\section{Introduction}\label[sec]{sec:Intro}

Tackling model uncertainty has become a popular research area of mathematical financ
, particularly over the past decade,
in great part due to
its providing
a more realistic description of financial markets and behaviours.

The classical paradigm in the literature has been that economic agents have an objective and unequivocal \emph{a priori} knowledge of the probability laws governing the evolution of market
, which is typically modelled by fixing a filtered probability space
that supports
an adapted stochastic process describing the prices of the underlying asset
\ (representative papers on expected utility maximisation within the classical ``model certainty'' theory include, but are 
far from limited to,
\citet{merton7
,pliska198
,karatzasls8
,coxhuang8
,karatzaslsx9
,cvitanickaratzas199
,zari9
,kramkov9
}
.
In
practic
, however
\ there is often insufficient available
information
to build completely accurate model
.

The dominant framework
to model this uncertainty or ambiguity
is to consider a non-empty family of priors
$\mathscr{P}$
representing
all the probability measures that are deemed possible on some canonical space where questions of hedgin
, pricin
, and optimal investment are addressed. The presence of multiple, possibly mutually singular probability measures expresses, in an intellectually satisfying way, that we are looking for solutions which work whatever the true model specification is (in the given class of models). The associated mathematical difficulties are tantalising, just like the connection to well-established theories such as that of optimal transpor
\ (we mention the work o
\ \citet{beiglbock2013,dolinskysoner2014
, among other

.

Looking
more closely
at
the
optimal investment
problem under uncertainty, the seminal paper of \citet{gilboaschmeidler1989} provides an extension to robust
preferences of the axiomat
c
characterisation
proposed
by \citet{neumann53} for standard
expected utility
without uncertaint
.
A significant body of literature has studied this problem under the assumption that the collection $\mathscr{P}$ is dominated, i.e.\ there is a reference measure with respect to which all
priors are absolutely continuous (the reader may consult e.g.\ the references cited in \citet{fsw2009,bck2019}). If, on the contrary, the set of scenario measures is non-dominated, the existence of a robust utility maximiser in discrete time has been established by: \citet{nutz2016}, for a bounded above utility function defined on the positive real line; \citet{neufeld-sikic2018}, also in the case of a utility function that is bounded from above, but in a market with frictions; \citet{bartl2017}, where a dual representation is derived for exponential utility; \citet{blanchard-carassus2017}, whose work appears to be the first treating the case of a general unbounded utility on the positive half-line; \citet{bck2019}, who rely on the axioms of ZFC set theory together with Martin's axiom (a cardinality principle weaker than the continuum hypothesis) to solve the problem for a bounded above utility on the whole real lin
.

This article addresses the utility maximisation
problem when uncertainty is present
in an alternative framework, which we regard as no less adequate and
allows to prove results that are currently unavailable in the mainstream settin
: the agents
consider a whole family $\mathcal{S}$ of stochastic processes, \emph{defined on the same fixed stochastic basis}, as the class of possible scenarios for the
price
, all of which are
seen
as equally plausible.
Put differently, while in our context investors know
the real-world likelihood of outcomes (in particular, what the null and sure events are), they are uncertain about the dynamics followed by the pric

, so the misspecification of the true model is represented by a collection of
processes to consider.
In short, the present setting deviates from the usual one in
that,
instead of fixing a process $S$ on a measurable space 
$\left(\Omega,\mathscr{F}\right)$ (in most cases $S$ consists of the coordinate mappings
on a convenient path space)
and considering a family $\mathscr{P}$ of probabilities on $\left(\Omega,\mathscr{F}\right)$, 
we fix the stochastic basis (i.e.\ the available information together with the true chances of events occurring) 
and vary the process $S\in\mathscr{S}$.

In the standard setting, robust utilities give rise to preference relations between 
all (suitably integrable) random variables (see e.g.\ the axiomatic treatment in \citet{gilboaschmeidler1989}), 
in particular preferences are defined for every possible portfolio terminal wealth.
In our setting, on the other hand,
preferences are actually defined on agents' \emph{decisions}
\ (specifically, the admissible controls/investment policies/portfolio strategies
, where the utility of a decision (portfolio strategy) is the infimum
of
its expected utility over the family of possible models. 
Apparently there is no way to extend these preferences to \emph{all} random variables.
There is no obvious relationship between these two approaches.
Ours emphasises that there is a class of models (e.g.\ corresponding to a parametrisation), 
the investor is uncertain about which is the right one and prefers strategies 
with higher worst-case utilities.
In the standard approach uncertainty is already present at the choice of 
the probability space, as usual in theoretical statistics, and any two random outcomes
can be compared, regardless of being a portfolio terminal value or not.

A
\ important aspect to notice is that this alternative uncertainty modelling approach
does not exclude the laws of these processes being e.g.\ mutually singular.
One obvious comparative advantag
\ at the core of our wor
\ is tha
\ we are still able to use results of classical probability theor
; namely, we are able to avoid the delicate measurability issues arising in the non-dominated quasi-sure framewor
\ (we point out that we do not impose any particular structure on our family of models $\mathscr{S}$
, and apply Koml\'{o}s-type arguments to construct candidates for optimal strategies.
In this way, we can cover certain cases which the standard setting cannot.
For example, the filtration may contain random variables (economic factors) other than the price process; then the strategies are not functionals of the price alone, hence the quasi-sure path space setting cannot be used while ours works without any problem.
There are also model specifications that elude our present approach, but are conveniently handled with the existing tools. Hence, our contribution complement
, without subsuming
\ the prevailing approach of e.g.\ \citet{nutz2016,bartl2017,blanchard-carassus2017,neufeld-sikic2018}; see \Cref{sec:Example} for more on this.

The aim of this paper is to
investigate
conditions under which the
robust utility maximisation problem from terminal wealth for a risk-averse investo
\ in a discrete-time financial marke
\ admits a solutio
.
The agents adopt a worst-case approach in evaluating a given payoff by first minimising the welfare performance of each portfolio over all the stock prices that can materialise, and only then selecting the investment strategies that
offer the best of
such worst-scenario utilities.
In the aforementioned paper
\ in the non-dominated quasi-sure settin
, a main mathematical tool for proving the existence of optimal strategies is dynamic programming, which allows to reduce the multi-period robust portfolio choice problem to a sequence of one-period decision problem
, but leads to difficult measurability and analyticity issue
\ (for instance, at each time step it is necessary to consider a suitable family of possible transition probability measures, and subsequently take their concatenation; this is referred to as ``time-consistency'' in the terminology of \citet{bck2019}
. In the present work, we instead use an
approach
that deals with the primal problem directl
, by constructing optimal portfolios from optimising sequences (i.e., whose worst-case expected utility becomes arbitrarily close to the supremum over all strategies
.
We need neither the time-consistency assumption (present in all related papers except
\citet{bck2019}) nor extra set-theoretical hypotheses (present in \citet{bck2019}).

Our results are obtained under the assumption that there are no arbitrage opportunities for at least \emph{some}
price evolution
in the collection of all possible models and that model is non-degenerate in a suitable sense.
We compare this condition to the conventional robust no-arbitrage condition
in Remark \ref{obs:RNA} below.
This condition enables us to work
with an equivalent martingale measure (whose existence would otherwise be unknown at this time
; it is also intimately connected to other existing notions of absence of arbitrage, such as the ``weak no-arbitrage'' first proposed by \citet{davishobson2007
\ (see \Cref{as:N
} and the discussion in \Cref{obs:RNA} below
.

One
main contributio
\ of this work is that
utility functions on the positive axis and on the whole real line are both treated. In addition, we consider bounded above functions as well as functions that can grow without bound; in the latter case, we must replace boundedness of the utility with certain integrability conditions
on quantities related to the price process.
Random endowments or liabilities could be admitted at little cost, but this is not pursued in the present work.

The structure of the paper is as follows: \Cref{sec:Model} presents the model and the
mathematical formulation of the utility maximisation problem taking into account model uncertainty; \Cref{sec:Main} states and discusses our main results on the existence of robust optimal strategies; \Cref{sec:Example} contains some
illustrative examples highlighting
important differences, advantages, and drawbacks of the present framework in relation to the quasi-sure framework; \Cref{sec:Conc}
outlines
future
directions of research and
concludes. For fluidity, all
proofs are collected
in the
\hyperref[app:Main]
Appendix}
.


