# Cost-efficient Payoffs under Model Ambiguity

Carole Bernard,
Gero Junike
Thibaut Lux
Steven Vanduffel Corresponding author: Carole Bernard, Department of
Accounting, Law and Finance at Grenoble Ecole de Management (email:
<EMAIL_ADDRESS>and Department of Economics and Political
Sciences at Vrije Universiteit Brussel (VUB). Institute of Mathematics, Carl
von Ossietzky University, 26111 Oldenburg, Germany (e-mail:
gero.junike@uol.de). Helvetia Insurance Group (e-mail:
thibaut.lux@helvetia.ch). Steven Vanduffel, Department of Economics and
Political Sciences at Vrije Universiteit Brussel (VUB). (e-mail:
steven.vanduffel@vub.ac.be). We thank Ruodu Wang for constructive comments on
an earlier version. The views expressed in this manuscript do not represent
the opinions of Helvetia Insurance Group. This work was supported by the FWO
projects G015320N and S006721N.

(August 27, 2024)

###### Abstract

Dybvig (1988a, b) solves in a complete market setting the problem of finding a
payoff that is cheapest possible in reaching a given target distribution
(“cost-efficient payoff”). In the presence of ambiguity, the distribution of a
payoff is, however, no longer known with certainty. We study the problem of
finding the cheapest possible payoff whose worst-case distribution
stochastically dominates a given target distribution (“robust cost-efficient
payoff”) and determine solutions under certain conditions. We study the link
between “robust cost-efficiency” and the maxmin expected utility setting of
Gilboa and Schmeidler, as well as more generally with robust preferences in a
possibly non-expected utility setting. Specifically, we show that solutions to
maxmin robust expected utility are necessarily robust cost-efficient. We
illustrate our study with examples involving uncertainty both on the drift and
on the volatility of the risky asset.

KEYWORDS: Cost-efficient payoffs, model ambiguity, maxmin utility, robust
preferences, drift and volatility uncertainty
JEL classification: C02, C63, D80
AMS classification: 91B30, 62E17

##  1 Introduction

In a (complete) market without ambiguity, Dybvig (1988a, b) characterizes
optimal payoffs for agents having law-invariant increasing preferences (e.g.,
expected utility maximizers). His result is based on the observation that any
optimal payoff $X$ must be cost-efficient in the sense that there cannot exist
another payoff with the same probability distribution that is strictly cheaper
than $X$. He then derives, for a given target distribution of terminal wealth,
the payoff that achieves this target distribution at the lowest possible cost
(cost-efficient payoff). Optimal portfolios are thus driven by distributional
constraints rather than appearing as a solution to some optimal expected
utility problem. In this regard, Brennan and Solanki (1981) note that “from a
practical point of view it may well prove easier for the investor to choose
directly his optimal payoff function than it would be for him to communicate
his utility function to a portfolio manager.” Sharpe et al. (2000) and
Goldstein et al. (2008) introduce a tool called the distribution builder,
which makes it possible for investors to analyze distributions of terminal
wealth and to choose their preferred one among alternatives with equal cost;
see also Sharpe (2011) and Monin (2014). Moreover, Goldstein et al. (2008)
argue that such a tool makes it possible to better elicit investor’s
preferences.

Our main objective is to extend Dybvig’s results when there is uncertainty on
the real-world probability measure. Uncertainty has become a prime issue in
many academic domains, from economics to environmental science and psychology.
Model ambiguity refers to random phenomena or outcomes whose probabilities are
themselves unknown. For instance, the random outcome of a coin toss is subject
to model uncertainty when the probability of the coin showing either a head or
a tail is not or is at most partially known. This notion of model ambiguity
goes back to Knight (1921) and is therefore commonly referred to as Knightian
uncertainty.

In the presence of ambiguity, the probability distribution of a payoff is not
anymore determined. Thus, looking for a minimum cost payoff with a given
probability distribution is no longer possible. However, investors may still
determine a desired distribution function that they would like to achieve “at
least”. In this paper, we look for a minimum cost payoff that dominates a
target distribution for a chosen stochastic integral order under any plausible
real-world probability distribution. Our contributions are three-fold. First,
we solve this problem explicitly for a general stochastic ordering under
certain assumptions. Solutions to this problem are called “robust cost-
efficient.” Second, we draw connections between such a minimum cost payoff and
the problem of finding an optimal portfolio under ambiguity for general sets
of robust preferences. Third, we present a number of examples, including one
on the robust portfolio choice in the presence of volatility uncertainty.

Our results generalize the results on cost-efficiency given in Dybvig (1988a,
b), Cox and Leland (2000) and Bernard et al. (2014, 2015). When there is no
ambiguity on the real-world probability, the robust cost efficient payoffs
coincide with the cost-efficient payoffs studied in the literature. To derive
our results, we build on the so-called quantile approach to solve the
optimization of a law invariant increasing functional, e.g., Schied (2004),
Carlier and Dana (2006, 2008, 2011), Jin and Zhou (2008), He and Zhou (2011a,
b), Bernard et al. (2014), Xu (2016) and Rüschendorf and Vanduffel (2020).

Specifically, we consider a static setting but we have incomplete markets and
are thus able to address uncertainty about volatility. We show that, under
certain conditions, the solution to a general robust portfolio maximization
problem is equal to the solution of a classical portfolio maximization problem
under a _least favorable measure_ $\mathbb{P}^{\ast}$ with respect to some
stochastic ordering. This was already shown by Schied (2005) for the case of
robust expected utility theory and using first order stochastic dominance
ordering; here, however, we show that these results extend to the case of more
general preferences. Specifically, we focus on the case of first order and
second order stochastic dominance.

Furthermore, we show that there is a natural correspondence between optimal
portfolios in the maxmin utility setting of Gilboa and Schmeidler (1989), with
a concave increasing utility, and robust cost-efficient payoffs: for any
robust cost-efficient payoff $X^{\ast}$, there is a utility function such that
$X^{\ast}$ solves the maxmin expected utility maximization problem. We further
show that the solution to a robust maximization problem with respect to a
general family of preferences is cost-efficient. This result implies that
instead of solving a robust maximization problem with respect to a general
family of preferences one could solve an expected utility maximization problem
under the single measure $\mathbb{P}^{\ast}$ for a suitable concave utility
function.

The literature on optimal payoff choice under ambiguity includes the seminal
setting of Gilboa and Schmeidler (1989) that is, the so-called “maxmin
expected utility,” which was later referred to as robust utility functional by
Schied et al. (2009). Specifically, these authors characterize preferences
that have a robust utility numerical representation
$\underset{\mathbb{P}\in\mathcal{P}}{\min}\,E_{\mathbb{P}}[u(X)]$ for some set
of probabilities $\mathcal{P}$. Gundel (2005) provides a dual characterization
of the solution for robust utility maximization in both a complete and an
incomplete market model. Klibanoff et al. (2005) distinguish between
subjective beliefs, e.g., the definition of the set of possible or plausible
subjective probability measures, and ambiguity attitude, i.e., a
characterization of the agent’s behavior toward ambiguity. Based on Klibanoff
et al. (2005), Gollier (2011) analyzes the effect of ambiguity aversion on the
demand for the uncertain asset in a portfolio choice problem.

Schied (2005) solves the maximization problem of maxmin expected utility of
Gilboa and Schmeidler (1989) in a general complete market model with dynamic
trading, provided there is a least favorable measure with respect to FSD
ordering. Specifically, he finds that the optimum for the maxmin utility
setting of Gilboa and Schmeidler (1989) can be derived in the standard
expected utility setting under the least favorable measure. Schied (2005)
works with a complete market model and mainly in a static setting; dynamics
only come into play when the martingale method is applied to the static
solutions. A survey on robust preferences and robust portfolio choice can be
found in Schied et al. (2009).

The paper is organized as follows. The robust cost-efficiency problem is
described in Section 2. In Section 3, we solve the robust cost-efficiency
problem, and we include two examples in a log-normal market with uncertainty
on the drift and the volatility along with another example in a Lévy market in
which the physical measure is obtained by the Esscher transform. In Section 4,
we develop the correspondence between robust cost-efficient payoffs and
strategies that solve a robust optimal portfolio problem, including the maxmin
utility setting of Gilboa and Schmeidler (1989) as a special case. In Section
5, we show that the solution to a general robust optimal portfolio problem can
also be obtained as a solution to the maximization of the maxmin utility
setting of Gilboa and Schmeidler (1989) for a well-chosen concave utility
function. Section 6 concludes.

##  2 Problem statement

We assume a static market setting in which trading only takes place today and
at the end of the planning horizon $T>0$. There is a bank account earning the
continuously compounded risk-free interest rate $r\in\mathbb{R}$. Let
$\mathbb{R}_{+}=[0,\infty)$. Let $S_{T}:\Omega\rightarrow\mathbb{R}_{+}$
represent the random value of a risky asset at maturity. We denote by
$S_{0}>0$ its current value and by $\mathcal{F}$ the $\sigma$-algebra $S_{T}$
generates. Let $\mathcal{P}$ be a set of equivalent real-world probability
measures on $(\Omega,\mathcal{F})$. The set $\mathcal{P}$ can be thought of as
a collection of probability measures that the investor deems plausible for the
market. We define the set of _payoffs_
$\mathcal{X}=\left\\{g(S_{T}),\,g:\mathbb{R}_{+}\to\mathbb{R}_{+}\text{ is
measurable and }E_{\mathbb{Q}}[|g(S_{T})|]<\infty\right\\},$ where
$\mathbb{Q}$ is a martingale measure, equivalent to all
$\mathbb{P}\in\mathcal{P}.$ Furthermore, for any $X\in\mathcal{X}$, its price
is given by $e^{-rT}E_{\mathbb{Q}}[X]$.

######  Remark 2.1.

Under the assumption that all call options $(S_{T}-K)^{+}$, $K\geqslant 0$ are
traded and under the condition that the function mapping every $K\geqslant 0$
to the price of $(S_{T}-K)^{+}$ is twice differentiable with respect to $K$,
Breeden and Litzenberger (1978) show that any $X\in\mathcal{X}$ can almost
surely be replicated using a static portfolio of calls.

Consider an investor with a finite budget and planning horizon $T>0$ who
wishes to invest in the market whilst having ambiguous views on the real-world
probability measure. How can she find her optimal investment strategy? As in
Schied (2005), she could maximize some robust expected utility à la Gilboa and
Schmeidler (1989). The basic idea is then to look for a payoff that maximizes
the worst case expected utility, reflecting the idea that the investor aims to
protect against the worst whilst hoping for the best. However, it seems easier
for investors to specify the desired probability distribution of the terminal
wealth rather than a utility function (Brennan and Solanki (1981), Sharpe et
al. (2000), Goldstein et al. (2008)).

As in Dybvig (1988), Sharpe et al. (2000), Vrecko and Langer (2013), and
Bernard et al. (2014), we thus assume in this paper that the investor
specifies a desired (cumulative) distribution function $F_{0}$ of future
terminal wealth. Once the investor understands which distribution function
$F_{0}$ is acceptable to her, the natural question arises as to how to find,
under ambiguity, the cheapest portfolio with a distribution function at
maturity that is “at least as good” as $F_{0}$. This is the robust cost-
efficiency problem formalized hereafter. In this regard, we need to recall the
concept of integral stochastic ordering, see e.g., Denuit et al. (2005). In
this paper, we denote by $\mathbb{F}$ a set of measurable functions from
$\mathbb{R}_{+}$ to $\mathbb{R}$.

######  Definition 2.2 (Stochastic integral ordering).

Let $G$ and $F$ be two distribution functions with support on
$\mathbb{R_{+}}$. The distribution function $G$ dominates the distribution
function $F$ in _integral stochastic ordering with respect to the set
$\mathbb{F}$_, in notation $F\preceq_{\mathbb{F}}G$, if

$\forall
f\in\mathbb{F},\quad\int_{\mathbb{R_{+}}}f(x)dF\leqslant\int_{\mathbb{R_{+}}}f(x)dG,$

such that expectations are finite.

Let $\mathbb{F}_{FSD}$ denote the set of all non-decreasing functions from
$\mathbb{R}_{+}$ to $\mathbb{R}$. The corresponding stochastic integral
ordering is called _first order stochastic dominance_ (FSD). Furthermore, let
$\mathbb{F}_{SSD}$ denote the set of all non-decreasing and concave functions
from $\mathbb{R}_{+}$ to $\mathbb{R}$. The corresponding stochastic integral
ordering is called _second order stochastic dominance_ (SSD). It is well-known
that FSD reflects the common agreement of all investors with law-invariant
increasing preferences (Bernard et al., 2015, Theorem 1), whereas SSD order
reflects the common agreement of those who have law-invariant increasing and
diversification loving preferences (risk averse investors) (Bernard and Sturm,
2023, Corollary 2.6).

######  Problem 1 (Robust cost-efficiency problem).

The $\mathbb{F}-$_robust cost-efficiency problem for a distribution function
$F_{0}$_ is defined as

$\underset{X\in\mathcal{B}_{F_{0}}^{\mathbb{F}}}{\inf}e^{-rT}E_{\mathbb{Q}}[X],$
(2.1)

in which $\mathcal{B}_{F_{0}}^{\mathbb{F}}$ denotes a class of admissible
payoffs defined as

$\mathcal{B}_{F_{0}}^{\mathbb{F}}=\left\\{X\in\mathcal{X}\,:\,\forall\mathbb{P}\in\mathcal{P}:\,F_{0}\preceq_{\mathbb{F}}F_{X}^{\mathbb{P}}\right\\}.$

A solution to (2.1) is called a _$\mathbb{F}-$ robust cost-efficient payoff_.

As discussed above, the target distribution function of the investor is
$F_{0}$. That is, we are interested in all payoffs that have a distribution
function at maturity that is at least as good as $F_{0}$ under all plausible
scenarios $\mathbb{P\in\mathcal{P}}$. For example, when
$\mathbb{F}=\mathbb{F}_{FSD}$, we care about payoffs having distribution
functions $F_{X}^{\mathbb{P}},$ $\mathbb{P}\in\mathcal{P}$ that dominate
$F_{0}$ in FSD. In order not to “throw away investors’ money”, see Dybvig
(1988), we then aim to determine the cheapest payoff among the payoffs in the
admissible set $\mathcal{B}_{F_{0}}^{\mathbb{F}}$. In Theorem 3.1 we provide
solutions to the $\mathbb{F}-$robust cost-efficiency problem (2.1) under
regularity conditions on the set $\mathbb{F}$ and $F_{0}$.

