# On the Martingale Representation with Respect to the super-Brownian
Filtration

Christian Mandler Mathematisches Institut, Justus-Liebig-Universität Giessen
(e-mail<EMAIL_ADDRESS>Financial support from the
Deutsche Forschungsgemeinschaft is gratefully acknowledged.  Ludger Overbeck
Mathematisches Institut, Justus-Liebig-Universität Giessen (e-mail:
<EMAIL_ADDRESS>
###### Abstract

We derive the explicit form of the martingale representation for square-
integrable processes that are martingales with respect to the natural
filtration of the super-Brownian motion. This is done by using a weak
extension of the Dupire derivative for functionals of superprocesses.

## Introduction

Dupire’s landmark work on the functional Itō-formula [Dupire, 2009] gave rise
to a completely new approach to numerous questions in the field of stochastic
calculus. In this paper, we use it to derive a martingale representation of
functionals of the super-Brownian motion, a well-studied, infinite-
dimensional, namely measure-valued, stochastic process.

Cont and Fournié ([Cont and Fournié, 2010], [Cont and Fournié, 2013], [Cont,
2016]) as well as Levental et al. ([Levental et al., 2013]) used different
formalizations of Dupire’s idea to derive versions of a functional Itō-formula
for real-valued semi-martingales using different derivatives. In addition,
Cont used the functional Itō-formula to derive a martingale representation
formula for Brownian martingales and extended the functional derivative to a
weak derivative for square-integrable functionals (see [Cont and Fournié,
2013], [Cont, 2016]).

In [Mandler and Overbeck, 2021] the functional Itō-formula for
$B(A,c)$-superprocesses is derived using derivatives extending the ones
introduced by Cont and Fournié to functionals on the infinite-dimensional
space of measures. In this paper, we extend the vertical derivative to a weak
one and use it to derive a martingale representation for functionals of the
super-Brownian motion.

The martingale representation of functionals of superprocesses has been
studied in [Overbeck, 1995], [Evans and Perkins, 1994] and [Evans and Perkins,
1995], where the authors proved the existence of a martingale representation
and derive the explicit form for the richer class of so-called historical
processes using their cluster representation, from which one can derive the
representation for superprocesses as a projection. The approach presented in
Section 2 is based on the ideas presented by Cont and Fournié in [Cont and
Fournié, 2013] and [Cont, 2016], does not use the concept of historical
processes and only considers the super-Brownian motion as a prototype of more
general measure-valued diffusions.

##  1 The setting

Consider a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$, where
$\mathbb{P}$ is the law of the super-Brownian motion on $\mathbb{R}^{d}$,
$d\geq 1$, starting at $m\in M_{F}(\mathbb{R}^{d})$, $\mathcal{F}$ is the
Borel-$\sigma$-field corresponding to $\Omega=C([0,T],M_{F}(\mathbb{R}^{d}))$
and $(\mathcal{F}_{t})_{t\in[0,T]}$ is the canonical filtration satisfying the
usual conditions. Also, denote the Laplacian operator by $\Delta$. From
[Dawson, 1993], we know that the probability measure $\mathbb{P}$ is a
solution to the following martingale problem:

$\displaystyle\mathbb{P}(X(0)=m)=1\text{ and for all $\phi\in
D(\textstyle{\frac{1}{2}}\Delta)$ the process}$ $\displaystyle
M(t)(\phi)=\langle X(t),\phi\rangle-\langle
X(0),\phi\rangle-\int_{0}^{t}\langle X(s),\frac{1}{2}\Delta\phi\rangle
ds,\quad t\in[0,T]$ is a $(\mathcal{F}_{t})_{t}$-martingale $\mathbb{P}$-a.s.
and has quadratic variation $\displaystyle[M(\phi)]_{t}=\int_{0}^{t}\langle
X(s),\phi^{2}\rangle ds\quad\forall t\in[0,T]\quad\mathbb{P}-a.s..$

As the super-Brownian motion is a $B(A,c)$-superprocess, the process $M(\phi)$
in the martingale problem gives rise to a martingale measure in the sense of
[Walsh, 1986] which we denote by $M_{X}$. This martingale measure is defined
on $\Omega\times\mathcal{B}[0,T]\times\mathcal{E}$ and plays in a crucial role
in the Itō-formulas presented in [Mandler and Overbeck, 2021] as well as the
results introduced below.

Denote by $\mathbb{S}(\mathbb{R}^{d})$ the Schwartz space on $\mathbb{R}^{d}$.
Functions in $\mathbb{S}(\mathbb{R}^{d})$ are infinitely continuously
differentiable. Thus $\mathbb{S}(\mathbb{R}^{d})\subset D(\frac{1}{2}\Delta)$
and for any $h\in\mathbb{S}(\mathbb{R}^{d})$ it holds $\frac{1}{2}\Delta h\in
D(\frac{1}{2}\Delta)$.

In addition, let $\mathcal{S}$ be the space of simple functions, i.e. the
space of functions that are linear combinations of functions
$f:\Omega\times[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ of form

$f(\omega,s,x)=X(\omega)1_{(a,b]}(s)1_{A}(x)$

with $0\leq a<b\leq t$, $X$ bounded and $\mathcal{F}_{a}$-measurable and
$A\in\mathcal{B}(\mathbb{R}^{d})$. Denote by $\mathcal{P}$ the predictable
$\sigma$-field on $\Omega\times[0,T]\times\mathbb{R}^{d}$, i.e. the
$\sigma$-field generated by $\mathcal{S}$. If a function is
$\mathcal{P}$-measurable, we say it is predictable.