\section{Model}\label[sec]{sec:Model}


Consider a financial market
consisting of
$d\in\mathbb{N}$ traded risky asset
\ plus an additional risk-free asse
\ with constant unit pric
, and fix a time horizon $T\in\mathbb{N}$.
Let $(\Omega,\mathscr{F},\mathbb{F}=\left\{\mathscr{F}_{t}\right\}_{t\in\left\{0,1,\ldots,T\right\}},\mathbb{P})$ be a discrete-time stochastic basis, and assume that the $\sigma$-algebras occurring in this paper contain all $\mathbb{P}$-null sets.
For every $n\in\mathbb{N}$ and $\mathscr{G}\subseteq\mathscr{F}$, denote by $\Lspace[n]{0}{}{\mathscr{G}}$ the space of $\mathscr{G}$-measurable, $\mathbb{R}^{n}$-valued random variables (as usual, we identify random variables coinciding outside a $\mathbb{P}$-null se
), endowed with the
metrisable
topology
of convergence in probability.

Let $\mathscr{S}$ be
a non-empty family
of
adapted, $\mathbb{R}^{d}$-valued processes on this stochastic basis, and
define the topological product space $\mathscr{L}\coloneqq\otimes_{t=1}^{T}L^{0}(\mathscr{F}_{t-1};\mathbb{R}^{d})
.
When the (discounted) prices of the risky assets evolve according to the process $S\in\mathscr{S}$,
the (discounted)
wealth at time $t\in\left\{0,1,\ldots,T\right\}$ of a self-financing
portfolio $\phi\in\mathscr{L}$
starting from
initial capital $\iw\in\mathbb{R}$
is given by
\begin{equation*}
	W_{t}^{S}\!\left(\iw,\phi\right)=\iw+\sum_{s=1}^{t}\innerp{\phi_{s}}{\Delta S_{s}}.
\end{equation*}
Here, $\Delta S_{t}\coloneqq S_{t}-S_{t-1}$
is the price change at trading period
$t\in\left\{1,\ldots,T\right\}$, while $\innerp{\cdot}{\cdot}$ denotes the Euclidean inner product in $\mathbb{R}^{d}$ with the corresponding norm $\norm{\cdot}$.
A strategy is admissible if it is feasible for every particular
possible
pric
; more precisely,
the set of admissible trading strategies with initial capital $\i
$ is
	$\adm{\iw}\coloneqq\bigcap_{S\in\mathscr{S}}\mathscr{L}\!\left(\iw,S\right
	

\ for some $\mathscr{L}\!\left(\iw,S\right)\subseteq\mathscr{L}
\ to
be specified later, depending on the domain of the utility function.

Nex
, for every $S\in\mathscr{S}$ and
$t\in\left\{1,\ldots,T\right\}$, let $\rcd[t-1]{\Delta S_{t}}{F}\!\left(\cdot,\cdot\right)$ be a regular conditional distribution of $\Delta S_{t}$ with respect to $\mathscr{F}_{t-1}$. By redefining on a $\mathbb{P}$-null set,
$\rcd[t-1]{\Delta S_{t}}{F}\!\left(\cdot,\omega\right)$ is a probability measure on $(\mathbb{R}^{d},\mathscr{B}(\mathbb{R}^{d}))$ for all $\omega\in\Omega
, and we
denote by $D_{t}^{S}\!\left(\omega\right)$ the smallest affine subset of $\mathbb{R}^{d}$ containing the support of $\rcd[t-1]{\Delta S_{t}}{F}\!\left(\cdot,\omega\right)$.

Throughout the paper,
assume that
the collection of stock prices contains at least one process for which there
is no chance of making a riskless profit out of nothing.
In addition,
\ADD{1}{the redundant assets in the trading mechanism
for such price
mode
\ must also be redundant for all the other price processes that can materialise
. These
requirements are
formalised as follows.
\begin{assumption}\label[as]{as:NA}
	
	
	
	
	
		There exists $S^{*}\in\mathscr{S
		$ satisfying both
	
	
		\begin{equation*}\label{eq:NA}\labeltarget{eq:NAS}
		
			\text{for all }\phi\i
			\mathscr{L}\!\left(\iw,S^{\ast
			\right
			,\ \text{if }W^{S^{\ast
			}_{T}\!\left(0,\phi\right)\geq0\ \mathbb{P}\text{-almost surely (a.s.)},\ \text{then }W^{S^{\ast
			}_{T}\!\left(0,\phi\right)=0\ \mathbb{P}\text{-a.s.
			
			\tag{NA($S^{\ast}$)}
		\end{equation*}
	
		and for each $S\in\mathcal{S}
		\begin{equation}\label{eq:nonred}
			D_{t}^{S}\subseteq D_{t}^{S^{\ast}}



			\text{almost surely, for all }t\in\left\{1,\ldots,T\right\}.
		\end{equation}
		The set of such $S^{*}$ will be denoted by $\NAset$.
	
\end{assumption}

	
	






The following alternative characterisation of
the
arbitrage-free
and ``non-redundant''
price processes
is
provided
by \citet[Proposition~3.3]{rs05} (see also \citet[Proposition~2.1]{carassusrasonyi2009}).
\begin{samepage}
\begin{proposition}\label[prop]{prop:NA}
	Let $S\in\mathscr{S}$.
	The two statements below are equivalent:\nopagebreak
	\begin{enumerate}[label=\emph{(\roman*)}]
		\item \hyperlink{eq:NAS}{NA($S$)} holds.
	
	
	
	
		
		\item
		For every $t\in\left\{1,\ldots,T\right\}$, there exist $\mathscr{F}_{t-1}$-measurable random variables $\beta_{t}^{S}>
		$ $\mathbb{P}$-a.s.\ and $\kappa_{t}^{S}>0$ $\mathbb{P}$-a.s.\ such that
		\begin{equation}\label{eq:altNA}
			\pushQED{\qed
			\essinf_{\xi\in\Xi_{t}^{S}
			\mathbb{P}\!\left\{\left.\innerp{\xi}{\Delta S_{t}}\leq-\beta_{t}^{S}\norm{\xi}\right\vert\mathscr{F}_{t-1}\right\}\geq\kappa_{t}^{S}\ \mathbb{P}\text{-a.s.\ on
			} \left\{D_{t}^{S}\neq\left\{\vect{0}\right\}\right\
		
			,
		\end{equation}
		where $\Xi_{t}^{S}\coloneqq\left\{\xi\in\Lspace[d]{0}{}{\mathscr{F}_{t-1}}:~\xi\!\left(\omega\right)\in D_{t}^{S}\!\left(\omega\right)\text{ for }\mathbb{P}\text{-a.e.\ }\omega\in\Omega\right\}$.
	\end{enumerate}
\end{proposition}
\end{samepage}

Furthermor
, we make the assumption
that the investors' risk preferences are continuous and non-satiabl
, with declining marginal utility.
These are fairly weak conditions on the utility function: we
require neither strict concavity nor
smoothness

in particular,
no Inada-type condition
\ need to be satisfie

.
\begin{assumption}\label[as]{as:U}
	The \emph{utility} function $U:~\mathbb{R}\rightarrow\mathbb{R}\cup\left\{-\infty\right\}$ is
non-decreasing, concave and upper semicontinuous.

\end{assumption}

We define
\begin{equation}\label{eq:domU}
	\dom\!\left(U\right)\coloneqq\left\{x\in\mathbb{R}:~U\!\left(x\right)>-\infty\right\},
\end{equation}
which is an interval of the form $\left(b,+\infty\right)$ or 
$\left[\left.b,+\infty\right)\right.$ 
for some $b\in\mathbb{R}\cup\{\pm\infty\}$.
Upper semicontinuity means that, if $b\in\mathbb{R}$, then $\lim_{x\rightarrow b^{+}}U\!\left(x\right)=U\!\left(b\right)$ (where the latter may be finite or $-\infty$).