Dybvig (1988a, b) introduced the standard cost-efficiency problem without
ambiguity on the set of physical measures; that is, when
$\mathcal{P}=\\{\mathbb{P}\\}$. Specifically, for some fixed
$\mathbb{P}\in\mathcal{P}$, the problem he considered reads as

$\underset{X\in\mathcal{A}_{F_{0}}^{\mathbb{P}}}{\inf}e^{-rT}E_{\mathbb{Q}}[X],$
(2.2)

where

$\mathcal{A}_{F_{0}}^{\mathbb{P}}=\left\\{X\in\mathcal{X}\,:\,F_{0}=F_{X}^{\mathbb{P}}\right\\}.$

We refer to this problem as the _standard cost-efficiency problem_.
Furthermore, we say that a payoff $X$ that is distributed with
$F_{X}^{\mathbb{P}}$ is _$\mathbb{P}-$ cost-efficient_ if $X$ solves the
standard cost-efficiency problem (2.2) under $\mathbb{P}$ w.r.t.
$F_{0}=F_{X}^{\mathbb{P}}$. By Bernard et al. (2014), a payoff $X$ is
$\mathbb{P}-$cost-efficient if and only if $X$ is non-increasing in the state
price $\mathbb{P}-$a.s., see also Schied (2004, Proposition 2.5).

The standard cost-efficiency problem (2.2) has been solved in Dybvig (1988a,
b) and in Bernard et al. (2014), see also Lemma B.1 in the Appendix. In
Corollary 3.5 we show that in the case in which $\mathcal{P}$ is a singleton,
the solution to the $\mathbb{F}-$robust cost-efficiency problem is unique and
coincides with the solution to the standard cost-efficiency problem.

######  Remark 2.3.

For a fixed $X\in\mathcal{X}$, we have that in general $F_{X}^{\mathbb{P}}$
cannot be equal to $F_{0}$ for all $\mathbb{P}\in\mathcal{P}$. Therefore, we
replace the condition $F_{0}=F_{X}^{\mathbb{P}}$ in the standard cost-
efficiency problem with $F_{0}\preceq_{\mathbb{F}}F_{X}^{\mathbb{P}}$ in the
robust setting.

The next example anticipates Sections 3.1.1 and 3.1.2 and is designed to help
distinguish between the standard and the robust cost-efficiency problem.

######  Example 2.4.

(Cost-efficient payoffs). Assume the real-world distribution function of
$S_{T}$ is log-normal. There are three investors: one investor assumes that
the drift of $S_{T}$ under the physical measure, denoted by
$\mathbb{P}^{\mu_{1}}$, is equal to $\mu_{1}>r$. Another investor assumes that
the drift is given by $\mu_{2}>\mu_{1}$ under the physical measure, denoted by
$\mathbb{P}^{\mu_{2}}$. A third investor has ambiguity and assumes that the
drift lies in the interval $[\mu_{1},\mu_{2}]$, and thus considers a set
$\mathcal{P}=\\{\mathbb{P}^{\mu},\,\mu\in[\mu_{1},\mu_{2}]\\}$ as the set of
all plausible probability measures on $(\Omega,\mathcal{F})$. The cheapest
payoffs to obtain a fixed target distribution function $F_{0}$ are well-known
for investor one and two and are given by

$X_{1}^{\ast}:=F_{0}^{-1}\left(F_{S_{T}}^{\mathbb{P}^{\mu_{1}}}(S_{T})\right)\quad\text{and}\quad
X_{2}^{\ast}:=F_{0}^{-1}\left(F_{S_{T}}^{\mathbb{P}^{\mu_{2}}}(S_{T})\right),$

respectively; see Proposition 3 in Bernard et al. (2014). Within the set
$\mathcal{P}$, $\mathbb{P}^{\mu_{1}}$ corresponds to a pessimistic view of the
stock price behavior, and we will see in Section 3.1.1 that $X_{1}^{\ast}$
also solves the robust cost-efficiency problem for $F_{0}$ if
$\mathbb{F}=\mathbb{F}_{FSD}$. In the case in which
$\mathbb{F}=\mathbb{F}_{SSD}$, $X_{1}^{\ast}$ also solves the robust cost-
efficiency problem if additionally $F_{0}^{-1}\circ
F_{S_{T}}^{\mathbb{P}^{\mu_{1}}}$ is concave; see Section 3.1.2. This example
illustrates that the solution to the standard cost-efficiency problem for
arbitrarily $\mathbb{P}\in\mathcal{P}$ and the solution to the robust cost-
efficiency problem generally do not coincide.

######  Remark 2.5 (Uncertainty on target distribution $F_{0}$).

As in Rüschendorf and Wolf (2016), we could also consider uncertainty on the
target distribution function $F_{0}.$ Specifically, in Rüschendorf and Wolf
(2016) it is assumed that the investor specifies finitely many acceptable
distribution functions $F^{1}_{0},...,F^{N}_{0}$. As all $N$ distribution
functions are acceptable to the investor, she could solve the robust cost-
efficiency problem $N$ times and buy the cheapest solution among the $N$
solutions. It is also possible to consider a continuum of acceptable
distribution functions: generalizing Rüschendorf and Wolf (2016) slightly, we
could consider the set

$\mathcal{G}=\left\\{F\ |F\ \text{is a cdf},\quad
F_{0}\preceq_{\mathbb{F}}F\right\\},$

as the set of distribution functions that are acceptable to the investor or
client. We could then consider the cost-efficiency problem with uncertainty on
the physical measure _and_ the target distribution by

$\underset{X\in\mathcal{B}_{\mathcal{G}}^{\mathbb{F}}}{\inf}e^{-rT}E_{\mathbb{Q}}[X],\quad\mathcal{B}_{\mathcal{G}}^{\mathbb{F}}=\left\\{X\in\mathcal{X}\,|\,\
\forall\mathbb{P}\in\mathcal{P}\,\ \exists
F\in\mathcal{G},\,F\preceq_{\mathbb{F}}F_{X}^{\mathbb{P}}\right\\}.$ (2.3)

However, if $\preceq_{\mathbb{F}}$ is transitive, it holds that
$\mathcal{B}_{\mathcal{G}}^{\mathbb{F}}=\mathcal{B}_{F_{0}}^{\mathbb{F}}$,
i.e., problem (2.1) without uncertainty on the target distribution and problem
(2.3) are equivalent.

###  2.1  Assumptions

In order to solve the robust cost efficiency problem (2.1), we need some
regularity conditions on the set $\mathbb{F}$ and on the target distribution
$F_{0}$. In this regard, we define some concepts.

Recall first the concept of a _least favorable measure_ introduced by Schied
(2005) for the case $\mathbb{F}=\mathbb{F}_{FSD}.$ For
$\mathbb{P}\in\mathcal{P}$, we define the corresponding _likelihood ratio_ 111
The random variable $\frac{e^{-rT}}{\ell^{\mathbb{P}}}$ is also called _state
price_ because the price of a payoff $X\in\mathcal{X}$ can be expressed by
$e^{-rT}E_{\mathbb{Q}}[X]=E_{\mathbb{P}}\left[\frac{e^{-rT}}{\ell^{\mathbb{P}}}X\right],\quad\mathbb{P}\in\mathcal{P}.$
by $\ell^{\mathbb{P}}=\frac{d\mathbb{P}}{d\mathbb{Q}}$.

######  Definition 2.6 (Least favorable measure with respect to $\mathbb{F}$).

A measure $\mathbb{P}^{\ast}\in\mathcal{P}$ with corresponding likelihood
ratio $\ell^{\ast}:=\frac{d\mathbb{P}^{\ast}}{d\mathbb{Q}}$ is called a _least
favorable measure with respect to $\mathbb{F}$ _if
$F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\preceq_{\mathbb{F}}F_{\ell^{\ast}}^{\mathbb{P}}$
for all $\mathbb{P}\in\mathcal{P}$.

Definition 2.6 generalizes Definition 2.1 of Schied (2005), who assumed the
existence of a least favorable measure w.r.t. $\mathbb{F}_{FSD}$ to determine
payoffs that solve the robust expected utility problem of Gilboa-Schmeidler.
We also need the following definition.

######  Definition 2.7 (Composition-consistency of $\mathbb{F}$).

The set $\mathbb{F}$ is said to be _composition-consistent_ if for
$f,g\in\mathbb{F}$ also $f\circ g\in\mathbb{F}$.

Note that the sets $\mathbb{F}_{FSD}$ and $\mathbb{F}_{SSD}$ are composition-
consistent. This follows from the fact that the composition of non-decreasing
(resp. non-decreasing and concave) functions is again non-decreasing (resp.
non-decreasing and concave).

The following proposition provides conditions that guarantee the existence of
a least favorable measure and turns out to be very useful for applications.

######  Proposition 2.8 (Sufficient conditions for the existence of a least
favorable measure).

Assume that $\mathbb{F}$ is composition consistent. If
$F_{S_{T}}^{\mathbb{P}^{\prime}}\preceq_{\mathbb{F}}F_{S_{T}}^{\mathbb{P}}$
for some $\mathbb{P}^{\prime}\in\mathbb{P}$ and all
$\mathbb{P}\in\mathcal{P},$ and $\ell^{\mathbb{P}^{\prime}}=f(S_{T})$ for some
$f\in\mathbb{F}$, then $\mathbb{P}^{\prime}$ is a least favorable measure
w.r.t. $\mathbb{F}$. If, additionally, $S_{T}$ is continuously distributed
under $\mathbb{P}^{\prime}$ and $f$ is strictly increasing, then
$\ell^{\mathbb{P}^{\prime}}$ is continuously distributed under
$\mathbb{P}^{\prime}$.

###### Proof.

Let $\mathbb{P},\mathbb{P}^{\prime}\in\mathcal{P}$. Let $X$ be a payoff and
$f\in\mathbb{F}$. Note that
$F_{X}^{\mathbb{P}^{\prime}}\preceq_{\mathbb{F}}F_{X}^{\mathbb{P}}$ if and
only if $E_{\mathbb{P}^{\prime}}[g(X)]\leqslant E_{\mathbb{P}}[g(X)]$ for all
$g\in\mathbb{F}$ such that expectations are finite. Because $\mathbb{F}$ is
composition-consistent, it follows that

$F_{X}^{\mathbb{P}^{\prime}}\preceq_{\mathbb{F}}F_{X}^{\mathbb{P}}\Rightarrow
F_{f(X)}^{\mathbb{P}^{\prime}}\preceq_{\mathbb{F}}F_{f(X)}^{\mathbb{P}}.$
(2.4)

The expression
$F_{\ell^{\mathbb{P}^{\prime}}}^{\mathbb{P}^{\prime}}\preceq_{\mathbb{F}}F_{\ell^{\mathbb{P}^{\prime}}}^{\mathbb{P}}$
then follows by Equation (2.4). Let $f$ be strictly increasing. By Embrechts
and Hofert (2013), the generalized inverse $f^{-1}$ of $f$ is continuous on
the range of $f$. Thus, it holds that

$\displaystyle\mathbb{P}^{\prime}\left(\ell^{\mathbb{P}^{\prime}}\leqslant
x\right)$ $\displaystyle=\mathbb{P}^{\prime}\left(S_{T}\leqslant
f^{-1}(x)\right)=F_{S_{T}}^{\mathbb{P}^{\prime}}\left(f^{-1}(x)\right),\quad
x\in\mathbb{R},$

which implies that $\ell^{\mathbb{P}^{\prime}}$ is continuously distributed
under $\mathbb{P}^{\prime}$. ∎

We also need a definition that is, to the best of our knowledge, new to the
literature.

######  Definition 2.9 (Cost-consistency of $\mathbb{F}$).

The set $\mathbb{F}$ is called _cost-consistent_ if for all
$X,Y\in\mathcal{X}$ and all $\mathbb{P}\in\mathcal{P}$ such that $X,Y$ are
$\mathbb{P}-$cost-efficient,
$F_{X}^{\mathbb{P}}\preceq_{\mathbb{F}}F_{Y}^{\mathbb{P}}$ implies
$E_{\mathbb{Q}}[X]\leqslant E_{\mathbb{Q}}[Y]$ and, additionally,
$F_{X}^{\mathbb{P}}\neq F_{Y}^{\mathbb{P}}$ implies
$E_{\mathbb{Q}}[X]<E_{\mathbb{Q}}[Y]$.

As the set $\mathbb{F}_{SSD}$ is contained in $\mathbb{F}_{FSD}$, the
following proposition implies that $\mathbb{F}_{FSD}$ and $\mathbb{F}_{SSD}$
are cost-consistent. In Example 3.6 we discuss a set that is not cost-
consistent.

######  Proposition 2.10.

$\mathbb{F}_{SSD}$ is cost-consistent. Moreover, if
$\mathbb{F}_{SSD}\subset\mathbb{F}$, then $\mathbb{F}$ is cost-consistent.

###### Proof.

The cost-consistency of $\mathbb{F}_{SSD}$ can be proven along the lines of
the proof of Lemma 2 in Bernard et al. (2019). Furthermore,
$F_{X}^{\mathbb{P}}\preceq_{\mathbb{F}}F_{Y}^{\mathbb{P}}$ implies
$F_{X}^{\mathbb{P}}\preceq_{\mathbb{F}_{SSD}}F_{Y}^{\mathbb{P}}$, which
finishes the proof. ∎

We now list a series of conditions that we often use to derive our main
results:

######  Condition C1.

$F_{0}^{-1}$ is square integrable, i.e.,
$\int_{0}^{1}(F_{0}^{-1}(u))^{2}du<\infty$, and $F_{0}(x)=0$ for $x<0$.

######  Condition C2.

The set $\mathbb{F}$ is composition-consistent and cost-consistent.

######  Condition C3.

The set $\mathcal{P}$ contains a least favorable measure with respect to
$\mathbb{F}.$ We denote this least favorable measure by $\mathbb{P}^{\ast}.$

######  Condition C4.

Denote by $\ell^{\ast}$ the likelihood ratio of the least favorable measure
$\mathbb{P}^{\ast}.$ We assume that $x\mapsto
F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}(x)$ is continuous and that
$\frac{1}{\ell^{\ast}}$ has finite variance under $\mathbb{P}^{\ast}$.

Condition C1 is technical and ensures that the robust cost-efficiency problem
is well-posed, i.e., $\mathcal{B}_{F_{0}}^{\mathbb{F}}$ is not empty, see
Theorem 3.1.

Condition C3 can also be found in Schied (2005) for the case
$\mathbb{F}=\mathbb{F}_{FSD}.$ Note that when $\mathbb{F}$ becomes larger, the
condition C3 becomes stronger. Specifically, requiring a least favorable
measure $\mathbb{P}^{\ast}\in\mathcal{P}$ with respect to
$\mathbb{F}=\mathbb{F}_{FSD}$ is more stringent than in the case
$\mathbb{F}=\mathbb{F}_{SSD}$. In particular, Proposition 2.8 provides
sufficient conditions for the existence of a least favorable measure
${P}^{\ast}\in\mathcal{P}$ with respect to $\mathbb{F}$.

