# Representation of random variables as Lebesgue integrals

Sara Biagini  Sara Biagini, Department of Economics and Finance, LUISS
University, Rome, Italy  and  Gordan Žitković  Gordan Žitković, Department of
Mathematics, University of Texas at Austin, Texas, US

###### Abstract.

We study representations of a random variable $\xi$ as an integral of an
adapted process with respect to the Lebesgue measure. The existence of such
representations in two different regularity classes is characterized in terms
of the quadratic variation of (local) martingales closed by $\xi$.

The authors would like to thank Fausto Gozzi for suggesting some related
literature. They are also grateful to the anonymous referees, and the Editors
for a number of constructive comments that improved the quality of this paper.

##  1\. Introduction

This paper focuses on the representation of a random variable as an adapted
Lebesgue - as opposed to stochastic - integral. We start the analysis with
statements of our main results and then place them in the extant literature
while offering motivation for their study.
Let $(\Omega,(\mathcal{F}_{t})_{t\leq T},\mathcal{F},\mathbb{P})$ be a
filtered probability space. Given an $\mathcal{F}_{T}$-measurable random
variable $\xi$, we ask whether there exists a progressively-measurable process
$\beta$ such that

(1.1) $\displaystyle\xi=\int_{0}^{T}\beta_{u}\,du,\text{ a.s.}$

with $\beta$ in a given integrability class. We focus on the Lebesgue measure
on a finite time-horizon $[0,T]$ because other settings (alternative measures
instead of the Lebesgue measure, alternative horizons, or the discrete time on
an infinite horizon instead of the continuous time) lead to a similar
analysis.
Our main results apply to two integrability classes for $\beta$, but we
discuss interesting features of some other classes, too, in Section 4. We say
that $\beta$ is _weakly regular_ if

$\int_{0}^{T}\beta_{u}^{2}\,du<\infty\ \ a.s.,$

and _strongly regular_ if

$\mathbb{E}\left[\int_{0}^{T}\beta_{u}^{2}\,du\right]<\infty.$

Assuming throughout that all $\mathcal{F}$-local martingales are continuous,
we show in Theorem 2.1 that the representation (1.1) holds for some strongly
regular $\beta$ if and only if $\xi\in\mathbb{L}^{1}$ and

$\displaystyle\mathbb{E}\left[\int_{0}^{T}\frac{1}{T-t}\,d\left\langle
M\right\rangle_{t}\right]<\infty\text{ where
}M_{t}=\mathbb{E}\left[\xi\nonscript\>\middle|\nonscript\>\mathopen{}\mathcal{F}_{t}\right].$

In a less restrictive, weakly regular case, our Theorem 3.1 states that (1.1)
holds for a weakly regular $\beta$ if and only if there exists a probability
measure $\mathbb{Q}$ equivalent to $\mathbb{P}$ and a $\mathbb{Q}$ local
martingale $M$ with $M_{T}=\xi$ such that

(1.2) $\displaystyle\int_{0}^{T}\frac{1}{T-t}\,d\left\langle
M\right\rangle_{t}<\infty,\text{ a.s.}$

Intuitively, an absolutely continuous representation of the form (1.1) with a
weakly regular $\beta$ exists if and only if $\xi$ closes a local martingale
whose quadratic variation grows slowly enough at $T$. This problem has a
interesting link with the so-called “fundamental theorem of asset pricing”
(see Theorem 1.1, p. 487 of [DS94]). As is well known in the Mathematical
Finance community, this Theorem states that a locally-bounded semimartingale
$M$ is a local martingale under some measure $\mathbb{Q}$ equivalent to
$\mathbb{P}$ if and only if it satisfies the condition of No Free Lunch with
Vanishing Risk (NFLVR in the sequel). NFLVR is a slightly stronger version of
the classical NA (No Arbitrage) condition of Mathematical Finance. We may
think, informally, of a process that satisfies NFLVR as a measure-free version
of a local martingale, or, similarly, as a semimartingale whose local-
martingale part is everywhere more active than its finite-variation part.
When focusing on the representation (1.1) of $\xi$ under the weaker,
probability-free, condition on $\beta$, that question boils down to the
relationship between $\xi$, the set of null events, and the filtration.
Rephrased in financial terms, what we show is that (1.1) holds if and only if
$\xi$ closes a price process which has the property and moreover is a “slow”
local martingale under a suitable $\mathbb{Q}$ \- in the sense of (1.2). Such
“slow” local martingale that converges to $\xi$ can be used as a proxy for the
good approximability of $\xi$ by $\mathcal{F}_{t}$-adapted random variables as
$t\nearrow T$.
Unlike in the case of martingale representation, the question of uniqueness of
an absolutely continuous representation admits a trivially negative answer in
many interesting integrability classes, including both weak and strong
regularity discussed above. That fact served as a prompt to look for a
canonical, rather than unique $\beta$. When
$\mathbb{E}[\int_{0}^{T}\beta_{u}^{2}\,du]<\infty$ is required, the $\beta$
that minimizes $\mathbb{E}[\int_{0}^{T}\beta_{u}^{2}\,du]$ admits an easy-to-
verify explicit form, namely

$\hat{\beta}_{t}=\frac{1}{T}M_{0}+\int_{0}^{t}\frac{1}{T-u}\,dM_{u},\
t\in[0,T),$

where
$M_{t}=\mathbb{E}\left[\xi\nonscript\>\middle|\nonscript\>\mathopen{}\mathcal{F}_{t}\right]$.
Unfortunately, we could not identify an analogous natural notion of canonicity
in the weakly regular case.

Absolutely continuous representation issues arise quite easily in
applications. For instance, in [AB23] the authors deal with a linear-quadratic
stochastic control problem on the Wiener space, arising from carbon
regulation. In that problem, the controls are square integrable _rates_ ,
i.e., state dynamics involve integrals of these controls with respect to $dt$.
Furthermore, the objective function contains a terminal penalty term which is
a function of an integral of one of the controls, $\beta$, so that the random
variable $\xi=\int_{0}^{T}\beta_{t}\,dt$ appears in the objective function.
Since the problem is not strictly convex in $\beta$, the authors of [AB23]
were only able to obtain an explicit expression for the optimal $\hat{\xi}$,
and for the associated martingale
$\hat{M}_{t}=\mathbb{E}[\hat{\xi}\nonscript\>|\nonscript\>\mathopen{}\mathcal{F}_{t}]$.
They left the problem of finding an optimal, square integrable, rate
$\hat{\beta}$ that _represents_ the optimal $\hat{\xi}$ open (see [AB23],
Remark 4.1).

Integrable-enough absolutely continuous representations come in handy in other
contexts, as well. For example, they provide useful estimates when proving
existence of solutions to stochastic differential equations. The interested
reader can consult Chapter 6 of the [FGS17] for a general treatment, or
[BGZ22] for an application to stochastic delayed differential equations in an
optimal investment problem.
The only existing result concerning absolutely-continuous representation we
are aware of is the “factorization formula” of Da Prato and Zabczyk (see
Theorem 5.2.5, p. 58 in [DPZ96]). Set on an abstract Wiener space, it provides
an explicit absolutely continuous representation of a random variable given by
a stochastic integral. It relies on a version of a stochastic Fubini theorem
(see Theorem 4.18 of [DPZ14]) but does not address the regularity of the
representation itself, or provide any necessary conditions. A deeper
discussion of why their approach, based on the stochastic Fubini theorem, does
not lead to the kinds of results we are interested in is given in Remark 3.2.

Our results extend the existing ones in several directions. First, we give
_necessary and sufficient_ conditions on the random variable $\xi$ for the
representation to exist under both weak and strong regularity. Furthermore, in
the strongly regular case we show that the unique _martingale_ solution of the
representation problem arises as the $\mathbb{L}^{2}$-norm minimizer on the
product space.

The paper is organized as follows: Section 2 treats the strongly regular and
Section 3 the weakly regular case; Section 4 contains further examples,
results and comments.
Setup and notation. We consider a measurable space $(\Omega,\mathcal{F})$,
together with a maximal family $\mathcal{P}$ of mutually equivalent
probability measures on $\mathcal{F}$, as well as a right-continuous
filtration $\mathbb{F}=\\{\mathcal{F}_{t}\\}_{t\in\left[0,T\right]}$, with
$\mathcal{F}_{0}$ $\mathcal{P}$-trivial. When we write that a filtration is
_generated by a Brownian motion W_ , we always have the usual right-continuous
and complete augmentation of the natural filtration in mind. On the other
hand, a filtration $\\{\mathcal{F}_{t}\\}_{t\in\left[0,T\right]}$ is said to
_support_ a Brownian motion if there exists an
$\\{\mathcal{F}_{t}\\}_{t\in\left[0,T\right]}$-Brownian motion $W$.

We say that a process $M$ is a _$\mathcal{P}$ -local martingale_ if it is a
local martingale under some $\mathbb{P}\in\mathcal{P}$, and we denote the set
of all $\mathcal{P}$-local martingales by $\mathcal{M}^{loc}$. We impose the
following, standing, assumption throughout:

######  Assumption 1.1.

Each $\mathcal{P}$-local martingale is continuous.

In particular, the above assumption implies that for each
$M\in\mathcal{M}^{loc}$ there exists a unique process $\langle M\rangle$ such
that $M^{2}-\langle M\rangle$ is also in $\mathcal{M}^{loc}$.

######  Remark 1.2.

According to Theorem 5.38, p. 155 in [HWY92], continuity of all martingales on
a filtered probability space is equivalent to the requirement that all
$\\{\mathcal{F}_{t}\\}_{t\in\left[0,T\right]}$-stopping times be predictable.
Since this property stays invariant under equivalent measure changes, we
conclude that Assumption 1.1 holds if we only ask that there _exists_ a single
probability measure $\mathbb{P}\in\mathcal{P}$ such that all
$\mathbb{P}$-local martingales are continuous.

For $\mathbb{P}\in\mathcal{P}$, $\mathbb{L}^{p}(\mathbb{P})$ is a shorthand
for $\mathbb{L}^{p}(\Omega,\mathcal{F},\mathbb{P})$ while
$\mathbb{L}^{p,q}(\mathbb{P})$, $q\in[0,\infty)$ denotes the set of all
$\\{\mathcal{F}_{t}\\}_{t\in\left[0,T\right]}$-predictable processes $\beta$
with
$\int_{0}^{T}\left\lvert\beta_{u}\right\rvert^{p}\,du\in\mathbb{L}^{q}(\mathbb{P})$.
When $p\geq 1$, the space $\mathbb{L}^{p,1}(\mathbb{P})$ comes with the norm:

$\left\lVert\beta\right\rVert_{\mathbb{L}^{p,1}(\mathbb{P})}=\mathbb{E}^{\mathbb{P}}\left[\
\int_{0}^{T}\left\lvert\beta_{u}\right\rvert^{p}\,du\right]^{1/p},$

while no topology on $\mathbb{L}^{p,0}(\mathbb{P})$ will be needed. Since the
spaces $\mathbb{L}^{0}(\mathbb{P}),\mathbb{L}^{p,0}(\mathbb{P})$,
$\mathbb{P}\in\mathcal{P}$ coincide, we omit the probability measure from the
notation and simply write $\mathbb{L}^{0},\mathbb{L}^{p,0}$.

For $\xi\in\mathcal{F}_{T}$ and $\mathbb{P}\in\mathcal{P}$, we set

$\displaystyle\mathcal{B}^{p,q}(\xi,\mathbb{P})$
$\displaystyle:=\Big{\\{}\beta\in\mathbb{L}^{p,q}(\mathbb{P})\,:\,\textstyle\int_{0}^{T}\beta_{u}\,du=\xi\text{
a.s.}\Big{\\}}.$

When $q=0$, we omit the measure $\mathbb{P}$ and write only
$\mathcal{B}^{p,0}(\xi)$.

