\end{split}$

Before we investigate the well posedness of (19) we want to verify that it
defines a consistent mean-field representation of the posterior $\eta$. As a
first step towards this goal we verify that (19) reproduces the correct
posterior mean and covariance that are given by the Kalman–Bucy filter (14).

######  Lemma 19 (moment consistency).

Let $\bar{m}$ and $\bar{P}$ denote the mean and covariance of $\bar{u}$, the
solution to (19). Then $\bar{m}$ and $\bar{P}$ satisfy the Kalman–Bucy
equations (14) and are thus identical to the mean and covariance of the
posterior.

###### Proof.

Taking the expectation in (19) gives us the Kalman equation (14a). We thus
only have to verify that the covariance matrix satisfies the Riccati equation
(14b). Just as in the proof of Lemma 5, applying Itô’s rule to the product
$\left\langle v,\bar{u}_{t}-\bar{m}_{t}\right\rangle_{\mathscr{H}}\left\langle
w,\bar{u}_{t}-\bar{m}_{t}\right\rangle_{\mathscr{H}}$ gives us by taking the
(conditional) expectation

$\displaystyle\begin{split}\partial_{t}\left\langle
v,\bar{P}_{t}w\right\rangle_{\mathscr{H}}&=\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}\bar{P}_{t}w,v\right\rangle_{\mathscr{V}}}+\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}\bar{P}_{t}v,w\right\rangle_{\mathscr{V}}}-\left\langle
v,\bar{P}_{t}H^{\prime}R^{-1}_{t}H\bar{P}_{t}w\right\rangle_{\mathscr{H}}\\\
&\phantom{=}+\left\langle
v,\mathcal{B}\sqrt{\mathcal{Q}}\left(\mathcal{B}\sqrt{\mathcal{Q}}\right)^{\prime}w\right\rangle_{\mathscr{H}}.\end{split}$

Thus $\bar{P}$ indeed satisfies the Riccati equation (14b). ∎

We can use the moment consistency to prove the well posedness of (19) as the
following Lemma shows. For the finite dimensional setting this can also be
found in [20].

######  Lemma 20.

Under the assumptions of Theorem (15) there exists a unique solution of (19)

###### Proof.

The standard method for proving well posedness of McKean–Vlasov equations uses
a fixed point argument with respect to the law. For (19) this argument
simplifies substantially as we can use (14) to guess the right fixed point. To
this end let $P$ be the unique solution to (14b) and define $\tilde{u}$ to be
the unique solution to the linear equation

$\displaystyle\mathrm{d}\tilde{u}_{t}=\mathcal{A}\tilde{u}_{t}\mathrm{d}t+\mathcal{B}\mathrm{d}\bar{W}_{t}+P_{t}H^{\prime}R^{-1}_{t}\left(\mathrm{d}Y_{t}-H\frac{\tilde{u}_{t}+\mathbb{E}_{Y}\left[\tilde{u}_{t}\right]}{2}\mathrm{d}t\right).$
(20)

Then the covariance $\tilde{P}$ of $\tilde{u}$ satisfies the linear equation

$\displaystyle\frac{\mathrm{d}P_{t}}{\mathrm{d}t}=\mathcal{A}\tilde{P}_{t}+\tilde{P}_{t}\mathcal{A}^{\prime}-\frac{1}{2}P_{t}H^{\prime}R^{-1}_{t}H\tilde{P}_{t}-\frac{1}{2}\tilde{P}_{t}H^{\prime}R^{-1}_{t}HP_{t}+\left(\mathcal{B}\sqrt{\mathcal{Q}}\right)\left(\mathcal{B}\sqrt{\mathcal{Q}}\right)^{\prime}.$

However, by definition, this equation must also be satisfied by the solution
$P$ to the Riccati equation (14b). Thus by uniquenss of solutions to the
linear problem above, we conclude that $P=\tilde{P}$ and $\tilde{u}$ is thus a
solution to the EnKBF.

Uniqueness of solutions follows from the uniqueness of the Riccati equation.
Given two solutions $\bar{u}^{i},i=1,2$, their covariances
$\bar{P}^{i},\leavevmode\nobreak\ i=1,2$ must both satisfy the Riccati
equation and thus by uniqueness coincide $\bar{P}=\bar{P}^{1}=\bar{P}^{2}$.
Therefore $\bar{u}^{1}$ and $\bar{u}^{2}$ both satisfy the same linear
equation (where $\bar{P}$ is taken as a covariable/coefficient of the
equation) and by uniqueness they too must coincide. ∎

The fixed point argument in the proof of Lemma (20) can also be used to verify
that the EnKBF is not just consistent with respect to the first two moments,
but also with respect to its law. To this end we note that for given operator
valued process $P$, equation (20) defines an Ornstein-–Uhlenbeck process, and
thus for Gaussian initial condition its solution $\bar{u}$ will have Gaussian
(time-)marginals. Therefore, since $\bar{u}$, the solution of (19), has
consistent mean and covariance processes, its marginals are also consistent.
We state this fact in the following Corollary.

######  Corollary 21 (consistency of the law).

Denote by $\bar{\eta}$ the time-marginals of the law of $\bar{u}$, i.e.
$\bar{u}_{t}\sim\bar{\eta}_{t}$ for all times $t\geq 0$. Then, in the linear
Gaussian setting, $\bar{\eta}_{t}=\eta_{t}$ for all times $t\geq 0$.

With the fixed point equation (20) one can also derive an explicit formula for
the solution to (19). As for given $P$, the equation (20) is a linear
McKean–Vlasov equation we can use a Duhamel/Variation of Constants formula to
derive that (for the fixed point $P=\bar{P}$) it holds that

$\displaystyle\begin{split}\bar{u}(t)&=\exp(\mathcal{A}_{P1}(t))\left(\bar{u}_{0}-\mathbb{E}_{Y}\left[\bar{u}_{0}\right]\right)+\exp\left(\mathcal{A}_{P2}(t)\right)\mathbb{E}_{Y}\left[\bar{u}_{0}\right]\\\
&\phantom{=}+\int_{0}^{t}\exp\left(\mathcal{A}_{P1}(s,t)\right)\mathcal{B}\mathrm{d}W_{s}+\int_{0}^{t}\exp\left(\mathcal{A}_{P2}(s,t)\right)\bar{P}_{s}H^{\prime}R^{-1}_{s}\mathrm{d}Y_{s},\end{split}$
(21)

where

$\displaystyle\mathcal{A}_{P1}(s,t)$
$\displaystyle:=\int_{s}^{t}\mathcal{A}-\frac{\bar{P}_{r}H^{\prime}_{r}R^{-1}_{r}H}{2}\mathrm{d}r$
$\displaystyle\mathcal{A}_{P2}(s,t)$
$\displaystyle:=\int_{s}^{t}\mathcal{A}-\bar{P}_{r}H^{\prime}R^{-1}_{r}H\mathrm{d}r,$

and $\exp\left(\mathcal{A}_{Pi}(s,t)\right),\leavevmode\nobreak\ i=1,2$ are
the corresponding solution semigroups.

######  Remark 22.

One could also use (21) to formulate a mild solution theory for the EnKBF.
This is out of the scope of this paper and we focus instead on a variational
theory.

##  5 The Feedback Particle Filter

In the linear Gaussian setting the mean field EnKBF (19) describes a diffusion
process with the remarkable property, that its (time) marginal laws are given
by the desired posterior distribution. In the general setting this attribute
is achieved by the Feedback Particle Filter (FPF), it is given by

$\displaystyle\begin{split}\mathrm{d}\hat{u}_{t}&=\mathcal{A}(\hat{u}_{t})\mathrm{d}t+\mathcal{B}(\hat{u}_{t})\mathrm{d}\bar{W}_{t}\\\
&\phantom{=}+K(\hat{u}_{t},\hat{\eta}_{t})\left(\mathrm{d}Y_{t}-\frac{H(\hat{u}_{t})+\mathbb{E}_{Y}\left[H(\hat{u}_{t})\right]}{2}\mathrm{d}t\right)+\frac{1}{2}\xi(\hat{u}_{t},\hat{\eta}_{t})\mathrm{d}t,\end{split}$
(22)

where $\hat{\eta}_{t}$ denotes the conditional distribution of $\hat{u}_{t}$,
the so called gain term
$K(\cdot,\hat{\eta}_{t}):\mathscr{H}\to\mathscr{H}\times\mathbb{R}^{d_{y}}$ is
(not uniquely) determined by the weak differential equation

$\displaystyle\hat{\eta}_{t}\left(\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi\leavevmode\nobreak\
,\leavevmode\nobreak\
K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}\right)=\hat{\eta}_{t}\left(\phi\left(H-\hat{\eta}_{t}(H)\right)^{\prime}\right)R^{-1}_{t}\leavevmode\nobreak\
\text{for all Itô testfunctions}\leavevmode\nobreak\ \phi$ (23)

and the correctional drift term
$\xi(\cdot,\hat{\eta}_{t}):\mathscr{H}\to\mathscr{H}$ is given by

$\displaystyle\xi(\hat{u}_{t},\hat{\eta}_{t}):=\left(\left\langle
K\left(\hat{u}_{t},\hat{\eta}_{t}\right),\nabla\right\rangle_{\mathscr{H}}R_{t}K\left(\hat{u}_{t},\hat{\eta}_{t}\right)^{{}^{\prime}}\right)^{{}^{\prime}}:=\sum_{j\in\mathbb{N}}\sum_{k\in\mathbb{N}}\left\langle\nu_{j},\partial_{\nu_{k}}K(\hat{u}_{t},\hat{\eta}_{t})\right\rangle_{\mathscr{H}}R_{t}\left\langle\nu_{k},K(\hat{u}_{t},\hat{\eta}_{t})\right\rangle_{\mathscr{H}}^{\mathrm{T}}\nu_{j}.$

######  Remark 23.

In the one dimensional setting, i.e.
$\mathscr{V}=\mathscr{H}=\mathbb{R},d_{y}=1$ the Kalman gain $K$ can be
determined explicitly. Let us not distinguish between $\hat{\eta}$ and its
density function (assuming it exists) and denote its cumulative distribution
function (cdf) by
$\hat{\Xi}_{t}(x):=\int_{-\infty}^{x}\hat{\eta}_{t}(y)\mathrm{d}y$, then the
gain is given by

$\displaystyle\begin{split}K(x,\hat{\eta}_{t})&=\frac{-\int_{-\infty}^{x}\left(H(y)-\hat{\eta}(H)\right)\hat{\eta}_{t}(y)\mathrm{d}y}{\hat{\eta}_{t}(x)}R^{-1}_{t}\\\
&=\left(\mathbb{E}_{Y}\left[H(\hat{u}_{t})\right]-\mathbb{E}_{Y}\left[H\leavevmode\nobreak\
(\hat{u}_{t})\leavevmode\nobreak\ |\leavevmode\nobreak\ \hat{u}_{t}\leq
x\leavevmode\nobreak\
\right]\right)\left(\partial_{x}\log\hat{\Xi}_{t}(x)\right)^{-1}R^{-1}_{t}.\end{split}$
(24)

In this setting the Itô correction $\xi$ also simplifies drastically, as

$\displaystyle\partial_{x}K(x,\hat{\eta}_{t})=-R^{-1}_{t}\left(H(x)-\hat{\eta}(H)\right)-K(x,\hat{\eta}_{t})\partial_{x}\log\hat{\eta}_{t}(x),$

and therefore

$\displaystyle\begin{split}K(\hat{u}_{t},\hat{\eta}_{t})R_{t}\partial_{x}K(\hat{u}_{t},\hat{\eta}_{t})&=-K(\hat{u}_{t},\hat{\eta}_{t})\left(H(\hat{u}_{t})-\hat{\eta}(H)\right)-K(\hat{u}_{t},\hat{\eta}_{t})^{2}R_{t}\partial_{x}\log\hat{\eta}_{t}(\hat{u}_{t}),\end{split}$

which in turn gives us the 1D FPF

$\displaystyle\begin{split}\mathrm{d}\hat{u}_{t}&=\mathcal{A}(\hat{u}_{t})\mathrm{d}t+\mathcal{B}(\hat{u}_{t})\mathrm{d}\bar{W}_{t}+K(\hat{u}_{t},\hat{\eta}_{t})\left(\mathrm{d}Y_{t}-\left(H\left(\hat{u}_{t}\right)+\frac{R_{t}K(\hat{u}_{t},\hat{\eta}_{t})\partial_{x}\log\hat{\eta}_{t}\left(\hat{u}_{t}\right)}{2}\right)\mathrm{d}t\right).\end{split}$
(25)

In the finite dimensional setting the FPF was first derived in [51] with an
optimal control approach, independently and prior to this work a similar mean-
field optimal filter for smoothed noise and finite dimensional signals has
been found in [16]. In [41] various finite dimensional consistent mean field
filters, among them the original FPF, have been derived by matching the strong
Fokker–Planck equation of a diffusion process to the strong form of the KSE
(9). Building on this work we now extend the FPF to infinite dimensions by
showing that it describes the optimal filter, in the sense that the
(conditional) law of (22) propagates in time exactly according to the KSE (9).
However, since we are working in the infinite setting, we do so by matching
the weak Fokker–Planck equation to the weak KSE (9).