The condition in C4 that $x\mapsto F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}(x)$ is
continuous distribution function under $\mathbb{P}^{\ast}$ is also made in a
setting without ambiguity in e.g., Jin and Zhou (2008), He and Zhou (2011a,
b), Bernard et al. (2014) and Xu (2016) among many others. It is a strong
assumption in the sense that we essentially exclude discrete settings.

##  3 Solution of the robust cost-efficiency problem

In the next theorem, we make the assumption that $F_{0}^{-1}\circ
F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\in\mathbb{F}$. Note that this assumption
is always true if $\mathbb{F}=\mathbb{F}_{FSD}$. The assumption is also true
if $\mathbb{F}=\mathbb{F}_{SSD}$, provided that $F_{0}^{-1}\circ
F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}$ is concave.

######  Theorem 3.1 ($\mathbb{F}-$robust cost-efficient payoff).

Assume that the conditions C1, C2, C3 and C4 hold and that $F_{0}^{-1}\circ
F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\in\mathbb{F}.$ Then, the
$\mathbb{F}-$robust cost-efficiency problem for $F_{0}$ has a
$\mathbb{P}^{\ast}-$a.s. unique solution given by

$F_{0}^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\left(\ell^{\ast}\right)\right).$

###### Proof.

Recall that $\ell^{\ast}$ denotes the likelihood ratio that corresponds to
$\mathbb{P}^{\ast}$. Let

$\displaystyle
X^{\ast}=F_{0}^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}(\ell^{\ast})\right).$

As $F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\left(\ell^{\ast}\right)$ is uniformly
distributed under $\mathbb{P}^{\ast}$ (Condition C4), it follows by Lemma B.1
that $F_{X^{\ast}}^{\mathbb{P}^{\ast}}=F_{0}$; and, by condition C1 it holds
that

$E_{\mathbb{P}^{\ast}}\left[\left(F_{0}^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\left(\ell^{\ast}\right)\right)\right)^{2}\right]=\int_{0}^{1}(F_{0}^{-1}(u))^{2}du<\infty.$

Condition C4 thus implies that $E_{\mathbb{Q}}[X^{\ast}]<\infty$ because

$\displaystyle E_{\mathbb{Q}}[|X^{\ast}|]$
$\displaystyle=E_{\mathbb{P}^{\ast}}\left[\frac{1}{\ell^{\ast}}F_{0}^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\left(\ell^{\ast}\right)\right)\right]$
$\displaystyle\leqslant\sqrt{E_{\mathbb{P}^{\ast}}\left[\frac{1}{(\ell^{\ast})^{2}}\right]E_{\mathbb{P}^{\ast}}\left[\left(F_{0}^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\left(\ell^{\ast}\right)\right)\right)^{2}\right]}$
$\displaystyle<\infty.$

Therefore, $X^{\ast}\in\mathcal{X}$. By conditions C2 and C3 and as
$F_{0}^{-1}\circ F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\in\mathbb{F}$, it follows
from Equation (2.4) that
$F_{0}=F_{X^{\ast}}^{\mathbb{P^{\ast}}}\preceq_{\mathbb{F}}F_{X^{\ast}}^{\mathbb{P}}$
for all $\mathbb{P}\in\mathcal{P}$; hence,
$X^{\ast}\in\mathcal{B}_{F_{0}}^{\mathbb{F}}$. Let
$Y\in\mathcal{B}_{F_{0}}^{\mathbb{F}}$ and define

$Y^{\ast}=\left[F_{Y}^{\mathbb{P}^{\ast}}\right]^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}(\ell^{\ast})\right).$

Then $Y^{\ast}$ is $\mathbb{P}^{\ast}-$cost-efficient for
$F_{Y}^{\mathbb{P}^{\ast}}$ and we have
$F_{X^{\ast}}^{\mathbb{P}^{\ast}}=F_{0}\preceq_{\mathbb{F}}F_{Y}^{\mathbb{P}^{\ast}}=F_{Y^{\ast}}^{\mathbb{P}^{\ast}}$.
By condition C2, $\mathbb{F}$ is cost-consistent, which implies that
$E_{\mathbb{Q}}[X^{\ast}]\leqslant E_{\mathbb{Q}}[Y^{\ast}]\leqslant
E_{\mathbb{Q}}[Y]$. Hence, every admissible payoff is more expensive than
$X^{\ast}$. We now show uniqueness. Let $\hat{X}$ be another solution to the
robust cost-efficiency problem. It holds that
$F_{0}\preceq_{\mathbb{F}}F_{\hat{X}}^{\mathbb{P}^{\ast}}$. If
$F_{\hat{X}}^{\mathbb{P}^{\ast}}=F_{0}$ and
$E_{\mathbb{Q}}[X^{\ast}]=E_{\mathbb{Q}}[\hat{X}]$, then $X^{\ast}=\hat{X}$,
$\mathbb{P}^{\ast}-$a.s. by Lemma B.1 because the solution $X^{\ast}$
corresponds to the solution of the standard $\mathbb{P}^{\ast}-$cost-
efficiency problem for $F_{0}$, which has a unique solution. If $F_{0}\neq
F_{\hat{X}}^{\mathbb{P}^{\ast}}$, then
$E_{\mathbb{Q}}[X^{\ast}]<E_{\mathbb{Q}}[\hat{X}]$ because $\mathbb{F}$ is
cost-consistent. Hence, $X^{\ast}$ is the unique solution to the robust cost-
efficiency problem. ∎

######  Remark 3.2.

Instead of requiring that $F_{0}^{-1}$ is square integrable, the proof of
Theorem 3.1 shows that it is sufficient to assume that
$F_{0}^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\left(\ell^{\ast}\right)\right)$
has finite price.

######  Remark 3.3.

Does ambiguity increase costs? Let the assumptions of Theorem 3.1 be in force.
Let us compare two investors. Investor A has ambiguity and considers the set
$\mathcal{P}$ as the set of possible real-world measures. Investor B has,
e.g., based on a deep market analysis or insider knowledge, no ambiguity and
knows that $\mathbb{P}\in\mathcal{P}$ is the true real-world measure. Both
investors consider $F_{0}$ as the target distribution function. Investor A
buys
$X^{\ast}=F_{0}^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\left(\ell^{\ast}\right)\right)$
according to Theorem 3.1, whereas investor B buys
$X=F_{0}^{-1}(F_{\ell^{\mathbb{P}}}^{\mathbb{P}}(\ell^{\mathbb{P}}))$ (see
Lemma B.1). As $X^{\ast}\in\mathcal{B}_{F_{0}}^{\mathbb{F}}$, it holds that
$F_{X}^{\mathbb{P}}=F_{0}\leqslant_{\mathbb{F}}F_{X^{\ast}}^{\mathbb{P}}$. As
the set $\mathbb{F}$ is cost-consistent, it follows that
$E_{\mathbb{Q}}[X]\leqslant E_{\mathbb{Q}}[X^{\ast}]$. If we additionally have
$F_{X}^{\mathbb{P}}\neq F_{X^{\ast}}^{\mathbb{P}}$, then it follows that
$E_{\mathbb{Q}}[X]<E_{\mathbb{Q}}[X^{\ast}]$. In the robust setting we end up
with a payoff $X^{\ast}$ whose distribution $F_{X^{\ast}}^{\mathbb{P}}$,
$\mathbb{P}\in\mathcal{P}$ dominates $F_{0}$ in stochastic ordering for all
$\mathbb{P}$. That is, under ambiguity, the preferred payoff has a (strictly)
higher price and the optimal robust choice $X^{\ast}$ typically will not match
the choice $X$ without uncertainty.

######  Remark 3.4.

The condition in Theorem 3.1 that the function $F_{0}^{-1}\circ
F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}$ must be concave in the case in which
$\mathbb{F}=\mathbb{F}_{SSD}$ means that the target distribution function
$F_{0}$ is required to be lighter-tailed than the distribution function
$F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}$. Specifically,
$F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}$ must dominate $F_{0}$ in the sense of
transform convex order (Shaked and Shanthikumar (2007)).

The next corollary shows that the standard and the robust cost-efficiency
problem coincide in a setting without uncertainty. Note that we do not require
$F_{0}^{-1}\circ F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\in\mathbb{F}$ as in
Theorem 3.1.

######  Corollary 3.5.

Assume that the conditions C1 and C4 hold and that
$\mathcal{P}=\\{\mathbb{P}\\}$ is a singleton. The solutions to the
$\mathbb{F}-$robust cost-efficiency problem for $F_{0}$ and the standard cost-
efficiency problem for $F_{0}$ are unique and identical.

###### Proof.

Note that when $\mathcal{P}=\\{\mathbb{P}\\}$, then $\mathbb{P}$ is a least
favorable measure. By Lemma B.1,
$X^{\ast}=F_{0}^{-1}(F_{\ell^{\mathbb{P}}}^{\mathbb{P}}(\ell^{\mathbb{P}})$)
is the unique solution to the standard cost-efficiency problem. As in the
proof of Theorem 3.1, one can show that $X^{\ast}\in\mathcal{X}$. Then
$X^{\ast}\in\mathcal{B}_{F_{0}}^{\mathbb{F}}$ follows immediately because
$F_{X^{\ast}}^{\mathbb{P}}=F_{0}$. As in the proof of Theorem 3.1, one can
show that $X^{\ast}$ is the only admissible payoff solving the
$\mathbb{F}-$robust cost-efficiency problem. ∎

The sets $\mathbb{F}_{FSD}$ and $\mathbb{F}_{SSD}$ are cost- and composition-
consistent. We provide examples of sets $\mathbb{F}$ of functions that are not
cost- or composition-consistent so that Theorem 3.1 cannot be applied to find
robust optimal payoffs.

######  Example 3.6.

Third order stochastic dominance is the stochastic integral ordering that
arises from the set $\mathbb{F}_{TSD}$, containing all functions
$f:\mathbb{R}_{+}\to\mathbb{R}$ such that $f^{\prime}>0$,
$f^{\prime\prime}\leqslant 0$ and $f^{\prime\prime\prime}\geqslant 0$. The set
$\mathbb{F}_{TSD}$ is composition-consistent but is in general not cost-
consistent: see Appendix A.

######  Example 3.7.

Müller et al. (2017) introduced the $(1+\gamma)-$stochastic dominance order
for $\gamma\in(0,1)$, which lies between FSD and SSD ordering. The set induced
by $(1+\gamma)-$stochastic dominance order is in general not composition-
consistent, but it is cost-consistent in light of Proposition 2.10.

######  Example 3.8.

Rothschild and Stiglitz (1970) introduced concave stochastic order, which is
defined via the set of all concave (but not necessarily non-decreasing)
functions. Concave stochastic order coincides with SSD if we compare two
payoffs with the same mean (Föllmer and Schied, 2011, Remark 2.63). The set of
all concave functions is cost-consistent but not composition-consistent.

In the following section we illustrate Theorem 3.1 in a log-normal market
setting with uncertainty on the drift _and_ volatility, whereas in Section 3.2
we deal with a more general market setting.

###  3.1  Robust cost-efficient payoffs in lognormal markets

We assume that under the pricing measure $\mathbb{Q}$, $S_{T}$ has a log-
normal distribution function with parameters
$\log(S_{0})+(r-\frac{{s}^{2}}{2})T$ and ${s}^{2}T$ with stock price $S_{0}>0$
today, interest rate $r\in\mathbb{R}$, time horizon $T>0$ and volatility
$s>0$. Under $\mathbb{Q}$, $S_{T}$ is log-normally distributed with density
$f^{r,s}$, where for $m\in\mathbb{R}$ and $\varsigma>0$ we define

$f^{m,\varsigma}(x)=\frac{1}{x\varsigma\sqrt{T}\sqrt{2\pi}}\exp\left(-\frac{\left(\ln\left(x\right)-\ln(S_{0})-\left(m-\frac{\varsigma^{2}}{2}\right)T\right)^{2}}{2\varsigma^{2}T}\right),\quad
x>0.$ (3.1)

####  3.1.1 Drift uncertainty: $\mathbb{F}_{FSD}-$robust cost-efficient payoff
(Schied, 2005)

The real-world distribution function of $S_{T}$ is assumed to be log-normal
with parameters $\log(S_{0})+(\mu-\frac{s^{2}}{2})T$ and $s^{2}T$, but there
is uncertainty about the precise level of the drift parameter $\mu$. In
particular, the agent only expects the true drift parameter $\mu$ to lie in
the interval
$\mathcal{D}^{\mu_{1}}=\left\\{\mu\in\mathbb{R}\,:\,\mu\geqslant\mu_{1}\right\\}$
for $\mu_{1}>r$, and thus she considers
$\mathcal{P}=\left(\mathbb{P}^{\mu}\right)_{\mu\in\mathcal{D}^{\mu_{1}}}$ as
the set of all plausible probability measures on $(\Omega,\mathcal{F})$. Under
$\mathbb{P}^{\mu}$, $S_{T}$ is log-normal with density $f^{\mu,s}$. It follows
that
$F_{S_{T}}^{\mathbb{P}^{\ast}}\preceq_{\mathbb{F}_{FSD}}F_{S_{T}}^{\mathbb{P}^{\mu}}$
for all $\mu\geqslant\mu_{1}$, where
$\mathbb{P}^{\ast}:=\mathbb{P}^{\mu_{1}}$. Let
$h^{\mu,\varsigma}(x)=\frac{f^{\mu,\varsigma}(x)}{f^{r,s}(x)}$, $x>0$. A
straightforward computation shows that

$h^{\mu,s}(x)=\left(\frac{x}{S_{0}}\right)^{\frac{\left(\mu-r\right)}{s^{2}}}\exp\left(\frac{r^{2}-\mu^{2}+s^{2}(\mu-r)}{2s^{2}}T\right),\quad
x>0.$ (3.2)

As $\ell^{\mathbb{P}^{\mu_{1}}}=h^{\mu_{1},s}(S_{T})$ and $\mu_{1}>r,$
$\ell^{\mathbb{P}^{\mu_{1}}}$ is a strictly increasing function of $S_{T}$.
Furthermore, $\frac{1}{\ell^{\mathbb{P}^{\mu_{1}}}}$ has finite variance. By
Proposition 2.8, $\mathbb{P}^{\ast}$ is a least favorable measure with
corresponding likelihood ratio $\ell^{\ast}:=\ell^{\mathbb{P}^{\mu_{1}}}$,
i.e., conditions C3 and C4 are satisfied. Theorem 3.1 shows that the
$\mathbb{F}_{FSD}-$robust cost-efficient payoff for a distribution function
$F_{0}$ satisfying condition C1 is given by

$X^{\ast}=F_{0}^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\left(\ell^{\ast}\right)\right)=F_{0}^{-1}\left(F_{S_{T}}^{\mathbb{P}^{\mu_{1}}}(S_{T})\right).$

The second equality follows from the increasingness of
$\ell^{\mathbb{P}^{\mu_{1}}}$ in $S_{T}$. The agent thus chooses the optimal
payoff as if she believes that the worst-case plausible value for the drift
parameter $\mu$, i.e., $\mu_{1}$, will materialize. This finding is consistent
with the results obtained by Schied (2005, Section 3.1) on the impact of drift
uncertainty on optimal payoff choice in a Black-Scholes setting.