In Section 2 we consider functions $F$ of the path of the super-Brownian
motion $X$. More precisely, we consider the stopped path $X_{t}$ of the
process $X$. To formalize the notion of (stopped) paths, consider an arbitrary
path $\omega\in D([0,T],M_{F}(\mathbb{R}^{d}))$, the space of right continuous
functions with left limits. We equip $D([0,T],M_{F}(\mathbb{R}^{d}))$ with the
metric $\tilde{d}$ given by

$\tilde{d}(\omega,\omega^{\prime})=\sup_{u\in[0,T]}d_{P}(\omega(u),\omega^{\prime}(u))$

for $\omega$, $\omega^{\prime}\in D([0,T],M_{F}(\mathbb{R}^{d}))$, where
$d_{P}$ is the Prokhorov metric on $M_{F}(\mathbb{R}^{d})$. For such a path,
define the path stopped at time $t$ by $\omega_{t}(u)=\omega(t\wedge u)$.
Using this, we define an equivalence relation on the space $[0,T]\times
D([0,T],M_{F}(\mathbb{R}^{d}))$ by

$(t,\omega)\sim(t^{\prime},\omega^{\prime})\quad\Leftrightarrow\quad
t=t^{\prime}\ \text{and}\ \omega_{t}=\omega^{\prime}_{t^{\prime}}.$

This relation gives rise to the quotient space

$\displaystyle\Lambda_{T}$ $\displaystyle=\\{(t,\omega_{t})\ |\
(t,\omega)\in[0,T]\times D([0,T],M_{F}(\mathbb{R}^{d}))\\}$
$\displaystyle=([0,T]\times D([0,T],M_{F}(\mathbb{R}^{d})))/\sim.$

Next, define a metric $d_{\infty}$ on $\Lambda_{T}$ by

$d_{\infty}((t,\omega),(t^{\prime},\omega^{\prime}))=\sup_{u\in[0,T]}d_{P}(\omega(u\wedge
t),\omega^{\prime}(u\wedge t^{\prime}))+|t-t^{\prime}|.$

A functional $F:\Lambda_{T}\rightarrow\mathbb{R}$ is continuous with respect
to $d_{\infty}$ if for all $(t,\omega)\in\Lambda_{T}$ and every
$\varepsilon>0$ there exists an $\eta>0$ such that for all
$(t^{\prime},\omega^{\prime})\in\Lambda_{T}$ with
$d_{\infty}((t,\omega),(t^{\prime},\omega^{\prime}))<\eta$ we have

$|F(t,\omega)-F(t^{\prime},\omega^{\prime})|<\varepsilon.$

Further, a functional $F$ on $[0,T]\times D([0,T],M_{F}(\mathbb{R}^{d}))$ is
called _non-anticipative_ if it is a measurable map on the space of stopped
paths, i.e. $F:\Lambda_{T}\rightarrow\mathbb{R}$. In other words, $F$ is non-
anticipative if $F(t,\omega)=F(t,\omega_{t})$ holds for all $\omega\in
D([0,T],M_{F}(\mathbb{R}^{d}))$.

For continuous non-anticipative functionals, we can define the following
derivative. For this, denote by $\delta_{x}$, $x\in\mathbb{R}^{d}$, the Dirac
measure with unit mass at $x$.

######  Definition 1.

A continuous non-anticipative functional $F:\Lambda_{T}\rightarrow\mathbb{R}$
is called _vertically differentiable_ at $(t,\omega)\in\Lambda_{T}$ in
direction $\delta_{x}$, $x\in\mathbb{R}^{d}$, if the limit

$\mathcal{D}_{x}F(t,\omega)=\lim_{\varepsilon\rightarrow
0}\frac{F(t,\omega_{t}+\varepsilon\delta_{x}1_{[t,T]})-F(t,\omega_{t})}{\varepsilon}$

exists. If this is the case for all $(t,\omega)\in\Lambda_{T}$, we call
$\mathcal{D}_{x}F$ the _vertical derivative of $F$ in direction $\delta_{x}$_.
Higher order vertical derivatives are defined iteratively.

##  2 The Explicit Form of the Martingale Representation for the Super-
Brownian Motion

Let $\mathcal{L}^{2}(M_{X})$ be the space of predictable functions
$\phi:\Omega\times[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ such
that111This space coincides with the space $\mathcal{P}_{M}$ in [Walsh, 1986].
Therefore the stochastic integral with respect to the martingale measure
$M_{X}$ is well defined for all $\phi\in\mathcal{L}^{2}(M_{X})$.

$\|\phi\|_{\mathcal{L}^{2}(M_{X})}^{2}=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\phi^{2}(s,x)X(s)(dx)ds\right]<\infty,$

$\mathcal{M}^{2}$ be the space of square-integrable
$(\mathcal{F}_{t})_{t}$-martingales with initial value zero and with norm

$\|Y\|^{2}_{\mathcal{M}^{2}}=\mathbb{E}[Y(T)^{2}].$

Further define the space $U$ which is the linear span of functions of form

$\phi_{\Gamma,a,h}(\omega,t,x)=\Gamma{(\omega)}\cdot h(x)1_{(a,T]}(t)$

with $\Gamma$ bounded, $a\in[0,T)$ and $\mathcal{F}_{a}$-measurable and
$h\in\mathbb{S}(\mathbb{R}^{d})$. As functions in $U$ can be expressed as
pointwise limits of functions in $\mathcal{S}$, these functions are
predictable.

######  Proposition 1.

The space $U$ is a dense subset of $\mathcal{L}^{2}(M_{X})$.

###### Proof.

From the above, we already know that functions $\phi\in U$ are predictable.
Hence to show that $U$ is indeed a subspace of $\mathcal{L}^{2}(M_{X})$, we
have to prove that $\|\phi\|^{2}<\infty$ holds for all $\phi\in U$.