######  Remark 24.

The well posedness of the FPF (22) is an open problem, even in the much
simpler finite dimensional case, and is thus just assumed in the following.

######  Lemma 25.

Denote by $\left(\hat{\eta}_{t}\right)_{t\geq 0}$ the law of the FPF
$\left(\hat{u}_{t}\right)_{t\geq 0}$, given by (22). We assume that for all
times $t\geq 0$ both $K(\cdot,\hat{\eta}_{t})$ and $\xi(\cdot,\hat{\eta}_{t})$
are well defined functions from $\mathscr{H}$ into $\mathscr{H}$, that are
integrable with respect to $\hat{\eta}_{t}$.
Then $\hat{\eta}$ satisfies the weak form of the KSE (9) for all Itô
testfunctions (according to Definition 4) $\phi:\mathscr{H}\to\mathbb{R}$
satisfying the following properties

  * •

$\phi$ is integrable with respect to $\hat{\eta}_{t}$ for all $t\geq 0$.

  * •

For all $v\in H$ the Hessian $\mathrm{D}^{2}_{\mathscr{H}}\phi(v)$ is a self
adjoint operator on $\mathscr{H}$.

  * •

The map $\hat{\Phi}:\mathscr{H}\to\mathbb{R}^{d_{y}}$, defined by
$\hat{\Phi}(v):=\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi(v),K(v,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}R_{t}$,
is an Itô function (componentwise) that is also integrable with respect to
$\eta_{t}$ for all $t\geq 0$.

###### Proof.

Let $\phi:\mathscr{H}\to\mathbb{R}$ be arbitrary, satisfying the properties
specified above. We note by Itô’s formula, that the Kolmogorov forward
equation describing the evolution of $\hat{\eta}$ is given by

$\displaystyle\begin{split}\mathrm{d}\hat{\eta}_{t}(\phi)&=\eta_{t}(\mathcal{L}\phi)\mathrm{d}t+\hat{\eta}_{t}\left(\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi,K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}\right)\left(\mathrm{d}Y_{t}-\hat{\eta}_{t}(H)\mathrm{d}t\right)\\\
&\phantom{=}+\frac{\hat{\eta}_{t}\left(\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi,\xi(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}\right)+\hat{\eta}_{t}\left(\mathrm{tr}_{\mathscr{H}}\left[\left(\mathrm{D}^{2}_{\mathscr{H}}\phi\right)K(\cdot,\hat{\eta}_{t})RK(\cdot,\hat{\eta}_{t})^{\prime}\right]\right)}{2}\mathrm{d}t\\\
&\phantom{=}-\frac{\hat{\eta}_{t}\left(\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi,K(\cdot,\hat{\eta}_{t})(H-\hat{\eta}_{t}(H))\right\rangle_{\mathscr{H}}\right)}{2}\mathrm{d}t\end{split}$
(26)

Due to (23), the first line of (26) is exactly the KSE and thus, to show
consistency, we only have to prove that the second line is zero. To this end
we note that by Parseval we have

$\displaystyle\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi,\xi(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}+\mathrm{tr}_{\mathscr{H}}\left[\left(\mathrm{D}^{2}_{\mathscr{H}}\phi\right)K(\cdot,\hat{\eta}_{t})RK(\cdot,\hat{\eta}_{t})^{\prime}\right]$
$\displaystyle=\sum_{j\in\mathbb{N}}\left\langle\nu_{j},\mathrm{D}^{1}_{\mathscr{H}}\phi\right\rangle_{\mathscr{H}}\left\langle\nu_{j},\xi(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}+\sum_{k\in\mathbb{N}}\left\langle\nu_{k},\left(\mathrm{D}^{2}_{\mathscr{H}}\phi\right)K(\cdot,\hat{\eta}_{t})RK(\cdot,\hat{\eta}_{t})^{\prime}\nu_{k}\right\rangle_{\mathscr{H}}$
$\displaystyle=\sum_{j\in\mathbb{N}}\sum_{k\in\mathbb{N}}\left\langle\nu_{j},\mathrm{D}^{1}_{\mathscr{H}}\phi\right\rangle_{\mathscr{H}}\left\langle\nu_{j},\partial_{\nu_{k}}K(\hat{u}_{t},\hat{\eta}_{t})\right\rangle_{\mathscr{H}}R_{t}\left\langle\nu_{k},K(\hat{u}_{t},\hat{\eta}_{t})\right\rangle_{\mathscr{H}}^{\mathrm{T}}$
$\displaystyle\phantom{=}+\sum_{k\in\mathbb{N}}\left\langle\nu_{k},\left(\mathrm{D}^{2}_{\mathscr{H}}\phi\right)K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}R_{t}\left\langle\nu_{k},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}^{\mathrm{T}}$

Next we now that since $\mathrm{D}^{2}_{\mathscr{H}}\phi$ is self adjoint and
by Parseval we have

$\displaystyle\left\langle\nu_{k},\mathrm{D}^{2}_{\mathscr{H}}\phi
K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}$
$\displaystyle=\left\langle\mathrm{D}^{2}_{\mathscr{H}}\phi\nu_{k},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}=\left\langle\partial_{\nu_{k}}\mathrm{D}^{1}_{\mathscr{H}}\phi,K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}$
$\displaystyle=\sum_{j\in\mathbb{N}}\left\langle\nu_{j},\partial_{\nu_{k}}\mathrm{D}^{1}_{\mathscr{H}}\phi\right\rangle_{\mathscr{H}}\left\langle\nu_{j},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}.$

This gives us by again using Parseval

$\displaystyle\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi,\xi(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}+\mathrm{tr}_{\mathscr{H}}\left[\left(\mathrm{D}^{2}_{\mathscr{H}}\phi\right)K(\cdot,\hat{\eta}_{t})RK(\cdot,\hat{\eta}_{t})^{\prime}\right]$
$\displaystyle=\sum_{k\in\mathbb{N}}\sum_{j\in\mathbb{N}}\left\langle\nu_{j},\mathrm{D}^{1}_{\mathscr{H}}\phi\right\rangle_{\mathscr{H}}\left\langle\nu_{j},\partial_{\nu_{k}}K(\hat{u}_{t},\hat{\eta}_{t})\right\rangle_{\mathscr{H}}R_{t}\left\langle\nu_{k},K(\hat{u}_{t},\hat{\eta}_{t})\right\rangle_{\mathscr{H}}^{\mathrm{T}}$
$\displaystyle\phantom{=}+\sum_{k\in\mathbb{N}}\sum_{j\in\mathbb{N}}\left\langle\nu_{j},\partial_{\nu_{k}}\mathrm{D}^{1}_{\mathscr{H}}\phi\right\rangle_{\mathscr{H}}\left\langle\nu_{j},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}R_{t}\left\langle\nu_{k},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}^{\mathrm{T}}$
$\displaystyle=\sum_{k\in\mathbb{N}}\left(\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi,\partial_{\nu_{k}}K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}+\left\langle\partial_{\nu_{k}}\mathrm{D}^{1}_{\mathscr{H}}\phi,K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}\right)R_{t}\left\langle\nu_{k},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}^{\mathrm{T}}$

Using the product formula in Hilbert spaces444More precisely the formula for
the directional derivative of the scalar product of two differentiable
functions., we thus derive

$\displaystyle\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi,\xi(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}+\mathrm{tr}_{\mathscr{H}}\left[\left(\mathrm{D}^{2}_{\mathscr{H}}\phi\right)K(\cdot,\hat{\eta}_{t})RK(\cdot,\hat{\eta}_{t})^{\prime}\right]$
$\displaystyle=\sum_{k\in\mathbb{N}}\partial_{\nu_{k}}\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi,K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}R_{t}\left\langle\nu_{k},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}^{\mathrm{T}}.$

The map $\hat{\Phi}$, defined in the statement of the Lemma, then allows us to
again use Parseval to derive

$\displaystyle\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi,\xi(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}+\mathrm{tr}_{\mathscr{H}}\left[\left(\mathrm{D}^{2}_{\mathscr{H}}\phi\right)K(\cdot,\hat{\eta}_{t})RK(\cdot,\hat{\eta}_{t})^{\prime}\right]$
$\displaystyle=\sum_{k\in\mathbb{N}}\left(\partial_{\nu_{k}}\hat{\Phi}\right)\left\langle\nu_{k},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}^{\mathrm{T}}=\sum_{k\in\mathbb{N}}\left\langle\nu_{k},\mathrm{D}^{1}_{\mathscr{H}}\hat{\Phi}\right\rangle_{\mathscr{H}}\left\langle\nu_{k},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}^{\mathrm{T}}$
$\displaystyle=\sum_{i=1}^{d_{y}}\sum_{k\in\mathbb{N}}\sum_{i=1}^{d_{y}}\left\langle\nu_{k},\mathrm{D}^{1}_{\mathscr{H}}\hat{\Phi}\right\rangle_{\mathscr{H}}\delta_{i}\delta_{i}^{\mathrm{T}}\left\langle\nu_{k},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}^{\mathrm{T}}=\sum_{i=1}^{d_{y}}\left\langle\mathrm{D}^{1}_{\mathscr{H}}\hat{\Phi}\delta_{i},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}\delta_{i},$

where $\delta_{i},\leavevmode\nobreak\ i=1,\cdots,d_{y}$ denotes the canonical
basis of $\mathbb{R}^{d_{y}}$. Thus, by the assumed regularity of
$\hat{\Phi}$, we derive by using (23)

$\displaystyle\hat{\eta}_{t}\left(\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi,\xi(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}\right)+\hat{\eta}_{t}\left(\mathrm{tr}_{\mathscr{H}}\left[\left(\mathrm{D}^{2}_{\mathscr{H}}\phi\right)K(\cdot,\hat{\eta}_{t})RK(\cdot,\hat{\eta}_{t})^{\prime}\right]\right)=\sum_{i=1}^{d_{y}}\hat{\eta}_{t}\left(\left\langle\mathrm{D}^{1}_{\mathscr{H}}\hat{\Phi}\delta_{i},K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}\right)\delta_{i}$
$\displaystyle=\sum_{i=1}^{d_{y}}\hat{\eta}_{t}\left(\hat{\Phi}\delta_{i}\left(H-\hat{\eta}(H)\right)^{\prime}\right)R^{-1}_{t}\delta_{i}=\sum_{i=1}^{d_{y}}\hat{\eta}_{t}\left(\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi(v),K(v,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}R_{t}\delta_{i}\left(H-\hat{\eta}(H)\right)^{\prime}\right)R^{-1}_{t}\delta_{i}$
$\displaystyle=\sum_{i=1}^{d_{y}}\hat{\eta}_{t}\left(\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi(v),K(v,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}R_{t}\delta_{i}\delta_{i}^{\mathrm{T}}R^{-1}_{t}\left(H-\hat{\eta}(H)\right)^{\prime}\right)=\hat{\eta}_{t}\left(\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi(v),K(v,\hat{\eta}_{t})\left(H-\hat{\eta}(H)\right)\right\rangle_{\mathscr{H}}\right),$

which in turn lets us conclude that (26) coincides with the KSE (9) and thus
the FPF is indeed consistent. ∎

The FPF is a true generalization of the EnKBF to general filtering problems
and it even provides a connection between the EnKBF and the true posterior
even in inconsistent setting as the following Lemma shows.