######  Example 3.9.

We consider next the exponential distribution for the distribution function
$F_{0}$, i.e., $F_{0}(x)=1-e^{-x}$, $x\geqslant 0$, which satisfies condition
C1. Panel A of Figure 1 displays the price of the robust cost-efficient payoff
for varying levels of the parameter $\mu_{1}$, which describes the ambiguity
that the agent faces (consistently with Remark 3.3). The higher $\mu_{1}$, the
smaller the set $\mathcal{D}^{\mu_{1}}$, i.e., the lower the degree of
ambiguity, and the cheaper $X^{\ast}$. In panel B of Figure 1 we display, for
several values of $\mu_{1}$, the robust cost-efficient payoff normalized for
its initial price as a function of realizations $s$ of $S_{T}$; i.e., we
display the curve

$s\mapsto\frac{F_{0}^{-1}\left(F_{S_{T}}^{\mathbb{P}^{\mu_{1}}}(s)\right)}{\pi_{\mu_{1}}},$

where
$\pi_{\mu_{1}}=e^{-rT}E_{\mathbb{Q}}\left[F_{0}^{-1}\left(F_{S_{T}}^{\mathbb{P}^{\mu_{1}}}(S_{T})\right)\right]$.
We observe that the curve is flatter when $\mu_{1}$ is smaller, i.e., more
ambiguity gives rise to payoffs that reflect a higher degree of conservatism.

|
---|---
Panel A | Panel B
Figure 1:  We use the parameters $S_{0}=1$, $r=0$, $T=1$ and $s=0.9$. The
reference distribution function is the exponential distribution function
$F_{0}(x)=1-e^{-x}$. Panel A: Price of the cost-efficient payoff $X^{\ast}$
depending on the value of the ambiguity parameter $\mu_{1}$. Panel B: Cost-
efficient payoff per unit of investment for various values of $\mu_{1}$.

####  3.1.2 Drift and volatility uncertainty: $\mathbb{F}_{SSD}-$robust cost-
efficient payoff

The real-world distribution function of $S_{T}$ is assumed to be log-normal
with parameters $\log(S_{0})+(\mu-\frac{{\sigma}^{2}}{2})T$ and $\sigma^{2}T$,
but now the agent faces uncertainty about the precise level of the parameters
$\mu$ and $\sigma$. In particular, the agent only expects the true parameters
to lie within the cube

$\mathcal{D}^{\mu_{1},\mu_{2},\sigma_{1},s}=\left\\{(\mu,\sigma)\subset\mathbb{R}^{2}\,:\,\mu_{1}\leqslant\mu\leqslant\mu_{2},\,\sigma_{1}\leqslant\sigma\leqslant
s\right\\}$

for $r<\mu_{1}<\mu_{2}$ and $0<\sigma_{1}\leqslant s$, and thus she considers
$\mathcal{P}=\left(\mathbb{P}^{\mu,\sigma}\right)_{(\mu,\sigma)\in\mathcal{D}^{\mu_{1},\mu_{2},\sigma_{1},s}}$
as the set of all plausible probability measures on $(\Omega,\mathcal{F})$.
Under $\mathbb{P}^{\mu,\sigma}$, $S_{T}$ is log-normal with density
$f^{\mu,\sigma}$, defined in Equation (3.1). In this regard, note that while
$r<\mu_{1}$ is a natural assumption, there is some empirical evidence for the
hypothesis that $\sigma\leqslant s$; see Table 1 in Christensen and Prabhala
(1998) and Table 1 in Christensen and Hansen (2002).

######  Remark 3.10.

In contrast to the dynamic Black-Scholes model, in which the stock price
$S_{T}$ is also log-normally distributed, we work in a static market setting.
In a dynamic Black-Scholes framework where continuous trading is allowed at
zero transaction cost, the absence of arbitrage opportunities implies that the
volatility of the stock does not change when moving from the real-world
measure to the risk-neutral measure, i.e., there does not exist uncertainty
about the volatility in a dynamic Black-Scholes model. Here, however, we do
not assume dynamic trading. Hence, even when call option prices reflect a risk
neutral distribution function for $S_{T}$ that is log-normally distributed,
the agent may have a view on the real-world distribution that is different
from a log-normal and, in particular, may be unsure about the exact values for
drift and volatility.

In the next proposition, we assume that

$\frac{\mu_{1}-r}{s^{2}}\in(0,1].$ (3.3)

For example, $s\in[0.2,\infty)$ and $(\mu_{1}-r)\in(0,0.04]$ or
$s\in[0.35,\infty)$ and $(\mu_{1}-r)\in(0,0.1]$ imply Equation (3.3), that is:
there are economically reasonable environments such that Equation (3.3) holds.

######  Proposition 3.11 ($\mathbb{F}_{SSD}-$robust cost-efficient payoff).

If $\frac{\mu_{1}-r}{s^{2}}\in(0,1]$, then it holds that
$F_{S_{T}}^{\mathbb{P}^{\mu_{1},s}}\preceq_{\mathbb{F}_{SSD}}F_{S_{T}}^{\mathbb{P}^{\mu,\sigma}}$,
$(\mu,\sigma)\in\mathcal{D}^{\mu_{1},\mu_{2},\sigma_{1},s}$ and conditions C3
and C4 are satisfied for the set $\mathbb{F}_{SSD}$. The least favorable
measure is $\mathbb{P}^{\ast}=\mathbb{P}^{\mu_{1},s}$ with corresponding
likelihood ratio $\ell^{\ast}=\ell^{\mathbb{P}^{\mu_{1},s}}$. The
$\mathbb{F}_{SSD}-$robust cost-efficient payoff for the distribution function
$F_{0}$ satisfying condition C1 such that $F_{0}^{-1}\circ
F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}$ is concave is then given by

$X^{\ast}:=F_{0}^{-1}\left(F_{S_{T}}^{\mathbb{P}^{\mu_{1},s}}(S_{T})\right).$
(3.4)

###### Proof.

For a log-normal distribution function $F$ with parameters $M$ and $V$, it
holds that

$\int_{0}^{q}F^{-1}(p)dp=\frac{e^{M+\frac{V}{2}}}{q}\
\Phi\left(\Phi^{-1}(q)-\sqrt{V}\right),\quad q\in(0,1),$

where $\Phi$ denotes the distribution function of a standard normal random
variable. It follows that

$\displaystyle\int_{0}^{q}\left[F_{S_{T}}^{\mathbb{P}^{\mu_{1},s}}\right]^{-1}(p)dp$
$\displaystyle=\frac{e^{\mu_{1}T}}{q}\
\Phi\left(\Phi^{-1}(q)-s\sqrt{T}\right)$ $\displaystyle\leqslant\frac{e^{\mu
T}}{q}\
\Phi\left(\Phi^{-1}(q)-\sigma\sqrt{T}\right),\,\,\,q\in(0,1),\,\,\,\mu_{1}\leqslant\mu,\,\,\,\sigma\leqslant
s.$

Hence,
$F_{S_{T}}^{\mathbb{P}^{\mu_{1},s}}\preceq_{\mathbb{F}_{SSD}}F_{S_{T}}^{\mathbb{P}^{\mu,\sigma}}$,
$(\mu,\sigma)\in\mathcal{D}^{\mu_{1},\mu_{2},\sigma_{1},s}$. As in Section
3.1.1, let

$\ell^{\mathbb{P}^{\mu_{1},s}}=\frac{f^{\mu_{1},s}(S_{T})}{f^{r,s}(S_{T})}=h^{\mu_{1},s}(S_{T}).$

Hence, the likelihood ratio $\ell^{\mathbb{P}^{\mu_{1},s}}$ is strictly
increasing and concave in $S_{T}$ if (3.3) is satisfied. By Proposition 2.8,
conditions C3 and C4 are satisfied for the set $\mathbb{F}_{SSD}$ with least
favorable measure $\mathbb{P}^{\ast}=\mathbb{P}^{\mu_{1},s}$ with likelihood
ratio $\ell^{\ast}=\ell^{\mathbb{P}^{\mu_{1},s}}.$ As in Section 3.1.1, some
simple calculations and Theorem 3.1 show that the robust cost-efficient payoff
for the distribution function $F_{0}$ is given by (3.4). ∎

We provide an example for $F_{0}$ that make it possible to apply Proposition
3.11 to determine ${\mathbb{F}_{SSD}}-$robust cost-efficient payoffs. In this
regard, observe that $F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}$ in Proposition 3.11
is the log-normal distribution with parameters $\frac{1}{2}\theta^{2}$ and
$\theta$ for $\theta:=\sqrt{T}\frac{\mu_{1}-r}{s}>0$.

######  Example 3.12.

If $F_{0}$ is the log-normal distribution with parameters $M\in\mathbb{R}$ and
$V>0$, then $F_{0}^{-1}\circ F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}$ in
Proposition 3.11 is concave if $V\leqslant\theta$ because

$F_{0}^{-1}\circ
F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}(x)=x^{\frac{V}{\theta}}\exp(-\frac{1}{2}\theta
V+M),\quad x>0.$

###  3.2 Robust cost-efficient payoffs in general markets using Esscher
transform

Inspired by Corcuera et al. (2009), let $S_{0}>0$ and $s>0$ and $Z$ be a
payoff with mean zero and variance one. Under $\mathbb{Q}$, assume that $Z$
has density $f_{Z}^{\mathbb{Q}}(x)>0$, $x\in\mathbb{R}$ and model the future
stock price at date $T$ by

$S_{T}=S_{0}e^{(r+\omega)T+s\sqrt{T}Z},$

where $\omega\in\mathbb{R}$ is a mean correcting term, i.e., $\omega$ is
chosen such that

$e^{-rT}E_{\mathbb{Q}}\left[S_{T}\right]=S_{0}.$

The density of $X=\log(S_{T})$ under $\mathbb{Q}$ is

$f_{X}^{\mathbb{Q}}(x)=\frac{1}{s\sqrt{T}}f_{Z}^{\mathbb{Q}}\left(\frac{x-\log(S_{T})-(r+\omega)T}{s\sqrt{T}}\right),\quad
x>0.$

The corresponding density of $S_{T}$ under $\mathbb{Q}$ is denoted by
$f_{S_{T}}^{\mathbb{Q}}$, and it holds that

$f_{S_{T}}^{\mathbb{Q}}(x)=f_{X}^{\mathbb{Q}}(\log(x))\frac{1}{x},\quad x>0.$

Let $h^{\ast}>0$ and $\mathcal{H}\subset[h^{\ast},\infty)$ be a set containing
$h^{\ast}$ such that $E_{\mathbb{Q}}\left[(S_{T})^{h}\right]$ exists for all
$h\in\text{$\mathcal{H}$}$. Define a family of probability measures
$\mathcal{P}=\left(\mathbb{P}^{h}\right)_{h\in\mathcal{H}}$ as follows:
$\mathbb{P}^{h}$ is a measure such that $X$ has density
$f_{X}^{\mathbb{P}^{h}}$ under $\mathbb{P}^{h}$, where
$f_{X}^{\mathbb{P}^{h}}$ is obtained from $f_{X}^{\mathbb{Q}}$ by applying the
Esscher transform. The use of the Esscher transform can be supported by a
utility maximizing argument; see Gerber and Shiu (1996). In particular, we
define $\mathbb{P}^{h}$ such that

$f_{X}^{\mathbb{P}^{h}}(x)=\frac{e^{hx}f_{X}^{\mathbb{Q}}(x)}{\int_{\mathbb{R}}e^{hy}f_{X}^{\mathbb{Q}}(y)dy}=\frac{e^{hx}f_{X}^{\mathbb{Q}}(x)}{E_{\mathbb{Q}}\left[S_{T}^{h}\right]},\quad
x>0.$

It follows that

$f_{S_{T}}^{\mathbb{P}^{h}}(x)=\frac{x^{h}}{x}\frac{f_{X}^{\mathbb{Q}}(\log(x))}{E_{\mathbb{Q}}\left[S_{T}^{h}\right]},\quad
x>0.$

The density $f_{S_{T}}^{\mathbb{P}^{h^{\ast}}}$ crosses
$f_{S_{T}}^{\mathbb{P}^{h}}$ only once from above for $h^{\ast}<h$; hence, by
Denuit et al. (2005, Property 3.3.32), it follows that

$F_{S_{T}}^{\mathbb{P}^{h^{\ast}}}\preceq_{\mathbb{F}_{FSD}}F_{S_{T}}^{\mathbb{P}^{h}}\Rightarrow
F_{S_{T}}^{\mathbb{P}^{h^{\ast}}}\preceq_{\mathbb{F}_{SSD}}F_{S_{T}}^{\mathbb{P}^{h}},\quad
h\in\mathcal{H}.$

For the likelihood ratio it, holds that

$\ell^{\mathbb{P}^{h^{\ast}}}=\frac{f_{S_{T}}^{\mathbb{P}^{h^{\ast}}}(S_{T})}{f_{S_{T}}^{\mathbb{Q}}(S_{T})}=\frac{(S_{T})^{h^{\ast}}}{E_{\mathbb{Q}}\left[(S_{T})^{h^{\ast}}\right]},$

which is strictly increasing in $S_{T}$ as $h^{\ast}>0$ and concave if
$h^{\ast}\in(0,1]$. We can apply Proposition 2.8 to show that conditions C3
and C4 are satisfied for the sets $\mathbb{F}_{FSD}$ and $\mathbb{F}_{SSD}$
with least favorable measure $\mathbb{P}^{\ast}=\mathbb{P}^{h^{*}}$ and
corresponding likelihood ratio $\ell^{\ast}=\ell^{\mathbb{P}^{h^{*}}}.$ We can
use Theorem 3.1 to compute the cost-efficient payoff of a distribution
function $F_{0}$.