It is enough to show that
$\|\phi_{\Gamma,a,h}\|_{\mathcal{L}^{2}(M_{X})}^{2}<\infty$ for all functions
$\phi_{\Gamma,a,h}$ generating $U$. Assuming that the bounds of $\Gamma^{2}$
and $h^{2}$ are given by $C_{\Gamma^{2}}$ and $C_{h^{2}}$, respectively, we
get

$\displaystyle\|\phi_{\Gamma,a,h}\|_{\mathcal{L}^{2}(M_{X})}^{2}$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}(\Gamma
h(x)1_{(a,T]}(s))^{2}X(s)(dx)ds\right]$ $\displaystyle\leq
C_{\Gamma^{2}}\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}h^{2}(x)1_{(a,T]}(s)X(s)(dx)ds\right]$
$\displaystyle\leq
C_{\Gamma^{2}}\,C_{h^{2}}\mathbb{E}\left[\int_{a}^{T}X(s)(\mathbb{R}^{d})ds\right]$
$\displaystyle\leq
C_{\Gamma^{2}}\,C_{h^{2}}\,(T-a)\,\mathbb{E}\left[\max_{t\in[a,T]}X(t)(\mathbb{R}^{d})\right].$

As the total mass of $X$ is a critical Feller continuous state branching
process, we know that

$\mathbb{E}[X(t)(\mathbb{R}^{d})]=X(0)(\mathbb{R}^{d})<\infty,$

holds for all $t\in[0,T]$, which yields that $\phi_{\Gamma,a,h}$ has finite
$\|\cdot\|_{\mathcal{L}^{2}(M_{X})}$-norm. Consequently
$\phi\in\mathcal{L}^{2}(M_{X})$ and thus $U\subset\mathcal{L}^{2}(M_{X})$.

To obtain the density of $U$ in $\mathcal{L}^{2}(M_{X})$, note that
$\mathcal{S}$ is dense in $\mathcal{L}^{2}(M_{X})$, i.e.
$\bar{\mathcal{S}}=\mathcal{L}^{2}(M_{X})$ holds. Since the closures of
$\mathcal{S}$ and $U$ coincide, we obtain
$\bar{U}=\bar{\mathcal{S}}=\mathcal{L}^{2}(M_{X})$ and thus that $U$ is dense
in $\mathcal{L}^{2}(M_{X})$. ∎

In the following, we derive the martingale representation formula for all
square-integrable $(\mathcal{F}_{t})_{t}$-martingales. We do so by extending
the notion of vertical derivatives to obtain an operator $\nabla_{M}$ on
$\mathcal{M}^{2}$ which plays a crucial role in the martingale representation.
To get the general representation formula, we start by deriving the formula
and defining the operator $\nabla_{M}$ for martingales in a subset of
$\mathcal{M}^{2}$. As the considered subspace is dense in $\mathcal{M}^{2}$,
we can then extend the operator as well as the martingale representation
formula to all $Y\in\mathcal{M}^{2}$ and thus obtain a martingale
representation for all square-integrable $(\mathcal{F}_{t})_{t}$-martingales.

######  Definition 2.

A linear operator $\Pi$ mapping from its domain $D(\Pi)$ into a Hilbert space
$\mathcal{H}$ is called an _extension_ of the linear operator
$\tilde{\Pi}:D(\tilde{\Pi})\rightarrow\mathcal{H}$ if $D(\tilde{\Pi})\subset
D(\Pi)$ and $\tilde{\Pi}v=\Pi v$ for all $v\in D(\tilde{\Pi})$.

To construct the just mentioned subspace of $\mathcal{M}^{2}$, we consider the
stochastic integral $I_{M_{X}}(\phi)$ of a function
$\phi\in\mathcal{L}^{2}(M_{X})$ with respect to the martingale measure $M_{X}$
corresponding to $X$. This allows us to define the set

$I_{M_{X}}(U)=\\{Y\ |\
Y(t)=\int_{0}^{t}\int_{\mathbb{R}^{d}}\phi(s,x)M_{X}(ds,dx),\ \phi\in U,\
t\in[0,T]\\}.$

To prove that this is indeed a subset of $\mathcal{M}^{2}$, we need parts of
the proof of the following result.

######  Proposition 2.

The mapping

$\displaystyle\begin{split}I_{M_{X}}:\quad\mathcal{L}^{2}(M_{X})\quad&\rightarrow\quad\mathcal{M}^{2}\\\
\phi\quad&\mapsto\quad\int_{0}^{\cdot}\int_{\mathbb{R}^{d}}\phi(s,x)M_{X}(ds,dx)\end{split}$
(1)

is an isometry.

###### Proof.

Let $Q_{M_{X}}$ be the covariation of $M_{X}$ and $\phi$,
$\psi\in\mathcal{L}^{2}(M_{X})$. Then, by adapting the proof of Theorem 2.5 in
[Walsh, 1986] to our setting, we get that

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}^{d}}\phi(s,x)M_{X}(ds,dx)\int_{0}^{t}\int_{\mathbb{R}^{d}}\psi(s,x)M_{X}(ds,dx)$
$\displaystyle-\quad$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\phi(s,x)\psi(s,y)Q_{M_{X}}(ds,dx,dy)$

is a martingale for all $\phi$, $\psi\in\mathcal{L}^{2}(M_{X})$ and thus

$\displaystyle\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{d}}\phi(s,x)M_{X}(ds,dx)\int_{0}^{t}\int_{\mathbb{R}^{d}}\psi(s,x)M_{X}(ds,dx)\right]$
$\displaystyle=\quad$
$\displaystyle\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\phi(s,x)\psi(s,y)Q_{M_{X}}(ds,dx,dy)\right].$

In the scenario we consider $M_{X}$ is orthogonal. For orthogonal martingale
measures, the covariation $Q_{M_{X}}$ simplifies to
$Q([0,t],B,B)=\nu([0,t],B)$ with $\nu$ being a measure on
$\mathbb{R}^{d}\times[0,T]$. From Example 7.1.3 in [Dawson, 1993] we know that
$\nu$ has the following form if $M_{X}$ is the martingale measure associated
with the $B(A,c)$-superprocess:

$\nu(ds,dx)=cX(s)(dx)ds.$

As $c=1$ in our setting, this yields

$\displaystyle\begin{split}&\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{d}}\phi(s,x)M_{X}(ds,dx)\int_{0}^{t}\int_{\mathbb{R}^{d}}\psi(s,x)M_{X}(ds,dx)\right]\\\
=\quad&\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{d}}\phi(s,x)\psi(s,x)X(s)(dx)ds\right]\end{split}$
(2)

for all $t\in[0,T]$ and therefore

$\displaystyle\|I_{M_{X}}(\phi)\|_{\mathcal{M}^{2}}^{2}$
$\displaystyle=\mathbb{E}\left[\left(\int_{0}^{T}\int_{\mathbb{R}^{d}}\phi(s,x)M_{X}(ds,dx)\right)^{2}\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\phi^{2}(s,x)X(s)(dx)ds\right]$
$\displaystyle=\|\phi\|_{\mathcal{L}^{2}(M_{X})}^{2}.$