######  Lemma 26.

Let again $\left(\hat{\eta}\right)_{t\geq 0}$ be the (conditional) marginal
laws to the FPF (22). Assuming integrability of $K(\cdot,\hat{\eta}_{t})$,
then it holds that

$\displaystyle\mathbb{E}_{Y}\left[K(\hat{u}_{t},\hat{\eta}_{t})\right]=\hat{\eta}_{t}\left(K(\cdot,\hat{\eta}_{t})\right)=\mathbb{Cov}_{\hat{\eta}_{t}}\left[\mathrm{id}_{\mathscr{H}},H\right]R^{-1}_{t}=\mathbb{Cov}_{Y}\left[\hat{u}_{t},H(\hat{u}_{t})\right]R^{-1}_{t}.$
(27)

If $H$ is linear and $\hat{\eta}_{t}$ is Gaussian, one can even choose the
gain term $K$ such that
$K(\cdot,\hat{\eta}_{t})=\mathbb{Cov}_{\hat{\eta}_{t}}\left[\mathrm{id}_{\mathscr{H}},H\right]R^{-1}_{t}$.
In the linear Gaussian setting the EnKBF is thus just a special case of the
FPF.

###### Proof.

For any $i\in\mathbb{N}$ we set
$\phi_{i}(v):=\left\langle\nu_{i},v\right\rangle_{\mathscr{H}}$ as a
testfunction in the gain equation (23), then we have

$\displaystyle\left\langle\nu_{i},\hat{\eta}_{t}\left(K(\cdot,\hat{\eta}_{t})\right)\right\rangle_{\mathscr{H}}=\hat{\eta}_{t}\left(\left\langle\mathrm{D}^{1}_{\mathscr{H}}\phi_{i}\leavevmode\nobreak\
,\leavevmode\nobreak\
K(\cdot,\hat{\eta}_{t})\right\rangle_{\mathscr{H}}\right)=\hat{\eta}_{t}\left(\phi_{i}\left(H-\hat{\eta}(H)\right)^{\prime}\right)R^{-1}_{t}$
$\displaystyle=\left\langle\nu_{i},\mathbb{Cov}_{\hat{\eta}_{t}}\left[\mathrm{id}_{H},H\right]R^{-1}_{t}\right\rangle_{\mathscr{H}}.$

Since this holds for any $i\in\mathbb{N}$ this indeed shows the validity of
(27). The second claim follows from Gaussian integration by parts as in [11].
∎

.

Identity (27) is the reason why the extension of the mean field EnKBF to
nonlinear signals is sometimes referred to in the literature as the constant
gain approximation (to the FPF) [48]. The next section is concerned with the
basic properties of this McKean–Vlasov equation, showing existence and
uniqueness of solutions.

##  6 The mean field EnKBF for nonlinear signals

We consider the extension of the EnKBF to nonlinear signals

$\displaystyle\begin{split}\mathrm{d}\bar{u}_{t}&=\mathcal{A}(\bar{u}_{t})\mathrm{d}t+\mathcal{B}(\bar{u}_{t})\mathrm{d}\bar{W}_{t}+\mathbb{Cov}_{Y}\left[\bar{u}_{t},H(\bar{u}_{t})\right]R^{-1}_{t}\left(\mathrm{d}Y_{t}-\frac{H(\bar{u}_{t})+\mathbb{E}_{Y}\left[H(\bar{u}_{t})\right]}{2}\mathrm{d}t\right),\end{split}$
(28)

with not necessarily Gaussian initial condition.

Note that for nonlinear signals, the mean field EnKBF (28) does not allow for
a seperate description of its covariance matrix via a Riccati equation. Thus
we can not simply apply the same argument as in Lemma 20 for well posedness.
Indeed due to missing growth conditions, as well as only local Lipschitz
properties of the coefficients involved, the question of well posedness of
(28) is non trivial.

######  Remark 27 (Literature).

In a finite dimensional setting well posedness of the nonlinear EnKBF (28) was
shown in [15] for possibly correlated observation noises and bounded
observation functions. For linear observation functions [21] showed well
posedness of finite dimensional mean-field EnKBFs that may also include
singular correction terms in the presence of correlated noise. This was done
by a combination of a fixed point and a stopping argument with respect to the
covariance $\mathbb{Cov}_{Y}\left[\bar{u}_{t},H(\bar{u}_{t})\right]$. The main
tool was a law of total variance (30) that was made robust with respect to the
fixed point argument via stopping times.
In the infinite dimensional setting this argument does not work due to missing
equivalence of norms. While using a Galerkin argument would thus seem
tempting, it would also not imply the desired uniqueness of solutions, which
is a property that is difficult to show, and sometimes does not even hold for
McKean–Vlasov equations under local Lipschitz conditions [44].
So instead we use an adapted fixed point argument, that makes use of the same
variance bounds as in section 3, that also hold for (28) and seem to be the
only consistency of its distribution with respect to the actual posterior.

First we investigate the covariance structure in (28). We do this however in a
more general form. Let $(h_{t})_{t\geq 0}$ be a given $R^{d_{y}}$-valued,
adapted stochastic process and assume that $\xi^{Y}$ is an
$\mathbb{R}^{d_{y}}$-valued semimartingale that is adapted to the natural
filtration generated by $Y$. Then for a $\tilde{u}$ satisfying

$\displaystyle\begin{split}\mathrm{d}\tilde{u}_{t}&=\mathcal{A}(\tilde{u}_{t})\mathrm{d}t+\mathcal{B}(\tilde{u}_{t})\mathrm{d}\bar{W}_{t}+\mathbb{Cov}_{Y}\left[\tilde{u}_{t},h_{t}\right]R^{-1}_{t}\left(\mathrm{d}\xi^{Y}_{t}-\frac{h_{t}+\mathbb{E}_{Y}\left[h_{t}\right]}{2}\mathrm{d}t\right)\end{split}$
(29)

with mean $\tilde{m}_{t}:=\mathbb{E}_{Y}\left[\tilde{u}_{t}\right]$, it holds
that

$\displaystyle\begin{split}&\partial_{t}\left\langle
v,\mathbb{Cov}_{Y}\left[\tilde{u}_{t}\right]w\right\rangle_{\mathscr{H}}\\\
&=\mathbb{E}_{Y}\left[\left\langle
v,\tilde{u}_{t}-\tilde{m}_{t}\right\rangle_{\mathscr{H}}\prescript{}{\mathscr{V}^{\prime}}{\left\langle
w,\mathcal{A}(\tilde{u}_{t})-\mathcal{A}(\tilde{m}_{t})\right\rangle_{\mathscr{V}}}+\left\langle
w,\tilde{u}_{t}-\tilde{m}_{t}\right\rangle_{\mathscr{H}}\prescript{}{\mathscr{V}^{\prime}}{\left\langle
v,\mathcal{A}(\tilde{u}_{t})-\mathcal{A}(\tilde{m}_{t})\right\rangle_{\mathscr{V}}}\right]\mathrm{d}t\\\
&\phantom{=}+\mathbb{E}_{Y}\left[\left\langle
v,\mathcal{B}(\tilde{u}_{t})\sqrt{\mathcal{Q}}\left(\mathcal{B}(\tilde{u}_{t})\sqrt{\mathcal{Q}}\right)^{\prime}w\right\rangle_{\mathscr{H}}\right]-\left\langle
v,\mathbb{Cov}_{Y}\left[\tilde{u}_{t},h_{t}\right]R^{-1}_{t}\mathbb{Cov}_{Y}\left[h_{t},\tilde{u}_{t}\right]w\right\rangle_{\mathscr{H}}.\end{split}$

for every $v,w\in\mathscr{V}$. Thus by the positivity of $\left\langle
v,\mathbb{Cov}_{Y}\left[\tilde{u}_{t},h_{t}\right]R^{-1}_{t}\mathbb{Cov}_{Y}\left[h_{t},\tilde{u}_{t}\right]v\right\rangle_{\mathscr{H}}$
for every $v\in\mathscr{V}$ we immediately derive

$\displaystyle\partial_{t}\mathbb{E}_{Y}\left[\left\|\tilde{u}_{t}-\tilde{m}_{t}\right\|_{\mathscr{H}}^{2}\right]\leq\lambda\leavevmode\nobreak\
\mathbb{E}_{Y}\left[\left\|\tilde{u}_{t}-\tilde{m}_{t}\right\|_{\mathscr{H}}^{2}\right]+\beta,$
(30)

and therefore

$\displaystyle\mathrm{tr}_{\mathscr{H}}\mathbb{Cov}_{Y}\left[\tilde{u}_{t}\right]=\mathbb{E}_{Y}\left[\left\|\tilde{u}_{t}-\tilde{m}_{t}\right\|_{\mathscr{H}}^{2}\right]\leq\beta
e^{\lambda t}.$ (31)

Thus the EnKBF satisfies the variance bound that is implied by the Bayesian
filtering problem, it does so even in a stronger sense, as taking the
expectation is not required. As implied by the law of total variance for the
optimal filter, this bound is robust with respect to perturbations of both the
modelled observation function $H$ and the actual observation data $Y$.

Next we show that the robust variance bound (31) can be used to show well
posedness of the EnKBF via a Picard argument as the following Lemma shows.

######  Theorem 28.

If the conditions in Assumption (2) and (7) are satisfied, there exists a
unique (strong) solution to the nonlinear mean-field EnKBF (28).

###### Proof.

For proving well posedness it is enough to restrict ourselves to a small time
frame $[0,T]$ with $T$ chosen later on. The extension to arbitrary time frames
can then be easily achieved by standard glueing arguments

The proof is separated into two steps. First we introduce partially stopped
dynamics and show their well posedness via a fixed point argument. Next we
show that these stopped dynamics must always coincide with solutions to the
EnKBF on events that cover the whole probability space almost surely.

In the following we will make use of the semimartingale decomposition of the
observation process $Y$. For better highlighting that the true signal process
$u$ plays the role of a parameter to the EnKBF and to easily distinguish it
from other processes encountered in the proof we will denote it by
$u^{\mathrm{ref}}$. The observation process $Y$ is thus given by

$\displaystyle\mathrm{d}Y_{t}=H(u^{\mathrm{ref}}_{t})\mathrm{d}t+\Gamma_{t}\mathrm{d}V_{t}.$
(32)

Step 1: Well posedness for partially stopped dynamics.