##  4 Robust portfolio selection

Gilboa and Schmeidler (1989) provide axioms that justify a maxmin expected
utility framework to make robust decisions when there is ambiguity on the
probability measure $\mathbb{P}$, i.e., when $\\#{\mathcal{P}}>1.$ In this
framework, Schied (2005) shows that when a least favorable measure
$\mathbb{P}^{\ast}\in\mathcal{P}$ with respect to FSD ordering (e.g., the
stochastic integral ordering induced by the set $\mathbb{F}_{FSD}$ as defined
in Section 3) exists, an optimal portfolio can be derived. In this section, we
extend the work of Schied (2005) in two different ways. First, we account for
preferences beyond expected utility. Specifically, we derive optimal
portfolios for robust preferences that are in accord with expected utility
theory, rank dependent utility theory and Yaari’s dual theory. Second,
assuming the existence of a least favorable measure $\mathbb{P}^{\ast}$ with
respect to a general stochastic integral ordering induced by some set
$\mathbb{F}$, not necessarily identical to $\mathbb{F}_{FSD}$, we derive the
optimal portfolio. Specifically, we derive optimal portfolios when a least
favorable measure $\mathbb{P}^{\ast}\in\mathcal{P}$ with respect to SSD
ordering exists (see Proposition 2.8 for a sufficient condition) and the
target distribution $F_{0}$ is sufficiently light tailed (see Remark 3.4).

###  4.1 Family consistent preferences

A _preference_ $W$ is defined as a functional from the set of payoffs
$\mathcal{X}$ to the real line (He et al. (2017), Assa and Zimper (2018)).
Under preference $W$, the payoff $Y$ is preferred to $X$ if $W(X)\leqslant
W(Y)$. In general, $W$ may depend on the different measures
$\mathbb{P}\in\mathcal{P}$ in a complicated way. In what follows, we denote a
preference that depends solely on some $\mathbb{P}\in\mathcal{P}$ by
$W_{\mathbb{P}}$.

######  Definition 4.1.

Let $\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$ be a family of
preferences. The preference $W_{\mathbb{P}}$, $\mathbb{P}\in\mathcal{P}$ is
called _$\mathbb{P}-$ law invariant_ if
$F_{X}^{\mathbb{P}}=F_{Y}^{\mathbb{P}}$ implies that
$W_{\mathbb{P}}(X)=W_{\mathbb{P}}(Y)$. The family
$\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$ is called _law
invariant_ if each individual preference $W_{\mathbb{P}}$ is _$\mathbb{P}-$_
law invariant.

######  Example 4.2.

A standard example of a $\mathbb{P}-$law invariant preference is
$W_{\mathbb{P}}(X)=E_{\mathbb{P}}[u(X)]$ for some increasing utility function
$u$. In this case,
$W(X)=\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(X)$ amounts to
the worst-case expected utility, commonly called _robust expected utility_ ,
which was introduced in Gilboa and Schmeidler (1989). It is also referred to a
_robust utility functional_ in Schied et al. (2009).

To the best of our knowledge, the next definition is new to the literature. It
will be helpful in solving robust portfolio choice problems.

######  Definition 4.3.

Let $(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}$ be a family of preferences.
Let $\mathcal{Y}\subset\mathcal{X}$. The family of preferences
$(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}$ is called _$\mathbb{F}-$ family
consistent on_ $\mathcal{Y}$ _with respect to_
$\mathbb{P}^{\ast}\in\mathcal{P}$ if for all $Y\in\mathcal{Y}$ the inequality

$F_{Y}^{\mathbb{P}^{\ast}}\preceq_{\mathbb{F}}F_{Y}^{\mathbb{P}},\quad\mathbb{P}\in\mathcal{P}$

implies that

$W_{\mathbb{P}^{\ast}}(Y)\leqslant
W_{\mathbb{P}}(Y),\quad\mathbb{P}\in\mathcal{P}.$

$\mathbb{F}-$family consistency of
$(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}$ with respect to some
$\mathbb{P}^{\ast}\in\mathcal{P}$ has the following interpretation: if a
measure $\mathbb{P}^{\ast}$ yields the most pessimistic view of any payoff $Y$
w.r.t. the stochastic ordering induced by some set $\mathbb{F}$, then the
preference under that measure is the lowest as well.

Next, we discuss some examples. Let $\mathcal{Y}\subset\mathcal{X}$ be a set
of payoffs and let $\mathcal{D}$ be the set of cumulative distribution
functions induced by $\mathcal{Y}$, i.e.,

$\mathcal{D}=\\{F_{Y}^{\mathbb{P}}:\,Y\in\mathcal{Y},\,\mathbb{P}\in\mathcal{P}\\}.$

Let us consider an agent taking into account a family of law invariant
preferences $(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}},$ i.e.,

$W_{\mathbb{P}}(Y)=w(F_{Y}^{\mathbb{P}})$ (4.1)

for some well defined $w:\mathcal{D}\rightarrow\mathbb{R}$. If $w$ respects
integral stochastic ordering, i.e.,

$F\preceq_{\mathbb{F}}G\Rightarrow w(F)\leqslant w(G),\quad
F,G\in\mathcal{D},$ (4.2)

then $(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}$ is $\mathbb{F}-$family
consistent on $\mathcal{Y}$ with respect to $\mathbb{P}^{\ast}\in\mathcal{P}$.
We provide some specific examples in the contexts of expected utility theory,
Yaari’s dual theory of choice and rank-dependent expected utility theory:

######  Example 4.4.

Let $u:\mathbb{R}_{+}\to\mathbb{R}$. Let $\phi:[0,1]\rightarrow[0,1]$ with
$\phi(0)=0$ and $\phi(1)=1$. For a given distribution function $F$, define

$\displaystyle w^{\text{EUT}}(F)$ $\displaystyle=\int_{\mathbb{R_{+}}}u(x)dF$
$\displaystyle w^{\text{Yaari}}(F)$
$\displaystyle=\int_{\mathbb{R_{+}}}\phi(1-F(x))dx$ $\displaystyle
w^{\text{RDEU}}(F)$ $\displaystyle=\int_{\mathbb{R_{+}}}u(x)d(1-\phi(1-F(x)),$

where we tacitly assume that all integrals exist. It is straightforward to
show that when $u$ and $\phi$ are non-decreasing, it holds that the family of
preferences induced by $w^{\text{EUT}}$, $w^{\text{Yaari}}$ or
$w^{\text{RDEU}}$ as in (4.1) is _ $\mathbb{F}_{FSD}-$_family consistent on
$\mathcal{Y}$ where $\mathcal{Y}$ is restricted to contain random variables
such that all relevant integrals exist. Furthermore, if $u$ is strictly
increasing and concave and $\phi$ is strictly increasing, continuously
differentiable and convex, we obtain that such a family is a _
$\mathbb{F}_{SSD}-$_family consistent on $\mathcal{Y}$; see Yaari (1987), Wang
and Young (1998), He et al. (2017) and Ryan (2006).

######  Remark 4.5.

One could allow the function $w$ in (4.1) to depend on $\mathbb{P}$, i.e.,
define $W_{\mathbb{P}}(Y)=w_{\mathbb{P}}(F_{Y}^{\mathbb{P}})$ for some
$w_{\mathbb{P}}:\mathcal{D}\rightarrow\mathbb{R}$, $\mathbb{P}\in\mathcal{P}$.
The family of preferences is then $\mathbb{F}-$family consistent on
$\mathcal{Y}$ with respect to $\mathbb{P}^{\ast}\in\mathcal{P}$ if both

$w_{\mathbb{P}^{\ast}}(F)\leqslant w_{\mathbb{P}}(F),\quad
F\in\mathcal{D},\quad\mathbb{P}\in\mathcal{P}$

and (4.2) holds. To see this, let $Y\in\mathcal{Y}$ with
$F_{Y}^{\mathbb{P}^{\ast}}\leqslant F_{Y}^{\mathbb{P}}$ for all
$\mathbb{P}\in\mathcal{P}$. It follows that

$W_{\mathbb{P}^{\ast}}(Y)=w_{\mathbb{P}^{\ast}}\left(F_{Y}^{\mathbb{P}^{\ast}}\right)\leqslant
w_{\mathbb{P}}\left(F_{Y}^{\mathbb{P}^{\ast}}\right)\leqslant
w_{\mathbb{P}}\left(F_{Y}^{\mathbb{P}}\right)=W_{\mathbb{P}}(Y).$

###  4.2 Optimal portfolio for robust preferences

Inspired by Gilboa and Schmeidler (1989) and Schied (2005), we consider the
following problem:

######  Problem 2.

Let $x_{0}>0$ be the initial wealth. Let
$\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$ be a family of
preferences. We consider the robust maximization problem

$\max_{X\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}\inf_{\mathbb{P}\in\mathcal{P}}W_{\mathbb{P}}(X),$
(4.3)

where
$\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}=\bigcap_{\mathbb{P}\in\mathcal{P}}\mathcal{Y}_{W_{\mathbb{P}}}^{x_{0}}$
and

$\mathcal{Y}_{W_{\mathbb{P}}}^{x_{0}}:=\left\\{X\in\mathcal{X}\,:W_{\mathbb{P}}[X]\in\mathbb{R},\,e^{-rT}E_{\mathbb{Q}}\left[X\right]\leqslant
x_{0}\right\\},\quad\mathbb{P}\in\mathcal{P}.$ (4.4)

It turns out that under certain conditions a solution to the robust
optimization problem (4.3) can be found as a solution to a _maximization
problem under a single measure_ $\mathbb{P}\in\mathcal{P}$.

######  Problem 3.

Let $x_{0}>0$ be the initial wealth. Let $W_{\mathbb{P}}$,
${\mathbb{P}\in\mathcal{P}}$ be a preference. We consider the _maximization
problem_

$\max_{X\in\mathcal{Y}_{W_{\mathbb{P}}}^{x_{0}}}W_{\mathbb{P}}(X).$ (4.5)

Under the assumption of the existence of a least favorable measure with
respect to FSD ordering, Schied (2005) showed that in order to solve the
robust maximization problem (4.3) for preferences
$\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$,
${W}_{\mathbb{P}}(x)=E_{\mathbb{P}}[u(X)]$, it actually suffices to solve the
single measure maximization problem (4.5). The following theorem generalizes
this result beyond the expected utility setting to a general law invariant
family of preferences
$\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$. The theorem is
illustrated in Section 4.3, where we consider a robust rank-dependent expected
utility maximization problem for an investor with ambiguity on the trend
and/or volatility of the risky asset.

######  Theorem 4.6.

Let $\mathbb{F}=\mathbb{F}_{FSD}$. Given conditions C3 and C4, assume that
$\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$ is a law invariant
and $\mathbb{F}_{FSD}-$consistent family of preferences on
$\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}$ with
respect to $\mathbb{P}^{\ast}\in\mathcal{P}$. Assume that the maximization
problem (4.5) under $\mathbb{P}^{\ast}$ has a solution
$\tilde{X}\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}.$
Then it holds that

$\underset{X\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}{\max}\,\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(X)=\underset{X\in\mathcal{Y}_{W_{\mathbb{P^{\ast}}}}^{x_{0}}}{\max}\,W_{\mathbb{P}^{\ast}}(X).$

###### Proof.

Let $h\in\mathbb{F}_{FSD}$ such that
$h(\ell^{\ast})\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}$.
Then, it holds by the $\mathbb{F}_{FSD}-$family consistency, condition C3 and
(2.4) that

$W_{\mathbb{P}^{\ast}}(h(\ell^{\ast}))\leqslant\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(h(\ell^{\ast})).$
(4.6)

Let

$X^{\ast}=\left[F_{\tilde{X}}^{\mathbb{P}^{\ast}}\right]^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}(\ell^{\ast})\right).$

Then $X^{\ast}$ solves the standard cost-efficiency problem for
$F_{\tilde{X}}^{\mathbb{P}^{\ast}}$ and thus
$E_{\mathbb{Q}}[X^{\ast}]\leqslant E_{\mathbb{Q}}[\tilde{X}]$ and
$F_{\tilde{X}}^{\mathbb{P}^{\ast}}=F_{X^{\ast}}^{\mathbb{P}^{\ast}}$; hence,
by the law invariance of
$\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$, it holds that
$X^{\ast}\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}$.
It further holds that $X^{\ast}$ is a non-decreasing function of
$\ell^{\ast}$. It follows by (4.6) that

$\displaystyle\underset{X\in\mathcal{Y}_{W_{\mathbb{P^{\ast}}}}^{x_{0}}}{\max}\,W_{\mathbb{P}^{\ast}}(X)$
$\displaystyle=W_{\mathbb{P}^{\ast}}(\tilde{X})=W_{\mathbb{P}^{\ast}}(X^{\ast})$
$\displaystyle\leqslant\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(X^{\ast})$
$\displaystyle\leqslant\underset{X\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}{\max}\,\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(X)$
$\displaystyle\leqslant\underset{X\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}{\max}\,W_{\mathbb{P}^{\ast}}(X)$
$\displaystyle\leqslant\underset{X\in\mathcal{Y}_{W_{\mathbb{P^{\ast}}}}^{x_{0}}}{\max}\,W_{\mathbb{P}^{\ast}}(X),$

where the last inequality follows by
$\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}\subset\mathcal{Y}_{W_{\mathbb{P^{\ast}}}}^{x_{0}}$.
∎

From Theorem 4.6, it follows immediately that solving robust preference
maximization problems may reduce to solving an optimization problem under a
single probability measure. The following example illustrates this
consequence.

######  Example 4.7.

Assume $\mathbb{F}=\mathbb{F}_{FSD}$ and that conditions C3 and C4 are
satisfied. Let $W_{\mathbb{P}}(F)=w(F_{Y}^{\mathbb{P}})$ as in (4.1), where
$w\in\\{w^{\text{EUT}},w^{\text{Yaari}},w^{\text{RDEU}}\\}$ as in Example 4.4.
Assuming a solution to (4.5) under $\mathbb{P}^{\ast}\in\mathcal{P}$ exists,
then it follows that

$\underset{X\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}{\max}\,\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(X)=\underset{X\in\mathcal{Y}_{W_{\mathbb{P^{\ast}}}}^{x_{0}}}{\max}\,W_{\mathbb{P}^{\ast}}(X).$

The main assumption in Theorem 4.6 that is needed to solve the robust
maximization problem (4.3) in the case of a family of law invariant
preferences $\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$ is the
existence of a least favorable measure $\mathbb{P}^{\ast}$ with respect to
$\mathbb{F}_{FSD}$. In the following theorem we show that it is possible to
weaken this assumption in that we only require existence of a least favorable
measure $\mathbb{P}^{\ast}$ with respect to some
$\mathbb{F}\subset\mathbb{F}_{FSD}$, e.g., $\mathbb{F}=\mathbb{F}_{SSD}$. The
theorem is illustrated in Section 4.3, where we consider a robust rank-
dependent expected utility maximization problem for an investor who faces
ambiguity on expected return and volatility of the risky asset.