In the presence
of uncertainty,
the investors' objective is to choose an admissible portfolio that maximises their
expected utility from terminal wealth
in the worst possible concretisation of the asset price
, where the same plausibility is attached to every price process in $\mathscr{S}
.
This
model uncertainty version of the optimal portfolio problem
is
defined
precisely below.
\begin{definition}[Robust optimal portfolio]\label[def]{def:rEU}
	A portfolio $\opt\in\adm{\iw}$ is \emph{optimal} under model uncertainty for an investor with initial wealth $\iw$ if
	\begin{equation}\label{eq:rEU}
		u\!\left(\iw\right)\coloneqq\sup_{\varphi\in\adm{\iw}}\inf_{S\in\mathscr{S}}\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{S}\!\left(\iw,\varphi\right)\right)\right]=\inf_{S\in\mathscr{S}}\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{S}\!\left(\iw,\opt\right)\right)\right].
	\end{equation}
	Here, we adopt the usual convention that, if a random variable is such that its positive and negative parts both have infinite expectation, then its expectation is defined to be $-\infty$ 
	i.e
	, $+\infty-\infty\coloneqq-\infty$).
\end{definition}

\begin{remark}\label[obs]{obs:u>-infty}

	It is trivial that

		
		\inf_{S\in\mathscr{S}}\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{S}\!\left(\iw,\safe\right)\right)\right]=U\!\left(\iw\right)

	\ for all $\iw\in\dom\!\left(U\right)
	, where $\safe\equiv\left(\vect{0},\ldots,\vect{0}\right
	$ denotes the safe portfolio allocating all wealth to the safe asset.
	Therefore, $u\!\left(\iw\right)>-\infty$ provided that

	the safe portfolio is admissible.
\end{remark}

The next section
demonstrates
the existence of an optimiser for \eqref{eq:rEU} in two settings: initially,
investors
are subject to the usual no-bankruptcy requiremen
; subsequently, we relax this
restriction by treating
the case of finite utility on the whole real lin
.

\begin{remark}\label[obs]{obs:RNA}
	\mbox{}
	\begin{enumerate}[label=\emph{(\roman*)}]
	
	\item
	We establish

	our results under \Cref{as:NA}, which implies that there is at least \emph{one} model $S^{\ast
	\in\mathscr{S}$ that is
	arbitrage-fre
	\ in the classical sens
	; in other words,
	\hyperlink{eq:NAS}{NA($S^{\ast}$)}
	corresponds to the classical
	absence of arbitrage opportunities for the price process $S^{\ast
	$.
	Incidentally, this condition is the counterpart of the conventional
	concept
	of ``weak no-arbitrage'' of \citet{davishobson2007
	\ (itself a particular case of
	the
	``no $\mathscr{R}$-arbitrage'' property
	formulated in
	\citet[Definition~5.1]{bfm2016}).
	
	\item


	Another
	pertinent comparison is
	to the standard ``robust no-arbitrage condition''
	introduced by \citet[Definition~1.1]{bouchardnutz2015
	, which in the present
	framework

	read
	\ as:
	\begin{equation*}\label{eq:RNA}
		\text{For all }S\in\mathscr{S},\ W_{T}^{S}\!\left(0,\phi\right)\geq0\ \mathbb{P}\text{-a.s.}\Rightarrow \text{For all }S\in\mathscr{S},\ W_{T}^{S}\!\left(0,\phi\right)=0\ \mathbb{P}\text{-a.s.}.\tag
		{NA($\mathscr{S}$)
		}
	\end{equation*}
	
	In the multiple priors setting, it is shown by 
	\citet[Theorem 3.8]{bc19} that \ref{eq:RNA} implies \hyperlink{eq:NAS}{NA($S$)} for some $S\in\mathscr{S}$,
	but the converse direction fails; refer to \citet[statement (5) of Lemma~3.7]{bc19}.
	Theorem~3.30 of \citet{bc19} shows that Assumption \ref{as:NA} is actually equivalent to 
	the robust no-arbitrage condition of \citet{bouchardnutz2015}, hence it seems to be
	a fairly natural requirement. 
	Note, however, that the multiple priors framework is different from the current one, 
	therefore it is not possible to directly transfer results from either one to the other.
	
	Getting back to our framework, we show that
	\Cref{as:NA} is stronger than
	condition~\ref{eq:RNA


	.
	To see this, let $\phi\in\adm{\iw}$ such that, for all $S\in\mathscr{S}$, it holds $\mathbb{P}$-a.s.\ 
	that $W_{T}^{S}\!\left(0,\phi\right)\geq0$. Then, for all $t\in\left\{1,\ldots,T\right\}$, we get 
	$W_{t}^{S^{\ast}}\!\left(0,\phi\right)=0$ $\mathbb{P}$-a.s., which in turn implies that 
	$\phi_{t}\!\left(\omega\right)$ belongs to the orthogonal complement of $D_{t}^{S^{\ast}}\!\left(\omega\right)$ 
	for $\mathbb{P}$-a.e.\ $\omega\in\Omega$. Hence, $\phi_{t}$ is also orthogonal to $D_{t}^{S}\subseteq D_{t}^{S^{\ast}}$
	for all $S\in\mathcal{S}$, and
	we conclude $W_{T}^{S}\!\left(0,\phi\right)=0$ $\mathbb{P}$-a.s.
	Unlike in the multiple priors case,	we do not know whether 
	\ref{eq:RNA} implies Assumption \ref{as:NA} or \hyperlink{eq:NAS}{NA($S^{\ast}$)} for some $S^{\ast}$
	in the present framework.
	
	The reason why we
	work under the
	``weak no-arbitrage'' assumption
	rather than the
	``robust no-arbitrage'' condition
	is that
	the former
	provides us with an equivalent martingale measure, by the Dalang-Morton-Willinger theorem~\citep{dalang1990}; 
	obtaining a version of the Fundamental Theorem of Asset Pricing associated with \ref{eq:RNA} is a project 
	worth further pursuing 
	in the current setting (as already done in the quasi-sure framework; see \citet{bouchardnutz2015,bc19}).
	
	\item
	Compare also to \cite

	{blanchard-carassus2017}, where unbounded from above utilities
	defined on the positive half-real line
	are treated under a (strengthened) no-arbitrage condition for \emph{every} model $S\in\mathscr{S}
	\ (see

	Definition~2.4 therein



	.
	We point out that we obtain our existence \Cref{th:UrealU} below, for 
	unbounded utilities over the whole real line, under a similar (strong) absence of 
	arbitrage assumption for all possible price processes. This 
	trivially implies the robust no-arbitrage condition~\ref{eq:RNA}, 
	as also observed by \citet[Lemma~3.7(4)]{bc19} in the multi-priors setting; additionally,
	see
	Theorem~3.8 in the same paper, establishing the equivalence between robust no-arbitrage and the existence of some subfamily of priors (having the same ``relevant'' events as the original one) which are all arbitrage-free.

	It is still unclear
	what
	the
	suitable
	notion
	of
	no-arbitrage
	should be
	in addressing the unbounded robust optimal portfolio problem.
	Recall that, in utility maximisation ``under certainty'',
	absence of arbitrage is a necessary condition for the existence of an optimiser,
	as shown e.g.\ by \citet[Proposition~3.1]{rs05}.
	\end{enumerate}
\end{remark}




\section{Main Results}\label[sec]{sec:Main}

The proofs of the
existence theorems in this section all share a common structure: First, we verify the finiteness of the value function for the optimisation problem~\eqref{eq:rEU
, so as to ensure that investors cannot attain near bliss from the strategies available in the marke
. We can then take a maximising sequence of portfolios, and obtain from it a candidate for an optimiser via a compactness argument. Lastly, we use the upper semicontinuity of the worst-case expected utility with respect to the strategies to
check
that the portfolio found in the preceding step is indeed
optima
.

Under the upper boundedness assumption on risk preferences, both well-posedness and up\-per semi\-con\-ti\-nu\-i\-ty are straightforwar
---the only point remaining to show being that a suitable collection of strategies satisfies some form of compactness.
Once we drop the requirement that the utility should be bounded from above, however, we start running into some integrability problems (namely, we no longer have an obvious integrable upper bound for the positive part of terminal utility, guaranteeing the convergence of certain integrals), which we handle by introducing additional hypotheses.

When the non-bankruptcy constraint is in force, the classical absence of arbitrage combined with
trading
``non-redundancy''
in \emph{one} of the realisable models is enough to establish bounds on the
admissible strategies (given by the auxiliary, almost surely finit
\ random variables $G^{S}_{t}$ found in \Cref{lem:key} below).
For the reasons
just set forth
\ with unbounded utilities we need these bounds to be
integrable as well, which is
achieved through
a strengthening of the no-arbitrage conditio
, in the sense of~\eqref{eq:intW}.

Turning to a setting where wealth is allowed to take on positive as well as negative values with positive probability, we again see that boundedness of preferences together with risk aversion suffices to have compactness
\ the arguments are more lengthy and involved than those of its positive real line counterpart, since the bounds of \Cref{lem:key} are no longer available.
Finally, for the particularly delicate case of an unbounded utility defined on the whole real line, the proof becomes even more intricate; as it turns out, absence of arbitrage in just one model is not enough anymore, and we must actually assume that \emph{every} possible model admits no arbitrage opportunities. On top of it, even though we allow the utility to grow arbitrarily large, we must require that it does so
in a strictly sublinear way in the sense of 
\eqref{maruszja} belo
. These conditions ensure the upper semicontinuity properties that are
needed
for the proof.