For any $k\in\mathbb{N}$ we denote by $\tilde{\mathbb{1}}_{k}$ a smoothed
version of the indicator function $\mathbb{1}_{[0,k]}$, such that
$\mathbb{1}_{[0,k]}\leq\tilde{\mathbb{1}}_{k}\leq\mathbb{1}_{[0,k+1]}$.

Define for $k,l\in\mathbb{N}$ and any $\mathscr{H}$-valued random variable $v$
the stopped observation function $H^{k}$ by

$\displaystyle
H^{k}(v):=\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[H(v)\right]\right|\right)H(v)$

as well as the stopped observation process $Y^{l}$ by

$\displaystyle\mathrm{d}Y^{l}_{t}:=\tilde{\mathbb{1}}_{l}\left(\mathbb{E}_{Y}\left[\left|H(u^{\mathrm{ref}}_{t})\right|^{2}\right]\right)\mathrm{d}Y_{t}.$

In this step we show that there exists a unique solution $\bar{u}^{k}$ of

$\displaystyle\begin{split}\mathrm{d}\bar{u}^{k}_{t}&=\mathcal{A}(\bar{u}^{k}_{t})\mathrm{d}t+\mathcal{B}(\bar{u}^{k}_{t})\mathrm{d}\bar{W}_{t}\\\
&\phantom{=}+\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]R^{-1}_{t}\left(\mathrm{d}Y^{k}_{t}-\frac{H^{k}\left(\bar{u}^{k}_{t}\right)+\mathbb{E}_{Y}\left[H^{k}\left(\bar{u}^{k}_{t}\right)\right]}{2}\mathrm{d}t\right),\end{split}$
(33)

via a fixed point argument with respect to the stopped modelled observations
$\left(H(\bar{u}^{k}_{t})\right)_{t\in[0,T]}$.

To this end we consider for a given process $h$ the unique solution
$\tilde{u}$ of

$\displaystyle\begin{split}\mathrm{d}\tilde{u}_{t}&=\mathcal{A}(\tilde{u}_{t})\mathrm{d}t+\mathcal{B}(\tilde{u}_{t})\mathrm{d}\bar{W}_{t}\\\
&\phantom{=}+\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h_{t}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}_{t},h_{t}\right]R^{-1}_{t}\left(\mathrm{d}Y^{l}_{t}-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h_{t}\right]\right|\right)\frac{h_{t}+\mathbb{E}_{Y}\left[h_{t}\right]}{2}\mathrm{d}t\right).\end{split}$
(34)

Well posedness of (34) is assured by the standard (global) Lipschitz
conditions. We define the map $\Xi$ by

$\displaystyle\Xi(h):=H(\tilde{u}).$

The existence and uniqueness of solutions to (33) corresponds to the existence
and uniqueness of fixed points of $\Xi$. We prove this via a Banach fixed
point argument and thus have to show the contractivity of $\Xi$. Due to the
Lipschitz continuity of $H$, this further reduces to the problem of showing
that the solution map $h\mapsto\tilde{u}$ defined by the equation (29) is
Lipschitz, with constant strictly smaller than $1/\mathrm{Lip}(H)$.

Since (34) is of the form (29), the process $\tilde{u}$ must also satisfy the
uniform variance bound (31) corresponding to the law of total variance.
Therefore, by the Lipschitz continuity of $H$, we can assume that any
potential fixed point $h$ satisfies

$\displaystyle\mathbb{Var}_{Y}\left[h_{t}\right]=\mathrm{tr}_{\mathbb{R}^{d_{y}}}\mathbb{Cov}_{Y}\left[h_{t}\right]\leq\mathrm{Lip}(\mathscr{H})\beta
e^{\lambda t}.$ (35)

To show the contractivity of $\Xi$, let $h^{i},\leavevmode\nobreak\ i=1,2$ be
two given processes and denote by $\tilde{u}^{i},\leavevmode\nobreak\ i=1,2$
the corresponding solutions to (34). Using the uniform variance bounds (35) as
well as the Lipschitz continuity of $\tilde{\mathbb{1}}_{k}$ and its
boundedness $0\leq\tilde{\mathbb{1}}_{k}\leq 1$, we derive first the following
bound for the gain difference

$\displaystyle\begin{split}&\left\|\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{t}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{t},h^{1}_{t}\right]-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{t}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{t},h^{2}_{t}\right]\right\|_{L(\mathbb{R}^{d_{y}},\mathscr{H})}\\\
&\leq\left|\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{t}\right]\right|\right)-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{t}\right]\right|\right)\right|\leavevmode\nobreak\
\left\|\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{t},h^{1}_{t}\right]\right\|_{L(\mathbb{R}^{d_{y}},\mathscr{H})}\\\
&\phantom{=}+\sqrt{\mathbb{E}_{Y}\left[\left\|\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]}\sqrt{\mathbb{E}_{Y}\left[\left\|h^{1}_{s}-\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right\|_{\mathbb{R}^{d_{y}}}^{2}\right]}\\\
&\phantom{=}+\sqrt{\mathbb{E}_{Y}\left[\left\|\tilde{u}^{2}_{t}-\mathbb{E}_{Y}\left[\tilde{u}^{2}_{t}\right]\right\|_{\mathscr{H}}^{2}\right]}\sqrt{\mathbb{E}_{Y}\left[\left\|h^{1}_{t}-h^{2}_{t}\right\|_{\mathbb{R}^{d_{y}}}^{2}\right]}\\\
&\leq\left(\left(\mathrm{Lip}(\tilde{\mathbb{1}}_{k})+1\right)\sqrt{\mathrm{Lip}(H)}+1\right)\beta
e^{\lambda
t}\left(\sqrt{\mathbb{E}_{Y}\left[\left\|\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]}+\sqrt{\mathbb{E}_{Y}\left[\left\|h^{1}_{s}-h^{2}_{s}\right\|_{\mathbb{R}^{d_{y}}}^{2}\right]}\right).\end{split}$
(36)

Using Itô’s formula for the squared norm, we derive

$\displaystyle\left\|\tilde{u}^{1}_{t}-\tilde{u}^{2}_{t}\right\|_{\mathscr{H}}^{2}=2\int_{0}^{t}\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(\tilde{u}^{1}_{s})-\mathcal{A}(\tilde{u}^{2}_{s}),\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s}\right\rangle_{\mathscr{V}}}\mathrm{d}s$
$\displaystyle+2\int_{0}^{t}\left\langle\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s},\left(\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{t}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{s},h^{1}_{s}\right]-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{t}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]\right)R^{-1}_{s}\mathrm{d}Y^{k}_{s}\right\rangle_{\mathscr{H}}$
$\displaystyle-2\int_{0}^{t}\left\langle\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s},\left(\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{s},h^{1}_{s}\right]-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]\right)\right.$
$\displaystyle\left.\phantom{2\int_{0}^{t}\langle
u^{1}-u^{2},++}R^{-1}_{s}\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\frac{h^{1}_{s}+\mathbb{E}_{Y}\left[h^{1}_{s}\right]}{2}\right\rangle_{\mathscr{H}}\mathrm{d}s$
$\displaystyle-2\int_{0}^{t}\left\langle\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s},\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]R^{-1}_{s}\right.$
$\displaystyle\left.\phantom{2\int_{0}^{t}\langle
u^{1}-u^{2},++}\frac{\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\left(h^{1}_{s}+\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right)-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\left(h^{2}_{s}+\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right)}{2}\right\rangle_{\mathscr{H}}\mathrm{d}s$
$\displaystyle+2\int_{0}^{t}\left\langle\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s},\left(\mathcal{B}(\tilde{u}^{1}_{s})-\mathcal{B}(\tilde{u}^{2}_{s})\right)\mathrm{d}W_{s}\right\rangle_{\mathscr{H}}$
$\displaystyle+\sum_{k\in\mathbb{N}}\int_{0}^{t}\left\langle\nu_{k},\left(\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{s},h^{1}_{s}\right]-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]\right)\right\rangle_{\mathscr{H}}R^{-1}_{s}$
$\displaystyle\phantom{=\sum_{k\in\mathbb{N}}\int_{0}^{t}}\left\langle\nu_{k},\left(\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{s},h^{1}_{s}\right]-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]\right)\right\rangle_{\mathscr{H}}^{\mathrm{T}}\mathrm{d}s$
$\displaystyle+\sum_{k\in\mathbb{N}}\sum_{n\in\mathbb{N}}\int_{0}^{t}q_{n}\left\langle\nu_{k},(\mathcal{B}(\tilde{u}^{1}_{s})-\mathcal{B}(\tilde{u}^{2}_{s}))e_{n}\right\rangle_{\mathscr{H}}^{2}\mathrm{d}s.$

Now we note that by Parseval and the onesided Lipschitz condition (2) we have

$\displaystyle\begin{split}&\sum_{k\in\mathbb{N}}\sum_{n\in\mathbb{N}}q_{n}\left\langle\nu_{k},(\mathcal{B}(\tilde{u}^{1}_{s})-\mathcal{B}(\tilde{u}^{2}_{s}))e_{n}\right\rangle_{\mathscr{H}}^{2}=\left\|\left(\mathcal{B}(u)-\mathcal{B}(v)\right)\circ\sqrt{\mathcal{Q}}\right\|_{\mathrm{L}_{2}\left(\mathscr{U};\mathscr{H}\right)}^{2}\leq\lambda\leavevmode\nobreak\
\left\|\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s}\right\|_{\mathscr{H}}^{2},\end{split}$
(37)

as well as

$\displaystyle\begin{split}&\sum_{k\in\mathbb{N}}\left\langle\nu_{k},\left(\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{s},h^{1}_{s}\right]-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]\right)\right\rangle_{\mathscr{H}}R^{-1}_{s}\\\
&\phantom{=\times}\left\langle\nu_{k},\left(\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{s},h^{1}_{s}\right]-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]\right)\right\rangle_{\mathscr{H}}^{\mathrm{T}}\\\
&=\left\|\left(\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{s},h^{1}_{s}\right]-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]\right)R^{-1/2}_{s}\right\|_{L(\mathbb{R}^{d_{y}},\mathscr{H})}^{2}\\\
&\leq\left(\left(\mathrm{Lip}(\tilde{\mathbb{1}}_{k})+1\right)\sqrt{\mathrm{Lip}(H)}+1\right)^{2}\beta^{2}e^{{}^{2}\lambda
t}\left|R^{-1/2}_{s}\right|^{2}\left(\mathbb{E}_{Y}\left[\left\|\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]+\mathbb{E}_{Y}\left[\left\|h^{1}_{s}-h^{2}_{s}\right\|_{\mathbb{R}^{d_{y}}}^{2}\right]\right).\end{split}$
(38)

Furthermore we note that

$\displaystyle\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\mathbb{E}_{Y}\left[\left|\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)h^{1}_{s}-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)h^{2}_{s}\right|^{2}\right]$
$\displaystyle\leq
2\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\mathbb{E}_{Y}\left[\left|h^{1}_{s}-h^{2}_{s}\right|^{2}\right]$
$\displaystyle\phantom{=}+2\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\mathbb{E}_{Y}\left[\left|h^{2}_{s}\right|^{2}\right]\left|\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)-\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\right|^{2}$
$\displaystyle\leq
2\mathbb{E}_{Y}\left[\left|h^{1}_{s}-h^{2}_{s}\right|^{2}\right]+2\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|\right)\left(\left|\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|^{2}+\mathbb{E}_{Y}\left[\left|h^{2}_{s}-\mathbb{E}_{Y}\left[h^{2}_{s}\right]\right|^{2}\right]\right)\mathrm{Lip}\left(\tilde{\mathbb{1}}_{k}\right)\left|\mathbb{E}_{Y}\left[h^{1}_{s}-h^{2}_{s}\right]\right|^{2}$
$\displaystyle\leq 2\left(1+\left((k+1)^{2}+\mathrm{Lip}(H)\beta e^{\lambda
t}\right)\mathrm{Lip}\left(\tilde{\mathbb{1}}_{k}\right)\right)\mathbb{E}_{Y}\left[\left|h^{1}_{s}-h^{2}_{s}\right|^{2}\right],$

where we used that $\tilde{\mathbb{1}}_{k}\leq\mathbb{1}_{[0,k+1]}$ and the
variance bound (35) to derive the last inequality.