######  Theorem 4.8.

Consider a given set $\mathbb{F}.$ Given conditions C3 and C4, assume that the
maximization problem (4.5) under $\mathbb{P}^{\ast}$ has a solution
$\tilde{X}\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}$,
which can $\mathbb{P}^{\ast}-$a.s. be expressed as $f(\ell^{\ast})$ for some
$f\in\mathbb{F}$. Further, assume that
$\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$ is
$\mathbb{F}-$family consistent on
$\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}$ with
respect to $\mathbb{P}^{\ast}$. Then it holds that

$\underset{X\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}{\max}\,\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(X)=\underset{X\in\mathcal{Y}_{W_{\mathbb{P^{\ast}}}}^{x_{0}}}{\max}\,W_{\mathbb{P}^{\ast}}(X).$

###### Proof.

Let $h\in\mathbb{F}$ such that
$h(\ell^{\ast})\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}$.
Then, (4.6) holds true by the $\mathbb{F}-$family consistency and (2.4). By
assumption, $\tilde{X}=f(\ell^{\ast})$, $\mathbb{P}^{\ast}-$a.s. for some
$f\in\mathbb{F}$. Hence, it holds that

$\displaystyle\underset{X\in\mathcal{Y}_{W_{\mathbb{P^{\ast}}}}^{x_{0}}}{\max}\,W_{\mathbb{P}^{\ast}}(X)$
$\displaystyle=W_{\mathbb{P}^{\ast}}(\tilde{X})$
$\displaystyle\leqslant\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(\tilde{X})$
$\displaystyle\leqslant\underset{X\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}{\max}\,\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(X)$
$\displaystyle\leqslant\underset{X\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}{\max}\,W_{\mathbb{P}^{\ast}}(X)$
$\displaystyle\leqslant\underset{X\in\mathcal{Y}_{W_{\mathbb{P^{\ast}}}}^{x_{0}}}{\max}\,W_{\mathbb{P}^{\ast}}(X),$

in which the last inequality follows by
$\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}\subset\mathcal{Y}_{W_{\mathbb{P^{\ast}}}}^{x_{0}}$.
∎

Note that, as compared to the statements in Theorem 4.6, in Theorem 4.8 the
$W_{\mathbb{P}}$ do not need to be law-invariant. Moreover, as long as the
solution can be expressed as a certain function of the likelihood ratio
$\ell^{\ast}$, the preferences do not need to be increasing, i.e., $X\leqslant
Y$ does not need to imply $W_{\mathbb{P}}(X)\leqslant W_{\mathbb{P}}(Y).$ As
pointed out, Theorem 4.8 is applicable, in particular, for the case
$\mathbb{F}=\mathbb{F}_{SSD}$. However, the requirement that
$\tilde{X}\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}$
can be $\mathbb{P}^{\ast}-$a.s. expressed as $h(\ell^{\ast})$ for some
$h\in\mathbb{F}_{SSD}$ is equivalent to $h$ being increasing and concave. This
property is difficult to verify ex-ante. Hereafter, we show that in the case
in which $W_{\mathbb{P}}$ is an expected utility, this condition translates
into an easy-to-verify condition on the utility function. We formulate the
following theorem.

######  Theorem 4.9.

Let $\mathbb{P}\in\mathcal{P}$ with likelihood ratio $\ell^{\mathbb{P}}$. Let
$u:\mathbb{R}_{+}\rightarrow\mathbb{R}$ be a differentiable, concave and
strictly increasing utility function such that $u^{\prime}$ is strictly
decreasing. If the maximization problem (4.5) under $\mathbb{P}$ has a
solution, then the solution is a non-decreasing and concave function of
$\ell^{\mathbb{P}}$ if and only if $\frac{1}{u^{\prime}}$ is convex. If $u$ is
three times differentiable, $\frac{1}{u^{\prime}}$ is convex if and only if

$a(x)\geqslant\frac{p(x)}{2},$ (4.7)

in which $a(x):=-\frac{u^{\prime\prime}(x)}{u^{\prime}(x)}$ refers to the
absolute risk aversion measure and
$p(x):=-\frac{u^{\prime\prime\prime}(x)}{u^{\prime\prime}(x)}$ to the absolute
prudence.

###### Proof.

By Lemma 2 in Bernard et al. (2015), the solution to (4.5) is unique and given
by $[u^{\prime}]^{-1}\left(\frac{c_{0}}{\ell^{\mathbb{P}}}\right)$ for some
$c_{0}>0$. See also Merton (1971) for a proof in a context in which Inada’s
conditions are satisfied. Note that $u^{\prime}>0$ and that
$\frac{1}{u^{\prime}}$ is strictly increasing. Observe that the inverse of
$\frac{1}{u^{\prime}}$ is $x\mapsto[u^{\prime}]^{-1}(\frac{1}{x})$, which is
hence also strictly increasing. The inverse of a convex (concave) and strictly
increasing function is concave (convex). For the second assertion, observe
that $u^{\prime\prime}<0$ and that a function is convex on an open interval if
and only if its second derivative is non-negative. Then (4.7) follows
immediately. ∎

######  Remark 4.10.

Maggi et al. (2006) have shown that $a(x)>p(x)$ if and only if the utility has
increasing absolute risk aversion, which is somewhat unusual (it is typically
assumed that agents have decreasing absolute risk aversion given that they
become less risk averse as their wealth increases). Here, our condition (4.7)
is not incompatible with decreasing absolute risk aversion due to the factor
$\frac{1}{2}$.

Condition (4.7) has appeared several times in the literature. It has been
found to play a role in the context of insurance models in Bourlès (2017), but
it also appeared as a condition in the opening of a new asset market (Gollier
and Kimball (1996)), when there is uncertainty on the size (Gollier et al.
(2000)) or the probability of losses (Gollier (2002)) and under contingent
auditing (Sinclair-Desgagné and Gabel (1997)). Further interpretation of this
condition and, in particular, of the degree of concavity of the inverse of the
marginal utility can be found in Bourlès (2017). This condition also appears
in Varian (1985) in the context of portfolio selection under ambiguity.

######  Example 4.11.

As an illustration of Theorem 4.9, we provide two utility functions, which are
differentiable, concave and strictly increasing functions such that one over
the marginal utility is convex.

  * •

The exponential utility for risk-averse agents:
$u:\mathbb{R}_{+}\rightarrow\mathbb{R},\,x\mapsto 1-e^{-\lambda x},$ for
$\lambda>0$. It holds that $\frac{1}{u^{\prime}(x)}=e^{\lambda x}$, which is
strictly increasing and convex.

  * •

CRRA utility:
$u:\mathbb{R}_{+}\rightarrow\mathbb{R},\,x\mapsto\frac{x^{1-\eta}}{1-\eta},$
for $\eta>1$. It holds that $\frac{1}{u^{\prime}(x)}=x^{\eta}$, which is
strictly increasing and convex.

###  4.3 Rank-dependent utility in log-normal markets

We now discuss some examples to illustrate Section 4.2 in a log-normal market
setting with uncertainty on the drift and volatility. In particular, we
explicitly solve a robust rank-dependent expected utility problem using
Theorems 4.6 and 4.8. As in Section 3.1.2, we assume that the real-world
distribution of $S_{T}$ is log-normal with parameters
$\log(S_{0})+(\mu-\frac{\sigma^{2}}{2})T$ and $\sigma^{2}T$ and that the
investor has uncertainty on the trend and potentially also the volatility. She
may expect the true parameters to lie within the cube

$\mathcal{D}^{\mu_{1},\mu_{2},\sigma_{1},s}=\left\\{(\mu,\sigma)\subset\mathbb{R}^{2}\,:\,\mu_{1}\leqslant\mu\leqslant\mu_{2},\,\sigma_{1}\leqslant\sigma\leqslant
s\right\\}$

for $r<\mu_{1}<\mu_{2}$ and $0<\sigma_{1}\leqslant s$. The investor thus
considers
$\mathcal{P}=\left(\mathbb{P}^{\mu,\sigma}\right)_{(\mu,\sigma)\in\mathcal{D}^{\mu_{1},\mu_{2},\sigma_{1},s}}$
as the set of all plausible probability measures on $(\Omega,\mathcal{F})$.
Note that if $\sigma_{1}=s,$ the investor only faces drift ambiguity;
otherwise, she considers ambiguity on both trend and volatility. Under
$\mathbb{P}^{\mu,\sigma}$, $S_{T}$ is log-normal with density
$f^{\mu,\sigma}$, defined in Equation (3.1). In the next example, $\Phi$ is
the standard normal distribution function.

######  Example 4.12.

Let $U(x)=\frac{x^{1-\eta}}{1-\eta}$, $x\geqslant 0$ for $\eta\in(0,1)$ be the
CRRA utility function. Let $\gamma\in\mathbb{R}$ and let
$w(u)=\Phi(\Phi^{-1}(u)+\gamma)$, $u\in[0,1]$ denote the so-called Wang
transform, which is increasing concave if $\gamma>0$ and increasing convex if
$\gamma<0$. Consider the following portfolio choice problem; in which the
investor maximizes her expected rank-dependent utility:

$\max_{X\in\mathcal{Y}_{W_{\mathbb{P}^{\mu,\sigma}}}^{x_{0}}}\int_{0}^{\infty}U(x)d(1-w(1-F_{X}^{\mathbb{P}^{\mu,\sigma}}(x)))$
(4.8)

where $x_{0}>0$ is the initial wealth and
$(\mu,\sigma)\in\mathcal{D}^{\mu_{1},\mu_{2},\sigma_{1},s}$. Let
$\theta:=\sqrt{T}\frac{\mu-r}{\sigma}$. The solution to (4.8) is given by

$X_{\mu,\sigma}^{\ast}:=\begin{cases}\lambda^{-\frac{1}{\eta}}\exp\left(\frac{rT}{\eta}-\frac{1}{2}\frac{\gamma}{\eta}(\theta+\gamma)\right)\left(\ell^{\mathbb{P}}\right)^{\frac{\gamma}{\theta\eta}+\frac{1}{\eta}}&,\gamma>-\theta\\\
\lambda^{-\frac{1}{\eta}}&\text{, otherwise,}\end{cases}$ (4.9)

where $\lambda$ depends on $\eta$, $\gamma$ and $\theta$, see Equations
(4.11), (4.12) and (4.13). The solution to the robust rank-dependent utility
problem

$\max_{X\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}\inf_{\mathbb{P}\in\mathcal{P}}\int_{0}^{\infty}U(x)d(1-w(1-F_{X}^{\mathbb{P}}(x)))dx$
(4.10)

is given by $X_{\mu_{1},s}^{\ast}$ if there is no ambiguity on the volatility,
i.e., when $\sigma_{1}=s$. If there is ambiguity on the volatility, $\gamma<0$
and $\frac{\mu_{1}-r}{s^{2}}\in(0,1]$, then $X_{\mu_{1},s}^{\ast}$ still
solves (4.10).

###### Proof.

We first prove that $X_{\mu,\sigma}^{\ast}$ solves (4.8). Let
$\mathbb{P}=\mathbb{P}^{\mu,\sigma}$. The state price
$\xi^{\mathbb{P}}:=\frac{e^{-rT}}{\ell^{\mathbb{P}}}$ is log-normally
distributed with parameters $-rT-\frac{1}{2}\theta^{2}$ and $\theta>0$, see
Equation (3.2). Hence, as $\mu>r$, it holds that

$\displaystyle F_{\xi^{\mathbb{P}}}^{\mathbb{P}}(x)$
$\displaystyle=\mathbb{P}\left(\xi^{\mathbb{P}}\leqslant
x\right)=\Phi\left(\frac{\log(x)+rT+\frac{1}{2}\theta^{2}}{\theta}\right),\quad
x>0$

and

$\left[F_{\xi^{\mathbb{P}}}^{\mathbb{P}}\right]^{-1}(p)=\exp\left(\Phi^{-1}(p)\theta-
rT-\frac{1}{2}\theta^{2}\right),\quad p\in(0,1).$

Let

$H(z)=-\int_{0}^{w^{-1}(1-z)}\left[F_{\xi^{\mathbb{P}}}^{\mathbb{P}}\right]^{-1}(t)dt,\quad
z\in[0,1].$

The solution to the classical rank-utility problem (4.8) is well-known (see
for instance Theorem 4.1 in Xu (2016) or Section 3.2 in Rüschendorf and
Vanduffel (2020)), and is given by

$X_{\mu,\sigma}^{\ast}=[U^{\prime}]^{-1}\left(\lambda\hat{H}^{\prime}\left(1-w\left(F_{\xi^{\mathbb{P}}}^{\mathbb{P}}\left(\xi^{\mathbb{P}}\right)\right)\right)\right),$

where $\lambda$ is determined by
$E_{\mathbb{P}}\left[\xi^{\mathbb{P}}X_{\mu,\sigma}^{\ast}\right]=x_{0}$ and
$\hat{H}$ is the concave envelope of $H$. Using
$w^{\prime}(u)=\frac{\Phi^{\prime}(\Phi^{-1}(u)+\gamma)}{\Phi^{\prime}(\Phi^{-1}(u))}$
, after some calculations, we obtain that

$\displaystyle H^{\prime}(z)$
$\displaystyle=\frac{\left[F_{\xi^{\mathbb{P}}}^{\mathbb{P}}\right]^{-1}(w^{-1}(1-z))}{w^{\prime}(w^{-1}(1-z))}=\exp\left(\Phi^{-1}(1-z)(\gamma+\theta)-rT-\frac{1}{2}(\gamma+\theta)^{2}\right).$

We distinguish two cases to find a more explicit expression for
$X_{\mu,\sigma}^{\ast}$. _Case 1_ : $\gamma+\theta>0$. Then, $H^{\prime}$ is
non-increasing and hence $H$ is concave and equal to $\hat{H}$. As
$[U^{\prime}]^{-1}(y)=y^{-\frac{1}{\eta}}$, it is easy to see that

$\displaystyle X_{\mu,\sigma}^{\ast}$
$\displaystyle=\lambda^{-\frac{1}{\eta}}\exp\left(-\frac{rT\gamma}{\theta\eta}-\frac{1}{2}\frac{\gamma}{\eta}(\theta+\gamma)\right)\left(\xi^{\mathbb{P}}\right)^{-\frac{\gamma}{\theta\eta}-\frac{1}{\eta}}$
$\displaystyle=\lambda^{-\frac{1}{\eta}}\exp\left(\frac{rT}{\eta}-\frac{1}{2}\frac{\gamma}{\eta}(\theta+\gamma)\right)\left(\ell^{\mathbb{P}}\right)^{\frac{\gamma}{\theta\eta}+\frac{1}{\eta}}.$