∎

For any function in $\phi=\phi_{\Gamma,a,h}\in U$ we have

$\displaystyle I_{M_{X}}(\phi)(t)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{d}}\phi_{\Gamma,a,h}(s,x)M_{X}(ds,dx)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{d}}\Gamma\cdot
h(x)1_{(a,T]}(s)M_{X}(ds,dx)$
$\displaystyle=\Gamma\cdot(M(t)(h)-M(a)(h))1_{t>a}$
$\displaystyle=\Gamma\cdot\left(\vphantom{\int_{0}^{2}}\langle
X(t),h\rangle-\langle
X(a),h\rangle-\int_{a}^{t}\int_{\mathbb{R}^{d}}\frac{1}{2}\Delta
h(y)X(s)(dy)ds\right)1_{t>a}.$

From the martingale problem defining the super-Brownian motion, we get that
$(M(t)(h))_{t\in[0,T]}$ is a martingale as
$h\in\mathbb{S}(\mathbb{R}^{d})\subset D(\frac{1}{2}\Delta)$ and thus
$(I_{M_{X}}(\phi)(t))_{t\in[0,T]}$ is also a martingale for any $\phi\in U$.
The process is also square-integrable as we have for all $t\in[0,T]$

$\displaystyle\mathbb{E}[(I_{M_{X}}(\phi)(t))^{2}]$
$\displaystyle=\mathbb{E}\left[\left(\int_{0}^{t}\int_{\mathbb{R}^{d}}\phi(s,x)M_{X}(ds,dx)\right)^{2}\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{d}}\phi^{2}(s,x)X(s)(dx)ds\right]$
$\displaystyle\leq\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\phi^{2}(s,x)X(s)(dx)ds\right]$
$\displaystyle<\infty,$

since $\phi\in\mathcal{L}^{2}(M_{X})$. Hence $I_{M_{X}}(\phi)$ is square-
integrable and thus $I_{M_{X}}(U)\subset\mathcal{M}^{2}$.

Next, consider the function $F$ of form

$\displaystyle F:\quad\Lambda_{T}\quad$
$\displaystyle\rightarrow\quad\mathbb{R}$ $\displaystyle(t,\omega)\quad$
$\displaystyle\mapsto\quad\Gamma{(\omega)}\left(\vphantom{\int_{0}^{2}}\langle\omega(t),h\rangle-\langle\omega(a),h\rangle-\int_{a}^{t}\int_{\mathbb{R}^{d}}\frac{1}{2}\Delta
h(y)\omega(s)(dy)ds\right)1_{t>a}.$

Plugging $X_{t}$ into $F$ for $\omega$ yields

$\displaystyle F(t,X_{t})$ $\displaystyle=\Gamma(X_{t})\left(\langle
X_{t}(t),h\rangle-\langle
X_{t}(a),h\rangle-\int_{a}^{t}\int_{\mathbb{R}^{d}}\frac{1}{2}\Delta
h(x)X_{t}(s)(dx)ds\right)1_{t>a}$ $\displaystyle=\Gamma(X_{t})\left(\langle
X(t),h\rangle-\langle
X(a),h\rangle-\int_{a}^{t}\int_{\mathbb{R}^{d}}\frac{1}{2}\Delta
h(x)X(s)(dx)ds\right)1_{t>a}$

and, as $X_{t}(\omega)=\omega_{t}$ and $X(t)(\omega)=\omega(t)$, we get

$\displaystyle
F(t,X_{t})(\omega)=\Gamma(\omega_{a})\left(\langle\omega(t),h\rangle-\langle\omega(a),h)-\int_{a}^{t}\int_{\mathbb{R}^{d}}\langle\omega(s),\frac{1}{2}\Delta
h(x)\omega(s)(dx)ds\right),$

from which we get, as $\Gamma$ is $\mathcal{F}_{a}$-measurable, that
$F(t,X_{t})=I_{M_{X}}(\phi_{\Gamma,a,h})(t)$.

As, in addition, for any path $\omega\in C([0,T],M_{F}(E))$, it holds

$\displaystyle\lim_{\varepsilon\downarrow
0}\frac{1}{\varepsilon}\left(\vphantom{\int_{0}^{1}}\langle(\omega+\varepsilon\delta_{x}1_{[t,T]})(t),h\rangle-\langle(\omega+\varepsilon\delta_{x}1_{[t,T]})(a),h\rangle\right.$
$\displaystyle\hphantom{\lim_{\varepsilon\downarrow
0}\frac{1}{\varepsilon}\quad\langle(\omega+\varepsilon\delta_{x}1_{[t,T]})(t),h\rangle}-\int_{a}^{t}\int_{\mathbb{R}^{d}}\frac{1}{2}\Delta
h(y)(\omega+\varepsilon\delta_{x}1_{[t,T]})(r)(dy)dr$
$\displaystyle\hphantom{\quad\lim_{\varepsilon\downarrow
0}\frac{1}{\varepsilon}}\left.-\,\langle\omega(t),h\rangle+\langle\omega(a),h\rangle+\int_{a}^{t}\int_{\mathbb{R}^{d}}\frac{1}{2}\Delta
h(y)\omega(r)(dy)dr\right)$ $\displaystyle=\ $
$\displaystyle\lim_{\varepsilon\downarrow
0}\frac{1}{\varepsilon}\left(\varepsilon
h(x)-\varepsilon\int_{a}^{t}\int_{\mathbb{R}^{d}}\frac{1}{2}\Delta
h(y)1_{[t,T]}(r)\delta_{x}(dy)dr\right)$ $\displaystyle=\ $ $\displaystyle
h(x)$

and, as $\Gamma$ is $\mathcal{F}_{a}$-measurable, for $t\in(a,T]$ it holds

$\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon}\left(\Gamma(\omega+\varepsilon\delta_{x}1_{[t,T]})-\Gamma(\omega)\right)=0,$

we obtain, for all $(t,\omega)\in\Lambda_{T}$,

$\mathcal{D}_{x}F(t,\omega)=\Gamma{(\omega)}1_{(a,T]}(t)h(x)=\phi_{\Gamma,a,h}(t,x).$