##  2\. The strongly regular case

In this section we choose and fix a probability measure
$\mathbb{P}\in\mathcal{P}$ and use it as the underlying measure in all
probabilistic statements. In particular, we write: $\mathbb{L}^{2}$ and
$\mathbb{L}^{2,1}$ for $\mathbb{L}^{2}(\mathbb{P})$ and
$\mathbb{L}^{2,1}(\mathbb{P})$ respectively; and $\mathcal{B}^{2,1}(\xi)$ in
place of $\mathcal{B}^{2,1}(\xi,\mathbb{P})$.

######  Theorem 2.1.

For $\xi\in\mathbb{L}^{1}$, let $\\{M_{t}\\}_{t\in[0,T]}$ and
$\\{\hat{\beta}_{t}\\}_{t\in[0,T)}$ be defined by

(2.1) $\displaystyle M_{t}$
$\displaystyle=\mathbb{E}\left[\xi\nonscript\>\middle|\nonscript\>\mathopen{}\mathcal{F}_{t}\right],\
t\in[0,T]$ (2.2) $\displaystyle\hat{\beta}_{t}$
$\displaystyle=\frac{1}{T}M_{0}+\int_{0}^{t}\frac{1}{T-u}\,dM_{u},\
t\in[0,T).$

The following statements are equivalent under Assumption 1.1:

  1. 1.

$\mathcal{B}^{2,1}(\xi)\neq\emptyset$.

  2. 2.

$\hat{\beta}\in\mathcal{B}^{2,1}(\xi)$.

  3. 3.

$\hat{\beta}\in\mathbb{L}^{2,1}$.

  4. 4.

$\mathbb{E}[\int_{0}^{T}\frac{1}{T-t}\,d\left\langle
M\right\rangle_{t}]<\infty$.

When $\mathcal{B}^{2,1}(\xi)\neq\emptyset$, the process $\hat{\beta}$ given by
(2.2) is, up to a version,

  1. (a)

the unique martingale on $[0,T)$ in $\mathcal{B}^{2,1}(\xi)$

  2. (b)

the minimal $\mathbb{L}^{2,1}$-norm element in $\mathcal{B}^{2,1}(\xi)$.

###### Proof.

_1. $\to$ 2._ Assuming that $\mathcal{B}^{2,1}(\xi)$ is nonempty consider the
minimization problem

(2.3)
$\displaystyle\inf_{\beta\in\mathcal{B}^{2,1}(\xi)}\mathbb{E}\left[\int_{0}^{T}\beta_{u}^{2}\,du\right]=\inf_{\beta\in\mathcal{B}^{2,1}(\xi)}\left\lVert\beta\right\rVert_{\mathbb{L}^{2,1}}^{2}.$

The set $\mathcal{B}^{2,1}(\xi)$ is convex and closed in $\mathbb{L}^{2,1}$.
By intersecting it with a large-enough ball in $\mathbb{L}^{2,1}$, we may
assume that it is also bounded in $\mathbb{L}^{2,1}$. The Banach Alaoglu
theorem ensures then that such restricted subset of $\mathcal{B}^{2,1}(\xi)$
is _weakly compact_. Since the $\mathbb{L}^{2,1}$ norm is a weakly lower
semicontinuous function, so is its square and thus there exists a
$\tilde{\beta}$ which attains the minimum in (2.3). This minimizer is also
unique by strict convexity of the objective function.
The rest of the proof of this implication is organized as follows. We start by
showing that the minimality of $\tilde{\beta}$ implies that $\tilde{\beta}$ is
orthogonal in $\mathbb{L}^{2,1}$ to a sufficiently rich class of processes.
Using this result, we prove that $\tilde{\beta}$ is a martingale. Finally, we
apply Itô’s formula and find that (a modification of) $\tilde{\beta}$
coincides with $\hat{\beta}$.
In order to show martingality of $\tilde{\beta}$, we perturb it in the
direction of a process $\gamma\in\mathbb{L}^{2,1}$ with
$\textstyle\int_{0}^{T}\gamma_{u}\,du=0$, a.s. By construction, the processes
$\tilde{\beta}\pm\varepsilon\gamma$ belong to $\mathcal{B}^{2,1}(\xi)$, which
implies that:

$\lVert\tilde{\beta}\pm\varepsilon\gamma\rVert^{2}_{{\mathbb{L}^{2,1}}}\geq\lVert\tilde{\beta}\rVert^{2}_{{\mathbb{L}^{2,1}}},$

for each $\varepsilon\in{\mathbb{R}}_{+}$. Writing down the relevant
expectations, the inequality becomes

$\lVert\tilde{\beta}\rVert^{2}_{{\mathbb{L}^{2,1}}}\pm\,2\varepsilon\mathbb{E}\left[\int_{0}^{T}\tilde{\beta}_{u}\gamma_{u}\,du\right]+\varepsilon^{2}\mathbb{E}\left[\int_{0}^{T}\gamma_{u}^{2}\,du\right]\geq\lVert\tilde{\beta}\rVert^{2}_{{\mathbb{L}^{2,1}}}.$

Simplifying and sending $\varepsilon$ to zero, we get the following set of
“first-order conditions”

(2.4)
$\displaystyle\mathbb{E}\left[\int_{0}^{T}\tilde{\beta}_{u}\gamma_{u}\,du\right]=0,\
\forall\,\gamma\in\mathbb{L}^{2,1}\text{ with
}\int_{0}^{T}\gamma_{u}\,du=0,\text{ a.s.}$

Given $t<s$ in $[0,T)$, for each $\mathcal{F}_{t}$-measurable and random
variable $\chi\in\mathbb{L}^{2}$ we define

$\displaystyle\gamma^{\chi}_{u}=\begin{cases}{0}&{u\in[0,t]}\\\
\chi\tfrac{1}{s-t},&u\in(t,s],\\\ -\chi\tfrac{1}{T-s},&u\in(s,T].\end{cases}$

so that $\int_{0}^{T}\gamma^{\chi}_{u}\,du=0$ for each $\chi$. By applying the
equality in (2.4) to $\gamma^{\chi}$ for all $\mathcal{F}_{t}$-measurable
$\chi\in\mathbb{L}^{2}$ we obtain

$\displaystyle\mathbb{E}\Big{[}\frac{\tilde{A}_{s}-\tilde{A}_{t}}{s-t}\nonscript\>\Big{|}\nonscript\>\mathopen{}\mathcal{F}_{t}\Big{]}=\mathbb{E}\Big{[}\frac{\tilde{A}_{T}-\tilde{A}_{s}}{T-s}\nonscript\>\Big{|}\nonscript\>\mathopen{}\mathcal{F}_{t}\Big{]}\,\text{
a.s.}\,,$

where $\tilde{A}_{t}=\int_{0}^{t}\tilde{\beta}_{u}\,du$. Since
$\tilde{\beta}\in\mathcal{B}^{2,1}(\xi)$, $\tilde{A}_{T}=\xi=M_{T}$, with $M$
given by (2.1). Slightly rearranging, we obtain:

(2.5)
$\displaystyle\frac{1}{s-t}\Big{(}\mathbb{E}\big{[}\tilde{A}_{s}\nonscript\>\big{|}\nonscript\>\mathopen{}\mathcal{F}_{t}\big{]}-\tilde{A}_{t}\Big{)}=\frac{1}{T-s}\Big{(}M_{t}-\mathbb{E}\big{[}\tilde{A}_{s}\nonscript\>\big{|}\nonscript\>\mathopen{}\mathcal{F}_{t}\big{]}\Big{)}\text{
a.s.}$

Since the right-hand side of (2.5) is a martingale in $t$ on $[0,s)$, so is
the left-hand side. In particular, the finite-variation part in its
semimartingale decomposition, given via integration by parts by

$\displaystyle\int_{0}^{t}\left(\frac{1}{(s-u)^{2}}\big{(}\mathbb{E}\big{[}\tilde{A}_{s}\nonscript\>\big{|}\nonscript\>\mathopen{}\mathcal{F}_{u}\big{]}-\tilde{A}_{u}\big{)}\,-\frac{1}{s-u}\tilde{\beta}_{u}\right)\,du,$

must vanish for all $t<s$, a.s. Consequently,

$\tilde{\beta}_{t}=\frac{\mathbb{E}\big{[}\tilde{A}_{s}\nonscript\>\big{|}\nonscript\>\mathopen{}\mathcal{F}_{t}\big{]}-\tilde{A}_{t}}{s-t}\text{
a.s., for almost all $t<s<T$}.$

Passing to the limit $s\uparrow T$ on the right hand side above, we obtain

(2.6)
$\displaystyle\tilde{\beta}_{t}=\frac{\mathbb{E}\big{[}\tilde{A}_{T}\nonscript\>\big{|}\nonscript\>\mathopen{}\mathcal{F}_{t}\big{]}-\tilde{A}_{t}}{T-t}=\frac{M_{t}-\tilde{A}_{t}}{T-t}=\frac{M_{t}-\int_{0}^{t}\tilde{\beta}_{u}\,du}{T-t}.$

It follows that $\tilde{\beta}$ has a continuous version on $[0,T)$, which we,
from now on, adopt. Furthermore, the right-hand side of (2.6) is a
semimartingale on $[0,T)$ so we can use Itô’s formula once more to conclude
that

$\tilde{\beta}_{t}=\frac{1}{T}M_{0}+\int_{0}^{t}\frac{1}{T-u}\,dM_{u}\text{
for }t\in[0,T).$

Therefore, $\hat{\beta}=\tilde{\beta}$ and statement _2._ follows immediately.

_2._ $\to$ _3._ Immediate. _3._ $\to$  _4._ With $\tau_{n}=\inf\\{t\geq
0\,:\,\left\langle M\right\rangle_{t}\geq n\\}$, Fubini’s theorem implies that

(2.7) $\begin{split}\mathbb{E}\left[\int_{0}^{T}(\hat{\beta}_{\tau_{n}\wedge
u}-\hat{\beta}_{0})^{2}\,du\right]&=\int_{0}^{T}\mathbb{E}\left[\int_{0}^{\tau_{n}\wedge
u}\frac{1}{(T-t)^{2}}\,d\left\langle M\right\rangle_{t}\right]\,du\\\
&=\mathbb{E}\left[\int_{0}^{T}\int_{t}^{T}1_{\\{t\leq\tau_{n}\\}}\frac{1}{(T-t)^{2}}\,du\,d\left\langle
M\right\rangle_{t}\right]\\\
&=\mathbb{E}\left[\int_{0}^{\tau_{n}}d\left\langle
M\right\rangle_{t}\int_{t}^{T}\frac{1}{(T-t)^{2}}\,du\right]=\mathbb{E}\left[\int_{0}^{\tau_{n}}\frac{1}{T-t}\,d\left\langle
M\right\rangle_{t}\right].\end{split}$

Since $\hat{\beta}^{\tau_{n}}$ is an $\mathbb{L}^{2}$-bounded martingale on
$[0,t]$ for each $t<T$, $(\hat{\beta}^{\tau_{n}}-\hat{\beta}_{0})^{2}$ is a
submartingale on the same domain, and the optional sampling theorem implies
that

$\displaystyle\mathbb{E}\left[(\hat{\beta}_{\tau_{n}\wedge
t}-\hat{\beta}_{0})^{2}\right]\leq\mathbb{E}\left[(\hat{\beta}_{t}-\hat{\beta}_{0})^{2}\right]\text{
for each }t<T.$

Thus, by (2.7),

$\mathbb{E}\left[\int_{0}^{\tau_{n}}\frac{1}{T-t}\,d\left\langle
M\right\rangle_{t}\right]=\int_{0}^{T}\mathbb{E}\left[(\hat{\beta}_{\tau_{n}\wedge
u}-\hat{\beta}_{0})^{2}\right]\,du\leq\int_{0}^{T}\mathbb{E}\left[(\hat{\beta}_{u}-\hat{\beta}_{0})^{2}\right]\,du=\lVert\hat{\beta}-\hat{\beta}_{0}\rVert_{\mathbb{L}^{2,1}}^{2}<\infty,$

and it suffices to let $n\to\infty$ and use the monotone convergence theorem.

