text:
'I'm happy when I play football' – Neymar offers no assurances over PSG future
Neymar wanted to leave Paris Saint-Germain for Barcelona over the summer and claims to be merely "happy" playing football for now, with no assurances offered over his long-term future in France.
The last transfer window was dominated by speculation regarding the Brazil international, as he made it clear that he was looking to secure a return to Camp Nou.
No deal was done and the 27-year-old has been forced to play his way back into favour with a fan base at Parc des Princes that continues to question his commitment to the cause.
Unleash Foden! Guardiola can't hold back Man City star any longer
Solskjaer's Ferguson tribute act will only take Man Utd so far
How McTominay has allowed Man Utd to win again
'Valverde's going to win the Ballon d'Or!' – How Modric lost his Real Madrid spot to an unknown Uruguayan
And the former Barcelona star insists that being out on the field is the most important thing at this stage of his career, with the intention being to make the most of every opportunity he gets.
Neymar told RMC Sport after helping PSG to a 5-0 Champions League win over Galatasaray: "Am I happy in Paris? I'm happy when I play football.
"What I like best is being out on the field, no matter where I am.
"Where there are two posts, a ball and team-mates, I am happy."
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Neymar would likely be forgiven by many of his detractors in Paris if he were to taste European glory with PSG this season.
He was acquired in 2017 with the intention of helping a dominant force in Ligue 1 to make more of an impression on the continental scene.
That has not happened so far, with PSG having failed to make it past the round of 16 in three successive campaigns.
Neymar has not figured prominently in the knockout stages for them so far, with untimely injuries hindering his cause.
There is, however, a desire on the South American's part to help spearhead a quest for Champions League glory this season alongside the likes of Kylian Mbappe, Edinson Cavani and Mauro Icardi.
Neymar added: "It is important for us to still be in the competition come February, and it is very important for me too.
"I was not there last year and we have to think about the fans and do all we can to be at our best."
PSG are safely through to the last-16 this season and will discover their opponents for the first knockout round when the draw is held on Monday.


Started by three young brothers from Auckland, Sly Guild was born to create quality goods alongside amazing craftsmanship. The brand first started as a hobby but turned into something big once they created their own quality made t-shirt. Since no brand at the time did not have any plain quality made t-shirts, the group of friends decided to create a line dedicated to simplicity and basic wear. From that point on local stores in Auckland started to support their movement and their hand crafted goods became a business.
Being exposed to the cutting of garments at a young age gave them the advantage on seeing how manufacturing and production is executed. Taking their previous knowledge and applying it to Sly Guild has caused them to start production the right away. Their drive for the brand branches from going against the grain and rebelling against the normal fashion scene, resulting in minimalistic garments at amazing quality.
Sly Guild can be worn by anyone. From a 9-5 weekend warrior to a CEO of a corporation, the New Zealand fashion house caters to all sorts of individuals who appreciate great design and well crafted clothing. Check out our Sly Guild collection at the shop today!


text:
Vinny Marseglia (né le ) est un catcheur américain plus connu sous le nom de Vincent.

Il est connu pour avoir travaillé à la Ring of Honor où il a été à quatre reprises ROH Six World Tag Team Champion.

Biographie

Ring of Honor (2012-2022)

The Kingdom (2016-2019)
Le 5 novembre 2016, il rejoint The Kingdom aux côtés de Matt Taven et TK O'Ryan en battant Bullet Club (Adam Cole, Matt Jackson et Nick Jackson) pour avancer à la demi-finale du ROH 6-Man Tag Team Championship Tournament. Le , ils battent Team CMLL (Hechicero, Okumura et Último Guerrero) et se qualifient pour la finale du tournoi. Lors de , ils battent Jay White, Kushida et Lio Rush en finale du tournoi et deviennent les premiers ROH World Six-Man Tag Team Champions. Lors de , ils conservent les titres contre Jax Dane et War Machine.