Summing up, if we consider utilities that are bounded above, then we are able to show that a robust optimal portfolio exists without having to impose any conditions other than \Cref{as:NA}; by contrast, the robust portfolio choice with an unbounded utility involves increased difficulties that call for considerably more stringent assumption
, not only on arbitrage but also on price dynamics and utility growt
. These are not surprising, see the related discussions in \citet{nutz2016}, and \citet{blanchard-carassus2017}.

\subsection{Utility on the positive real line}\label[subsec]{subsec:positiverl}
We start by
focusin
\ on the case where
debt is disallowe
\ under every scenario for the evolution of stock price
, so the utility function has effective domain of definition equal to the positive half-line.
\begin{assumption}\label[as]{as:positiverl}
	For all $S\in\mathscr{S}$,
	\begin{equation}\label{eq:LS}
	
		\mathscr{L}\!\left(\iw,S\right)\coloneqq\left\{\phi\in\mathscr{L}:
		\forall\
		t\in\left\{0,1,\ldots,T\right\},\ W_{t}^{S}\!\left(\iw,\phi\right)\geq0\ \mathbb{P
		\text{-a.s.
		\right\}.
	\end{equation}
	Moreover, $\inter\!\left(\dom\!\left(U\right)\right)=\left(0,+\infty\right)$.
\end{assumption}
\begin{remark}
Due to
monotonicity,
$U\!\left(\cdot\right)$ extends
in a natural way to $\left[\left.0,+\infty\right)\right.$ by setting
	$U\!\left(0\right)\coloneqq\lim_{x\rightarrow0^{+}}U\!\left(x\right)
, which is possibly $-\infty$.
Suppose further, without loss of generality, that $U\!\left(+\infty\right)\coloneqq\lim_{x\rightarrow+\infty}U\!\left(x\right)>0$.
Finally
note that
\ in addition to being
non-empty
and convex,
$\adm{\iw}$ is
closed
with respect to the subspace topology $\tau$ of the product topology on $\mathscr{L}
.
\end{remark}

Both 
existence theorems
of the current subsection
rely
on
the following
key lemm
, whose proof
proceeds
along the lines of that of \citet[Lemma~2.1]{rsch06}.
This resul
\ comes into play
in the construction of an
optimal portfoli
\ as it implies the boundedness of the admissible set $\adm{\iw}
, therefore
making it possible
to
employ
a compactness argument
to
extract
a candidate solution to~\eqref{eq:rEU
.
\begin{samepage}
\begin{lemma}\label[lem]{lem:key}
	Let $\iw>0$, and let \Cref{as:NA} hold. Moreover, let \Cref{as:positiver
	} hold.




	Then,
	for every
	$S\in\NAset$ and
	$t\in\left\{1,\ldots,T\right\}$, there exists an $\mathscr{F}_{t-1}$-measurable
	random variable $G^{S}_{t}>0$


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

	such tha
	\begin{equation}\label{eq:etoile}
		\esssup_{\phi\i
		\mathscr{L}\!\left(\iw,
		\right
		
		\norm
		\hat{\phi}_{t}^{S
		}\leq G^{S}_{t}<+\inft
		\ \mathbb{P}\text{-a.s.
	
		,
	\end{equation}
	where $\hat{\phi}_{t}^{S}\!\left(\omega\right)$ denotes the orthogonal projection of $\phi_{t}\!\left(\omega\right)$ on $D_{t}^{S}\!\left(\omega\right)$ for all $\omega\in\Omega$.
\end{lemma}

\begin{proof}
	See~\Cref{app:Main
	.
\end{proof}
\end{samepage}

\begin{remark}\label[obs]{obs:proj}
	For all $\phi\in\mathscr{L}\!\left(\iw,S\right)$ and $t\in\left\{1,\ldots,T\right\}$, we have 
	$\hat{\phi}_{t}^{S}\in\Lspace[d]{0}{}{\mathscr{F}_{t-1}}$ as well as 
	$\innerp{\hat{\phi}_{t}^{S}}{\Delta S_{t}}=\innerp{\phi_{t}}{\Delta S_{t}}$ $\mathbb{P}$-a.s.\ 
	(refer to \citet[Remark~3.4]{carassusrasonyi2012}), so replacing a portfolio with its orthogonal 
	projection does not alter its value. More importantly, when Assumption \ref{as:NA} holds
	and $S^{\ast}\in\mathcal{S}^{\ast}$
	then, for all $S\in\mathcal{S}$, we have $\innerp{\hat{\phi}_{t}^{S^{\ast}}}{\Delta S_{t}}=\innerp{\phi_{t}}{\Delta S_{t}}$
	almost surely since $D^{S}\subseteq D^{S^{\ast}}$.
\end{remark}

Our first main result states that, whenever the investors' utility is bounded from above,
not only is the robust utility maximisation problem~\eqref{eq:rEU} trivially well-pose
\ (i.e., $u\!\left(\iw\right)<+\infty$ for every initial capital $\i


, but an optimal strategy always exists.
\begin{theorem}\label[th]{th:Ubounded}
	Let $\iw>0$, and let \Cref{as:NA,as:U,as:positiver
	} hold. If

		$U\!\left(+\infty\right
	
		<+\infty$,

	then




	there exists $\opt\in\adm{\iw}$ such that
	\begin{equation*}
		u\!\left(\iw\right)=\inf_{S\in\mathscr{S}}\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{S}\!\left(\iw,\opt\right)\right)\right
		<+\inft
		.
	\end{equation*}
\end{theorem}

\begin{proof}
	See~\Cref{app:Main
	.
\end{proof}

Alternatively, suppose that investors can derive an arbitrarily high level of satisfaction as wealth grows arbitrarily larg
.
In order to still
be able to
control from above the robust expected utility of terminal wealth,
and thus obtain a solution for~\eqref{eq:rEU},
we
make the additional assumption~\eqref{eq:intW}
concerning
the dynamics of \emph{all} possible price
\ combined with
a stronger version of absence of arbitrag
\ in at least
\emph{one} of the model
.
\begin{theorem}\label[th]{th:intW}
	Let $\iw>0$, and let \Cref{as:NA,as:U,as:positiver
	} be valid.
	Also, define
	\begin{equation}\label{eq:W}
		\mathscr{W}\coloneqq\left\{X\in\Lspace[]{0}{}{\mathscr{F}_{T}}:~\forall\,p>0,\ \mathbb{E}_{\mathbb{P}}\!\left[\left\vert X\right\vert^{p}\right]<+\infty\right\}.
	\end{equation}
	Assume further that


	\begin{equation}\label{eq:intW}
	
	
	
	
		\left\{\norm{\Delta S_{t}}:~S\in\mathscr{S}\right\}\subseteq\mathscr{W
		\text{ and 
	
		\frac{1}{\beta^{S^{\ast}}_{t}
		\in\mathscr{W
	
	
		\ \text{for all }t\in\left\{1,\ldots,T\right\}
	\end{equation}
	holds
	for some $S^{\ast}\in\NAset$,
	where each $\beta^{S^{\ast
	}_{t}$ is the random variable given by \Cref{prop:NA}




	.
	Then




	there exists $\opt\in\adm{\iw}$ such that
	\begin{equation*}
		u\!\left(\iw\right)=\inf_{S\in\mathscr{S}}\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{S}\!\left(\iw,\opt\right)\right)\right
		<+\inft
		.
	\end{equation*}
\end{theorem}

\begin{proof}
	See~\Cref{app:Main
	.
\end{proof}

\begin{remark}
	We may wonder what happens if we relax the bankruptcy prohibition at all intermediate times by 
	imposing that the admissible strategies for each price process $S\in\mathscr{S}$ yield 
	non-negative wealth only at the terminal time.\footnote{We thank the referee for posing this question.
	\ In other words, instead of $\adm{\iw}$, we can consider the larger set
	\begin{equation*}
		\admE{\iw}\coloneqq\bigcap_{S\in\mathscr{S}}\tilde{\mathscr{L}}\!\left(\iw,S\right),
	\end{equation*}
	where
	\begin{equation}\label{eq:LE}
		\tilde{\mathscr{L}}\!\left(\iw,S\right)\coloneqq\left\{\phi\in\mathscr{L}:~W_{T}^{S}\!\left(\iw,\phi\right)\geq0\ \mathbb{P
		\text{-a.s.
		\right\}.
	\end{equation}
	It is not difficult to see that, under \Cref{as:NA},
	$\tilde{\mathscr{L}}\!\left(\iw,S\right)=\mathscr{L}\!\left(\iw,S\right)$ for all $S\in\NAset

	. Indeed, if we let $\phi\in\mathscr{L}$ with $W_{T}^{S}\!\left(\iw,\phi\right)\geq0$ $\mathbb{P}$-a.s., then we know by the Dalang-Morton-Willinger theorem~\citep{dalang1990} combined with \citet[Theorem~2]{jacod98} that the wealth process $\left\{W^{S^{\ast}}_{t}\!\left(\iw,\phi\right)\right\}_{t\in\left\{0,1,\ldots,T\right\}}$ is a true martingale under some equivalent martingale measure. The desired conclusion follows from the observation that a martingale with non-negative terminal value is non-negative.