_4._ $\to$  _3._ We let $n\to\infty$ in (2.7) and use Fatou’s lemma on the
left-hand side and the monotone convergence theorem on the right to conclude
that

$\displaystyle\lVert\hat{\beta}\rVert_{\mathbb{L}^{2,1}}\leq\lvert\hat{\beta}_{0}\rvert+\lVert\hat{\beta}-\hat{\beta}_{0}\rVert_{\mathbb{L}^{2,1}}\leq\lvert\hat{\beta}_{0}\rvert+\mathbb{E}\left[\int_{0}^{T}\frac{1}{T-t}\,d\left\langle
M\right\rangle_{t}\right]^{1/2}<\infty.$

_3._ $\to$  _1._ It suffices to show that
$\int_{0}^{T}\hat{\beta}_{t}\,dt=\xi$, a.s. The definition of $\hat{\beta}$ in
(2.2) and integration by parts imply that for $t\in[0,T)$ we have

(2.8)
$\displaystyle(T-t)\hat{\beta}_{t}=T\hat{\beta}_{0}+\int_{0}^{t}\,dM_{u}-\int_{0}^{t}\hat{\beta}_{u}\,du=M_{t}-\int_{0}^{t}\hat{\beta}_{u}\,du.$

Another round of integration by parts, but this time applied to the stochastic
integral $\int_{0}^{t}\frac{1}{T-u}\,dM_{u}$, implies that

(2.9)
$\displaystyle\hat{\beta}_{t}=\frac{1}{T-t}M_{t}+\int_{0}^{t}\frac{M_{u}}{(T-u)^{2}}\,du.$

Put together, identities (2.8) and (2.9), give

$\displaystyle\int_{0}^{t}\hat{\beta}_{u}\,du=\frac{\int_{0}^{t}\frac{M_{u}}{(T-u)^{2}}\,du,}{\frac{1}{T-t}}\text{
for }t\in[0,T),$

and the final step is to use l’Hôpital’s rule and the fact that $M_{t}\to\xi$,
as $t\to T$.
Concerning the last part of the statement of the theorem, (b) was established
in the course of the proof of _1._ $\to$  _2._ above. For (a), we assume that
there exists another martingale $\beta^{*}$ in $\mathcal{B}^{2,1}(\xi)$ so
that

$M_{t}=\mathbb{E}\left[\int_{0}^{T}\beta^{*}_{u}\,du\nonscript\>\middle|\nonscript\>\mathopen{}\mathcal{F}_{t}\right]=\int_{0}^{t}\beta^{*}_{u}\,du+\beta^{*}_{t}(T-t).$

The equality (2.8) above implies that

$0=\int_{0}^{t}(\beta^{*}_{u}-\hat{\beta}_{u})\,du+(\beta^{*}_{t}-\hat{\beta}_{t})(T-t)\text{
for all }t\in[0,T),\text{ a.s.}$

It follows that $\beta^{*}_{t}-\hat{\beta}_{t}$ is continuously differentiable
for $t\in[0,T)$, and the conclusion $\beta^{*}=\hat{\beta}$ follows by
differentiation. ∎

######  Remark 2.2.

As the anonymous referee observes, equation (2.6) can be interpreted as a
(pathwise) Volterra-type equation of the second kind:

$\displaystyle u(t)=f(t)+\int_{0}^{T}K(t,s)u(s)\ ds,\text{ where
}f(t)=\frac{M_{t}}{T-t}\text{ and }K(t,s)=-\frac{1}{T-t}1_{\\{s\leq t\\}}.$

A _formal_ iterative solution (obtained by repeatedly replacing $u(\cdot)$ on
the right-hand side by the whole right-hand side and then taking the limit)
can be written as $\tilde{\beta}_{t}=\lim_{n}u^{(n)}(t)$, where

$\displaystyle u^{(n)}(t)=f(t)+\sum_{i=0}^{{n}}\int K^{i}(t,s)f(s)\,ds$

and $K^{i}$ is the $i$-th composition power of $K$, i.e., $K^{0}=K$ and

(2.10) $\displaystyle K^{i}(t,s)$
$\displaystyle=\int_{0}^{T}\dots\int_{0}^{T}K(t,s_{1})K(s_{1},s_{2})\dots
K(s_{i},s)\,ds_{1}\,\dots\,ds_{i}\text{ for }i\geq 1.$

Substituting the explicit expression $K(t,s)=-(T-t)^{-1}1_{\\{s\leq t\\}}$
into (2.10) above, we obtain

$K^{i}(t,s)=(-1)^{i}(T-t)^{-1}\int_{0}^{T}\dots\int_{0}^{T}\frac{1}{T-s_{1}}\dots\frac{1}{T-s_{i}}1_{\\{t\geq
s_{1}\geq\dots\geq s_{i}\geq s\\}}\,ds_{1}\,\dots\,ds_{i}.$

The last iterated integral is taken over the simplex
$\Delta=\\{(s_{1},\dots,s_{i})\in[s,t]^{i-1}\,:\,s_{1}\leq s_{2}\dots\leq
s_{i}\\}$, and the function $\prod_{j=1}^{i}(T-s_{k})^{-1}$ inside the
integral is symmetric in $s_{1},\dots,s_{i-1}$, so, for $s\leq t$ we have

$\displaystyle K^{i}(t,s)$
$\displaystyle=(-1)^{i}\frac{1}{T-t}\,\frac{1}{i!}\int_{s}^{t}\dots\int_{s}^{t}(T-s_{1})^{-1}\dots(T-s_{i})^{-1}\,ds_{1}\,\dots\,ds_{i}$
$\displaystyle=(-1)^{i}\frac{1}{T-t}\frac{1}{i!}\Big{(}\int_{s}^{t}(T-u)^{-1}\,du\Big{)}^{i}=\frac{1}{T-t}\frac{1}{i!}\Big{(}\log\frac{T-t}{T-s}\Big{)}^{i},$

and consequently,

$\sum_{i=0}^{\infty}K^{i}(t,s)=\frac{1}{T-s}1_{\\{s\leq t\\}}.$

This implies that the formal solution $\tilde{\beta}_{t}$ takes the form

(2.11)
$\displaystyle\tilde{\beta}_{t}=\frac{M_{t}}{T-t}+\int_{0}^{t}\frac{M_{s}}{(T-s)^{2}}\,ds,$

which, after integrating by parts in the last integral, matches (2.2).

##  3\. The weakly regular case

Fix an $\mathcal{F}_{T}$ measurable random variable $\xi\in\mathbb{L}^{0}$,
and let $\mathcal{M}^{loc}(\xi)$ denote the set of all $M\in\mathcal{M}^{loc}$
such that $M_{T}=\xi$. Let $\mathcal{P}^{1}(\xi)$ be the set of probabilities
in $\mathcal{P}$ which integrate $\xi$.111 The set $\mathcal{P}^{1}(\xi)$ is
not empty, and it can be proved as in the proof of the next Theorem, arrow
_1._ $\to$  _2._ , by setting $\beta=0$. For
$\mathbb{P}\in\mathcal{P}^{1}(\xi)$, we set

$M^{\mathbb{P},\xi}_{t}=\mathbb{E}^{#1}\left[\xi\nonscript\>\middle|\nonscript\>\mathopen{}\mathcal{F}_{t}\right],\
t\in[0,T],$

taken in its continuous version, so that $M^{\mathbb{P},\xi}$ is the unique
$\mathbb{P}$-martingale in $\mathcal{M}^{loc}(\xi)$. Finally, for
$M\in\mathcal{M}^{loc}$ we define

(3.1)
$\displaystyle\hat{\beta}^{M}_{t}=\frac{1}{T}M_{0}+\int_{0}^{t}\frac{1}{T-u}\,dM_{u},\
t\in[0,T).$

######  Theorem 3.1.

For an $\mathcal{F}_{T}$ measurable random variable $\xi\in\mathbb{L}^{0}$,
the following are equivalent:

  1. 1.

$\mathcal{B}^{2,0}(\xi)\neq\emptyset$.

  2. 2.

$\mathcal{B}^{2,1}(\xi,\mathbb{Q})\neq\emptyset$ for some
$\mathbb{Q}\in\mathcal{P}^{1}(\xi)$.

  3. 3.

$\hat{\beta}^{M}\in\mathbb{L}^{2,0}$ for some $M\in\mathcal{M}^{loc}(\xi)$.

  4. 4.

$\int_{0}^{T}\frac{1}{T-t}\,d\left\langle M\right\rangle_{t}<\infty\,\text{
a.s.}$, for some $M\in\mathcal{M}^{loc}(\xi)$.

  5. 5.

$\mathbb{E}^{#1}[\int_{0}^{T}\frac{1}{T-t}\,d\left\langle
M^{\mathbb{Q},\xi}\right\rangle_{t}]<\infty$, for some
$\mathbb{Q}\in\mathcal{P}^{1}(\xi)$.

###### Proof.

_1._ $\to$  _2._ We pick $\beta\in\mathcal{B}^{2,0}(\xi)$ and
$\mathbb{P}\in\mathcal{P}$, and define $\mathbb{Q}\in\mathcal{P}$ by

$\displaystyle\frac{d\mathbb{Q}}{d\mathbb{P}}=c\frac{1}{1+\left\lvert\xi\right\rvert+\int_{0}^{T}\beta_{u}^{2}\,du},$

where $c$ is the normalizing constant. This way
$\mathbb{Q}\in\mathcal{P}^{1}(\xi)$ and the process $\beta$ belongs to
$\mathbb{L}^{2,1}(\mathbb{Q})$, and, hence, also to
$\mathcal{B}^{2,1}(\mathbb{Q})$.

_2._ $\to$  _5._ This is the content of the implication _1._ $\to$  _4._ in
Theorem 2.1, but, possibly, under an equivalent probability measure.

_5._ $\to$  _4._ Immediate.

_4._ $\to$  _3._ Let $M\in\mathcal{M}^{loc}(\xi)$ be as in the statement, and
let $\mathbb{Q}\in\mathcal{P}$ be such that $M$ is a $\mathbb{Q}$-local
martingale. We define the nondecreasing sequence
$\\{T_{m}\\}_{m\in{\mathbb{N}}}$ of stopping times by

$T_{m}=\inf\left\\{t\geq 0\,:\,\int_{0}^{t}\frac{1}{T-u}\,d\left\langle
M\right\rangle_{u}\geq m\right\\},$

so that $\mathbb{Q}[T_{m}<T]\to 0$, as $m\to\infty$. The process
$\hat{\beta}^{M}$, given by (3.1) above, is a continuous local
$\mathbb{Q}$-martingale on $[0,T)$ so there exists another nondecreasing
sequence $\\{\tau_{n}\\}_{n\in{\mathbb{N}}}$ of stopping times with the
property that $\tau_{n}\to T$, a.s., such that $(\hat{\beta}^{M})^{\tau_{n}}$
is an $\mathbb{L}^{2}(\mathbb{Q})$-bounded martingale on $[0,T]$. In
particular,

$\mathbb{E}^{#1}\left[(\hat{\beta}^{M}_{u\wedge
T_{m}\wedge\tau_{n}})^{2}\right]=\mathbb{E}^{#1}\left[{\langle\hat{\beta}^{M}\rangle_{u\wedge
T_{m}\wedge\tau_{n}}}\right]\text{ for all $m,n\in{\mathbb{N}}$ and
$u\in[0,T)$.}$

We let $n\to\infty$ and use Fatou’s lemma together with the monotone
convergence theorem to conclude that

(3.2) $\displaystyle\mathbb{E}^{#1}\left[(\hat{\beta}^{M}_{T_{m}\wedge
u})^{2}\right]\leq\mathbb{E}^{#1}\left[\langle\hat{\beta}^{M}\rangle_{T_{m}\wedge
u}\right]=\mathbb{E}^{#1}\left[\int_{0}^{T_{m}\wedge
u}\frac{1}{(T-r)^{2}}\,d\langle M\rangle_{r}\right]$

for $u<T$ and $m\in{\mathbb{N}}$. By using the inequality

$\int_{0}^{T_{m}}(\hat{\beta}^{M}_{u})^{2}\,du\leq\int_{0}^{T}(\hat{\beta}^{M}_{T_{m}\wedge
u})^{2}\,du\ \ a.s.,$

integrating (3.2) above in $u$ over $[0,T]$, and applying Fubini’s Theorem
under the product measure $du\otimes d\left\langle M\right\rangle_{r}$ we
obtain

$\displaystyle\mathbb{E}^{#1}\left[\int_{0}^{T_{m}}\big{(}\hat{\beta}^{M}_{{u}}\big{)}^{2}\,du\right]$
$\displaystyle\leq\mathbb{E}^{#1}\left[\int_{0}^{T}\big{(}\hat{\beta}^{M}_{T_{m}\wedge
u}\big{)}^{2}\,du\right]=\mathbb{E}^{#1}\left[\int_{0}^{T}\int_{0}^{T}1_{\\{r\leq
u\wedge T_{m}\\}}\frac{1}{(T-r)^{2}}\,d\left\langle
M\right\rangle_{r}\,du\right]$
$\displaystyle=\mathbb{E}^{#1}\left[\int_{0}^{T_{m}}\frac{1}{(T-r)^{2}}d\left\langle
M\right\rangle_{r}\int_{r}^{T}\,du\,\right]=\mathbb{E}^{#1}\left[\int_{0}^{T_{m}}\frac{1}{T-r}\,d\left\langle
M\right\rangle_{r}\right]\leq m.$

Since $\mathbb{Q}[T_{m}=T]\to 1$ as $m\to\infty$, we conclude that
$\int_{0}^{T}(\hat{\beta}^{M}_{u})^{2}\,du<\infty$, a.s.