If $1-\frac{\gamma}{\theta\eta}-\frac{1}{\eta}=0$, then
$\xi^{\mathbb{P}}X_{\mu,\sigma}^{\ast}$ is constant and it holds that

$\lambda=x_{0}^{-\eta}\exp\left(-\frac{rT\gamma}{\theta}-\frac{1}{2}\gamma(\theta+\gamma)\right).$
(4.11)

Otherwise, $\xi^{\mathbb{P}}X_{\mu,\sigma}^{\ast}$ is log-normally distributed
and it follows that

$\lambda=x_{0}^{-\eta}\exp\left(rT(1-\eta)+\frac{1}{2}\left(\theta^{2}(1-\eta)+\gamma^{2}+(\theta\eta-(\gamma+\theta))^{2}\right)\right).$
(4.12)

_Case 2_ : Assume $\gamma+\theta\leqslant 0$. Then, $H$ is convex. Note that
$H(0)=-1$ and $H(1)=0$. As $H$ is convex, the concave envelope $\hat{H}$ of
$H$ is given by $\hat{H}(x)=x-1$. Then $\hat{H}^{\prime}\equiv 1$ and

$X_{\mu,\sigma}^{\ast}=[U^{\prime}]^{-1}\left(\lambda\right)=\lambda^{-\frac{1}{\eta}}.$

Therefore,
$\xi^{\mathbb{P}}X_{\mu,\sigma}^{\ast}=\lambda^{-\frac{1}{\eta}}\xi^{\mathbb{P}}$,
and hence

$\lambda=x_{0}^{-\eta}e^{-rT\eta}.$ (4.13)

Assume that there is no ambiguity on the volatility, i.e., $\sigma_{1}=s$.
Section 3.1.1 shows that the least favorable measure with respect to
$\mathbb{F}_{FSD}$ is given by $\mathbb{P}^{\ast}=\mathbb{P}^{\mu_{1},s}$ with
corresponding likelihood ratio $\ell^{\ast}=\ell^{\mathbb{P}^{\mu_{1},s}}$. By
Example 4.4, the preference in (4.10) is $\mathbb{F}_{FSD}-$family consistent
and Theorem 4.6 shows that $X_{\mu,\sigma}^{\ast}$ solves the robust rank-
dependent utility problem (4.10). Lastly, assume that there is ambiguity on
the volatility. If $\frac{\mu_{1}-r}{s^{2}}\in(0,1]$, Proposition 3.11 shows
that the least favorable measure with respect to $\mathbb{F}_{SSD}$ is also
$\mathbb{P}^{\ast}$. If $\gamma<0$, $X_{\mu,\sigma}^{\ast}$ is a concave and
non-decreasing function of $\ell^{\ast}$. By Example 4.4, the preference in
(4.10) is $\mathbb{F}_{SSD}-$family consistent. Apply Theorem 4.8 to show that
also in this case, $X_{\mu,\sigma}^{\ast}$ solves the robust rank-dependent
utility problem (4.10). ∎

To better understand the solution in Example 4.12, we “rationalize” the
solution as in Bernard et al. (2015), i.e., we show that the optimal
investment strategy in the robust rank-dependent setting also solves an
expected utility maximization problem. Example 4.13 shows that the solution to
the expected rank-dependent utility problems (4.8) and (4.10) involving a Wang
transform with parameter $\gamma$ and a CRRA utility function with parameter
$\eta$ can be rationalized by a CRRA utility with parameter
$\frac{\eta\theta}{\gamma+\theta}$.

######  Example 4.13.

Let $(\mu,\sigma)\in\mathcal{D}^{\mu_{1},\mu_{2},\sigma_{1},s}$,
$X_{\mu,\sigma}^{\ast}$, $\theta$, $x_{0}$ and
$\mathcal{Y}_{W_{\mathbb{P}^{\mu,\sigma}}}^{x_{0}}$ as in Example 4.12, such
that $\gamma>-\theta$ and $\eta\theta\neq\gamma+\theta$.
$X_{\mu,\sigma}^{\ast}$ solves the following expected utility maximization
problem:

$\max_{X\in\mathcal{Y}_{W_{\mathbb{P}^{\mu,\sigma}}}^{x_{0}}}\int_{0}^{\infty}u(x)dF_{X}^{\mathbb{P}^{\mu,\sigma}}(x)$

for the utility function

$u(x)=\frac{1}{1-\frac{\eta\theta}{\gamma+\theta}}x^{1-\frac{\eta\theta}{\gamma+\theta}}.$
(4.14)

The function $u:\mathbb{R}_{+}\to\mathbb{R}$ is non-decreasing and concave.

###### Proof.

Note that $\gamma>-\theta$ implies
$\frac{\gamma}{\theta\eta}+\frac{1}{\eta}\neq 0$. Let
$\mathbb{P}=\mathbb{P}^{\mu,\sigma}$. Let
$\xi^{\mathbb{P}}:=\frac{e^{-rT}}{\ell^{\mathbb{P}}}$. As in Bernard et al.
(2015), let $c>0$ and define

$\tilde{u}(x)=\int_{c}^{x}\left[F_{\xi^{\mathbb{P}}}^{\mathbb{P}}\right]^{-1}\left(1-F_{X_{\mu,\sigma}^{\ast}}^{\mathbb{P}}(y)\right)dy,\quad
x\in\mathbb{R}_{+}.$

$X_{\mu,\sigma}^{\ast}$ is log-normally distributed. It follows that

$\displaystyle\left[F_{\xi^{\mathbb{P}}}^{\mathbb{P}}\right]^{-1}\left(1-F_{X_{\mu,\sigma}^{\ast}}^{\mathbb{P}}(x)\right)$
$\displaystyle=\kappa x^{-\frac{\eta\theta}{\gamma+\theta}},$

where $\kappa>0$ is a suitable constant. Thus,

$\displaystyle\tilde{u}(x)$
$\displaystyle=\kappa\frac{\gamma+\theta}{\theta(1-\eta)+\gamma}\left(x^{-\frac{\eta\theta}{\gamma+\theta}+1}-c^{-\frac{\eta\theta}{\gamma+\theta}+1}\right).$

In summary, as $\tilde{u}$ is only determined up to positive affine
transformations, $X_{\mu,\sigma}^{\ast}$ solves the expected utility
maximization problem for the utility given in (4.14): see Theorem 2 in Bernard
et al. (2015). ∎

##  5 Rationalizing robust cost-efficient payoffs

When there is no ambiguity on the probability measure $\mathbb{P}$, there is a
close relationship between cost-efficiency and portfolio optimization: for any
cost-efficient payoff $X$, there exists a utility function $u$ (unique up to a
linear transformation) such that $X$ also solves the expected utility
maximization problem (Bernard et al. (2015)). In this section, we show that
this result can be generalized to the robust setting developed previously in
that robust cost-efficient payoffs can be rationalized in terms of the maxmin
utility framework introduced in Gilboa and Schmeidler (1989). Specifically, we
show $-$ under the same assumptions as in Theorems 4.6 and 4.8 $-$ that
payoffs maximize a robust utility functional as in Gilboa and Schmeidler
(1989) if and only if they are robust cost-efficient.

In the following theorem we distinguish two cases: a) we deal with law-
invariant preferences and FSD ordering and assume that the various (robust)
maximization problems have unique solutions. Or, b) we deal with general
preferences and stochastic ordering and we do not require uniqueness of the
solutions but assume that the solution $X^{\ast}$ of the various maximization
problems can be written as $X^{\ast}=f(\ell^{\ast})$ for some
$f\in\mathbb{F}$.

######  Theorem 5.1.

Assume $\mathbb{F}_{SSD}\subset\mathbb{F}\subset\mathbb{F}_{FSD}$. Let
conditions C2, C3 and C4 hold. Let $X^{\ast}\in\mathcal{X}$ be a payoff. Let
$x_{0}=e^{-rT}E_{\mathbb{Q}}[X^{\ast}]<\infty$. Let $c>0$, such that
$F_{X^{\ast}}^{\mathbb{P}^{\ast}}(c)>0,$ $\mathbb{P}^{\ast}\in\mathcal{P}$.
Let $\xi^{\ast}=\frac{e^{-rT}}{\ell^{\ast}}$ and define

$u(x)=\int_{c}^{x}F_{\xi^{\ast}}^{-1}\left(1-F_{X^{\ast}}^{\mathbb{P}^{\ast}}(y)\right)dy,\quad
x\in\mathbb{R}_{+}.$ (5.1)

Assume that $E_{\mathbb{P}}[u(X^{\ast})]<\infty$ for all
$\mathbb{P}\in\mathcal{P}$. We further assume that one of the following two
conditions is satisfied:

_(a)_

$\mathbb{F}=\mathbb{F}_{FSD}$

_(b)_

$X^{\ast}=f(\ell^{\ast})$, $\mathbb{P}^{\ast}-$a.s. for some $f\in\mathbb{F}$
and $\left[F_{X^{\ast}}^{\mathbb{P^{\ast}}}\right]^{-1}\circ
F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\in\mathbb{F}$.

Then, the following statements are equivalent:

_i)_

_$X^{\ast}$ is cost-efficient under $\mathbb{P}^{\ast}$._

_ii)_

_It holds that
$X^{\ast}=\left[F_{X^{\ast}}^{\mathbb{P^{\ast}}}\right]^{-1}\left(F_{\ell^{\mathbb{P^{\ast}}}}^{\mathbb{P^{\ast}}}\left(\ell^{\mathbb{P}^{\ast}}\right)\right)$,
$\mathbb{P}^{\ast}-$a.s._

_iii)_

_$X^{\ast}$ is $\mathbb{P}^{\ast}-$a.s. non-decreasing in $\ell^{\ast}$._

_iv)_

_$X^{\ast}$ solves the $\mathbb{F}-$robust cost-efficiency problem for
$F_{X^{\ast}}^{\mathbb{P}^{\ast}}$._

_v)_

_$X^{\ast}$ solves the expected utility maximization problem under
$\mathbb{P}^{\ast}$ for the utility function $u$_

$\underset{X\in\mathcal{Y}_{E_{\mathbb{P}^{\ast}}[u(.)]}^{x_{0}}}{\max}\,E_{\mathbb{P}^{\ast}}[u(X)].$

_vi)_

_$X^{\ast}$ solves the robust expected utility problem for the utility
function $u$_

$\underset{X\in\mathcal{Y}_{(E_{\mathbb{P}}[u(.)])_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}{\max}\,\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,E_{\mathbb{P}}[u(X)]$

_and the solution is unique if condition (a) is satisfied._

_vii)_

_There is a family of preferences
$\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$ that is
$\mathbb{F}-$family consistent on
$\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}$ with
respect to_ $\mathbb{P}^{\ast}$_such that
$X^{\ast}\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}$
and $X^{\ast}$ is the solution to the maximization problem under
$\mathbb{P}^{\ast}$_

$\underset{X\in\mathcal{Y}_{W_{\mathbb{P^{\ast}}}}^{x_{0}}}{\max}\,W_{\mathbb{P}^{\ast}}(X)$

_and the solution is unique and the family of preferences is law invariant if
condition (a) is satisfied._

_viii)_

_There is a family of preferences
$\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$ that is
$\mathbb{F}-$family consistent on
$\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}$ with
respect to_ $\mathbb{P}^{\ast}$_such that $X^{\ast}$ is the solution to the
robust maximization problem_

$\underset{X\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}{\max}\,\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(X)$

_and the solution is unique and the family of preferences is law invariant if
condition (a) is satisfied._

###### Proof.

The equivalence between i), ii) and iii) follows from Lemma B.1. The
equivalence between iv) and ii) follows from Theorem 3.1 and Remark 3.2. Note
that $\left[F_{X^{\ast}}^{\mathbb{P^{\ast}}}\right]^{-1}\circ
F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}\in\mathbb{F}_{FSD}$ is always true. By
Theorem 3 in Bernard et al. (2015), i) implies v). By Lemma B.3 and Lemma 3 in
Bernard et al. (2015), v) implies i). By Example 4.4, v) implies vii)
trivially, define $W_{\mathbb{P}}(.)=E_{\mathbb{P}^{\ast}}[u(.)]$ for all
$\mathbb{P}\in\mathcal{P}$. v) implies vi), if (a) holds, by Theorem 4.6 and
Example 4.4 as $E_{\mathbb{P}}[u(X^{\ast})]<\infty$ for all
$\mathbb{P}\in\mathcal{P}$ by assumption. v) implies vi), if (b) holds, by
Theorem 4.8. vi) implies viii) trivially. By Lemma B.3 and Lemma B.4, if (a)
holds, vi)$\Rightarrow$i) and vii)$\Rightarrow$i) and viii)$\Rightarrow$i).
Note that, if (b) holds, iii) is always true because
$f\in\mathbb{F}\subset\mathbb{F}_{FSD}$ is non-decreasing. ∎

######  Remark 5.2.

Assuming that all functions in $\mathbb{F}$ are non-decreasing, i.e., that
$\mathbb{F}\subset\mathbb{F}_{FSD}$, is not really a restriction. Otherwise,
there are two sure payoffs $x_{0},y_{0}\in\mathcal{X}$, i.e., $x_{0},y_{0}$
are constant, such that $x_{0}<y_{0}$ but the distribution of $y_{0}$ does not
dominate the distribution function of $x_{0}$ in integral stochastic ordering.

######  Example 5.3.

Assume $\mathbb{F}=\mathbb{F}_{SSD}$. Consider the robust rank-dependent
expected utility maximization problem in Example 4.12 with solution
$X_{\mu_{1},s}^{\ast}$ defined in (4.9). Let
$\mathbb{P}^{\ast}=\mathbb{P}^{\mu_{1},s}$ and
$\ell^{\ast}=\ell^{\mathbb{P}^{\ast}}$. With the help of the explicit
expressions of $F_{X_{\mu_{1},s}^{\ast}}^{\mathbb{P}^{\ast}}$ and
$F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}$ from the proofs of Examples 4.12 and
4.13, it is easy to see that (b) in Theorem 5.1 holds if $\gamma<0$.