The variance bounds (31) and (35) also imply that

$\displaystyle\left\|\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]\right\|_{L(R^{d_{x}},\mathscr{H})}\leq\sqrt{\mathrm{Lip}(H)}\beta
e^{\lambda t}.$ (39)

If we now take the supremum on the time interval $[0,T]$ and the conditional
expectation $\mathbb{E}_{Y}$, standard Cauchy–Schwarz inequalities, together
with (37), (38) and the one-sided Lipschitz condition (2), we derive that
there exists a constant $\kappa_{1}(T)$, that only depends on the timeframe
$T$, such that

$\displaystyle\begin{split}&\mathbb{E}_{Y}\left[\sup_{t\leq
T}\left\|\tilde{u}^{1}_{t}-\tilde{u}^{2}_{t}\right\|_{\mathscr{H}}^{2}\right]\\\
&\leq\kappa_{1}(T)\int_{0}^{T}\mathbb{E}_{Y}\left[\left\|\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]+\mathbb{E}_{Y}\left[\left\|h^{1}_{s}-h^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]\mathrm{d}s\\\
&\phantom{=}+\kappa_{1}(T)\int_{0}^{t}\left(\mathbb{E}_{Y}\left[\left\|\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]+\mathbb{E}_{Y}\left[\left\|h^{1}_{s}-h^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]\right)\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\mathbb{E}_{Y}\left[\left\|h^{1}_{s}+\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right\|_{\mathscr{H}}^{2}\right]\mathrm{d}s\\\
&\phantom{=}+2\mathbb{E}_{Y}\left[\sup_{t\leq
T}\left|\int_{0}^{t}\left\langle\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s},\left(\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{s},h^{1}_{s}\right]-\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]\right)R^{-1}_{s}\mathrm{d}Y_{s}\right\rangle_{\mathscr{H}}\right|\right]\\\
&\phantom{=}+2\mathbb{E}_{Y}\left[\sup_{t\leq
T}\left|\int_{0}^{t}\left\langle\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s},\left(B(\tilde{u}^{1}_{s})-B(\tilde{u}^{2}_{s})\right)\mathrm{d}W_{s}\right\rangle_{\mathscr{H}}\right|\right].\end{split}$
(40)

Note that due to (35) we get

$\displaystyle\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\mathbb{E}_{Y}\left[\left\|h^{1}_{s}+\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right\|_{\mathscr{H}}^{2}\right]$
$\displaystyle\leq
2\mathbb{E}_{Y}\left[\left\|h^{1}_{s}-\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right\|_{\mathscr{H}}^{2}\right]+8\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right|\right)\left\|\mathbb{E}_{Y}\left[h^{1}_{s}\right]\right\|_{\mathscr{H}}^{2}$
$\displaystyle\leq 2\mathrm{Lip}(H)\beta e^{\lambda t}+8(k+1)^{2}$

Thus if we now use the specific form of the observations (32) and take the
full expectation in (40) we derive

$\displaystyle\mathbb{E}\left[\sup_{t\leq
T}\left\|\tilde{u}^{1}_{t}-\tilde{u}^{2}_{t}\right\|_{\mathscr{H}}^{2}\right]$
$\displaystyle\leq\kappa_{2}(T,k)\int_{0}^{T}\mathbb{E}\left[\left\|\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]+\mathbb{E}\left[\left\|h^{1}_{s}-h^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]\mathrm{d}s$
$\displaystyle+2\mathbb{E}\left[\int_{0}^{T}\mathbb{E}_{Y}\left[\left|\left\langle\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s},\left(\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{s},h^{1}_{s}\right]-\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]\right)R^{-1}_{s}\tilde{\mathbb{1}}_{l}\left(\mathbb{E}_{Y}\left[\left|H(u^{\mathrm{ref}}_{s})\right|^{2}\right]\right)H(u^{\mathrm{ref}}_{s})\right\rangle_{\mathscr{H}}\right|\right]\right]$
$\displaystyle+2\mathbb{E}\left[\sup_{t\leq
T}\int_{0}^{t}\left\langle\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s},\left(\mathbb{Cov}_{Y}\left[\tilde{u}^{1}_{s},h^{1}_{s}\right]-\mathbb{Cov}_{Y}\left[\tilde{u}^{2}_{s},h^{2}_{s}\right]\right)R^{-1}_{s}\tilde{\mathbb{1}}_{l}\left(\mathbb{E}_{Y}\left[\left|H(u^{\mathrm{ref}}_{s})\right|^{2}\right]\right)\Gamma_{s}\mathrm{d}V_{s}\right\rangle_{\mathscr{H}}\right]$
$\displaystyle+2\mathbb{E}\left[\sup_{t\leq
T}\int_{0}^{t}\left\langle\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s},\left(\mathcal{B}(\tilde{u}^{1}_{s})-\mathcal{B}(\tilde{u}^{2}_{s})\right)\mathrm{d}W_{s}\right\rangle_{\mathscr{H}}\right],$

for some constant $\kappa_{2}(T,k)$, where we of course used that
$\mathbb{E}\left[\leavevmode\nobreak\
\mathbb{E}_{Y}\left[\cdot\right]\right]=\mathbb{E}\left[\cdot\right]$.

To dominate the second term on the right hand side of the inequality we use
(36) together with the fact that
$\tilde{\mathbb{1}}_{k}\left(\mathbb{E}_{Y}\left[\left|H(u^{\mathrm{ref}}_{s})\right|^{2}\right]\right)\mathbb{E}_{Y}\left[\left|H(u^{\mathrm{ref}}_{s})\right|^{2}\right]\leq(k+1)$.
For the other two terms we use the Burkholder–Davis–Gundy inequality together
with (37) and (38) to derive that there exists a constant
$\kappa_{3}\left(T,k,l\right)>0$ we have

$\displaystyle\mathbb{E}\left[\sup_{t\leq
T}\left\|\tilde{u}^{1}_{t}-\tilde{u}^{2}_{t}\right\|_{\mathscr{H}}^{2}\right]\leq\kappa_{3}\left(T,k,l\right)\int_{0}^{T}\mathbb{E}\left[\left\|\tilde{u}^{1}_{s}-\tilde{u}^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]+\mathbb{E}\left[\left\|h^{1}_{s}-h^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]\mathrm{d}s,$

which by the (deterministic) Grönwall Lemma implies

$\displaystyle\mathbb{E}\left[\sup_{t\leq
T}\left\|\tilde{u}^{1}_{t}-\tilde{u}^{2}_{t}\right\|_{\mathscr{H}}^{2}\right]\leq\kappa_{3}\left(T,k,l\right)\exp\left(T\leavevmode\nobreak\
\kappa_{3}\left(T,\left\|H\right\|_{\infty}\right)\right)\int_{0}^{T}\mathbb{E}\left[\left\|h^{1}_{s}-h^{2}_{s}\right\|_{\mathscr{H}}^{2}\right]\mathrm{d}s,$

and thus for $T$ small enough we indeed have the desired contraction property.

Step 2: The stopping argument.

First we define the stopping times which we use for our argument by

$\displaystyle\begin{split}\tau^{k}&:=\inf\left\\{\leavevmode\nobreak\ t\geq
0\leavevmode\nobreak\ :\leavevmode\nobreak\
\left|\mathbb{E}_{Y}\left[H(u^{k}_{t})\right]\right|^{2}>k\leavevmode\nobreak\
\right\\}\\\ \tau^{l}_{\mathrm{ref}}&:=\inf\left\\{\leavevmode\nobreak\ t\geq
0\leavevmode\nobreak\ :\leavevmode\nobreak\
\mathbb{E}_{Y}\left[\left|H(u^{\mathrm{ref}}_{t})\right|^{2}\right]>l\leavevmode\nobreak\
\right\\},\end{split}$ (41)

and note that both are stopping times with respect to the filtration generated
by $Y$, implying that for any stochastic process $(z_{t})_{t\geq 0}$ and any
(suitably integrable) functions $f,g$, the identities

$\displaystyle
g\left(\mathbb{E}_{Y}\left[f(z_{\min\\{\tau^{k},t\\}})\right]\right)$
$\displaystyle=\left.g\left(\mathbb{E}_{Y}\left[f(z_{s})\right]\right)\right|_{s=\min\\{\tau^{k},t\\}}$
$\displaystyle
g\left(\mathbb{E}_{Y}\left[f(z_{\min\\{\tau^{l}_{\mathrm{ref}},t\\}})\right]\right)$
$\displaystyle=\left.g\left(\mathbb{E}_{Y}\left[f(z_{s})\right]\right)\right|_{s=\min\\{\tau^{l}_{\mathrm{ref}},t\\}}$

hold and therefore $\bar{u}^{k}$ is a solution to the EnKBF (28) on the random
time interval $[0,\min\\{\tau^{k},\tau^{l}_{\mathrm{ref}}\\}]$. By the
uniqueness of (34), $\bar{u}^{k}$ and $\bar{u}^{k+1}$ must even coincide on
$[0,\min\\{\tau^{k},\tau^{l}_{\mathrm{ref}}\\}]$. Thus we can construct a
solution to (28) using the solutions to (34) and in order to conclude
existence and uniqueness of the EnKBF, we just have to show that

$\displaystyle\bigcup_{k,l\in\mathbb{N}}\left\\{\tau^{k}>T\right\\}\cap\left\\{\tau^{l}_{\mathrm{ref}}>T\right\\}$

defines a covering of the sample space almost surely.