_3._ $\to$  _1._ The last argument in the proof of Theorem 2.1 is based only
on the integration by parts formula and on the property
$\hat{\beta}\in\mathbb{L}^{1,0}$. Therefore it can be applied here, since
$\hat{\beta}^{M}\in\mathbb{L}^{2,0}\subseteq\mathbb{L}^{1,0}$. ∎

######  Remark 3.2.

As it aims for generality, but also operates within specific regularity
classes, our proof of Theorem 3.1 above does not use the _stochastic Fubini
theorem_ , a joint name for a class of statements about the permissibility of
the interchange of a Lebesgue and a stochastic integral under different sets
of conditions. We refer the reader to Theorem 4.18 in [DPZ14] or Theorem 2.2
of [Ver12] for two versions referred to later in this paper.

To provide a more detailed explanation, let us start with a brief description
of how an argument based on it would play out. Under Assumption 1.1, it would
start with a choice of a measure $\mathbb{Q}\in\mathcal{P}$ as in Theorem 3.1,
item _2._ and the associated martingale $M^{\mathbb{Q},\xi}$, where we assume,
without loss of generality, that $M_{0}=\mathbb{E}^{#1}\left[\xi\right]=0$.
When its conditions are satisfied, the stochastic Fubini theorem, applied to
the function $\psi(s,t,\omega)=\frac{1}{T-t}I_{[0,s]}(t)$ and with integrals
with respect to $dM$ and $ds$, yields

$\displaystyle\int_{0}^{T}ds\int_{0}^{s}\frac{1}{T-t}\,dM_{t}=\int_{0}^{T}dM_{t}\,\frac{1}{T-t}\int_{t}^{T}ds=M_{T}=\xi,$

making

(3.3) $\displaystyle\beta_{s}=\int_{0}^{s}\frac{1}{T-t}\,dM_{t}$

an absolutely-continuous representation of $\xi$. One of the weakest
conditions for the above to hold is due to Veraar (see Theorem 2.2 of
[Ver12]), and it can be stated in our case as

(3.4)
$\displaystyle\int_{0}^{T}\left(\int_{0}^{s}\frac{1}{(T-t)^{2}}\,d\left\langle
M\right\rangle_{t}\right)^{\tfrac{1}{2}}\,ds<\infty,\text{ a.s.}$

It is superficially related to our condition 4. of Theorem 3.1, but it does
not automatically insert $\beta$ into our (weak or strong) regularity classes.
For example, assume that we are on a filtration generated by a Brownian motion
$W$, and that $\xi=W_{T}$. In that case (3.4) is clearly satisfied, but, as we
will see in Proposition 4.1 below, $W_{T}$ does not admit an absolutely
continuous representation with a weakly (or strongly) regular $\beta$. Put
differently:

_regularity conditions for the validity of the stochastic
Fubini theorem do not correspond to our regularity classes._

A natural question is whether a condition such as:

  1. 4’.

$\int_{0}^{T}\frac{1}{T-t}\,d\left\langle M\right\rangle_{t}<\infty\text{
a.s.}$, for all $M\in\mathcal{M}^{loc}(\xi)$

can be inserted in Theorem 3.1. An equivalent question is whether the
condition $4.$ of Theorem 3.1 implies the condition $4^{\prime}.$ above. We
only have a partial (positive) answer to this problem. It states that under
certain regularity conditions, if $M^{\mathbb{P}}$ satisfies condition 4 in
Theorem 2.1, then all the probability measures $\mathbb{Q}\sim\mathbb{P}$ with
a finite relative entropy share the property, namely
$\mathbb{E}^{#1}[\int_{0}^{T}\frac{1}{T-t}\,d\left\langle
M^{\mathbb{Q}}\right\rangle_{t}]<\infty$.

######  Proposition 3.3.

Suppose that the filtration is generated by a $\mathbb{P}$-Brownian motion $W$
and that $\xi$ is of the form

$\xi=\int_{0}^{T}\sigma_{u}\,dW_{u},\text{ for some bounded $\sigma$.}$

If

  1. (1)

the martingale
$M^{\mathbb{P}}_{t}=\mathbb{E}^{#1}\left[\xi\nonscript\>\middle|\nonscript\>\mathopen{}\mathcal{F}_{t}\right]$
satisfies

$\mathbb{E}^{#1}\left[\int_{0}^{T}\frac{1}{T-t}\,d\langle
M^{\mathbb{P}}\rangle_{t}\right]<\infty,\text{ and }$

  2. (2)

$\mathbb{Q}$ is a probability measure equivalent to $\mathbb{P}$ with a finite
relative entropy, i.e.,

(3.5)
$\displaystyle\mathbb{E}^{#1}\left[\frac{d\mathbb{Q}}{d\mathbb{P}}\log\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right)\right]<\infty,$

then

(3.6) $\displaystyle\mathbb{E}^{#1}\left[\int_{0}^{T}\frac{1}{T-t}\,d\langle
M^{\mathbb{Q}}\rangle_{t}\right]<\infty.$

###### Proof.

With $\mathbb{P}\sim\mathbb{Q}$ as in the statement, let $\theta$ be such that
the dynamics of the density process
$Z_{t}=\mathbb{E}^{#1}\left[\tfrac{d\mathbb{Q}}{d\mathbb{P}}\nonscript\>\middle|\nonscript\>\mathopen{}\mathcal{F}_{t}\right]$
is given by

$dZ_{t}=-Z_{t}\,\theta_{t}\,dW_{t}.$

Let us show, first, that

(3.7)
$\displaystyle\mathbb{E}^{#1}\left[\int_{0}^{T}\theta^{2}_{u}\,du\right]<\infty.$

Condition (3.5) above, together with the fact that the function $x\mapsto
x\log(x)$ is convex and bounded from below, implies that the process
$Z\log(Z)$ is a continuous $\mathbb{P}$-submartingale on $[0,T]$. Hence, there
exists a finite constant $C$ such that

(3.8) $\displaystyle\mathbb{E}^{#1}\left[Z_{\tau}\log(Z_{\tau})\right]\leq
C\text{ for each $[0,T]$-valued stopping time $\tau$.}$

Itô’s formula applied to the semimartingale $Z\log(Z)$ yields

$\displaystyle
Z_{\tau}\log(Z_{\tau})=N_{\tau}+\tfrac{1}{2}\int_{0}^{\tau}Z_{u}\theta_{u}^{2}\,du\text{
where $N$ is a local $\mathbb{P}$-martingale. }$

Let $\\{\tau_{n}\\}_{n\in{\mathbb{N}}}$ is a sequence of stopping times that
reduces the process $N$. The upper bound of (3.8) above implies that

$\displaystyle
C\geq\mathbb{E}^{#1}\left[Z_{\tau_{n}}\log(Z_{\tau_{n}})\right]=\tfrac{1}{2}\mathbb{E}^{#1}\left[\int_{0}^{\tau_{n}}Z_{u}\theta_{u}^{2}\,du\right]=\tfrac{1}{2}\mathbb{E}^{#1}\left[\int_{0}^{\tau_{n}}\theta_{u}^{2}\,du\right],$

and it remains to use the monotone-convergence theorem to conclude that (3.7)
holds.

We continue the proof by using Girsanov’s theorem and the boundedness of
$\sigma$ to conclude that the process:

$M^{\prime}_{t}=\int_{0}^{t}\sigma_{u}\,(dW_{u}+\theta_{u}\,du)=M^{\mathbb{P}}_{t}+\int_{0}^{t}\sigma_{u}\theta_{u}\,du$

is a $\mathbb{Q}$-martingale. Boundedness of the process $\sigma$, together
with (3.7), implies that the random variable
$\xi=M^{\mathbb{P}}_{T}=M^{\prime}_{T}-\int_{0}^{T}\sigma_{u}\theta_{u}\,du$
is $\mathbb{Q}$-integrable, so that the $\mathbb{Q}$-martingale
$M^{\mathbb{Q}}_{t}=\mathbb{E}^{#1}\left[\xi\nonscript\>\middle|\nonscript\>\mathopen{}\mathcal{F}_{t}\right]$
is well defined. Moreover, we have

$\displaystyle M^{\mathbb{Q}}_{t}$
$\displaystyle=\mathbb{E}^{#1}\left[M^{\prime}_{T}-\int_{0}^{T}\sigma_{u}\theta_{u}\,du\nonscript\>\middle|\nonscript\>\mathopen{}\mathcal{F}_{t}\right]=M^{\mathbb{P}}_{t}+\int_{0}^{t}\sigma_{u}\theta_{u}\,du-
L_{t},$

where

$L_{t}=\mathbb{E}^{#1}\left[\int_{0}^{T}\sigma_{u}\theta_{u}\,du\nonscript\>\middle|\nonscript\>\mathopen{}\mathcal{F}_{t}\right].$

It follows that

$\displaystyle\langle M^{\mathbb{Q}}\rangle_{t}=\langle
M^{\mathbb{P}}-L\rangle_{t}.$

Furthermore, since

$\displaystyle\langle M^{\mathbb{P}}-L\rangle_{t}\leq\langle
M^{\mathbb{P}}-L\rangle_{t}+\langle M^{\mathbb{P}}+L\rangle_{t}=2\langle
M^{\mathbb{P}}\rangle_{t}+2\langle L\rangle_{t},$

our final goal, namely (3.6), will be reached if we can prove that

(3.9)
$\displaystyle\mathbb{E}^{#1}\left[\int_{0}^{T}\frac{1}{T-t}\,d\left\langle
L\right\rangle_{t}\right]<\infty.$

For that, we note that $\xi^{\prime}=L_{T}$ admits the absolutely continuous
representation $L_{T}=\textstyle\int_{0}^{T}\beta^{\prime}_{t}\,dt$, where
$\beta^{\prime}_{t}=\sigma_{t}\theta_{t}$, by its very definition. Since the
boundedness of $\sigma$ and (3.7) above imply that
$\beta^{\prime}\in\mathbb{L}^{2,1}(\mathbb{Q})$, equation (3.9) follows from
the implication $1.\to 4.$ of Theorem 2.1. ∎

##  4\. Examples and further remarks

###  4.1. Functions of the terminal value of a Brownian motion W

Our first subsection focuses on the (lack of) absolutely continuous
representation property for the random variables of the form $g(W_{T})$. In
contrast with what happens with the martingale representation, we show that
such a $\xi$ admits an absolutely continuous representation if and only if it
is constant. Informally speaking: in order to have regular-enough
representation, $\xi$ must be sufficiently path dependent.