	Additionally,
	we can easily check

	that, if we replace $\adm{\iw}$ by $\admE{\iw}$ in \Cref{th:Ubounded,th:intW}, the results remain valid with the same proof
	\ (the key fact being that the two sets of feasible portfolios actually coincide for all arbitrage-free price models, as shown above
	. However, the value of the robust utility maximization problem over these two domains may well differ; below we provide an example where this happens.

	Let $T\coloneqq2$,
	and let 
	
	$ be a standard Gaussian. Consider also the random variables $\varepsilon_{i}$, $i\in\left\{1
	
	\right\}$ such that





	\begin{align*}
		&\mathbb{P}\!\left\{\varepsilon_{1}=-1/2\right\}=\mathbb{P}\!\left\{\varepsilon_{1}=4\right\}=1/2,\\
		&\mathbb{P}\!\left\{\varepsilon_{2}=-1/2\right\}=\mathbb{P}\!\left\{\varepsilon_{2}=1/2\right\}=1/2,
	\end{align*}
	and


	$X,\varepsilon_{1},\varepsilon_{2}$
	are independent. Moreover, the filtration $\mathbb{F}$ is defined by setting $\mathscr{F}_{0}$ to be the family of $\mathbb{P}$-null sets, $\mathscr{F}_{1}\coloneqq\mathscr{F}_{0}\vee\sigma\!\left(X,\varepsilon_{1}\right)$, and $\mathscr{F}_{2}\coloneqq\mathscr{F}_{1}\vee \sigma(\varepsilon_{2})$.\footnote{Here, $\mathscr{G}\vee\mathscr{H}$ denotes the $\sigma$-algebra generated by the union of the two $\sigma$-algebras $\mathscr{G}$ and $\mathscr{H}$.

	We consider one risky asset, i.e.\ $d=1$.
	Our family of models has $3$ elements, $\mathscr{S}=\left\{\hat{S},\tilde{S},\bar{S}\right\}$ defined as follows,
	\begin{alignat*}{3}
		&\Delta S^{\ast}_{1}= \varepsilon_{1},\qquad &&\Delta S^{\ast}_{2}= \varepsilon_{2},\\
		&\Delta \tilde{S}_{1}= X,\qquad &&\Delta \tilde{S}_{2}= 3-X,\\
		&\Delta \bar{S}_{1}= 3,\qquad &&\Delta \bar{S}_{2}= 0.
	\end{alignat*}
	Clearly, $D_{1}^{S^{\ast}}=D_{2}^{S^{\ast}}=\mathbb{R}$ hence \Cref{as:NA}
	is testified by $S^{\ast}\in\NAset$.

	Take the bounded utility $U\!\left(x\right)\coloneqq\min\{\sqrt{x},2\}$, for all $x\i
	\left
	0,+\inft
	\right)$,
	and set the initial capital to $\iw=1$.
	Each portfolio strategy $\phi=(\phi_{1},\phi_{2})$
	consists of
	a deterministic number $\phi_{1}\in\mathbb{R}$, and
	an $\mathscr{F}_{1}$-measurable real-valued random variable $\phi_{2}$.

	Notice that $\phi\in\adm{1}$ implies $\phi_{1}=0$, since $X$ is unbounded both from above and from below.
	Hence,
	by the construction of $\bar{S}$,
	\begin{equation*}
		u\!\left(1\right)
		\sup_{\phi\in\adm{1}}\inf_{S\in\mathscr{S}}\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{S}\!\left(1,\phi\right)\right)\right
		\leq\sup_{\phi\in\adm{1}}\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{\bar{S}}\!\left(1,\phi\right)\right)\right
		=U\!\left(1\right)=1.
	\end{equation*}

	On the other hand, choosing the strategy $\tilde{\phi}$ with $\tilde{\phi}_{1}=\tilde{\phi}_{2}=1$ gives $W_{T}^{S}\!\left(1,\tilde{\phi}\right)\geq 0$ for all $S\in\mathscr{S}$, thus $\tilde{\phi}\in\admE{1}$. Moreover,
	we can directly check that, for all $S\in\mathscr{S}$,
	\begin{equation*}
		\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{S}\!\left(1,\tilde{\phi}\right)\right)\right]\geq\frac{5}{4
		,
	\end{equation*}

	which
	in turn results in
	\begin{equation*}
		\tilde{u}\!\left(1\right)
		\sup_{\phi\in\admE{1}}\inf_{S\in\mathscr{S}}\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{S}\!\left(1,\phi\right)\right)\right]>1.
	\end{equation*}
\end{remark}

\subsection{Utility on the whole real line}\label[subsec]{subsec:wholerl}

Consider now
the possibility that wealth can
become negativ
, meaning that
the utility function takes on finite values everywhere on the real lin
.
In other word
, no constraints
are
imposed on portfolio choice, hence any self-financing investment strategy is admissible.
\begin{assumption}\label[as]{as:wholerl}
	For all $S\in\mathscr{S}$,

	
		$\mathscr{L}\!\left(\iw,S\right)\coloneqq\mathscr{L}
		.

	Moreover, $\dom\!\left(U\right)=\mathbb{R}$.
\end{assumption


The next result
extends our \Cref{th:Ubounded} to the whole real lin
\ and
sho
s
that there is also
an optimal portfolio in this setting.
\begin{samepage}
\begin{theorem}\label[th]{th:Ureal}
	Let $\iw\in\mathbb{R}$, and let \Cref{as:NA,as:U,as:wholer
	} hold.
	If

		$U\!\left(+\infty\right
	
		<+\infty$,

	then there exists $\opt\i
	\mathscr{L
	$ such that
	\begin{equation*}
		u\!\left(\iw\right
		=\inf_{S\in\mathscr{S}}\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{S}\!\left(\iw,\opt\right)\right)\right
		<+\inft
		.
	\end{equation*}
\end{theorem}

\begin{proof}
	See~\Cref{app:Main
	.
\end{proof}
\end{samepage}

In our last theorem, we remove the hypothesis of $U\!\left(\cdot\right)$ being bounded from above. This comes at the expense of assuming that \emph{each} $S\in\mathscr{S}$ should be arbitrage-free, in a strong sens
;
we further point out that
\eqref{eq:lune} bears a close resemblance to the conditions of \Cref{th:intW} above as well as those of \citet[Theorem~3.6]{blanchard-carassus2017}.
Likewise,
dealing with the integrability issues that arise from $U\!\left(+\infty\right)=+\infty$ requires a
supplementary
growth condition on the utility for large positive values of wealt
---namely, that the function displays strictly sub-linear growt
.
Nonetheless,
\eqref{maruszja} is still quite mild; in fact, it is even slightly
weaker than the reasonable asymptotic elasticity property at $+\infty$ (see \citet{kramkov99}).
As far as we know, this is the first result in the literature that treats, in a robust setting,
an unbounded utility function defined on the whole real line.

\begin{remark}\label[obs]{obs:AdmR}
	A simple, yet important observation
	underpinning the proof of \Cref{th:UrealU} below
	is that $u\!\left(\iw\right)>-\infty$ (recall \Cref{obs:u>-infty}) along with the convention $+\infty-\infty\coloneqq-\infty$
	leads to the equality

		$u\!\left(\iw\right)=\sup_{\varphi\in\admR{\iw}}\inf_{S\in\mathscr{S}}\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{S}\!\left(\iw,\varphi\right)\right)\right]$,

	wher
	\footnote{Hereafter,
		\begin{equation*}
			U^{\pm}\!\left(x\right)\coloneqq\max\!\left\{\pm U\!\left(x\right),0\right\},\quad\text{for all }x\i

			\dom\!\left(U\right
			,
		\end{equation*}
		denote the positive and negative parts of $U\!\left(\cdot\right)$, respectively.}
	\begin{equation*
		\admR{\iw}\coloneqq\bigcap_{S\in\mathscr{S}}\left\{\phi\in\mathscr{L}:~\mathbb{E}_{\mathbb{P}}\left[U^{-}\!\left(W^{S}_{T}\!\left(w_{0},\phi\right)\right)\right]<+\infty\right\}\neq\varnothing
	\end{equation*}
	This means that any investment leading to infinite disappointment

	in at least one possible model is automatically excluded from the robust portfolio problem as it can never be optimal, thereby effectively restricting the set of admissible controls from $\mathscr{L}$ to $\admR{\iw}
	.\footnote{We thank the referee for drawing our attention to this point.
\end{remark}

\begin{samepage}
\begin{theorem}\label[th]{th:UrealU}
	Let $\iw\in\mathbb{R}$, and let \Cref{as:NA,as:U
	as:wholer
	}
	hol
	.