To this end we first note that

$\displaystyle\mathrm{d}\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}$
$\displaystyle=2\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(\bar{u}^{k}_{t}),\bar{u}^{k}_{t}\right\rangle_{\mathscr{V}}}\mathrm{d}t+\left\langle\bar{u}^{k}_{t},\mathcal{B}(\bar{u}^{k}_{t})\mathrm{d}\bar{W}_{t}\right\rangle_{\mathscr{H}}+\mathrm{tr}_{\mathscr{H}}\left[\mathcal{B}(\bar{u}^{k}_{t})\sqrt{\mathcal{Q}}\left(\mathcal{B}(\bar{u}^{k}_{t})\sqrt{\mathcal{Q}}\right)^{\prime}\right]\mathrm{d}t$
$\displaystyle\phantom{=}+2\left\langle\bar{u}^{k}_{t},\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]R^{-1}_{t}\left(\mathrm{d}Y^{l}_{t}-\frac{H^{k}\left(\bar{u}^{k}_{t}\right)+\mathbb{E}_{Y}\left[H^{k}\left(\bar{u}^{k}_{t}\right)\right]}{2}\mathrm{d}t\right)\right\rangle_{\mathscr{H}}$
$\displaystyle\phantom{=}+\mathrm{tr}_{\mathscr{H}}\left[\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]R^{-1}_{t}\mathbb{Cov}_{Y}\left[H^{k}\left(\bar{u}^{k}_{t}\right),\bar{u}^{k}_{t}\right]\right].$

Taking the conditional expectation thus gives us

$\displaystyle\begin{split}&\mathrm{d}\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]\\\
&=2\mathbb{E}_{Y}\left[\prescript{}{\mathscr{V}^{\prime}}{\left\langle\mathcal{A}(\bar{u}^{k}_{t}),\bar{u}^{k}_{t}\right\rangle_{\mathscr{V}}}\right]\mathrm{d}t+\mathbb{E}_{Y}\left[\mathrm{tr}_{\mathscr{H}}\left[\mathcal{B}(\bar{u}^{k}_{t})\sqrt{\mathcal{Q}}\left(\mathcal{B}(\bar{u}^{k}_{t})\sqrt{\mathcal{Q}}\right)^{\prime}\right]\right]\mathrm{d}t\\\
&\phantom{=}+2\mathbb{E}_{Y}\left[\left\langle\bar{u}^{k}_{t},\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]R^{-1}_{t}\left(\mathrm{d}Y^{l}_{t}-\frac{H^{k}\left(\bar{u}^{k}_{t}\right)+\mathbb{E}_{Y}\left[H^{k}\left(\bar{u}^{k}_{t}\right)\right]}{2}\mathrm{d}t\right)\right\rangle_{\mathscr{H}}\right]\\\
&\phantom{=}+\tilde{\mathbb{1}}_{l}\left(\mathbb{E}_{Y}\left[\left|H(u^{\mathrm{ref}}_{t})\right|^{2}\right]\right)^{2}\mathrm{tr}_{\mathscr{H}}\left[\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]R^{-1}_{t}\mathbb{Cov}_{Y}\left[H^{k}\left(\bar{u}^{k}_{t}\right),\bar{u}^{k}_{t}\right]\right]\mathrm{d}t.\end{split}$
(42)

The first two terms on the right hand side can be bounded using the growth
condition (3) and the diffusivity bound (7). The last term is just the squared
shadow 2-norm of the operator
$\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]R^{-1/2}_{t}$,
which we can estimate using Parseval and the robust variance bound (31) as

$\displaystyle\mathrm{tr}_{\mathscr{H}}\left[\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]R^{-1}_{t}\mathbb{Cov}_{Y}\left[H^{k}\left(\bar{u}^{k}_{t}\right),\bar{u}^{k}_{t}\right]\right]\leq\mathrm{tr}_{\mathscr{H}}\left[\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H\left(\bar{u}^{k}_{t}\right)\right]R^{-1}_{t}\mathbb{Cov}_{Y}\left[H\left(\bar{u}^{k}_{t}\right),\bar{u}^{k}_{t}\right]\right]$
$\displaystyle\leq\sum_{j\in\mathbb{N}}\mathbb{Cov}_{Y}\left[\left\langle\nu_{j},\bar{u}^{k}_{t}\right\rangle_{\mathscr{H}},H\left(\bar{u}^{k}_{t}\right)\right]R^{-1}_{t}\mathbb{Cov}_{Y}\left[H\left(\bar{u}^{k}_{t}\right),\left\langle\nu_{j},\bar{u}^{k}_{t}\right\rangle_{\mathscr{H}}\right]$
$\displaystyle\leq\sum_{j\in\mathbb{N}}\mathbb{E}_{Y}\left[\left\langle\nu_{j},\bar{u}^{k}_{t}-\mathbb{E}_{Y}\left[\bar{u}^{k}_{t}\right]\right\rangle_{\mathscr{H}}^{2}\right]\left|R^{-1}_{t}\right|\mathbb{E}_{Y}\left[\left|H(\bar{u}^{k}_{t})-\mathbb{E}_{Y}\left[H(\bar{u}^{k}_{t})\right]\right|^{2}\right]$
$\displaystyle\leq\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}-\mathbb{E}_{Y}\left[\bar{u}^{k}_{t}\right]\right\|_{\mathscr{H}}^{2}\right]\left|R^{-1}_{t}\right|\mathbb{E}_{Y}\left[\left|H(\bar{u}^{k}_{t})-\mathbb{E}_{Y}\left[H(\bar{u}^{k}_{t})\right]\right|^{2}\right]^{2}\leq\mathrm{Lip}(H)\left|R^{-1}_{t}\right|\beta^{2}e^{2\lambda
t}.$

Thus we can bound (42) by

$\displaystyle\begin{split}\mathrm{d}\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]&=\left(2\alpha_{H}\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]+2\alpha_{0}+\beta+\mathrm{Lip}(H)\left|R^{-1}_{t}\right|\beta^{2}e^{2\lambda
t}\right)\mathrm{d}t\\\
&\phantom{=}+2\mathbb{E}_{Y}\left[\left\langle\bar{u}^{k}_{t},\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]\right]R^{-1}_{t}\mathrm{d}Y^{l}_{t}\right\rangle_{\mathscr{H}}\\\
&\phantom{=}-2\mathbb{E}_{Y}\left[\left\langle\bar{u}^{k}_{t},\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]R_{t}^{-1}\frac{H^{k}(\bar{u}^{k}_{t})+\mathbb{E}_{Y}\left[H^{k}(\bar{u}^{k}_{t})\right]}{2}\right\rangle_{\mathscr{H}}\right]\mathrm{d}t.\end{split}$
(43)

We use (39) and $\tilde{\mathbb{1}}_{k}\leq 1$ to derive that

$\displaystyle\left|\mathbb{E}_{Y}\left[\left\langle\bar{u}^{k}_{t},\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]R_{t}^{-1}\frac{H^{k}(\bar{u}^{k}_{t})+\mathbb{E}_{Y}\left[H^{k}(\bar{u}^{k}_{t})\right]}{2}\right\rangle_{\mathscr{H}}\right]\right|$
$\displaystyle\leq
2\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]+\tilde{\mathbb{1}}_{k}\left(\left|\mathbb{E}_{Y}\left[H(\bar{u}^{k}_{t})\right]\right|\right)^{2}\left\|\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]\right\|_{\mathrm{L}\left(R^{d_{x}};\mathscr{H}\right)}^{2}\left|R^{-1}_{t}\right|\mathbb{E}_{Y}\left[|H(\bar{u}^{k}_{t})|^{2}\right]$
$\displaystyle\leq\left(2+\mathrm{Lip}(H)^{3}\beta^{2}e^{2\lambda
t}\left|R^{-1}_{t}\right|\right)\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]+\mathrm{Lip}(H)\beta^{2}e^{2\lambda
t}\left|R^{-1}_{t}\right|\left|H(0)\right|^{2}.$

Thus, we note that there exist constants $\kappa_{4}(T)$ and $\kappa_{5}(T)$,
only depending on the timeframe $T$, such that

$\displaystyle\begin{split}\mathrm{d}\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]&=\left(\kappa_{4}(T)\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]+\kappa_{5}(T)\right)\mathrm{d}t\\\
&\phantom{=}+2\mathbb{E}_{Y}\left[\left\langle\bar{u}^{k}_{t},\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]\right]R^{-1}_{t}\mathrm{d}Y^{l}_{t}\right\rangle_{\mathscr{H}},\end{split}$

which by using the explicit form of the observations $Y$ can be rewritten as

$\displaystyle\begin{split}\mathrm{d}\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]&=\left(\kappa_{4}(T)\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]+\kappa_{5}(T)\right)\mathrm{d}t\\\
&\phantom{=}+2\mathbb{E}_{Y}\left[\left\langle\bar{u}^{k}_{t},\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]\right]R^{-1}_{t}\tilde{\mathbb{1}}_{l}\left(\mathbb{E}_{Y}\left[\left|H(u^{\mathrm{ref}}_{t})\right|^{2}\right]\right)H(u^{\mathrm{ref}}_{t})\right\rangle_{\mathscr{H}}\mathrm{d}t\\\
&\phantom{=}+2\mathbb{E}_{Y}\left[\left\langle\bar{u}^{k}_{t},\mathbb{Cov}_{Y}\left[\bar{u}^{k}_{t},H^{k}\left(\bar{u}^{k}_{t}\right)\right]\right]R^{-1}_{t}\tilde{\mathbb{1}}_{l}\left(\mathbb{E}_{Y}\left[\left|H(u^{\mathrm{ref}}_{t})\right|^{2}\right]\right)\Gamma_{t}\mathrm{d}V_{t}\right\rangle_{\mathscr{H}}.\end{split}$
(44)

Again using (39) and the Burkholder–Davis–Gundy inequality we derive that
there exist constants $\kappa_{6}(T)$, depending solely on $T$, and
$\kappa_{7}(T,l)$, depending on $T$ and $l$, such that

$\displaystyle\begin{split}\mathbb{E}\left[\sup_{t\leq
T}\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]\right]&=\kappa_{6}(T)\int_{0}^{T}\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]\mathrm{d}t+\mathbb{E}_{Y}\left[\left\|u_{0}\right\|_{\mathscr{H}}^{2}\right]+T\kappa_{7}(T,l).\end{split}$
(45)

Thus, by the Grönwall Lemma, we derive that for fixed $l$ and $T$

$\displaystyle\mathbb{E}\left[\sup_{t\leq
T}\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]\right]\leq\exp\left(T\kappa_{6}(T)\right)\left(\mathbb{E}_{Y}\left[\left\|u_{0}\right\|_{\mathscr{H}}^{2}\right]+T\kappa_{7}(T,l)\right),$
(46)

which, implies that almost surely there exists a $k$ such that $\sup_{t\leq
T}\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{H}^{2}\right]\leq k$. By
the Lipschitz continuity of $H$ and the inequality
$\left|\mathbb{E}_{Y}\left[H(\bar{u}^{k}_{t})\right]\right|\leq|H(0)|+\mathrm{Lip}(H)\sqrt{\mathbb{E}_{Y}\left[\left\|\bar{u}^{k}_{t}\right\|_{\mathscr{H}}^{2}\right]}$
this then in turn implies that

$\displaystyle\bigcup_{k\in\mathbb{N}}\left\\{\tau^{k}>T\right\\}\cap\left\\{\tau^{l}_{\mathrm{ref}}>T\right\\}=\left\\{\tau^{l}_{\mathrm{ref}}>T\right\\}\leavevmode\nobreak\
\text{almost surely.}$

Since $\sup_{t\leq
T}\mathbb{E}_{Y}\left[\left|H(u^{\mathrm{ref}}_{t})\right|^{2}\right]$ is
finite almost surely, we can thus indeed conclude that there exists a solution
to the EnKBF (28), defined on every event
$\left\\{\tau^{k}>T\right\\}\cap\left\\{\tau^{l}_{\mathrm{ref}}>T\right\\}$ by
the sequence of $\bar{u}^{k}$.

To conclude uniqueness of solutions to (28) one can simply employ stopping
times (41) and the same Grönwall/contraction argument as in Step 1.

∎

######  Remark 29.

Note that even though in the proof above we used the specific form of the
observations $Y$, it actually does not matter that the true observation
function and the modelled observation function coincide, i.e. if
$\mathrm{d}Y_{t}=\mathfrak{C}(\bar{X}_{t})\mathrm{d}t+\Gamma_{t}\mathrm{d}V_{t}$
with $\mathfrak{C}\neq H$, then the proof would still hold, as long as
$\mathfrak{C}$ is assumed to be Lipschitz. Therefore, as an immediate
corollary of our chosen fixed point argument, one derives the continuity of
the EnKBF with respect to perturbations of the modelled observations $H$. The
continuous dependence on the signal parameters $\mathcal{A}$, $\mathcal{B}$
and the initial condition $u_{0}$ can easily be shown as well. Only the
robustness with respect to the observation stream $Y$ is a delicate matter due
to the discontinuity of the Itô–Lyons map.