We start with a simple argument of limited scope just to provide some
intuition. Let $\xi$ be the terminal value $W_{T}$ itself, and let $\beta$ be
its absolutely continuous representation, i.e.,
$\xi=\int_{0}^{T}\beta_{u}\,du$. Since the process
$W^{\beta}_{t}={W}_{t}-\int_{0}^{t}\beta_{u}\,du$ satisfies $W^{\beta}_{T}=0$,
it cannot be a martingale under any equivalent measure. This means that
$\beta$ cannot be too regular in the sense that the stochastic exponential
$\exp(\int_{0}^{t}\beta_{u}\,dW_{u}-\tfrac{1}{2}\int_{0}^{t}\beta_{u}^{2}\,du)$
cannot be a martingale, if it is well-defined at all. In fact, as our next
Proposition shows, such a $\beta$ cannot be weakly regular, i.e.,
$\int_{0}^{T}\beta_{u}^{2}\,du=+\infty$ with positive probability.

######  Proposition 4.1.

Suppose that the filtration supports a Brownian motion $W$, and that
$\xi=g(W_{T})$ for some $g\in C^{2}({\mathbb{R}})$. Then
$\mathcal{B}^{2,0}(\xi)\not=\emptyset$ if and only if $g$ is constant.

###### Proof.

The only implication which requires a proof is $(\Rightarrow)$.

We focus, first, on the case where the filtration is generated by the Brownian
motion. Since $g\in C^{2}(\mathbb{R})$, the process $g^{\prime\prime}(W)$ is
locally bounded so that
$\int_{0}^{T}(g^{\prime\prime}(W_{u}))^{2}\,du<\infty$, a.s. Itô’s formula
then implies that $\xi$ admits an absolutely continuous representation under
weak regularity, i.e., that $\mathcal{B}^{2,0}(\xi)\neq\emptyset$, if and only
if $\mathcal{B}^{2,0}(\bar{\xi})\neq\emptyset$, where

$\displaystyle\bar{\xi}=\int_{0}^{T}g^{\prime}(W_{t})\,dW_{t}.$

Fix a $\mathbb{Q}\in\mathcal{P}$ and call
$Z_{t}=\mathbb{E}\left[\tfrac{d\mathbb{Q}}{d\mathbb{P}}\nonscript\>\middle|\nonscript\>\mathopen{}\mathcal{F}_{t}\right]$
its density process. Since the filtration is assumed to be Brownian, there
exists a progressively measurable process $\theta$ such that
$Z=\mathcal{E}(\int_{0}^{\cdot}\theta_{t}dW_{t})$. Thus,

$\bar{\xi}=\int_{0}^{T}g^{\prime}(W_{t})\,dW_{t}=\int_{0}^{T}g^{\prime}(W_{t})\,dW^{Z}_{t}+\int_{0}^{T}g^{\prime}(W_{t})\theta_{t}\,dt$

where $W^{Z}=W-\int_{0}^{\cdot}\theta_{t}\,dt$ is a $\mathbb{Q}$-Brownian
motion. By the continuity of $g^{\prime}$ and the paths of $W$, we have
$\int_{0}^{T}(g^{\prime}(W_{t})\theta_{t})^{2}\,dt<\infty$, a.s. Being
deterministic, the quadratic variation of the Brownian motion has the same
distribution across all the probabilities in $\mathcal{P}$, so, by Theorem
3.1, $\bar{\xi}$ is representable if and only if:

(4.1)
$\displaystyle\int_{0}^{T}\frac{(g^{\prime}(W_{t}))^{2}}{T-t}\,dt<\infty,\text{
a.s.}$

Continuity of $g^{\prime}$ and $W$ force $g^{\prime}(W_{t})\to 0$ when
$t\rightarrow T$, i.e., $g^{\prime}(W_{T})=0$, a.s., whenever (4.1) holds.
Since $W_{T}$ has full support under any $\mathbb{P}\in\mathcal{P}$, we
conclude that $g^{\prime}\equiv 0$.
The next step in the proof is to relax the assumption that the filtration
$\\{\mathcal{F}_{t}\\}_{t\in\left[0,T\right]}$ is generated by a Brownian
motion. In preparation, let $\\{\mathcal{F}^{W}_{t}\\}_{t\in\left[0,T\right]}$
denote the subfiltration of $\\{\mathcal{F}_{t}\\}_{t\in\left[0,T\right]}$
generated by the Brownian motion $W$, and let $\operatorname{Prog}$ and
$\operatorname{Prog}^{W}$ denote the progressive $\sigma$-algebras on
$[0,T]\times\mathcal{F}$ corresponding to
$\\{\mathcal{F}_{t}\\}_{t\in\left[0,T\right]}$ and
$\\{\mathcal{F}^{W}_{t}\\}_{t\in\left[0,T\right]}$, respectively.

We argue by contradiction and assume that some $\xi=g(W_{T})$, with non
constant $g$, can be represented as $\int_{0}^{T}\beta_{u}\,du$ for some
$\beta\in\operatorname{Prog}$ with $\int_{0}^{T}\beta_{u}^{2}\,du<\infty$,
a.s. With $\mathbb{Q}$ denoting a probability measure equivalent to
$\mathbb{P}$ with the property that
$\mathbb{E}^{#1}[\int_{0}^{T}\beta_{u}^{2}\,du]<\infty$, we observe that
$\beta$, seen as a $\operatorname{Prog}$-measurable function on the product
space $[0,T]\times\Omega$, is square integrable with respect to the product
probability measure $\mu=\frac{1}{T}\lambda\otimes{\mathbb{Q}}$ (where
$\lambda$ denotes the Lebesgue measure on $[0,T]$). Hence, the conditional
expectation $\beta^{W}$ of $\beta$, taken on the probability space
$([0,T]\times\Omega,\operatorname{Prog},\mu)$, given the $\sigma$-algebra
$\operatorname{Prog}^{W}$, is well defined and satisfies

$\displaystyle\int_{B}\beta^{W}_{u}(\omega)\,d\mu(u,\omega)=\int_{B}\beta_{u}(\omega)\,d\mu(u,\omega)\text{
for all }B\in\operatorname{Prog}^{W}.$

If we set $B=[0,T]\times A$, for $A\in\mathcal{F}^{W}_{T}$ and remember that
$\xi\in\mathcal{F}^{W}_{T}$, we obtain immediately that
$\int_{0}^{T}\beta^{W}_{u}\,du=\xi$, a.s. Moreover, since the conditional
expectation preserves square integrability, we have

$\displaystyle\mathbb{E}^{#1}\left[\int_{0}^{T}(\beta^{W}_{u})^{2}\,du\right]<\infty,$

and, consequently, $\int_{0}^{T}(\beta^{W}_{u})^{2}\,du<\infty$, a.s. It
remains to observe that the existence of such $\beta^{W}$ contradicts the
conclusion of the first part of the proof, and completes the argument. ∎

###  4.2. Representations in $\mathbb{L}^{p,1}$ for $p<2$

The “factorization formula” of Da Prato and Zabczyk when specialized to the
case $S(t)=\operatorname{Id}$ and $U=H={\mathbb{R}}$ (see Theorem 5.2.5, p. 58
in [DPZ96] for the statement and the paragraph that precedes it for the
necessary definitions and notation) states the following: whenever
$\alpha\in[0,1)$, and ${\Psi}$ is a progressively measurable process such that

(4.2)
$\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\left(\int_{0}^{s}(s-r)^{-2\alpha}\mathbb{E}\left[\Psi(r)^{2}\right]\,dr\right)^{1/2}\,ds<\infty\text{
a.s.,}$

we have

(4.3)
$\displaystyle\int_{0}^{t}\Psi(s)\,dW_{s}=\frac{\sin(\alpha\pi)}{\pi}\int_{0}^{t}(t-s)^{\alpha-1}Y^{\Psi}_{\alpha}(s)\,ds\text{
a.s.},\text{ for all }t\in[0,T],$

where

$\displaystyle
Y^{\Psi}_{\alpha}(s)=\int_{0}^{t}(s-r)^{-\alpha}\Psi(r)\,dW_{r}.$

To see how (4.3) above leads to an interesting absolutely continuous
representation, we pick a bounded progressively measurable process $\sigma$
and apply the factorization formula (4.3) above with $t=T$ and $\Psi=\sigma$.
When $\alpha\in[0,1/2)$, so that (4.2) is satisfied, (4.3) yields directly the
following absolutely-continuous representation for the last element
$\xi=M_{T}$ of the martingale $M_{t}=\int_{0}^{t}\sigma_{u}\,dW_{u}$:

(4.4) $\displaystyle\xi=\int_{0}^{T}\beta_{t}\,dt\text{ where
}{\beta_{t}=\frac{\sin(\alpha\pi)}{\pi}(T-t)^{\alpha-1}R_{t}\text{ and
}R_{t}=\int_{0}^{t}(t-u)^{-\alpha}\,dM_{u}.}$

Note that we can interpret $R_{t}$ as a formal Riemann-Liouville fractional
integral of order $\alpha$ of the noise $dM$, up to a multiplicative constant.
To complete the factorization formula, we then integrate the result multiplied
by $(T-t)^{\alpha-1}$ with respect to $dt$, i.e., compute the Riemann-
Liouville integral of the complementary order $1-\alpha$ of the result. The
semigroup property of Riemann-Liouville integration would suggest that the
result should coincide with the integral of order $\alpha+(1-\alpha)=1$ of
$dM$, i.e., it should yield $M_{T}=\xi$, which is precisely the case. We refer
the reader to [SKM93] for details on fractional integration.

Since no restriction other than boundedness is imposed on $\sigma$ it may
appear at first glance that (4.4) contradicts Proposition 4.1 above, in that
it provides a representation for $\xi=W_{T}$, for example. The difference, as
in the discussion of the stochastic Fubini theorem in Remark 3.2 above, lies
in the regularity class. As an easy illustration, let us show that in the
special case $\sigma\equiv 1$, the representation $\beta$ given by (4.4) does
not even belong to the weak regularity class $\mathcal{B}^{2,0}(\xi)$. In
fact, we show below that it satisfies $\int_{0}^{T}\beta_{u}^{2}\,du=+\infty$,
a.s. To do that, we focus on the process $R$ from (4.4) which, in this case,
takes the form

(4.5) $\displaystyle R_{t}=\int_{0}^{t}(t-s)^{-\alpha}\,dW_{s}.$

Known as the _Riemann-Liouville process_ (up to a multiplicative constant),
the process $R$ from (4.5) above is Gaussian and admits a continuous
modification (see [MR99] for a survey of its properties and its relation to
the fractional Brownian motion). Its value $R_{T}$ at $T$ is normally
distributed with a nonzero variance, so that, by continuity, we have
$\lim_{t\to T}R_{t}=R_{T}\neq 0$, a.s. It follows that there exists a random
variable $K$ such that $K>0$, a.s., and

$\left\lvert\beta_{t}\right\rvert\geq K(T-t)^{\alpha-1},\text{ a.s.},$

for all $t$ in a (random) neighborhood of $T$. Since $\alpha<1/2$, i.e.,
$2\alpha-2<-1$, we necessarily have $\int_{0}^{T}\beta_{t}^{2}\,dt=\infty$,
a.s.
On the other hand, as our next result states, the factorization theorem allows
us to construct absolutely continuous representations of a wide variety of
random variables in any one of slightly less regular classes
$\mathbb{L}^{p,1}$, $p\in[1,2)$.

######  Proposition 4.2.

Suppose that the filtration supports a Brownian motion $W$ and that
$\xi=\int_{0}^{T}\sigma_{u}\,dW_{u}$, with $\sigma$ bounded. Then $\xi$ admits
an absolutely continuous representation in $\mathbb{L}^{p,1}$ for each
$p\in[1,2)$.