	If
	the following three
	conditions are all
	satisfied:
	\begin{enumerate}[label=\emph{(\roman*)}]
		\item
		$\mathscr{S}=\mathscr{S}^{\ast}$;
		
		\item
		there are $C>0$ and $\alpha\in\left[\left.0,1\right)\right.$ such that 
		\begin{equation}\label{maruszja}
			U\!\left(x\right)\leq C\left(x^{\alpha}+1\right)\
		
			\text{for all }x\geq
			;
		\end{equation}
	
		
		\item
	
		for all $S\in\mathscr{S}$,
		\begin{equation}\label{eq:lune}
		
			\frac{1}{\beta^S_t},\frac{1}{\kappa^S_t
			,\norm{\Delta S_{t}}\in\mathscr{W
			\ \text{for all }t\in\left\{1,\ldots,T\right\
			,
		\end{equation}
		where $\beta^S_t$, $\kappa^S_t$ are the random variables given by \Cref{prop:NA}, and $\mathscr{W}$ is as in~\eqref{eq:W
		;
	\end{enumerate}		
	then there exists $\opt\in\adm{\iw}$ such that
	\begin{equation*}
		u\!\left(\iw\right
		=\inf_{S\in\mathscr{S}}\mathbb{E}_{\mathbb{P}}\!\left[U\!\left(W_{T}^{S}\!\left(\iw,\opt\right)\right)\right
		<+\inft
		.
	\end{equation*}
\end{theorem}

\begin{proof}
	See~\Cref{app:Main
	.
\end{proof}
\end{samepage}

\begin{remark}
	At this point, let us briefly look at the case of ``model-free'' utility maximisation; see e.g.\ \citet{acciaio2013}, where $\mathscr{S}$ consists of all the adapted price processes (or just the ones satisfying no arbitrage). A moment's reflection shows that, in those cases, only the identically $0$ strategy is admissible in the case of $U\!\left(\cdot\right)$ on the positive real axis. In the case of $U\!\left(\cdot\right)$ defined on the whole real line, every strategy except $0$ yields utility $-\infty$. In other words, the problem
becomes trivial.

Another model-free approach has been outlined in \citet{bfm2016}.
However, it concentrates  uniquely on arbitrage, and probabilities do not appear
in the natural way (i.e.\ their set $\mathscr{S}$ does not determine a set of probabilities).
In the special case $\mathscr{S}=\{\Omega\}$, the set of \emph{all} probabilities
appears naturally, and we already addressed this case in the previous paragraph.
For all other cases of \citet{bfm2016} we do not see even how the utility maximization
problem could be properly formulated. 
\end{remark}


\section{Examples}\label[sec]{sec:Example}





\begin{example}\label[ex]{toxun}
When prices are assumed to be continuous-time It\^{o} processes, uncertainty in the drift and volatility are the classical setting for model robustness. This has been fervently studied, we refer to the seminal paper of \citet{epsteinji2014}. We sketch a simple discrete-time analogue below.

	Consider the $t$-fold Cartesian product $\Omega_{t}\coloneqq\left\{-1,1\right\}^{t}$ for all $t\in\left\{0,1,\ldots,T\right\}$, and denote by $\pow{\Omega_{t}}$ the power set of $\Omega_{t}$. 
	Set also $\Omega\coloneqq\Omega_T$.
	Define the mappings $\Pi_{t}:~\Omeg
	\rightarrow\Omega_{t}$ as
	\begin{equation*}
		\Pi_{t}\!\left(\omega\right)\coloneqq\left(\omega_{1},\ldots,\omega_{t}\right),\quad\text{for all }\omega=\left(\omega_{1},\ldots,\omega_{t},\ldots,\omega_{T}\right)\in\Omeg
		,
	\end{equation*}
	and the filtration $\mathbb{F}$ as $\mathscr{F}_{t}\coloneqq\Pi_{t}^{-1}\!\left(\pow{\Omega_{t}}\right)$.
	Fix $p\in\left(0,1\right)$, and equip the measurable space $\left(\Omega,2^\Omega\right)$ with the probability measure defined by
	\begin{equation}\label{ex:probP}
		\mathbb{P}\!\left(\left\{\omega\right\}\right)\coloneqq\prod_{t=1}^{T}\left(p\,\delta_{1}\!\left(\left\{\proj_{t}\!\left(\omega\right)\right\}\right)+\left(1-p\right)\delta_{-1}\!\left(\left\{\proj_{t}\!\left(\omega\right)\right\}\right)\right),\quad\text{for all 
		\omega\in\Omeg
		,
	\end{equation}


	where
	$\delta_{x}$ is the Dirac measure at $x\in\mathbb{R}
	,\footnote
	Given $x\in\mathbb{R}$, the \emph{Dirac measure} at $x$ is the probability measure $\delta_{x}:~\mathscr{B}\!\left(\mathbb{R}\right)\rightarrow\left\{0,1\right\}$ defined by
	\begin{equation*}
		\delta_{x}\!\left(B\right)\coloneqq\left\{
			\begin{array}{ll}
				1,& \text{if }x\in B,\\
				0,& \text{otherwise}.
			\end{array}
		\right.
	\end{equation*}
	}
	and $\proj_{t}$ is the $t$-th projection map from $\Omega$ to $\left\{-1,1\right\}$.\footnote{Given a Cartesian product $X_{1}\times\ldots\times X_{n}$ and $k\in\left\{1,\ldots,n\right\}$, the $k$-th \emph{projection map} $\proj_{k}:~X_{1}\times\ldots\times X_{n}\rightarrow X_{k}$ is defined by
	\begin{equation*}
		\proj_{k}\!\left(x_{1},\ldots,x_{n}\right)\coloneqq x_{k},\quad\text{for all }\left(x_{1},\ldots,x_{n}\right)\in X_{1}\times\ldots\times X_{n}.
	\end{equation*}}
	Moreover,
	let $s_{0}\in\mathbb{R}$ be given, and
	introduce the parameter space



	$\Theta\coloneqq\left[\ubar{\sigma},\bar{\sigma}\right]^{T}\times\mathbb{R}^{T}$ for some $0<\ubar{\sigma}<\bar{\sigma}$.
	For every $\theta=\left(\sigma,\mu\right)\in\Theta$,
	define the real-valued process $S^{\theta}=\left\{S^{\theta}_{t}\right\}_{t\in\left\{0,1,\ldots,T\right\}}$ as
	\begin{align*}
	
	
	
		S^{\theta}_{t}\!\left(\omega\right)\coloneqq
		s_{0}+\sum_{s=1}^{t}\left(\sigma\proj_{s}\!\left(\omega\right)+\mu\right
		,\quad\text{for all }\omega\in\Omega,\ t\in\left\{0
		1,\ldots,T\right\}.
	\end{align*}







	When $p=1/2$ then this model is a possible discrete-time analogue of the continuous-time Bachelier model.
	
	Even though arbitrage opportunities exist for certain price processe
	, it is easily seen that
	each $S^{\theta}\in\mathscr{S}$ with $\left\vert\mu\right\vert<\sigma$ 

	admits a unique equivalent martingale measure given by
	\begin{equation*}
		\mathbb{Q}^{\theta}\!\left(\left\{\omega\right\}\right)\coloneqq2^{-T}\prod_{t=1}^{T}\left(1-\proj_{t}\!\left(\omega\right)\frac{\mu}{\sigma}\right),\quad\text{for all }\omega\in\Omega.
	\end{equation*}
	In addition, $S^{\theta}$ has independent increments for every $\theta\in\Theta$, which
	entails that condition~\eqref{eq:intW} holds and
	\eqref{eq:lune} is satisfied by any arbitrage-free process in $\mathscr{S}
	, as observed by \citet[Proposition~7.1]{rs05
	.
	Note also that $\mathscr{S}$ may contain price processes whose laws are mutually singular.
	