##  7 The EnKBF as an interacting particle system

For the sake of brevity in formulas we make the following definition.

######  Definition 30.

For any
$\mathfrak{v}=(v_{1},\cdots,v_{N})\in\mathscr{H}^{N},\leavevmode\nobreak\
N\in\mathbb{N}$ we set

$\displaystyle\begin{split}\mathbb{E}^{N}\left[\mathfrak{v}\right]&:=\frac{1}{N}\sum_{i=1}^{N}v^{i},\leavevmode\nobreak\
\mathbb{E}^{N}_{H}\left[\mathfrak{v}\right]:=\frac{1}{N}\sum_{i=1}^{N}H(v^{i})\\\
\mathbb{C}^{N}_{H}[\mathfrak{v}]&:=\frac{1}{N}\sum_{i=1}^{N}\left(v^{i}-\mathbb{E}^{N}\left[\mathfrak{v}\right]\right)\leavevmode\nobreak\
\left(H(v^{i})-\mathbb{E}^{N}_{H}\left[\mathfrak{v}\right]\right)^{\prime}.\end{split}$
(47)

We use the normalization by $1/N$ for the empirical covariance instead of the
usual unbiased normalization by $1/(N-1)$ just for notational convenience in
the calculations that are to follow.

The mean field EnKBF (28) is naturally approximated by a system of interacting
S(P)DEs

$\displaystyle\begin{split}\mathrm{d}u^{i}_{t}&=\mathcal{A}(u^{i}_{t})\mathrm{d}t+\mathcal{B}(u^{i}_{t})\mathrm{d}\bar{W}^{i}_{t}\\\
&\phantom{=}+\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]R^{-1}_{t}\left(\mathrm{d}Y_{t}-\frac{H(u^{i}_{t})+\mathbb{E}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]}{2}\mathrm{d}t\right),\leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ i=1,\cdots,N\end{split}$ (48)

where $(\bar{W}^{i})_{i=1,\cdots,N}$ are independent copies of the Wiener
process $\bar{W}$, $u^{i}_{0}$ are independent copies of $u_{0}$ and
$\mathfrak{u}^{N}_{t}:=(u^{i}_{t})_{i=1,\cdots,N}$.

The interacting system of S(P)DEs (48) is often referred to as the
deterministic EnKBF, which is the continuous time counterpart to the filter
derived by Sakov and Oke [42].

###  7.1 Analysis of the particle approximations

While (48) is just a system of interacting ordinary SPDEs for which local one-
sided Lipschitz conditions are enough to derive well posedness [34], it does
not seem to satisfy the usual growth conditions for unbounded observation
functions $H$. In [32] the well posedness was proven for the finite
dimensional setting by showing that blow ups do not occur in finite times.
This is of course not sufficient to conclude well posedness in infinite
dimensions. Instead we employ a partial stopping argument, similar to the one
employed in the proof of Theorem 28. The law of total variance will again play
a key role, and so, to keep formulas simple we make the following definition.

######  Definition 31.

For any $N\in\mathbb{N}$ and any ensemble
$\mathfrak{v}=(v^{1},\cdots,v^{N})\in\mathscr{H}^{N}$ of $N$ elements of
$\mathscr{H}$, we define the empirical variance $\sigma^{N}[\mathfrak{v}]$ by

$\displaystyle\sigma^{N}[\mathfrak{v}]:=\frac{1}{N}\sum_{i=1}^{N}\left\|v^{i}_{s}-\mathbb{E}^{N}[\mathfrak{v}]\right\|_{\mathscr{H}}^{2}.$

And similarly we define the empirical observed variance by

$\displaystyle\sigma^{N,H}[\mathfrak{v}]:=\frac{1}{N}\sum_{i=1}^{N}\left|H(v^{i}_{s})-\mathbb{E}^{N}_{H}[\mathfrak{v}]\right|^{2}.$

We are now in the position to formulate and prove the following Lemma.

######  Lemma 32.

If the conditions in Assumption (2) and (7) are satisfied, there exists a
unique solution to the nonlinear EnKBF (48).

###### Proof.

First we note that for any fixed $k\in\mathbb{N}$, the system

$\displaystyle\begin{split}\mathrm{d}u^{i}_{t}&=\mathcal{A}(u^{i}_{t})\mathrm{d}t+\mathcal{B}(u^{i}_{t})\mathrm{d}\bar{W}^{i}_{t}\\\
&\phantom{=}+\tilde{\mathbb{1}}_{k}\left(\left\|\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]\right\|_{L(\mathbb{R}^{d_{y}},\mathscr{H})}^{2}\right)\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]R^{-1}_{t}\left(\mathrm{d}Y_{t}-\frac{H(u^{i}_{t})+\mathbb{E}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]}{2}\mathrm{d}t\right),\end{split}$
(49)

for $i=1,\cdots,N$, satisfies standard one-sided Lipschitz and growth
conditions and thus has a unique solution. To keep formulas simple we will
omit the argument of
$\tilde{\mathbb{1}}_{k}\left(\left\|\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]\right\|_{\mathscr{H}^{d_{y}}}^{2}\right)$
and instead simply write $\tilde{\mathbb{1}}_{k}$ in the rest of this proof.

We note that the ensemble mean
$\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right]$ satisfies

$\displaystyle\mathrm{d}\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right]=\frac{1}{N}\sum_{j=1}^{N}\mathcal{A}(u^{j}_{t})\mathrm{d}t+\frac{1}{N}\sum_{j=1}^{N}\mathcal{B}(u^{j}_{t})\mathrm{d}\bar{W}^{j}_{t}+\tilde{\mathbb{1}}_{k}\leavevmode\nobreak\
\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]R^{-1}_{t}\left(\mathrm{d}Y_{t}-\mathbb{E}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]\mathrm{d}t\right).$
(50)

This gives us the evolution equation for the centered particles

$\displaystyle\mathrm{d}\left(u^{i}_{t}-\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right]\right)$
$\displaystyle=\left(\mathcal{A}(u^{i}_{t})-\frac{1}{N}\sum_{j=1}^{N}\mathcal{A}(u^{j}_{t})\right)\mathrm{d}t+\left(\mathcal{B}(u^{i}_{t})\mathrm{d}\bar{W}^{i}_{t}-\frac{1}{N}\sum_{j=1}^{N}\mathcal{B}(u^{j}_{t})\mathrm{d}\bar{W}^{j}_{t}\right)$
$\displaystyle\phantom{=}-\tilde{\mathbb{1}}_{k}\leavevmode\nobreak\
\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]R^{-1}_{t}\frac{H(u^{i}_{t})-\mathbb{E}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]}{2}\mathrm{d}t.$

Note that by Parseval one easily verifies that

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\left\langle
u^{i}_{t}-\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right],\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]R^{-1}_{t}\left(H(u^{i}_{t})-\mathbb{E}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]\right)\right\rangle_{\mathscr{H}}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\sum_{k\in\mathbb{N}}\left\langle\nu_{k},u^{i}_{t}-\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right]\right\rangle_{\mathscr{H}}\left\langle\nu_{k},\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]R^{-1}_{t}\left(H(u^{i}_{t})-\mathbb{E}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]\right)\right\rangle_{\mathscr{H}}$
$\displaystyle=\mathrm{tr}_{\mathscr{H}}\left[\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]R^{-1}_{t}\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]^{\mathrm{T}}\right].$

Therefore, by Itô’s formula we derive the following equation for the average
deviation from the ensemble mean

$\displaystyle\begin{split}\mathrm{d}\sigma^{N}[\mathfrak{u}^{N}_{t}]&=\frac{2}{N}\sum_{i=1}^{N}\prescript{}{\mathscr{V}^{\prime}}{\left\langle\left(\mathcal{A}(u^{i}_{t})-\frac{1}{N}\sum_{j=1}^{N}\mathcal{A}(u^{j}_{t})\right),u^{i}_{t}-\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right]\right\rangle_{\mathscr{V}}}\mathrm{d}t+\mathrm{d}\mathfrak{m}^{N}_{t}\\\
&\phantom{=}-\tilde{\mathbb{1}}_{k}\leavevmode\nobreak\
\mathrm{tr}_{\mathscr{H}}\left[\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]R^{-1}_{t}\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]^{\prime}\right]\mathrm{d}t\\\
&\phantom{=}+\frac{1}{N}\sum_{i=1}^{N}\mathrm{tr}_{\mathscr{H}}\left[\mathcal{B}(u^{i}_{t})\sqrt{\mathcal{Q}}\left(\mathcal{B}(u^{i}_{t})\sqrt{\mathcal{Q}}\right)^{\prime}\right]\mathrm{d}t\\\
&\phantom{=}+\frac{1}{N^{3}}\sum_{i=1}^{N}\sum_{j\neq
i}\mathrm{tr}_{\mathscr{H}}\left[\mathcal{B}(u^{j}_{t})\sqrt{\mathcal{Q}}\left(\mathcal{B}(u^{j}_{t})\sqrt{\mathcal{Q}}\right)^{\prime}\right]\mathrm{d}t,\end{split}$
(51)

where $\mathfrak{m}^{N}$ denotes the local martingale given by

$\displaystyle\begin{split}\mathrm{d}\mathfrak{m}^{N}_{t}&:=\frac{2}{N}\sum_{i=1}^{N}\left\langle
u^{i}_{t}-\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right],\left(\mathcal{B}(u^{i}_{t})\mathrm{d}\bar{W}^{i}_{t}-\frac{1}{N}\sum_{j=1}^{N}\mathcal{B}(u^{j}_{t})\mathrm{d}\bar{W}^{j}_{t}\right)\right\rangle_{\mathscr{H}}\\\
&=\frac{2}{N}\sum_{i=1}^{N}\left\langle
u^{i}_{t}-\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right],\mathcal{B}(u^{i}_{t})\mathrm{d}\bar{W}^{i}_{t}\right\rangle_{\mathscr{H}},\leavevmode\nobreak\
\text{with}\leavevmode\nobreak\ \mathfrak{m}^{N}_{0}=0.\end{split}$ (52)

First we note that in (51) we can replace
$\frac{1}{N}\sum_{j=1}^{N}\mathcal{A}(u^{j}_{t})$ by
$\mathcal{A}\left(\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right]\right)$.
Thus by using the one-sided Lipschitz condition (2) and Assumption (7) as well
as the positivity of the trace
$\mathrm{tr}_{\mathscr{H}}\left[\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]R^{-1}_{t}\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]^{\prime}\right]$
we derive the inequality

$\displaystyle\mathrm{d}\sigma^{N}[\mathfrak{u}^{N}_{t}]\leq\left(2\lambda\leavevmode\nobreak\
\sigma^{N}[\mathfrak{u}^{N}_{t}]+\beta\right)\mathrm{d}t+\mathrm{d}\mathfrak{m}^{N}_{t}.$
(53)

Since $\mathfrak{m}^{N}$ is a real valued local martingale we can deduce by
the stochastic Grönwall Lemma [44, Theorem 4] that

$\displaystyle\mathbb{E}\left[\sup_{t\leq
T}\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left\|u^{i}_{t}-m^{N}_{t}\right\|_{\mathscr{H}}^{2}}\right]=\mathbb{E}\left[\sup_{t\leq
T}\sqrt{\sigma^{N}[\mathfrak{u}^{N}_{t}]}\right]\leq(\pi+1)\sqrt{\beta}e^{\lambda
T}.$ (54)

Due to the Lipschitz continuity of $H$, this also gives a uniform bound for
$\left\|\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]\right\|_{\mathscr{H}^{d_{y}}}^{2}$.
Since (49) coincide with (48) on
$\left\\{\left\|\mathbb{C}^{N}_{H}\left[\mathfrak{u}^{N}_{t}\right]\right\|_{\mathscr{H}^{d_{y}}}^{2}\leq
k\right\\}$ we have thus derived the well posedness of (48).