###### Proof.

Let $p\in[1,2)$ be given. We claim that that the representation (4.4) above,
with $M_{t}=\int_{0}^{t}\sigma_{u}\,dW_{u}$, and with a suitably chosen
$\alpha\in(0,1/2)$ belongs to $\mathbb{L}^{p,1}$. To see that we use Doob’s
maximal inequality followed by the Burkholder-Davis-Gundy theorem to obtain
the following:

$\displaystyle\mathbb{E}\left[\left\lvert\beta_{t}\right\rvert^{p}\right]$
$\displaystyle=(T-t)^{p(\alpha-1)}\mathbb{E}\left[\left\lvert\int_{0}^{t}(t-u)^{-\alpha}\,dM_{u}\right\rvert^{p}\right]$
$\displaystyle{\leq}(T-t)^{p(\alpha-1)}\mathbb{E}\left[\sup_{s\leq
t}\left\lvert\int_{0}^{s}(t-u)^{-\alpha}\,dM_{u}\right\rvert^{p}\right]$
$\displaystyle\lessapprox(T-t)^{p(\alpha-1)}\mathbb{E}\left[\left(\int_{0}^{t}(t-u)^{-2\alpha}\,d\left\langle
M\right\rangle_{u}\right)^{p/2}\right]$

where $a\lessapprox b$ is a shorthand for $a\leq C\,b$, for some constant
$C>0$ which depends only on $p$. Allowing $C$ to depend on $\sigma$ as well,
we can go on to conclude that

(4.6)
$\begin{split}\mathbb{E}\left[\int_{0}^{T}\left\lvert\beta_{t}\right\rvert^{p}\,dt\right]&\leq{\int_{0}^{T}(T-t)^{p(\alpha-1)}\mathbb{E}\left[\left(\int_{0}^{t}(t-u)^{-2\alpha}\,d\left\langle
M\right\rangle_{u}\right)^{p/2}\right]dt}\\\
&\lessapprox\int_{0}^{T}(T-t)^{p(\alpha-1)}\left(\int_{0}^{t}(t-u)^{-2\alpha}\,du\right)^{p/2}\,dt\\\
&=\int_{0}^{T}(T-t)^{p(\alpha-1)}\frac{t^{p(1/2-\alpha)}}{1-2\alpha}\,dt.\end{split}$

The last integral is finite if and only if $\alpha\in(1-1/p,1/2)$. Therefore,
the choice $\alpha=\tfrac{3}{4}-\tfrac{1}{2p}$ in (4.4) provides an absolutely
continuous representation of $\xi$ in $\mathbb{L}^{p,1}$ for $p\in(1,2)$. ∎

######  Remark 4.3.

The result of Proposition 4.2 above should be contrasted with the findings in
our Theorem 2.1, which implies that under the same assumptions, a
representation of $\xi=\int_{0}^{T}\sigma_{u}\,dW_{u}$ is possible in
$\mathbb{L}^{2,1}$ if and only if, additionally,

$\mathbb{E}\left[\int_{0}^{T}\frac{\sigma_{u}^{2}}{T-t}\,dt\right]<\infty.$

## References

  * [AB23] René Aïd and Sara Biagini, _Optimal dynamic regulation of carbon emissions market_ , Math. Finance 33 (2023), no. 1, 80–115.
  * [BGZ22] Sara Biagini, Fausto Gozzi, and Margherita Zanella, _Robust portfolio choice with sticky wages_ , SIAM J. Financial Math. 13 (2022), no. 3, 1004–1039.
  * [DPZ96] G. Da Prato and J. Zabczyk, _Ergodicity for infinite-dimensional systems_ , London Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996.
  * [DPZ14] Giuseppe Da Prato and Jerzy Zabczyk, _Stochastic equations in infinite dimensions_ , second ed., Encyclopedia of Mathematics and its Applications, vol. 152, Cambridge University Press, Cambridge, 2014.
  * [DS94] Freddy Delbaen and Walter Schachermayer, _A general version of the fundamental theorem of asset pricing_ , Math. Ann. 300 (1994), no. 3, 463–520.
  * [FGS17] Giorgio Fabbri, Fausto Gozzi, and Andrzej Świech, _Stochastic optimal control in infinite dimension_ , Probability Theory and Stochastic Modelling, vol. 82, Springer, Cham, 2017, Dynamic programming and HJB equations, With a contribution by Marco Fuhrman and Gianmario Tessitore.
  * [HWY92] Sheng Wu He, Jia Gang Wang, and Jia An Yan, _Semimartingale theory and stochastic calculus_ , Kexue Chubanshe (Science Press), Beijing; CRC Press, Boca Raton, FL, 1992.
  * [MR99] D. Marinucci and P. M. Robinson, _Alternative forms of fractional Brownian motion_ , J. Statist. Plann. Inference 80 (1999), no. 1-2, 111–122.
  * [SKM93] S. G. Samko, A. A. Kilbas, and O. I. Marichev, _Fractional integrals and derivatives: theory and applications_ , Gordon and Breach Science Publishers, 1993\.
  * [Ver12] Mark Veraar, _The stochastic Fubini theorem revisited_ , Stochastics 84 (2012), no. 4, 543–551.


# Experimental Demonstration of Network Convergence with Coherent and Analog
Radio-over-Fibre signals For Densified 5.5G/6G Small Cell Networks

Frank Slyne1, Colm Browning2, Amol Delmade2, Liam P. Barry 2 and Marco
Ruffini1
1 CONNECT centre, School of Computer Science and Statistics, Trinity College
Dublin (TCD), Ireland.
2 School of Engineering, Dublin City University (DCU), Ireland
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work we analyse and demonstrate the coexistence of digital coherent
and analogue radio over fibre signals over an access-metro transmission
network and field fibre. We analyse how the spectral proximity of the two
signals and the non-ideal filter alignment of typical telecomms-grade ROADMs
affect the signal performance. Our results show that coexistence is indeed
possible, although performance deteriorates with the increase in number of
ROADMs in the network topology. Thus, while today’s access-metro networks will
be able to support future 5.5 and 6G cell densification operating at mmWave
and THz frequency, using spectral efficient analogue radio over fibre
transmission, there will be trade-offs to be considered. In our experiment
setup, we show that the limit for ARoF accessible performance is reached after
transmission over 3 ROADMs and a total of 49 km of fibre.

###### Index Terms:

Signal coexistence, access-metro convergence, optical-wireless convergence,
ARoF.

##  I Introduction

The drive towards increasing densification of mobile network technology, has
led to much research on cost-effective fronthaul and backhaul technology [1,
2]. In addition to the large increase in capacity expected in the coming
years, the adoption of millimeter-wave (mmWave) and Terahertz (THz)
transmission technology will require site densification due to the limited
reach at these high carrier frequencies [3]. As 5.5 and 6G will develop higher
and higher capacity, the required transport data rate on the optical fibre
will continue to increase [5]. This is exacerbated by the use of RAN
disaggregation [6], functional split and centralisation of processing [7],
which further increases capacity and latency constraints on the fibre
transmission [8]. While transport of fronthaul signals is still feasible for
todays’ commercial base station capacity, the move towards much higher
wireless bandwidth at the base stations (e.g., of 1 GHz and above) for
mmWave/THz communications, will require high-cost transmission technology to
support data rates of hundreds Gb/s. In these scenarios, Analogue Radio over
Fibre (ARoF) transmission can satisfy the need to transport high-bandwidth
radio signals over fibre in a way that is both cost effective and spectrally
efficient [9].

With A-RoF fronthaul, the wireless signal is transmitted from the Distributed
Unit (DU), which may be co-located with optical networking hardware at a
Central Office (CO), through a fibre link or network to a remotely deployed
radio Unit (RU). The multi-carrier radio signal is transmitted on the RF
carrier or on an Intermediate Frequency (IF) (known as IF A-RoF). Transmission
in this manner allows the removal of costly Digital-to-Analog Converters and
Analog-to-Digital Converters from the RU, reducing hardware complexity. This
is highly advantageous in terms of operator Capital Expenditure (CAPEX), and
thereby helps to facilitate cell site densification into the 6G era.

Compared to Digital Radio-over-Fiber (D-RoF) fronthaul, Analog Radio-over-
Fiber (A-RoF) offers higher spectral efficiency, as there is no requirement
for quantization of the Radio Frequency (RF) signals. The inherent high Peak-
to-Average Power Ratio (PAPR) of multi-carrier modulation methods such as
Orthogonal Frequency Division Multiplexing (OFDM), widely employed in wireless
technologies, leads to a limitation in terms of optical power budget,
ultimately dictating the transmission distance and/or number of serviced
cells. A-RoF signals are known to be significantly degraded in the presence of
non-linearities, which are typical of optical fibre transmission [4]. However,
as wide-scale use of shorter range mmWave/THz communications is expected, in
order to provide ever higher wireless capacity, the motivations for ARoF
deployment are stronger than ever. The use of IF-ARoF requires the RU to
upconvert the received IF signal to the required RF for wireless transmission,
but provides spectral flexibility in the optical domain - a property which can
be advantageous where ARoF is provisioned over a shared optical transport
network, e.g. with Wavelength Division Multiplexing (WDM) technology. This
concept, known as network convergence, enables RoF provisioning to not only
exploit pre-existing fibre infrastructure (such as metro and access
topologies) for radio access transport, but also to avail of advances in
optical switching/routing technologies deployed in state-of-the-art optical
networks. Optimising the use of existing assets through infrastructure
sharing, providing higher bandwidth, new degrees of optical flexibility to the
radio domain and making use of highly spectrally efficient ARoF technologies
can be leveraged to form a truly sustainable platform for radio access
bandwidth and connectivity growth. But, to date, studies of converged radio
fronthaul and optical metro/access transport are largely confined to
relatively simplistic laboratory-based transmission experiments, often
focusing on niche applications.

Successful convergence of ARoF signals at 5 GHz with a single carrier 10 Gb/s
4-level Pulse Amplitude Modulation (PAM) optical access signal in a passive
optical network system scenario was demonstrated in [10]. Maintaining a focus
on access domain convergence, [11] demonstrates 56 Gb/s PAM-4 optical
broadband signal transmission in tandem with 10 $\times$ 400 MHz multi-carrier
signals at 28 GHz for mmWave ARoF transport. A network convergence experiment
was carried out in [12], where the authors demonstrated a dynamically
reconfigurable switching node with coherent and ARoF signals. The node, based
on Wavelength Selective Switch (WSS) technology, routes Digital Radio-over-
Fiber and Analog–Intermediate-Frequency over Fiber (A-IFoF) to support
densified 5G small-cell. However the experiment was carried out using
laboratory testbench equipment. Other work on fixed-mobile convergence in [13]
focused in stead on archictecture and SDN-driven orchestration.

Our experiment demonstrates the coexistence of digital coherent and ARoF
signal, using equipment that is typical of today’s operator networks. A-IFoF
coexists in close proximity to coherent signals in common channels providing
efficient and spectrum utilisation that complies with Fixed Grid scheme. The
experiment uniquely combines optical networking equipment located at the Open
Ireland testbed in Trinity College Dublin (TCD), and hybrid RF/Optical
research facilities located in a different lab at Dublin City University
(DCU).

##  II Experiment

The use case we target is that of high densification of small cells for 5.5
and 6G technology, where some of the terminations are connected through ARoF,
while others make use of digital transmission using standard coherent
transmission technology (i.e. for signals of 100Gb/s and above). The key point
is however that the ARoF and other coherent signals make use of existing metro
network infrastructure to be carried towards their destination (e.g., a telco
cloud). This is important in order to minimise the capital expenditure
required for the back haul network. In addition, different signals might be
multiplexed at different locations, as the metro network will carry several
signals of varying bandwidth across the given area. The reason we focus on
ARoF transmission is that these signals have much higher spectral efficiency
compared to digital RoF transmission, thus they can occupy a minimal part of
the fibre capacity, which is especially important as we move to large wireless
channel bandwidths from hundreds of MHz to a few GHz (i.e., for Thz
communications) [5]. The target of our analysis in to understand how the ARoF
signal deteriorates when it is transmitted over a metro networks, across
multiple ROADMs (with typical impairments found in telecomms-grade equipment),
and understand how the Error Vector Magnitude (EVM) for the analogue signal
increases as the signal moves through the network.