Now, for a process $Y$ defined by

$Y(t)=I_{M_{X}}(\phi_{\Gamma,a,h})(t)=F(t,X_{t})$

for $t\in[0,T]$, we can define an operator $\nabla_{M}$ of the form

$\displaystyle\begin{split}\nabla_{M}:\quad
I_{M_{X}}(U)\quad&\rightarrow\quad\mathcal{L}^{2}(M_{X})\\\
Y\quad&\mapsto\quad\nabla_{M}Y,\end{split}$ (3)

where $\nabla_{M}Y$ is given by

$\nabla_{M}Y:\quad(\omega,t,x)\quad\mapsto\quad\nabla_{M}Y(\omega,t,x):=\mathcal{D}_{x}F(t,X_{t}(\omega))=\mathcal{D}_{x}F(t,\theta)|_{\theta=X_{t}(\omega)}.$

Further, by the definition of $Y$, we get the martingale representation

$Y(t)=\int_{0}^{t}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)M_{X}(ds,dx),$ (4)

which is equal to

$F(t,X_{t})=\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathcal{D}_{x}F(s,X_{s})M_{X}(ds,dx).$

Thus, we derived a martingale representation formula for martingales in the
subspace $I_{M_{X}}(U)$ of $\mathcal{M}^{2}$ using the operator $\nabla_{M}$
defined by (3) on $I_{M_{X}}(U)$.

In [Evans and Perkins, 1994] the authors proved that, if $Y$ belongs to
$\mathcal{M}^{2}$, there exists a unique $\rho\in\mathcal{L}^{2}(M_{X})$ such
that

$Y(t)=\int_{0}^{t}\int_{\mathbb{R}^{d}}\rho(s,x)M_{X}(ds,dx)\quad\forall t\geq
0$ (5)

holds $\mathbb{P}$-a.s.. Consequently, the representation in (4) is unique.
Further, this yields that the mapping $I_{M_{X}}$ is bijective. These two
properties allow us to extend $\nabla_{M}$ to all processes
$Y\in\mathcal{M}^{2}$ by using the following result.

######  Proposition 3.

The space $\\{\nabla_{M}Y\ |\ Y\in I_{M_{X}}(U)\\}$ is dense in
$\mathcal{L}^{2}(M_{X})$ and the space $I_{M_{X}}(U)$ is dense in
$\mathcal{M}^{2}$.

###### Proof.

As $U$ is dense in $\mathcal{L}^{2}(M_{X})$ (see Proposition 1),

$U=\\{\nabla_{M}Y\ |\ Y\in I_{M_{X}}(U)\\}\subset\mathcal{L}^{2}(M_{X})$

yields that $\\{\nabla_{M}Y\ |\ Y\in I_{M_{X}}(U)\\}$ is dense in
$\mathcal{L}^{2}(M_{X})$. Further, as $I_{M_{X}}$ is a bijective isometry, we
get the density of $I_{M_{X}}(U)$ in $\mathcal{M}^{2}$.
∎

This density result allows to prove the following proposition that can be
interpreted as an integration by parts formula on
$[0,T]\times\mathbb{R}^{d}\times\Omega$.

######  Proposition 4.

If $Y\in I_{M_{X}}(U)$, then $\nabla_{M}Y$ is the unique element in
$\mathcal{L}^{2}(M_{X})$ such that

$\mathbb{E}\left[Y(T)Z(T)\right]=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)\nabla_{M}Z(s,x)X(s)(dx)ds\right]$
(6)

holds for all $Z\in I_{M_{X}}(U)$.

###### Proof.

From (2) and (4) we get that, for every $Y$, $Z\in I_{M_{X}}(U)$,

$\displaystyle\mathbb{E}[Y(T)Z(T)]=\ $
$\displaystyle\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)M_{X}(ds,dx)\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{M}Z(s,x)M_{X}(ds,dx)\right]$
$\displaystyle=\ $
$\displaystyle\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)\nabla_{M}Z(s,x)X(s)(dx)ds\right]$

holds.

To prove the uniqueness, let $\psi\in\mathcal{L}^{2}(M_{X})$ be another
process such that for $Z\in I_{M_{X}}(U)$ we have

$\mathbb{E}\left[Y(T)Z(T)\right]\\\
=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\psi(s,x)\nabla_{M}Z(s,x)X(s)(dx)ds\right].$

Then, by subtraction

$0=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}(\psi(s,x)-\nabla_{M}Y(s,x))\nabla_{M}Z(s,x)X(s)(dx)ds\right]$

for all $Z\in I_{M_{X}}(U)$, which yields the uniqueness of $\nabla_{M}Y$ in
$\mathcal{L}^{2}(M_{X})$ as $\\{\nabla_{M}Z\ |\ Z\in I_{M_{X}}(U)\\}$ is dense
in $\mathcal{L}^{2}(M_{X})$.
∎

The interpretation as an integration by parts formula becomes clear when we
consider the following alternative form of equation (6), which holds for all
$\phi\in\mathcal{L}^{2}(M_{X})$:

$\mathbb{E}\left[Y(T)\int_{0}^{T}\int_{\mathbb{R}^{d}}\phi(s,x)M_{X}(ds,dx)\right]=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)\phi(s,x)X(s)(dx)ds\right].$

Now, we have all the necessary results on hand to introduce the extension of
$\nabla_{M}$ to $\mathcal{M}^{2}$.

######  Theorem 1.

The operator defined in (3) can be extended to an operator

$\displaystyle\nabla_{M}:\quad\mathcal{M}^{2}\quad$
$\displaystyle\rightarrow\quad\mathcal{L}^{2}(M_{X})$ $\displaystyle Y\quad$
$\displaystyle\mapsto\quad\nabla Y.$

This operator is a bijection and the unique continuous extension of the
operator defined in (3) given by the following: For a given
$Y\in\mathcal{M}^{2}$, $\nabla_{M}Y$ is the unique element in
$\mathcal{L}^{2}(M_{X})$ such that

$\mathbb{E}[Y(T)Z(T)]=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)\nabla_{M}Z(s,x)X(s)(dx)ds\right]$
(7)

holds for all $Z\in I_{M_{X}}(U)$.