	Consider now $\mathscr{P}\coloneqq\left\{\Law\!\left(S\right):~S\in\mathscr{S}\right\}$, which is a subset of the space of all probability measures on $\left(\mathbb{R},\mathscr{B}\!\left(\mathbb{R}\right)\right)$.
	This simple model class
	is a typical example of model-misspecification where the ``volatility'' parameter $\sigma$ and the drift parameter $\mu$ are unknown. It can be treated in our framework.
	It does not, however, satisfy the condition on $\mathscr{P}$ in \citet[Section~3]{nutz2016} or \citet[Section~2]{bartl2017}; namely, the family of possible priors is not convex (as convexity here would mean being closed under taking mixtures of probabilities).

One could consider various generalizations of this example: $\mu_{t}$ and $\sigma_{t}$ could
be time-dependent and even $\mathscr{F}_{t-1}$-measurable; taking $\Omega:=\mathbb{R}^{T}$, we
could consider more general noise processes with possibly continuous laws, etc.
\end{example}

\begin{example}
	Most papers in the standard multiple-priors framework (e.g.\ \citet{nutz2016,bouchardnutz2015,bartl2017}) assume the ``time-consistency'' property.

	It is
	possible to construct an example where the corresponding $\mathscr{P}$ fails to satisfy

	this property, but can be treated in our framewor
	. Recall also \citet{bck2019}, who could drop time-consistency using additional set-theoretical axioms.
	
	Fix $T\coloneqq 2$ and $d=1$ (i.e., consider a single risky asset). Take $\Omega_{1}=\mathbb{R}$
	and $\Omega\coloneqq\Omega_{2}=\mathbb{R}^{2}$.
	Let $\varepsilon_{i}$, $i\in\left\{1,2\right\}$ be two independent and identically distributed random variables with
	$\mathbb{P}\!\left\{\varepsilon_{1}=\pm 1\right\}=1/2

	, and define the filtration $\mathbb{F}$ such that $\mathscr{F}_{0}$ is the $\sigma$-algebra of $\mathbb{P}$-zero sets, $\mathscr{F}_{1}\coloneqq\sigma\!\left(\varepsilon_{1}\right)$, and $\mathscr{F}_{2}\coloneqq\sigma\!\left(\varepsilon_{1},\varepsilon_{2}\right)$.
	
	Let $s_{0
	=0$, and
	for each $\theta=\left(\mu_{1},\mu_{2}\right)\in\mathbb{R}^{2}$ define the real-valued price process $S^{\theta}=\left\{S_{t}^{\theta}\right\}_{t\in\left\{0,1,2\right\}}$ as

	\begin{equation*}
		S_{t}^{\theta}\coloneqq s_{0}+\sum_{s=1}^{t}\left(\mu_{s}+\varepsilon_{s}\right),\quad\text{for all }t\in\left\{0,1,2\right\},
	\end{equation*}





	Assume further that there are only two possible price models; namely,
	$\mathscr{S}=\left\{S^{\theta}:~\theta\in\Theta\right\}$, where $\Theta=\left\{\theta_{1}=\left(0.1,0.3\right),\theta_{2}=\left(0.2,0.5\right)\right\}$. Note that $\mathscr{S}=\NAset$.
	Lastly,
	introduce the notation $\mathscr{P}\coloneqq\left\{\Law\!\left(
	\right):~
	\in\mathscr{S}\right\}$.


	Time-consistency means that there are collections of probabilities $\mathscr{P}_{0}$ and $\mathscr{P}_{1
	$ on $\mathbb{R}$ such that $\mathscr{P}$
	consists of
	all probability measures $\mathbb{P}$ on $\mathbb{R}^{2}$ admitting the decompositio
	\footnote{Here, $\mathscr{B}\!\left(\mathbb{R}^{2}\right)$ denotes the Borel $\sigma$-algebra of $\mathbb{R}^{2}$ (i.e., the smallest $\sigma$-algebra containing all open subsets of $\mathbb{R}^{2}$).}
	\begin{equation*}
		\mathbb{P}\!\left(A\right)=\int_{\mathbb{R}}\int_{\mathbb{R}}\bbOne_{A}\!\left(\omega_{1},\omega_{2}\right)\,P_{1}\!\left(\omega_{1},d\omega_{2}\right)\mathbb{P}_{0}\!\left(d\omega_{1}\right)\text{ for all }A\in\mathscr{B}\!\left(\mathbb{R}^{2}\right)
	\end{equation*}
	
	$\mathbb{P}=P_{0}\otimes P_{1}$ in abbreviated form
	
	\ for some
	$P_{0}\in\mathscr{P}_{0}$ and
	$P_{1
	$
	satisfying $P_{1}\!\left(\omega\right)\in\mathscr{P}_{1}\!\left(\omega\right)$ for all $\omega\in\mathbb{R}$;
	see \citet{bouchardnutz2015} 


	.
	
	Next, observe that, for every $\theta=\left(\mu_{1},\mu_{2}\right)\in\mathbb{R}^{2}$, the stock prices $S^{\theta}_{1}$ and $S^{\theta}_{2}$ have discrete laws with supports $\left\{s_{0}+\mu_{1}-1,s_{0}+\mu_{1}+1\right\}$ and $\left\{s_{0}+\mu_{1}+\mu_{2}-2,s_{0}+\mu_{1}+\mu_{2},s_{0}+\mu_{1}+\mu_{2}+2\right\}$, respectively; moreover,

	\begin{alignat*}{3}
		&\mathbb{P}\!\left\{S^{\theta}_{1}=s_{0}+\mu_{1}\pm1\right\}=\frac{1}{2},\qquad &&\\
		&\mathbb{P}\!\left\{S^{\theta}_{2}=s_{0}+\mu_{1}+\mu_{2}\pm2\right\}=\frac{1}{4},\qquad &&\mathbb{P}\!\left\{S^{\theta}_{2}=s_{0}+\mu_{1}+\mu_{2}\right\}=\frac{1}{2}.
	\end{alignat*}
	
	Assume, with a view to a contradiction, that time-consistency holds.
	Denoting by $\mathbb{P}_{i}$ the probability law of $S^{\theta_{i}}$ for $i\in\left\{1,2\right\}$,
	straightforward computations yield
	\begin{align*}
		&\mathbb{P}_{1}=P_{0}^{1}\otimes P_{1}^{1},\\
		&\mathbb{P}_{2}=P_{0}^{2}\otimes P_{1}^{2},
	\end{align*}
	where $P_{0}^{1}\in\mathscr{P}_{0}$ is the discrete distribution with
	$P_{0}^{1}\!\left\{0.1\pm1\right\}=1/2$;
	$P_{1}^{1}$ is such that
	$P_{1}^{1}\!\left(0.1\pm1\right)\in\mathscr{P}_{1}\!\left(0.1\pm1\right)$ are the discrete distributions with
	$P_{1}^{1}\!\left(0.1\pm1,\left\{0.1+0.3\pm2\right\}\right)=1/4$ and $P_{1}^{1}\!\left(0.1\pm1,\left\{0.1+0.3\right\}\right)=1/2$;
	$P_{0}^{2}\in\mathscr{P}_{0}$ is the discrete distribution with
	$P_{0}^{2}\!\left\{0.2\pm1\right\}=1/2$; and
	$P_{1}^{2}$ is such that
	$P_{1}^{2}\!\left(0.2\pm1\right)\in\mathscr{P}_{1}\!\left(0.2\pm1\right)$ are the discrete distributions with
	$P_{1}^{2}\!\left(0.2\pm1,\left\{0.2+0.5\pm2\right\}\right)=1/4$ and $P_{1}^{2}\!\left(0.1\pm1,\left\{0.2+0.5\right\}\right)=1/2$.
	
	However, $\bar{\mathbb{P}}\coloneqq P_{0}^{1}\otimes P_{1}^{2}$ is not an element of $\mathscr{P}$, a contradiction.