In our experiment, a wireless baseband signal of 488.28 Mhz bandwidth is first
modulated at Intermediate Frequency (IF) of 1.5 GHz. This signal is then used
to modulate an optical carrier, which is transmitted over fibre. The signal is
injected into a network of multiple ROADMs, where a 100Gb/s DP-QPSK coherent
signal is added to the system and co-propagates with the ARoF signal. The
coherent signal operates at a rate of 31.5 Gbaud, for which a channel of 37.5
GHz is allocated in the flexgrid ROADMs. The ARoF Dual Sideband (DSB) signal
we used only occupies 3.488 GHz (carrier plus 2 sidebands), and is allocated
to a 6.25 GHz slot in the ROADM (this is the smallest granularity that can be
allocated in typical telecomms-grade flexgrid ROADMs).

###  II-A Experiment setup

Figure 1: End to end architecture of the network convergence experiment Figure
2: ARoF Tx/Rx experiment setup

The experimental platforms used in this proposal are the OpenIreland testbed
located in Trinity College Dublin and the Dublin City University (DCU) Radio
and Optical Communications Laboratory. The two are linked through a dark fibre
link (field fibre) across the city of Dublin of 9 km length.

The coherent transponders, ROADMs and fibre spools are located in the
OpenIreland testbed. The transponder are ADVA Teraflex FSP3000C, that can
operate at variable baud rate (31.5 - 71.96 Gbaud) and modulation schemes (DP-
QPSK - DP-64QAM) to achieve bit rates from 100G through to 600Gbps. In this
experiment, we use the most basic configuration DP-QPSK at 31.5 Gbaud and use
a signal bandwidth of 37.5 Ghz. The line rate is 100 Gbps. The system
architecture is shown in Fig. 1. The coherent transponder is connected to an
8-way passive splitter to allow for signal monitoring (i.e., through an OSA)
and diagnostics. A booster EDFA amplifies the signal before transmission. This
is followed by VOA to allow dynamic selection of the output power. The overall
system is organised in a looped topology. The ARoF signal is generated at DCU
and sent to TCD, to then return to DCU for decoding after passing through a
number of ROADMS. The Coherent signal is generated at TCD, propagated across
the ROADMs, then sent to DCU and form there back to TCD to be decoded.

More in detail, the digital coherent signal is injected to the MUX of ROADM3
(R3), which performs the function of discriminating and combining it with the
ARoF signals (generated at DCU). An Add rule is applied the Mux WSS to add a
37.5 GHz wide band (194931.25 to 194968.75 GHz) for the coherent signal. An
Add rule is also applied to the Mux WSS for a 6.25 GHz wide band (194968.75 to
194975.00 GHz) for the RF signal. Equivalent rules are added to the Demux
side: the coherent signal is dropped towards the coherent receiver, while the
ARoF signal is sent back to DCU after passing through the chain of ROADMs.

The chain of ROADMs is composed by a series of two-degree ROADMs connected
through 25 km of SMF (notice that in this experiment we only consider a single
direction of communication). The number of traversed ROADMs varies during the
experiment to analyse how this affects the signal quality. After the last
ROADM, the signal goes through 9 km of dedicated dark fibre, leased from a
telecommunications operator, to reach the DCU Radio and Optical Communications
Laboratory.

Fig. 2 shows the laboratory experimental setup for the ARoF generation,
transmission and reception at DCU. An Intermediate Frequency (IF) Orthogonal
Frequency Division Multiplexing (OFDM) with 64-QAM is generated offline using
matlab and loaded into an Arbitrary Waveform Generator (AWG) operating at 60
GSa/s. The 2.93 Gb/s ARoF signal, whose numerology is shown in Table I, is
output from the AWG and used to drive a Dual-Drive Mach Zehnder Modulator (DD-
MZM) that modulates the light from an External Cavity Laser (ECL) operating at
194.97 THz (1537.6 nm). The output double side band optical signal is combined
with the coherent DP-QPSK signal originating from TCD via a 50/50 optical
coupler and the converged ARoF/Coherent services are transmitted at a total
launch power of +7 dBm over one fibre of the dark fibre pair, towards TCD
(denoted ‘To Network’ in Fig. 2).

TABLE I: ARoF signal numerologies. Property | Value | Unit
---|---|---
# Subcarriers | 250 | n/a
(I)DFT size | 1024 | n/a
QAM order | 64 | n/a
IF | 1.5 | GHz
Symbol Rate | 1.95 | MHz
Bandwidth | 488.28 | MHz
Cyclic Prefix | 6.25 | %
Raw Data Rate | 2.93 | Gb/s
Figure 3:  Optical spectrum of the digital coherent signal and analogue ROF
signal combined, measure in the DCU lab.

The ARoF receiver at DCU receives the multiplexed ARoF/coherent optical
signals from TCD via the second fibre of the dark fibre pair (denoted ‘From
Network’ in Fig. 2). An example of the optical signal spectrum captured at
this point is shown in Fig. 3. This received signal is first amplified using
an in-line EDFA and then sent to a WSS, whose programmable filter profile is
set to discriminate between the adjacent services, directing the ARoF signal
for local detection via one output port and recirculating the coherent signal
back to TCD via a second port as shown in Fig. 2. A Variable Optical
Attenuator (VOA) is used to set the ARoF optical power incident on a 20 GHz
Photo-Detector (PD). The output electrical signal is amplified and sampled
using a Real Time Oscilloscope operating at 50 GSa/s. The digitized ARoF
signal is then processed offline including IF down-conversion, demodulation
and Error Vector Magnitude (EVM) measurement.

Figure 4:  Error Vector Magnitude of the ARoF signal, for different
propagation distance and WSS counts (calculated across the different
topologies A, B, C and D). EVM 8% limit for LTE highlighted. Figure 5:
Optical spectra showing the DSB ARoF signal at the receiver for the different
topologies A, B, C and D.

###  II-B Procedure

To conduct the experiment, we first configure the physical lightpath topology
in Open Ireland and DCU. In Open Ireland, since all components are routed
through a large port-count fibre switch, physical paths can be set up
programmatically. This enables fast and easily reproducible experiment setup,
in addition to monitoring of power levels at all points of the topology, to
easily assess any change in power (all ports of the fibre switch have power
monitoring). Next, we configure the optimal transmit power levels of the
transponders, and the gain of each EDFA and attenuator in the topology. Within
each ROADM, the Mux WSS is followed by a booster EDFA, with gain configurable
between 5 and 25 dB. The Demux WSS is instead preceded by an EDFA preamp, with
gain also configurable between 5 and 25 dB. Each WSS add/drop port has a VOA
with attenuation configurable between 0 and 20 dB. Lastly, we iterate through
4 different topologies, referred to with A, B, C and D, with varying number of
ROADMs, as shown in Table II.

TABLE II:  Topologies used, with associated number of WSS and overal transmission distance. Topology | WSS Count |  | Fiber Length
---
km
A | 2 | 9
B | 4 | 34
C | 6 | 59
D | 8 | 84

Each ROADM (1x20 Lumentum ROADM Graybox) has 2 WSS’s (one for MUX and one for
DEMUX). For the first topology (A), we only use ROADM3 in TCD and the WSS in
DCU that discriminates the Coherent and RF signals. For the second topology
(B), a further ROADM is added in (thus, 2 more WSS), along with a length of 25
km fibre. For the third topology (C), we have 3 ROADMs (6 WSS), and finally,
in the fourth topology (D), 4 ROADMs (8 WSS) are used. The shortest combined
length (i.e., across the field fibre and lab reels) is 9 km, while the longest
is 84 km.

##  III Results and Discussion

The performance metrics we collect for the digital coherent transmission are
the real-time values for optical received power, Q-Factor, signal to noise
ratio, optical signal to noise ratio, pre-FEC bit error rate and the
uncorrected blocks, all of which are reported directly from the transponder.
Since the DP-QPSK signal is resilient to noise and has typically a reach above
1000km, it’s performance is not affected by the ARoF signal and are similar
across all tested topologies, with Q-factor of 16.5, SNR around 18.5 dB and
OSNR of around 28.5 dB, and negligible pre-FEC BER.

For the ARoF 64-QAM OFDM signal we instead measure the Error Vector Magnitude
(EVM), shown in Fig. 4 for the different topologies. The x axis of the figure
indicates the overall number of WSSs traversed. In the simplest transmission
case (Topology A) the received ARoF signal exhibits excellent performance with
an EVM of 4.7%. The received ARoF optical spectrum measured for this topology,
after signal discrimination, is shown in Fig. 5(a). The figure shows the DSB
ARoF signal alongside a portion of the DP-QPSK coherent signal which is
present due to spectral leakage owing to the proximity of the signals as they
pass through the non-ideal filter of the WSS signal discriminator.
Nevertheless, satisfactory performance is achieved with the received
constellation shown as an inset in Fig. 4 where the received optical power was
2 dBm.

The figure also shows ARoF degradation when the WSS count increases, as the
signal propagates through additional filters and longer distance. This
degradation can be understood by observing the respective measured ARoF
optical spectra for each topology in Figs. 5(a), (b), (c) and (d). The spectra
reveal that transmission through additional ROADM filters results in increased
attenuation of the ARoF signal’s upper sideband. This effect occurs due to the
non-perfect alignment of WSS filters in each ROADM and specifically limits
ARoF performance. The net result is an attenuation of the ARoF signal, leading
to lower received optical powers and a degradation in signal-to-noise ratio.
Fig. 5 shows EVMs of 7.05%, 8.2% and 13.9% for topologies B, C and D, with
these performances measured with received optical powers of -1 dBm, -3 dBm and
-6 dBm, respectively. The figure also shows the received 64-QAM constellation
measured after transmission over topology C as an inset.

These results are highly valuable as they reveal a new practical limitation on
the performance of ARoF transmission in the case of convergence with flexgrid
coherent services and networking equipment. The degradation highlighted by the
results could be alleviated through tailoring of the channel spacing dedicated
for adjacent ARoF fronthaul and coherent metro traffic, or altering the format
or bandwidth of the wireless signal, indicating an interesting trade-off
between spectral efficiency and the required complexity of the fronthaul
transmission path. The choice of IF for ARoF may also be varied here to
account for sideband attenuation. In the scenarios evaluated here, the results
shows that ARoF and coherent signal convergence is indeed a viable networking
option for future heterogenous optical networks, with results for topology C
(6 WSSs) exhibiting performance just at the current limit for 64-QAM OFDM 5G
NR signals.

##  IV Conclusions

In this work we have analysed the coexistence of digital coherent and analogue
radio over fibre signals in an access-metro network architecture. Our
experimental setup included telecomms-grade equipment (ROADMs, digital
transponders and amplifiers) and a 9 km field fibre (linking TCD and DCU
Universities). Transmission windows of 37.5 GHz and 6.25 GHz, respectively,
for the 31.5Gbaud digital coherent DP-QPSK and the 64-QAM OFDM ARoF signals,
where created by the ROADMs. In order to assess the degradation of the signals
across the metro networks, mostly generated by the non-ideal filter alignment
of different ROADM WSS, we have created 4 different topologies with increasing
number of ROADMs and transmission distance, to recreate a typical metro
transmission network. Our results show that filter misalignment that is
typical of telecomms-grade ROADMs does affect the ARoF signal quality,
limiting the transmission to a topology with 3 ROADMs. This demonstrates that
while transmission of ARoF signal is feasible in toady’s access-metro
networks, there are trade-offs that needs to be considered for topologies with
higher number of ROADMs. In our future work we intend to further investigate
these trade-offs, analysing the behaviour of ARoF signals with different
bandwidth and modulation formats, the use of ROADM windows with different
bandwidth and the effect of different channel loading configurations (i.e.,
loading the system with a higher number of digital coherent channels).