###### Proof.

As $\nabla_{M}:I_{M_{X}}(U)\rightarrow\mathcal{L}^{2}(M_{X})$ is a bounded
linear operator, $\mathcal{L}^{2}(M_{X})$ is a Hilbert space and
$I_{M_{X}}(U)$ is dense in $\mathcal{M}^{2}$, the the BLT theorem (see e.g.
Theorem 5.19 in [Hunter and Nachtergaele, 2001]) yields the existence of a
unique continuous extension
$\nabla_{M}:\mathcal{M}^{2}\rightarrow\mathcal{L}^{2}(M_{X})$. To prove that
(7) defines the extension, we have to prove that its restriction to
$I_{M_{X}}(U)$ is equal to the initial operator. As we immediately get this
from Proposition 4, the unique continuous extension is given by (7).

For $Y\in\mathcal{M}^{2}$, there exists a unique
$\rho\in\mathcal{L}^{2}(M_{X})$ such that (5) holds. This $\rho$ satisfies (7)
as we get from combining (5) with (4) and (2) that

$\displaystyle\mathbb{E}[Y(T)Z(T)]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\rho(s,x)M_{X}(ds,dx)Z(T)\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\rho(s,x)\nabla_{M}Z(s,x)X(s)(dx)ds\right]$

holds for all $Z\in I_{M_{X}}(U)$. The uniqueness of the integrand in (5) then
yields that $\rho$ and $\nabla_{M}Y$ coincide in $\mathcal{L}^{2}(M_{X})$.

Next, assume $Y$, $Y^{\prime}\in\mathcal{M}^{2}$ with
$\nabla_{M}Y=\nabla_{M}Y^{\prime}$ for $\nabla_{M}Y$,
$\nabla_{M}Y^{\prime}\in\mathcal{L}^{2}(M_{X})$. Then, for all $Z\in
I_{M_{X}}(U)$,

$\displaystyle 0$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}(\nabla_{M}Y(s,x)-\nabla_{M}Y^{\prime}(s,x))\nabla_{M}Z(s,x)X(s)(dx)ds\right]$
$\displaystyle=\mathbb{E}\left[\left(\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)M_{X}(ds,dx)ds-\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{M}Y^{\prime}(s,x)M_{x}(ds,dx)\right)Z(T)\right]$
$\displaystyle=\mathbb{E}[(Y(T)-Y^{\prime}(T))Z(T)],$

which implies that $Y=Y^{\prime}$ in $\mathcal{M}^{2}$ as $I_{M_{X}}(U)$ is
dense in $\mathcal{M}^{2}$. Therefore, the operator $\nabla_{M}$ is injective.
Further, as for every $\phi\in\mathcal{L}^{2}(M_{X})$ there exists the process

$Y=\int_{0}^{\cdot}\int_{\mathbb{R}^{d}}\phi(s,x)M_{X}(ds,dx)\in\mathcal{M}^{2}$

for which $\nabla_{M}Y=\phi$ holds, the operator is also surjective and thus
bijective. ∎

The extension of $\nabla_{M}$ from $I_{M_{X}}(U)$ to $\mathcal{M}^{2}$ now
immediately yields the following martingale representation for square-
integrable martingales.

######  Theorem 2.

For any square-integrable $(\mathcal{F}_{t})_{t}$-martingale $Y$ and every
$t\in[0,T]$ it holds

$Y(t)=Y(0)+\int_{0}^{t}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)M_{X}(ds,dx)\quad\text{$\mathbb{P}$-a.s.
.}$

###### Proof.

First, assume $Y\in\mathcal{M}^{2}$. Then there exists a unique
$\rho\in\mathcal{L}^{2}(M_{X})$ such that (5) holds and from the proof of
Theorem 1 we get that this $\rho$ satisfies (7). Therefore, by the uniqueness
of $\rho$ and the definition of $\nabla_{M}Y$, we have $\rho=\nabla_{M}Y$ in
$\mathcal{L}^{2}(M_{X})$ and thus

$Y(t)=\int_{0}^{t}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)M_{X}(ds,dx)$

holds $\mathbb{P}$-a.s..

Now, let $Y$ be a square-integrable $(\mathcal{F}_{t})_{t}$-integral. Then
$\tilde{Y}=Y-Y(0)\in\mathcal{M}^{2}$ and thus we can apply the above to
$\tilde{Y}$. Adding $Y(0)$ to both sides of the resulting representation for
$\tilde{Y}$ yields the desired martingale representation formula for $Y$. ∎

We conclude this paper with the two following properties of the extended
operator $\nabla_{M}$ defined on $\mathcal{M}^{2}$, which are worth
mentioning.

######  Lemma 1.

The operator $\nabla_{M}$ defined on $\mathcal{M}^{2}$ is an isometry and the
adjoint operator of $I_{M_{X}}$, the stochastic integral with respect to the
martingale measure $M_{X}$.

###### Proof.

Let $Y\in\mathcal{M}^{2}$. Then we get, using the same arguments as in
Proposition 2,

$\displaystyle\|\nabla_{M}Y\|_{\mathcal{L}^{2}(M_{X})}^{2}$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}(\nabla_{M}Y(s,x))^{2}X(s)(dx)ds\right]$
$\displaystyle=\mathbb{E}\left[\left(\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)M_{X}(ds,dx)\right)^{2}\right]$
$\displaystyle=\left\|\int_{0}^{\cdot}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)M_{X}(ds,dx)\right\|_{\mathcal{M}^{2}}^{2}$
$\displaystyle=\|Y\|_{\mathcal{M}^{2}}^{2}.$