Let us start from viii) in Theorem 5.1. Equation (4.9) implies that the
optimal solution $X_{\mu_{1},s}^{\ast}$ is a non-decreasing function of
$\ell^{\ast}$. Hence, iii) in Theorem 5.1 is satisfied. As shown by Example
4.13, $X_{\mu_{1},s}^{\ast}$ also solves an expected utility maximization
problem for the utility in (4.14), which illustrates v) in Theorem 5.1. One
can easily verify that ii) in Theorem 5.1 is respected by
$X_{\mu_{1},s}^{\ast}$ . Hence, Theorem 3.1 implies that
$X_{\mu_{1},s}^{\ast}$ solves the robust $\mathbb{F}_{SSD}-$cost-efficiency
problem as stated in iv) in Theorem 5.1.

The preferences in Theorem 5.1 for case (b) do not need to be law-invariant or
increasing, i.e., $X\leqslant Y$ does not need to imply
$W_{\mathbb{P}}(X)\leqslant W_{\mathbb{P}}(Y)$. We provide a simple example of
such a preference.

######  Example 5.4.

Define __ $X^{\ast}=f(\ell^{\ast})$ for some $f\in\mathbb{F}$. Let
$x_{0}=E_{\mathbb{Q}}[X^{\ast}]$. For $\mathbb{P}\in\mathcal{P}$, define

$W_{\mathbb{P}}(X)=\begin{cases}1&\text{if}\
X=X^{\ast},\,\,\,\mathbb{P}-\text{a.s.}\\\ 0&\text{if}\ \text{
otherwise},\end{cases}$

which is trivially $\mathbb{F}-$family consistent. An agent with such a
preference only likes $X^{\ast}$ and neglects everything else. She is not law-
invariant and does not prefer more to less. Someone interested only in the
market portfolio or in the risk-free bond might have such a preference. The
solution to the robust maximization problem ix) is $X^{\ast}$, which is cost-
efficient because $X^{\ast}$ is non-decreasing in $\ell^{\ast}$. A utility
function for v) can be constructed as in (5.1).

##  6 Final Remarks

In this paper we assume that the agent has Knightian uncertainty. She is
unsure about the precise physical measure describing the financial market and
knows only that the true physical measure lies within a set $\mathcal{P}$ of
probability measures. Given this ambiguity, it is no longer possible to target
a payoff with a given probability distribution function. In particular, the
close relation between payoffs that are the cheapest possible in reaching a
target distribution function and the optimality thereof under law-invariant
increasing preferences (Dybvig (1988a, b), Sharpe et al. (2000), Goldstein et
al. (2008), Bernard et al. (2015)) is a priori lost, as there is no consensus
regarding what probability distribution to adopt. For this reason, we
introduce the notion of a robust cost-efficient payoff.

For a given distribution function $F_{0}$, the robust cost-efficiency problem
aims at finding the cheapest payoff whose distribution function dominates
$F_{0}$ under all possible physical measures in some integral stochastic
ordering. We solve this problem under some conditions (namely, where there
exists a least favorable measure $\mathbb{P}^{\ast}$ and the integral
stochastic ordering $\preceq_{\mathbb{F}}$ is cost-consistent). The solution
is identical to the solution to the cost-efficiency problem without model
ambiguity under the physical measure $\mathbb{P}^{\ast}$ and given in closed-
form. We are thus able to reduce the problem formulated in a robust setting to
a problem formulated in a standard setting without model ambiguity.

Finally, we show that this notion of robust cost-efficiency plays a key role
in optimal robust portfolio selection and that a very general class of robust
portfolio selection problems (possibly in a non-expected utility setting) can
be reduced to the maxmin expected utility setting of Gilboa and Schmeidler
(1989) for a well-chosen concave utility function.

For this to hold, we make a relatively minor assumption on the family of
preferences, i.e., that it is family consistent: if the measure
$\mathbb{P}^{\ast}$ is the most pessimistic view of a payoff $Y$, then the
preference under that measure is the lowest as well. To the best of our
knowledge, family consistency is new to the literature, and we provide several
examples in the context of expected utility theory, Yaari’s dual theory and
rank-dependent utility theory.

We assume a static setting in which intermediate trading is not possible.
Whilst allowing for dynamic rebalancing may make it possible to achieve higher
levels for the objective at hand (e.g., robust expected utility), this
possibility is only clear when there are no transaction costs, which is not
realistic. In practice, transaction costs usually contain a fixed part, and
hence dynamic trading can only occur a finite number of times since otherwise
bankruptcy occurs. The study of optimal investments in the presence of fixed
costs is not yet very well understood. Recently, Belak et al. (2022) and
Bayraktar et al. (2022) provide optimal strategies in a Black-Scholes market
without ambiguity and assuming expected utility. By contrast our static
setting makes it possible to deal with ambiguity and to address fairly general
objectives.

Appendix

##  Appendix A  Proof for Example 3.6

######  Lemma A.1.

Let $F$ and $G$ be two cdfs. It holds that _ $G\preceq_{\mathbb{F}_{TSD}}F$_
if and only if

$\int_{-\infty}^{\eta}\int_{-\infty}^{\xi}F(x)dxd\xi\leqslant\int_{-\infty}^{\eta}\int_{-\infty}^{\xi}G(x)dxd\xi,\,\,\,\eta\in\mathbb{R}.$

###### Proof.

See Theorem 2.2 of Gotoh and Konno (2000). ∎

We are now ready to construct the counterexample stated in Example 3.6.

###### Proof.

Apply the chain rule to show that $\mathbb{F}_{TSD}$ is composition-
consistent. Next, we construct two distribution functions such that one
dominates the other in TSD but is cheaper.__
_Step 1: define some market setting as in Section 3.1.1:_ let $\mu_{1}=0.01$,
$r=0$, $T=1$ and $s=0.1$. Then, $\frac{\mu_{1}-r}{s^{2}}=1$. Choose $S_{0}$
such that it holds that
$\ell^{\ast}:=\ell^{\mathbb{P}^{\mu_{1}}}=h^{\mu_{1},s}(S_{T})=S_{T}$, i.e.,
$\log(S_{0})=-0.0025$. Under $\mathbb{P}^{\ast}:=\mathbb{P}^{\mu_{1}}$ the
stock is log-normally distributed with parameters
$(\mu_{1}-\frac{s^{2}}{2}+\log(S_{0}))=0.0025$ and $s^{2}=0.01$. Under
$\mathbb{Q}$, the stock is also log-normally distributed with parameters
$(r-\frac{s^{2}}{2}+\log(S_{0}))=-0.0075$ and $s^{2}=0.01$.
$\mathbb{P}^{\ast}$ is a least favorable measure with respect to the set
$\mathbb{F}_{FSD}$, and so hence also is $\mathbb{F}_{TSD}$ because
$\mathbb{F}_{TSD}\subset\mathbb{F}_{FSD}$.__
_Step 2: define two distribution functions:_ Let

$F(x)=\begin{cases}0&,x<0\\\ x&,0\leqslant x<1\\\ 1&,x\geqslant 1\end{cases}$

and, for $p_{0}\in(0,1)$, let

$G(x)=\begin{cases}0&,x<0\\\ p_{0}&,0\leqslant x<1\\\ 1&,x\geqslant
1.\end{cases}$

$F$ is the uniform distribution function and $G$ jumps at zero and at one. It
follows that $F^{-1}(p)=p$ and that

$G^{-1}(p)=\begin{cases}0&,p\in(0,p_{0}]\\\ 1&,p>p_{0}.\end{cases}$

_Step 3: show that $F$ dominates $G$ in TSD: _It holds for $\eta\in(0,1)$ that

$\int_{-\infty}^{\eta}\int_{-\infty}^{\xi}F(x)dxd\xi=\int_{0}^{\eta}\int_{0}^{\xi}xdxd\xi=\frac{1}{6}\eta^{3}.$

and that

$\int_{-\infty}^{\eta}\int_{-\infty}^{\xi}G(x)dxd\xi=\int_{0}^{\eta}\int_{0}^{\xi}p_{0}dxd\xi=\frac{1}{2}p_{0}\eta^{2}.$

Hence, if $\frac{1}{6}\eta^{3}\leqslant\frac{1}{2}p_{0}\eta^{2}$ or,
equivalently, $p_{0}\geqslant\frac{1}{3}$, it follows that
$G\preceq_{\mathbb{F}_{TSD}}F$.__
_Step 4: compute the lowest cost of both distribution functions:_ The cost-
efficient payoff for $F$ is

$X_{F}=F^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}(\ell^{\ast})\right)=F_{S_{T}}^{\mathbb{P}^{\ast}}(S_{T}).$

The lowest price of $F$ can be computed numerically:

$E_{\mathbb{Q}}[X_{F}]=\int_{0}^{\infty}F_{S_{T}}^{\mathbb{P}^{\ast}}(s)f_{S_{T}}^{r}(s)ds=0.472.$

The cost-efficient payoff for $G$ is

$X_{G}=G^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}(\ell^{\ast})\right)=\begin{cases}1&,S_{T}>\left[F_{S_{T}}^{\mathbb{P}^{\ast}}\right]^{-1}(p_{0})\\\
0&,\text{otherwise}\end{cases}.$

Its price is

$E_{\mathbb{Q}}[X_{G}]=\int_{\left[F_{S_{T}}^{\mathbb{P}^{\ast}}\right]^{-1}(p_{0})}^{\infty}f_{S_{T}}^{r}(s)ds=1-F_{S_{T}}^{\mathbb{Q}}\left(\left[F_{S_{T}}^{\mathbb{P}^{\ast}}\right]^{-1}(p_{0})\right).$

Under $\mathbb{P}^{\ast}$, $X_{F}$ is uniform distributed and $X_{G}$ is a
digital option. If $p_{0}=\frac{1}{3}$, the lowest price for $G$ is $0.63$,
which is greater than the lowest price to be paid for $F$. But in this case
$G\preceq_{\mathbb{F}_{TSD}}F$: hence, TSD is not cost-consistent. ∎

##  Appendix B Auxiliary results

######  Lemma B.1.

Fix $\mathbb{P}\in\mathcal{P}$. Let $\ell:=\frac{d\mathbb{P}}{d\mathbb{Q}}$ be
the Radon–Nikodym derivative of $\mathbb{P}$ with respect to $\mathbb{Q}$.
Assume that under $\mathbb{P}$ $\ell$ is continuously distributed and that
$1/\ell$ has finite variance. There is a $\mathbb{P}-$a.s. unique optimizer to
the standard cost-efficiency problem under the probability measure
_$\mathbb{P}$_ given by

$X^{\ast}=F_{0}^{-1}\left(F_{\ell^{\mathbb{P}}}^{\mathbb{P}}\left(\ell\right)\right).$

$X^{\ast}$ is left-continuous and non-decreasing $\mathbb{P}-$a.s.

###### Proof.

Let $\xi=\frac{e^{-rT}}{\ell}$. Then,
$1-F_{\xi}^{\mathbb{P}}(\xi)=F_{\ell^{\mathbb{P}}}^{\mathbb{P}}\left(\ell\right).$
This claim follows both from Dybvig (1988a, b) and from Corollary 2 in Bernard
et al. (2014). See also Schied (2004, Proposition 2.7) for the importance of
the continuity assumption of $\ell$ in obtaining the uniqueness.

∎

######  Lemma B.2.

Fix $\mathbb{P}\in\mathcal{P}$. Let $\ell:=\frac{d\mathbb{P}}{d\mathbb{Q}}$.
Assume that $\ell$ is continuously distributed under $\mathbb{P}$. A payoff
$X\in\mathcal{A}_{F_{0}}^{\mathbb{P}}$ is _$\mathbb{P}-$_ cost-efficient if
and only if it is non-decreasing in $\ell$, $\mathbb{P}-$almost surely.

###### Proof.

See Bernard et al. (2014, Corollary 2 and Proposition 2). ∎

In the following two lemmas we show that the solution of the single or robust
maximization problem is cost-efficient if it is unique.

######  Lemma B.3.

Let $\mathbb{P}\in\mathcal{P}$ with corresponding likelihood ratio
$\ell^{\mathbb{P}}$. Assume that $\ell^{\mathbb{P}}$ is continuously
distributed and that $W_{\mathbb{P}}$ is $\mathbb{P}-$law invariant and
$\tilde{X}$ is a $\mathbb{P}-$a.s. unique solution to the maximization problem
(4.5) under $\mathbb{P}$. Then, $\tilde{X}$ is $\mathbb{P}-$cost-efficient.

###### Proof.

Let

$X^{\ast}=\left[F_{\tilde{X}}^{\mathbb{P}}\right]^{-1}\left(F_{\ell^{\mathbb{P}}}^{\mathbb{P}}(\ell^{\mathbb{P}})\right).$

Then $X^{\ast}$ solves the standard cost-efficiency problem for
$F_{\tilde{X}}^{\mathbb{P}}$ and thus $E_{\mathbb{Q}}[X^{\ast}]\leqslant
E_{\mathbb{Q}}[\tilde{X}]$ and
$F_{\tilde{X}}^{\mathbb{P}}=F_{X^{\ast}}^{\mathbb{P}}$; hence, by the law
invariance of $\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$, it
holds that $X^{\ast}\in\mathcal{Y}_{W_{\mathbb{P}}}^{x_{0}}$. It follows by
law invariance that

$\underset{X\in\mathcal{Y}_{W_{\mathbb{P}}}^{x_{0}}}{\max}\,W_{\mathbb{P}}(X)=W_{\mathbb{P}}(\tilde{X})=W_{\mathbb{P}}(X^{\ast}).$

As $\tilde{X}$ is the unique solution, it must hold that $\tilde{X}=X^{\ast}$,
$\mathbb{P}-$a.s., and thus $\tilde{X}$ is cost-efficient. ∎

######  Lemma B.4.

Assume $\mathbb{F}=\mathbb{F}_{FSD}$. Given Assumptions C2, C3 and C4, assume
that the robust maximization problem (4.3) has a unique solution $\tilde{X}$
and that $\left(W_{\mathbb{P}}\right)_{\mathbb{P}\in\mathcal{P}}$ is law
invariant and $\mathbb{F}_{FSD}-$family consistent on
$\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}$ with
respect to $\mathbb{P}^{\ast}$. Then, $\tilde{X}$ is $\mathbb{P}^{\ast}-$cost-
efficient.

###### Proof.

The proof is similar to the one for Lemma B.3: let

$X^{\ast}=\left[F_{\tilde{X}}^{\mathbb{P}^{\ast}}\right]^{-1}\left(F_{\ell^{\ast}}^{\mathbb{P}^{\ast}}(\ell^{\ast})\right).$

It holds by law invariance that

$\underset{X\in\mathcal{Y}_{(W_{\mathbb{P}})_{\mathbb{P}\in\mathcal{P}}}^{x_{0}}}{\max}\,\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(X)=\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(\tilde{X})=\underset{\mathbb{P}\in\mathcal{P}}{\inf}\,W_{\mathbb{P}}(X^{\ast}).$

∎

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