## Acknowledgments

This work has been funded by DTIF EI grant DT 2019 0014B (Freespace), Science
Foundation Ireland projects OpenIreland 18/RI/5721, 13/RC/2077_p2 and
18/SIRG/5579. We acknowledge the technology support of ADVA.

## References

  * [1] T. R. Raddo, et al., Analog Radio-over-Fiber 5G Fronthaul Systems: blueSPACE and 5G-PHOS Projects Convergence. 2019 European Conference on Networks and Communications (EuCNC), 2019.
  * [2] C. Browning, et al. A Silicon Photonic Switching Platform for Flexible Converged Centralized Radio Access Networking. IEEE Journal of Lightwave Technology, Vol. 38, No. 19, April 2020.
  * [3] K. M. S. Huq, et al. Terahertz-Enabled Wireless System for Beyond-5G Ultra-Fast Networks: A Brief Survey. in IEEE Network, vol. 33, no. 4, July/August 2019
  * [4] D. Novak et al. Radio-Over-Fiber Technologies for Emerging Wireless Systems. in IEEE Journal of Quantum Electronics, vol. 52, no. 1, pp. 1-11, Jan. 2016
  * [5] T. Sizer et al. Integrated Solutions for Deployment of 6G Mobile Networks. in Journal of Lightwave Technology, vol. 40, no. 2, Jan 2022.
  * [6] Pedro Alvarez, Frank Slyne, Christian Bluemm, Johann M. Marquez-Barja, Luiz A. DaSilva, Marco Ruffini, Experimental Demonstration of SDN-controlled Variable-rate Fronthaul for Converged LTE-over-PON. Proc. of Optical Fibre Communications conference (OFC), paper Th2A.49, March 2018.
  * [7] M. Ruffini and F. Slyne. Moving the Network to the Cloud: the Cloud Central Office Revolution and its Implications for the Optical Layer. IEEE/OSA Journal of Lightwave Technology, Vol. 37, No. 7, April 2019.
  * [8] F. Saliou et al. Optical access network interfaces for 5G and beyond [Invited]. Journal of Optical Communications and Networking, Vol. 13, No. 8, June 2021.
  * [9] M. Gagnaire. Analog and Digitized Radio-over-Fiber. In Springer Fiber-Wireless Convergence in Next-Generation Communication Networks, 2017.
  * [10] C. Browning et al. 5G wireless and wired convergence in a passive optical network using UF-OFDM and GFDM. IEEE International Conference on Communications Workshops (ICC Workshops), 2017.
  * [11] L. Li et al. Digital-Analog Hybrid Optical Access Integrating 56-Gbps PAM-4 Signal and 5G mmWave Signal by Spectral Null Filling. in Journal of Lightwave Technology, vol. 39, no. 5, March 2021.
  * [12] K.Kanta et al. Demonstration of a Hybrid Analog–Digital Transport System Architecture for 5G and Beyond Networks. in Applied Sciences, vol. 12, No. 4, February 2022.
  * [13] R. Muñoz et al. Converged fixed-mobile access-metro optical network for 5G with programmable and elastic optical systems and SDN/NFV control. IEEE 5G Tech Focus: Vol. 2, No. 2, May 2018.

A-RoF

    Analog Radio-over-Fiber
ABNO

    Application-Based Network Operations
A-CPI

    Application Controller Plane Interface
ADC

    Analog-to-Digital Converter
API

    Application Programming Interface
AWG

    Arrayed Waveguide Grating
B-RAS

    Broadband Remote Access Servers
BB

    Base Band
BBU

    Base Band Unit
BER

    Bit Error Rate
BGP

    Border Gateway Protocol
BS

    Base Station
BtB

    Back to Back
BW

    Bandwidth
CAP

    Carierless Amplitude Phase
C-RAN

    Cloud Radio Access Networks
C-RoFN

    Cloud-based Radio over optical Fiber Network
CAPEX

    Capital Expenditure
CDN

    Content Distribution Network
CDR

    Clock Data Recovery
CIR

    Committed Information Rate
CO

    Central Office
CoMP

    Coordinated Multipoint
COP

    Control Orchestration Protocol
CPE

    Customer Premises Equipment
CPRI

    Common Public Radio Interface
CS

    Central Station
D-RoF

    Digital Radio-over-Fiber
DAC

    Digital-to-Analog Converter
DAS

    Distributed Antenna Systems
DBA

    Dynamic Bandwidth Allocation
DC

    Direct Current
DFB

    Distributed Feedback Laser
D-CPI

    D Controller Plane Interface
DL

    Downlink
DMT

    Discrete Multitone
DSB

    Double Side Band
DSLAM

    Digital Subscriber Line Access Multiplexer
DSL

    Digital Subscriber Line
DSP

    Digital Signal Processing
DSS

    Distributed Synchronization Service
DWDM

    Dense Wave Division Multiplexing
E-CORD

    Enterprise CORD
EDF

    Erbium-Doped Fiber
eICIC

    enhanced Inter-Cell Interference Coordination
EMBS

    Elastic Mobile Broadband Service
EON

    Elastic Optical Network
EPON

    Ethernet Passive Optical Network
EPC

    Evolved Packet Core
ETSI

    European Telecommunications Standards Institute
EVM

    Error Vector Magnitude
FBMC

    Filterbank Multicarrier
FEC

    Forward Error Correction
FDD

    Frequency Division Duplex
FFR

    Fractional Frequency Reuse
FFT

    Fast Fourier Transform
FPGA

    Field-Programmable Gate Array
FSO

    Free-Space-Optics
FSR

    Free Spectral Range
FTTB

    Fibre to the Building
FTTcab

    Fibre to the CAB
FTTH

    Fibre to the Home
FTTx

    Fibre to the x
G.Fast

    Fast Access to Subscriber Terminals
GFDM

    Generalized Frequency Division Multiplexing
GMPLS

    Generalized Multi-Protocol Label Switching
GPON

    Gigabit Passive Optical Network
GPP

    General Purpose Processor
GPRS

    General Packet Radio Service
GTP

    GPRS Tunneling Protocol
HA

    Hardware Accelerator
HARQ

    Hybrid-Automatic Repeat Request
I2RS

    Interface 2 Routing System
IA

    Interference Alignment
ICT

    Information and Communication Technology
I-CPI

    I Controller Plane Interface
IETF

    Internet Engineering Task Force
IF

    Intermediate Frequency
IoT

    Internet of Things
IP

    Internet Protocol
IQ

    In-phase/Quadrature
IRC

    Interference Rejection Combining
KVM

    Kernel-based Virtual Machine
LAN

    Local Area Network
LO

    Local Oscillator
LOS

    Line Of Sight
LR-PON

    Long Reach PON
LSP-DB

    Label Switched Path Database
LTE

    Long Term Evolution
LTE-A

    Long Term Evolution Advanced
MAC

    Medium Access Control
M-CORD

    Mobile CORD
MEC

    Multi-Access Edge Computing
MME

    Mobility Management Entity
mmWave

    millimeter Wave
MNO

    Mobile Network Operator
MIMO

    Multiple Input Multiple Output
MU-MIMO

    Multi-user MIMO
MPLS

    Multiprotocol Label Switching
MRC

    Maximum Ratio Combining
MSR

    Multi-Stratum Resources
MVNO

    Mobile Virtual Network Operator
MZI

    Mach-Zehnder Interferometer
MZM

    Mach-Zehnder Modulator
NFV

    Network Function Virtualization
NFVaaS

    Network Function Virtualization as a Service
NFV-O

    Network Functions Virtualisation Orchestrator
NG-PON2

    Next-Generation Passive Optical Network 2
NLOS

    None Line Of Sight
NMS

    Network Management System
NRZ

    Non Return-to-Zero
OBPF

    Optical BandPass Filter
OBSAI

    Open Base Station Architecture Initiative
OFC

    Optical Frequency Comb
OFDM

    Orthogonal Frequency Division Multiplexing
OLT

    Optical Line Termination
ONAP

    Open Network Automation Platform
ONF

    Open Networking Foundation
ONU

    Optical Network Unit
OOK

    On-off Keying
OpenCord

    Central Office Re-Architected as a Data Center
OPEX

    OPerating EXpense
OSI

    Open Systems Interconnection
OSS

    Operations Support Systems
OXM

    OpenFlow Extensible Match
PAM

    Pulse Amplitude Modulation
PAPR

    Peak-to-Average Power Ratio
PCE

    Path Computation Elements
PCEP

    Path Computation Element Protocol
PCF

    Photonic Crystal Fiber
PD

    Photodiode
PDCP

    RD Control Protocol
PGW

    Packet Gateway
PHY

    physical Layer
PIR

    Peak Information Rate
PMD

    Polarization Division Multiplexing
PON

    Passive Optical Network
PTP

    Precision Time Protocol
PWM

    Pulse Width Modulation
QAM

    Quadrature Amplitude Modulation
QoE

    Quality of Experience
QoS

    Quality of Service
QPSK

    Quadrature Phase Shift Keying
RAN

    Radio Access Network
R-CORD

    Residential CORD
RF

    Radio Frequency
RLC

    Radio Link Control
RN

    Remote Node
ROADM

    Reconfigurable Optical Add Drop Multiplexer
RoF

    Radio-over-Fiber
RRH

    Remote Radio Head
RRC

    Radio Resource Control
RRPH

    Remote Radio and PHY Head
RRU

    Remote Radio Unit
RSOA

    Reflective Semiconductor Optical Amplifier
Rx

    receiver
SD-RAN

    Software Defined Radio Access Network
SDMA

    Semi-Distributed Mobility Anchoring
SDN

    Software Defined Network
SDR

    Software Defined Radio
SFBD

    Single Fiber Bi-Direction
SGW

    Serving Gateway
SIMO

    Single Input Multiple Output
SMF

    Single Mode Fiber
SNR

    Signal-to-Noise Ratio
Split-PHY

    Split Physical Layer
TED

    Traffic Engineering Database
TEID

    Tunnel endpoint identifier
TDD

    Time Division Duplex
TD-LTE

    Time Division LTE
TDM

    Time Division Multiplexing
TDMA

    Time Division Multiple Access
TWDM

    Time and Wavelength Division Multiplexing
Tx

    transmitter
UD-CRAN

    Ultra-Dense Cloud Radio Access Network
UDP

    User Datagram Protocol
UE

    User Equipment
UFMC

    Universally Filtered Multicarrier
UF-OFDM

    Universally Filtered OFDM
UL

    Uplink
USRP

    Universal Software Radio Peripheral
vBBU

    virtualized BBU
vBS

    virtual Base station
vCPE

    virtual CPE
VDSL2

    Very-high-bit-rate digital subscriber line 2
VLAN

    Virtual Local Area Network
VNF

    Virtual Network Functions
VNTM

    Virtual Network Topology Manager
VPE

    virtual Provider Edge
VM

    Virtual Machine
vOLT

    virtual OLT
WAN

    Wide Area Network
WAP

    Wireless Access Point
WDM

    Wavelength Division Multiplexing
WDM-PON

    Wavelength Division Multiplexing - Passive Optical Network
WiMAX

    Worldwide Interoperability for Microwave Access
WRPR

    Wired-to-RF Power Ratio
XCI

    Crosshaul Control Infrastructure
XFE

    Crosshaul Packet Forwarding Element
XGPON

    10 Gigabit PON
10GEPON

    10 Gigabit EPON
XOS

    XaaS Operating System
5GPoA

    5G Points of Attachments
  *[CO]: Central Office
  *[CAPEX]: Capital Expenditure
  *[D-RoF]: Digital Radio-over-Fiber
  *[A-RoF]: Analog Radio-over-Fiber
  *[RF]: Radio Frequency
  *[PAPR]: Peak-to-Average Power Ratio
  *[OFDM]: Orthogonal Frequency Division Multiplexing