Further, let $\phi\in\mathcal{L}^{2}(M_{X})$. As

$\displaystyle\langle I_{M_{X}}(\phi),Y\rangle_{\mathcal{M}^{2}}$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\phi(s,x)M_{X}(ds,dx)Y(T)\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\phi(s,x)M_{X}(ds,dx)\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{M}Y(s,x)M_{X}(ds,dx)\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}^{d}}\phi(s,x)\nabla_{M}Y(s,x)X(s)(dx)ds\right]$
$\displaystyle=\langle\phi,\nabla_{M}Y\rangle_{\mathcal{L}^{2}(M_{X})}$

holds, we get that $\nabla_{M}$ is the adjoint operator of $I_{M_{X}}$. ∎

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# Object classification with Convolutional Neural Networks: from KiDS to
Euclid

G.A. Verdoes Kleijn  University of Groningen, Groningen, The Netherlands  C.A.
Marocico  University of Groningen, Groningen, The Netherlands  Y. Mzayek
University of Groningen, Groningen, The Netherlands  M. Pöntinen  University
of Helsinki, Finland  M. Granvik  University of Helsinki, Finland and Luleå
University of Technology, Luleå, Sweden  O. Williams  University of Groningen,
Groningen, The Netherlands  J.T.A. de Jong  University of Groningen,
Groningen, The Netherlands and Leiden Observatory, Leiden, The Netherlands  T.
Saifollahi  University of Groningen, Groningen, The Netherlands  L. Wang  SRON
Netherlands Institute for Space Research, Groningen, The Netherlands and
University of Groningen, Groningen, The Netherlands  B. Margalef-Bentabol
University of Groningen, Groningen, The Netherlands  A. La Marca  University
of Groningen, Groningen, The Netherlands and SRON Netherlands Institute for
Space Research, Groningen, The Netherlands  B. Chowdhary Nagam  University of
Groningen, Groningen, The Netherlands  L.V.E. Koopmans  University of
Groningen, Groningen, The Netherlands  E.A. Valentijn  University of
Groningen, Groningen, The Netherlands

###### Abstract

Large-scale imaging surveys have grown $\sim$1000 times faster than the number
of astronomers in the last 3 decades. Using Artificial Intelligence instead of
astronomer’s brains for interpretative tasks allows astronomers to keep up
with the data. We give a progress report on using Convolutional Neural
Networks (CNNs) to classify three classes of rare objects (galaxy mergers,
strong gravitational lenses and asteroids) in the Kilo-Degree Survey (KiDS)
and the Euclid Survey.

##  1 Needles in the Kilo-Degree Survey and the Euclid Survey haystacks

Figure 1: Growth of information technology, galaxy surveys and astronomers as
a function of time.

The typical number of transistors per CPU, the aperture area of telescopes
("glass"), the cumulative number of pixels in telescope cameras and number of
astronomers are shown. The plot is adapted from [Ty19]. The triangles show for
surveys the total number of galaxies in any filter with S/N $>$ 5 for flux
inside a 2 arcsec wide aperture per year. The arrow for the LSST triangle
indicates it is currently planned to start late 2024.

Figure 2: True positives for the three classes of objects in top row and false
positives in bottom row. The thumbnails are from KiDS for the classes of
strong lenses and galaxy mergers and from simulated Euclid data for the
asteroid class.

Figure 1 shows that observational digital information on galaxies has grown
roughly 1000 times faster than the number of astronomers since 1985.

The figure is taken from [Ty19] and data points for the Kilo-Degree Survey
[Ku19] and the Euclid Survey [ES22] have been added. The development of
information systems and their algorithmic software to analyze these digital
data has been critical to accommodate this growth. Information systems have
taken over many operational tasks performed earlier by astronomers. The next
step is that systems will take over from astronomers interpretative tasks such
as the classification of celestial objects based on their morphology, color
and context in images.

In the recent past classification relied on human judgement via the careful,
laborious creation of both customised catalogs with up to hundreds of features
extracted from pixels and also customised algorithms that took these features
as input and gave a classification as output. CNNs offer the option to let
information systems go from pixels straight to classification in a scalable,
reproducible and objective manner. They bypass the route of, possibly
subjective, human interpretation and the laborious task of feature cataloging
and classification algorithm creation.

Our teams of astronomers and data scientists focus on the detection and
classification of three classes of rare objects in KiDS and Euclid:
gravitational strong lenses, galaxy mergers and asteroids.

We expect 2.4E3 / 1E5 strong lenses, 1E4 1E4 / 1E7 galaxy mergers and 6.4E4 /
1.5E5 asteroids in KiDS / Euclid respectively.

They are the "needles in the haystack" of images. Only of the order of 1 in
1000 massive galaxies are a strong lens, only of the order of 10% of galaxies
are in major mergers, while the number of asteroids per square degree and
hence of the order of 1E5 source detections can be as low as a few.

For each class there are many more sources that resemble these classes, but
are "impostors" (e.g., Fig 2).

##  2 Classification with convolutional neural networks

CNNs have emerged as a powerful approach to classify massive datasets of
source images because they can learn from their own feature extraction and
optimise it for the desired classification (e.g., [HL22] for a review). The
feature extraction that is acquired through training for the properties of a
particular imaging survey can be transferred to other imaging surveys using
transfer learning. This makes the approach potentially very generalisable.

The first CNNs deployed to find strong lenses in KiDS had a ResNet
architecture ([Pe17], [Pe19], [Pe19a], [Li20], [Li21]). We have currently
developed a “DenseLens” approach based on a DenseNet architecture [Na23]. The
DenseNet architecture uses concatenation of all preceding feature maps and
this improves the flow of information from the first layer until the last
layer. This also helps to reduce the total number of parameters in the network
by an order of magnitude.

Each training step has an equal number of lenses and non-lenses. Simulated
lensing features were generated and added to thumbnails of elliptical galaxies
from KiDS to create lenses. Non-lenses are a combination of hand-picked
"impostors" and elliptical galaxies. In the training the Binary Cross Entropy
loss were minimized using an ADAM optimizer. The approach includes a technique
to rank-order strong lenses without human inspection by training on a metric
for the information content of the lensing features in the training data. The
automated rank-ordering is important for the Euclid Survey given the expected
105 true strong lenses from it.