∎

######  Remark 33 (Literature).

As already mentioned well posedness of the particle system (48) was proven in
[32]. An extension of this proof to the correlated noise framework, which
requires the control of singular terms can be found in [21].
In the thesis [28] it was shown that the discrete EnKF is well defined in an
infinite dimensional separable Hilbert space setting.
Finally we mention that the seminal paper [29] considered the well posedness
and accuracy of both discrete and continuous time EnKFs for a class of signals
that included the 2D Navier-Stokes equation. Existence of strong solutions to
the continuous time EnKF (48) with complete observations
($H=\mathrm{id}_{\mathscr{H}}$) was assumed and it was shown that solutions do
not blow up.

The key identity in the proof of Lemma 32 was (52), which is a stochastic
version of the law of total variance (30), i.e. the particle approximation
(48) also satisfies (an empirical version of) the law of total variance up to
martingale fluctuations. If (48) is to be a good approximation of (28) one
would expect these fluctations to become small as the number of particles is
increased sufficiently. Indeed, this is the case, as due to Assumption 7, we
derive

$\displaystyle\begin{split}\mathrm{d}\left[\mathfrak{m}^{N}\right]_{t}&=\frac{2}{N^{2}}\sum_{i=1}^{N}\left\langle
u^{i}_{t}-\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right],\mathcal{B}(u^{i}_{t})\mathcal{Q}\mathcal{B}(u^{i}_{t})^{\prime}(u^{i}_{t}-\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right])\right\rangle_{\mathscr{H}}\mathrm{d}t\\\
&\leq\frac{2\beta}{N}\frac{1}{N}\sum_{i=1}^{N}\left\|u^{i}_{t}-\mathbb{E}^{N}\left[\mathfrak{u}^{N}_{t}\right]\right\|_{\mathscr{H}}^{2}\mathrm{d}t=\frac{2\beta}{N}\sigma^{N}[\mathfrak{u}^{N}_{t}]\mathrm{d}t.\end{split}$
(55)

The last inequality is a consequence of the fact that the trace is invariant
under the change of the orthonormal basis and that for every nonzero vector
one can find an orthonormal basis that contains this vector.

Since we were able to bound $\sigma^{N}[\mathfrak{u}^{N}_{t}]$ uniformly in
time, the quadratic variation of $\mathfrak{m}^{N}$ will decrease to zero for
$N\to\infty$. Thus, an empirical version of the law of total variance is
almost satisfied for large ensemble sizes $N$. For our convergence proof we
will need a more rigorous quantification of this fact in the form of
exponential moment bounds of the empirical variance. Such bounds are delicate
as the ensemble $\mathfrak{u}^{N}$ will likely show some Gaussian (tail
behaviour) and thus $\mathbb{E}\left[\sup_{t\leq
T}\exp\left(r\sigma^{N}[\mathfrak{u}^{N}_{t}]\right)\right]$ might not be
finite for all values of $r\geq 0$. However, as $N\to\infty$ one would expect
$\sigma^{N}$ to become deterministic and as such any exponential moment should
exist for $N$ sufficiently large. We prove this fact in the following Lemma by
employing a Grönwall argument.

######  Lemma 34.

Let $q\geq 0$ be arbitrary. Then for any $N\in\mathbb{N}$ such that $N>2\beta
q\leavevmode\nobreak\ e^{(2\lambda+1)T}$ we have

$\displaystyle\mathbb{E}\left[\sup_{t\leq T}\exp\left(q\leavevmode\nobreak\
\sigma^{N}\left[\mathfrak{u}^{N}_{t}\right]\right)\right]\leq(\pi+1)\exp\left(\frac{q\left(e^{(2\lambda+1)T}-1\right)}{2(2\lambda+1)}\right)\mathbb{E}\left[\exp\left(2qe^{(2\lambda+1)T}\sigma^{N}\left[\mathfrak{u}^{N}_{0}\right]\right)\right].$

in particular the $q$-th exponential moment of the path of
$\sigma^{N}[\mathfrak{u}^{N}]$ exists up to time $T$, if the
$\left(2qe^{(2\lambda+1)T}\right)$-th exponential moment of the initial
empirical variance $\sigma^{N}[\mathfrak{u}^{N}]$ exists.

###### Proof.

Let $\mathfrak{a}:=2\lambda+1$ (see Assumption 2) and
$\mathfrak{b}:=qe^{\mathfrak{a}T}$. We define the process
$\mathfrak{s}_{t}:=2\mathfrak{b}\leavevmode\nobreak\
e^{-\mathfrak{a}t}\sigma^{N}[\mathfrak{u}^{N}_{t}]$. Then, using inequality
(53), we derive the inequality

$\displaystyle\mathrm{d}\mathfrak{s}_{t}=2\mathfrak{b}e^{-\mathfrak{a}t}\mathrm{d}\sigma^{N}[\mathfrak{u}^{N}_{t}]-2\mathfrak{a}\mathfrak{b}e^{-\mathfrak{a}t}\sigma^{N}[\mathfrak{u}^{N}_{t}]\mathrm{d}t$
$\displaystyle\leq(2\lambda-\mathfrak{a})2\mathfrak{b}e^{-\mathfrak{a}}\sigma^{N}\left[\mathfrak{u}^{N}_{t}\right]\mathrm{d}t+2\mathfrak{b}\beta
e^{-\mathfrak{a}}\mathrm{d}t+2\mathfrak{b}e^{-\mathfrak{a}t}\mathrm{d}\mathfrak{m}_{t}$
$\displaystyle=(2\lambda-\mathfrak{a})\mathfrak{s}_{t}\mathrm{d}t+2\mathfrak{b}\beta
e^{-\mathfrak{a}}\mathrm{d}t+2\mathfrak{b}e^{-\mathfrak{a}t}\mathrm{d}\mathfrak{m}_{t}.$

Furthermore we derive from (51) the form of the quadratic variation of
$\mathfrak{s}$ and from (55) the estimate

$\displaystyle\mathrm{d}\left[\mathfrak{s}\right]_{t}=(2\mathfrak{b})^{2}e^{-2\mathfrak{a}t}\mathrm{d}\left[\mathfrak{m}^{N}\right]_{t}\leq(2\mathfrak{b})^{2}e^{-2\mathfrak{a}t}\frac{2\beta}{N}\sigma^{N}[\mathfrak{u}^{N}_{t}]\mathrm{d}t=2\frac{2\beta\mathfrak{b}e^{-\mathfrak{a}t}}{N}\mathfrak{s}_{t}\mathrm{d}t.$

These inequalities together with Itô’s formula give us the following
inequality

$\displaystyle\mathrm{d}\exp(\mathfrak{s}_{t})$
$\displaystyle=\exp(\mathfrak{s}_{t})\mathrm{d}\mathfrak{s}_{t}+\exp(\mathfrak{s}_{t})\frac{1}{2}\mathrm{d}\left[\mathfrak{s}\right]_{t}$
$\displaystyle\leq(2\lambda-\mathfrak{a})\mathfrak{s}_{t}\exp(\mathfrak{s}_{t})\mathrm{d}t+2\mathfrak{b}\beta
e^{-\mathfrak{a}t}\exp(\mathfrak{s}_{t})\mathrm{d}t+2\mathfrak{b}e^{-\mathfrak{a}t}\exp(\mathfrak{s}_{t})\mathrm{d}\mathfrak{m}_{t}+\frac{2\beta\mathfrak{b}e^{-2\mathfrak{a}t}}{N}\mathfrak{s}_{t}\exp(\mathfrak{s}_{t})\mathrm{d}t$
$\displaystyle=\left(2\lambda-\mathfrak{a}+\frac{2\beta\mathfrak{b}e^{-\mathfrak{a}t}}{N}\right)\mathfrak{s}_{t}\exp(\mathfrak{s}_{t})\mathrm{d}t+2\mathfrak{b}\beta
e^{-\mathfrak{a}t}\exp(\mathfrak{s}_{t})\mathrm{d}t+2\mathfrak{b}e^{-\mathfrak{a}t}\exp(\mathfrak{s}_{t})\mathrm{d}\mathfrak{m}_{t}.$

Due to our assumptions we have
$\mathfrak{a}>2\lambda+\frac{2\beta\mathfrak{b}e^{-\mathfrak{a}t}}{N}$ and
thus derive the stochastic inequality

$\displaystyle\mathrm{d}\exp(\mathfrak{s}_{t})\leq 2\mathfrak{b}\beta
e^{-\mathfrak{a}t}\exp(\mathfrak{s}_{t})\mathrm{d}t+2\mathfrak{b}e^{-\mathfrak{a}t}\exp(\mathfrak{s}_{t})\mathrm{d}\mathfrak{m}_{t}.$

Since
$2\mathfrak{b}e^{-\mathfrak{a}t}\exp(\mathfrak{s}_{t})\mathrm{d}\mathfrak{m}_{t}$
defines a local martingale, the stochastic Grönwall inequality [44] gives us

$\displaystyle\mathbb{E}\left[\sup_{t\leq T}\exp\left(q\leavevmode\nobreak\
\sigma^{N}\left[\mathfrak{u}^{N}_{t}\right]\right)\right]$
$\displaystyle\leq\mathbb{E}\left[\sup_{t\leq
T}\sqrt{\exp\left(\mathfrak{s}_{t}\right)}\right]\leq(\pi+1)\exp\left(q/2e^{\mathfrak{a}T}\int_{0}^{T}e^{-\mathfrak{a}s}\mathrm{d}s\right)\mathbb{E}\left[\exp\left(\mathfrak{s}_{0}/2\right)\right]$
$\displaystyle\leq(\pi+1)\exp\left(\frac{q\left(e^{(2\lambda+1)T}-1\right)}{2(2\lambda+1)}\right)\mathbb{E}\left[\exp\left(2qe^{(2\lambda+1)T}\sigma^{N}\left[\mathfrak{u}^{N}_{0}\right]\right)\right],$

which concludes the proof. ∎

######  Remark 35.

In the proof of Lemma 34 we used a standard testfunction for our Grönwall
argument. Since we have good controls for the quadratic variation of
$\mathfrak{m}$, we also could have just used the standard
Burkholder–Davis–Gundy inequality in combination with a deterministic Grönwall
Lemma. The usage of the stochastic Grönwall inequality is not necessary in our
setting, however in [27] a similar testfunction and a novel stochastic
Grönwall–Lyapunov inequality were used to derive uniform exponential moment
bounds for SDEs satisfying an appropriate Lyapunov condition.

###  7.2 Quantitative propagation of chaos

Next we show propagation of chaos, i.e. that the system of interacting SPDEs
(48) indeed converges (in an appropriate sense) to the McKean–Vlasov SPDE
(28). For this we use a standard synchronous coupling approach, i.e. we
compare (48) to a tensorized version of (28) defined on the same probability
space. To this end we define conditionally555Conditioned on $Y$. independent
copies $\bar{u}^{i},\leavevmode\nobreak\ i\in\mathbb{N}$ of the mean field
process (28) to be the solutions of

$\displaystyle\mathrm{d}\bar{u}^{i}_{t}$
$\displaystyle=\mathcal{A}(\bar{u}^{i}_{t})\mathrm{d}t+\mathcal{B}(\bar{u}^{i}_{t})\mathrm{d}\bar{W}^{i}_{t}$