	A different example can be found in
	\citet[Appendix~D, Example~3]{riedel09}.
\end{example}

\begin{example}\label[ex]{horus}
This is a much simplified version of Example~2.7 of 
\citet{bartl2017}.
Let us fix $s_{0}\in\mathbb{R}$ and $\sigma>0$, set
	\begin{equation*}
		S_{t}\!\left(\omega\right)\coloneqq
		s_{0}+\sigma\sum_{s=1}^{t}\proj_{s}\!\left(\omega\right
		,\quad\text{for all }\omega\in\Omega,\ t\in\left\{0
		1,\ldots,T\right\},
	\end{equation*}
	and define $\mathscr{P}\coloneqq\left\{\mathbb{P}^{p}:~p\in\left(0,1\right)\right\}$
	where each $\mathbb{P}^{p}$ is of the form~\eqref{ex:probP}. While $\mathscr{P}$ satisfies 
	the conditions of \citet{nutz2016}, we cannot incorporate this example into our framework in a natural 
	way. Indeed, one would need an information filtration $\mathbb{F}=\left\{\mathscr{F}_{t}\right\}_{t\in\left\{0,1,\ldots,T\right\}}$ such that	the price increments (which can be constant times Bernoulli with arbitrary parameters)
	are measurable with respect to it. This is feasible only if $\mathbb{F
	$ is
	generated by continuous random variables. This is, however, unnatural, as
	the investment decision at time $t+1$ should be measurable with respect to the
	price movements up to $t$, which are \emph{discrete-valued} random variables.
	Hence there is no way of incorporating this example into our framework in a
	meaningful way.





\end{example}

\Cref{toxun,horus} show that, while the usual approach is capable of capturing uncertainty at the level of the 
``probabilistic skeleton'' of the process (e.g.\ probabilities on a tree), our approach can be successful 
for various parametrisations when the filtration (the information structure) is kept fixed.

Below
is
another, ``non-parametric'' example that can be covered by our method.

\begin{example}
	On a fixed probability space $\left(\Omega,\mathscr{F},\mathbb{P}\right)$, consider a collection $\left\{\varepsilon_{1},\ldots,\varepsilon_{T}\right\}$ of independent and standard uniform random variables. Define the filtration $\mathbb{F
	$ as $\mathscr{F}_{t}\coloneqq\sigma\left(\varepsilon_{1},\ldots,\varepsilon_{t}\right)$ for all $t\in\left\{0,1,\ldots,T\right\}$, let $s_{0}\in\mathbb{R}$ be given, and denote by $\Theta$ the set of $T$-tuples of $\mathscr{F}_{t-1}\otimes\mathscr{B}\!\left(\left[0,1\right]\right)$-measurable functions $\theta_{t}:~\Omega\times\left[0,1\right]\rightarrow\mathbb{R}$, $t\in\left\{1,\ldots,T\right\}$ that are non-decreasing in their second variable.

	For every $\theta=(\theta_{1},\ldots,\theta_{T})\in\Theta$, let $S^{\theta}=\{S^{\theta}_{t}\}_{t\in\left\{0,1,\ldots,T\right\}}$ be the real-valued process given by
	\begin{equation*}
		S^{\theta}_{t}\!\left(\omega\right)\coloneqq s_{0}+\sum_{s=1}^{t}\theta_{s}\!\left(\omega,\varepsilon_{s}\!\left(\omega\right)\right),\quad\text{for all }\omega\in\Omega, t\in\left\{0
		1,\ldots,T\right\}.
	\end{equation*}


	Let now $\mathscr{S}\subseteq\left\{S^{\theta}:\ \theta\in\Theta\right\}$. If $\mathscr{S}$ contains $S^{\theta^{\ast}}$ for some $\theta^{\ast}$ such that
	$\theta^{\ast}_{t}\!\left(\omega,\cdot\right)$ takes on positive as well as negative values with positive Lebesgue measure for all $t$ and $\omega
	, then $S^{l^{\ast}}$	clearly satisfies the no-arbitrage assumption as well as \eqref{eq:nonred}. Notice that an arbitrary probability law on $\mathbb{R}$ can be represented by a random variable that is a non-decreasing function of a standard uniform random variable.

We now show that this example subsumes Example~2.8 of \citet{bartl2017}, except that we prefer to use an additive dynamics instead of the multiplicative one of \citet{bartl2017}.
For every $t\in\left\{1,\ldots,T\right\}$,
fix $\mathscr{F}_{t-1}$-measurable real-valued random variables $\underline{\mu}_{t}\leq \overline{\mu}_{t}$,
and positive $\mathscr{F}_{t-1}$-measurable random variables $\sigma_{t},\epsilon_{t}$. Now we can define $\mathscr{S}$ as the set of price models $S=\left\{S_{t}\right\}_{t\in\left\{0,1,\ldots,T\right\}}$ satisfying $S_{0}\coloneqq s_{0}\in\mathbb{R}$ and
\begin{equation*}
	S_{t}=S_{t-1}+\mu_{t}+Y_{t},\quad\text{for all }t\in\left\{1,\ldots,T\right\},
\end{equation*}
where $\mu_{t}$ ranges over $\left[\underline{\mu}_{t},\overline{\mu}_{t}\right]$-valued $\mathscr{F}_{t-1}$-measurable random variables, and $Y_{t}$ ranges over the random variables whose conditional law with respect to $\mathscr{F}_{t-1}$ is $\epsilon_{t}\!\left(\omega\right)$-close to the Gaussian law with mean $0$ and variance $\sigma_{t}\!\left(\omega\right)$, for almost every $\omega$, in some suitable distance.
We do not enter into further technical detail

\end{example}

\begin{example}[Option trading]
Let $Z_{t}$, $t\in\left\{1,\ldots,T\right\}$ be $\mathbb{R}^{m}$-valued random variables on some probability space $\left(\Omega,\mathscr{F},\mathbb{P}\right)$, representing the prices of $m$ stocks, and let $Z_{0}\in\mathbb{R}^{m}$ be deterministic. Let $\mathscr{F}_{0}$ be the set of $\mathbb{P}$-zero sets, and let
\begin{equation*}
	\mathscr{F}_{t}\coloneqq\mathscr{F}_{0}\vee\sigma\!\left(Z_{s},\ s\in\left\{1,\ldots,t\right\}\right)
\end{equation*}
for $t\in\left\{1,\ldots,T\right\}$.
We assume that \hyperlink{eq:NAS}{NA($Z$)} holds, hence there is at least one probability, equivalent to $\mathbb{P}$, under which $Z$ is a martingale. Denote by $\mathscr{Q}$ the set of all such probabilities.
Let $G$ be an $\mathbb{R}^{d}$-valued $\mathscr{F}_{T}$-measurable random variable, representing the terminal value of $d$ options written on these stocks, and let $g\in\mathbb{R}^{d}$ be a vector representing the (known) prices of these options at time $0$.

Assume that $\mathscr{Q}\!\left(g\right)\coloneqq\left\{\mathbb{Q}\in\mathscr{Q}:~\mathbb{E}_{\mathbb{Q}}\!\left[G\right]=g\right\}\neq\varnothing$, and
fix some non-empty $\mathscr{R}\subset\mathscr{Q}\!\left(g\right)$. Define
\begin{equation*}
	\mathscr{S}\coloneqq\left\{\mathbb{E}_{\mathbb{Q}}\!\left[\left.G\right\vert\mathscr{F}_{t}\right],\ t\in\left\{0,\ldots,T\right\}:\ \mathbb{Q}\in\mathscr{R
	\right\},
\end{equation*}
the collection of possible option price processes under the various risk-neutral probabilities $\mathbb{Q}\in\mathscr{R}$.

Results of our paper apply to the above setting where there is uncertainty about the pricing measure and the investor with worst-case robust preferences tries to optimally trade in these options.
\end{example}






\section{Conclusion}\label[sec]{sec:Conc}


This paper provides
a positive answer to the question of existence of solutions to the worst-case 
robust utility maximisation problem in an alternative framework for model uncertainty.
It is desirable to extend the arguments of this
work
to continuous-time model
, as done in the companion paper of \citet{chaurasonyi2018} in a market where proportional transaction costs are presen
.

It
is also worth investigating
whether there exists a price process $S_{0}\in\mathscr{S}$ with respect to which the standard utility maximisation problem is equivalent to the robust utility maximisation proble
\ (by analogy with e.g.\ the least favourable measure in \citet{schied2005}).

The fundamental questions of arbitrage and superhedging have been intensively studied in the presence of
multiple priors, see \Cref{obs:RNA} above, as well as \cite{acciaio2013,bouchardnutz2015,bfm2016,bfhmo2018}
and the references therein. These questions have not yet been addressed in the setting of the present work, and will be the subject of future research.











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