The CNNs deployed on KiDS for galaxy mergers used transfer learning from a
pre-trained VGG19 CNN [WPR20]. Given that differences in appearance between
merging galaxies and normal galaxies can be subtle it is crucial to have a
comprehensive training sample. This is why hydrodynamical simulations are used
to construct our training sample. The 3D merging galaxies in the simulations
are converted to mock observations for the particular survey. Currently a new
custom CNN is being trained for survey data from the Hyper Suprime Camera on
Subaru as preparation for Euclid. Early results show an accuracy of 77 per
cent, with reliability on merger detection of 83 per cent.

The CNN for asteroids has six convolution layers and a desne layer. The
training is done with a YOLO loss function. For KiDS, the output layer has one
unit outputting a classification label with a sigmoid function. For KiDS the
training data consists of true asteroids in KiDS detected using a classical
detection pipeline [Ma18] and other detections as non-asteroids. For Euclid
the training examples are based on simulated Euclid VIS images generated with
the Euclid Consortium’s ELViS software with added simulated asteroid streaks
(courtesy L. Conversi and A. Nucita). For Euclid, the training includes an
Adam optimizer [KB14] and early stopping callback monitoring the validation
loss. For Euclid, the machine learning pipeline [Pö23] surpasses the asteroid
detection completeness (recall) achieved by a traditional segmentation-based
pipeline [Pö20], detecting both fainter and slower-moving asteroids.

##  3 Next steps

It is clear from the results published and from the work in progress discussed
above for KiDS and Euclid that CNNs provide a powerful way to address the
interpretative task of classification in current and future astronomical
surveys .

The workflow of deploying the CNNs above on surveys still includes the
creation of feature catalogs for detecting sources and classifying stars and
extended objects. To bypass the creation of any catalog for all three classes
we are experimenting to prepend the workflow with a discriminator CNN that
makes preliminary classification of objects into these three classes. The
output of the discriminator is then fed to the class-specific CNNs. Initial
results maintain 100% completeness and more than 50% purity in the output
ensembles using class weighting as hyperparameter.

Next steps also include attempting to let CNNs perform other types of
interpretative tasks. For example, for the asteroid CNN for the Euclid Survey
there is a more advanced version of the model under development, which has
five output units and a tailor-made loss function directly predicting asteroid
streak coordinates, and which can be used to detect and locate asteroids in
full CCDs with a sliding-window algorithm [Pö23]. This approach thus bypasses
not only the need for any detection catalog it also performs an additional
interpretative in addition to classification: astrometry.

## References

  * [ES22] Euclid Collaboration and R. al Scaramella  “Euclid preparation. I. The Euclid Wide Survey”  In _Astronomy and Astrophysics_ 662, 2022, pp. A112  DOI: 10.1051/0004-6361/202141938
  * [HL22] Marc Huertas-Company and François Lanusse  “The DAWES review 10: The impact of deep learning for the analysis of galaxy surveys”  In _arXiv e-prints_ , 2022, pp. arXiv:2210.01813  arXiv:2210.01813 [astro-ph.IM]
  * [KB14] Diederik P Kingma and Jimmy Ba  “Adam: A method for stochastic optimization”  In _arXiv preprint arXiv:1412.6980_ , 2014
  * [Ku19] K. Kuijken  “The fourth data release of the Kilo-Degree Survey: ugri imaging and nine-band optical-IR photometry over 1000 square degrees”  In _Astronomy and Astrophysics_ 625, 2019, pp. A2  DOI: 10.1051/0004-6361/201834918
  * [Li20] R. Li et al.  “New High-quality Strong Lens Candidates with Deep Learning in the Kilo-Degree Survey”  In _Astrophysical Journal_ 899.1, 2020, pp. 30  DOI: 10.3847/1538-4357/ab9dfa
  * [Li21] R. Li  “High-quality Strong Lens Candidates in the Final Kilo-Degree Survey Footprint”  In _Astrophysical Journal_ 923.1, 2021, pp. 16  DOI: 10.3847/1538-4357/ac2df0
  * [Ma18] M. Mahlke et al.  “Mining the Kilo-Degree Survey for solar system objects”  In _Astronomy and Astrophysics_ 610, 2018, pp. A21  DOI: 10.1051/0004-6361/201730924
  * [Na23] Bharath Chowdhary Nagam et al.  “DenseLens - Using DenseNet ensembles and information criteria for finding and rank-ordering strong gravitational lenses”  In _Monthly Notices of the Royal Astronomical Society_ 523.3, 2023, pp. 4188–4201  DOI: 10.1093/mnras/stad1623
  * [Pe17] C.. Petrillo et al.  “Finding strong gravitational lenses in the Kilo Degree Survey with Convolutional Neural Networks”  In _Monthly Notices of the Royal Astronomical Society_ 472.1, 2017, pp. 1129–1150  DOI: 10.1093/mnras/stx2052
  * [Pe19] C.. Petrillo et al.  “Testing convolutional neural networks for finding strong gravitational lenses in KiDS”  In _Monthly Notices of the Royal Astronomical Society_ 482.1, 2019, pp. 807–820  DOI: 10.1093/mnras/sty2683
  * [Pe19a] C.. Petrillo et al.  “LinKS: discovering galaxy-scale strong lenses in the Kilo-Degree Survey using convolutional neural networks”  In _Monthly Notices of the Royal Astronomical Society_ 484.3, 2019, pp. 3879–3896  DOI: 10.1093/mnras/stz189
  * [Pö20] Mikko Pöntinen et al.  “Euclid: Identification of asteroid streaks in simulated images using StreakDet software”  In _Astronomy & Astrophysics_ 644 EDP sciences, 2020, pp. A35
  * [Pö23] et al. Pöntinen  “”  In _in preparation_ , 2023
  * [Ty19] J. Tyson  “Cosmology data analysis challenges and opportunities in the LSST sky survey”  In _Journal of Physics Conference Series_ 1290, Journal of Physics Conference Series, 2019, pp. 012001  DOI: 10.1088/1742-6596/1290/1/012001
  * [WPR20] L. Wang, W.. Pearson and V. Rodriguez-Gomez  “Towards a consistent framework of comparing galaxy mergers in observations and simulations”  In _Astronomy and Astrophysics_ 644, 2020, pp. A87  DOI: 10.1051/0004-6361/202038084