Lors de , ils battent SoCal Uncensored (Christopher Daniels, Frankie Kazarian et Scorpio Sky) et remportent les ROH World Six-Man Tag Team Championship pour la seconde fois. Lors de , ils conservent les titres contre Los Ingobernables de Japón (Bushi, Evil et Sanada). Le 25 août, ils perdent les titres contre Bullet Club (Cody, Matt et Nick Jackson).

Lors de , ils battent Bullet Club (Cody, Matt et Nick Jackson) et remportent les ROH World Six-Man Tag Team Championship pour la troisième fois. Le 29 mars 2019, ils perdent les titres contre Villain Enterprises (Brody King, Marty Scurll et PCO).

Rivalité avec Matt Taven et The Righteous (2019-2022)

Début décembre, le Kingdom est étrangement pris pour cible par un assaillant inconnu. Celui-ci s'en est pris à Vinny Marseglia et TK O'Ryan, et Matt Taven cherche à connaître le responsable. Les lumières s'éteignent, Vinny Marseglia apparait derrière lui et l'attaque dans le dos. Révélant par la même occasion qu'il a mis en scène sa propre attaque et qu'il a attaqué son partenaire TK O'Ryan en coulisses.

Lors de , il bat Matt Taven. Après le combat, il lui brisera la cheville.

Lors de , lui, Bateman et Dutch battent Shane Taylor Promotions (Kaun, Moses et O'Shay Edwards) et remportent les ROH World Six-Man Tag Team Championship.

Impact Wrestling (2022)

Honor No More et départ (2022)
Lors de Hard to Kill (2022), il effectue ces débuts à Impact Wrestling aux côtés de Matt Taven, Mike Bennett, Maria Kanellis et PCO en attaquant Eddie Edwards, Heath, Rhino, Rich Swann et Willie Mack après le match de ces derniers.

Lors de No Surrender (2022), lui, Matt Taven, Mike Bennett, Kenny King et PCO battent la Team Impact (Chris Sabin,Rhino, Rich Swann, Steve Maclin et Willie Mack) avec l'aide de Eddie Edwards et obtiennent le droit de rester à Impact Wrestling.

Lors de Under Siege (2022), lui, Eddie Edwards, Matt Taven, Mike Bennett et Kenny King battent Bullet Club (Chris Bey, Doc Gallows, El Phantasmo, Jay White et Karl Anderson).

Le 8 octobre 2022, il quitte la compagnie, 9 mois après son arrivée.

Caractéristiques
 Prise de finition
 Redrum (Senton Atomico)
 The Shining

 Managers
 Vita VonStarr

 Equipes et Clans
 The Kingdom (2016-2019)
 The Righteous (2020-...)
 Honor No More (2022)

 Surnoms
 "INKsanity"
 "The Horror King"

 Thèmes musicaux

Palmarès
Canadian Wrestling's Elite
 1 fois CWE Championship

 Chaotic Wrestling
 1 fois Chaotic Wrestling Tag Team Champion avec Matt Taven

 Liberty States Pro Wrestling
 1 fois Liberty States Heavyweight Champion

 Northeast Wrestling
 2 fois NEW Tag Team Champion avec Frankie Arion (1) et TK O'Ryan (1)

 Premier Wrestling Federation Northeast
 1 fois PWF Northeast Lightning Cup Champion

 Ring of Honor
 4 fois ROH World Six-Man Tag Team Championship avec Matt Taven et TK O'Ryan (3) et Bateman et Dutch (1)
 ROH World Six-Man Tag Team Championship Tournament (2016) avec Matt Taven et TK O'Ryan

 Top Rope Promotions
 1 fois TRP Heavyweight Champion
 1 fois TRP Interstate Champion
 Kowalski Cup Tournament (2012, 2015)

 Xtreme Wrestling Alliance
 1 fois XWA Firebrand Champion

Notes et références

Liens externes

Naissance en janvier 1986
Catcheur américain
Catcheur de la Ring of Honor

redpajama_set_name: RedPajamaWikipedia


A photographer friend called me one day and asked if I could meet with a client of his regarding a portrait of her daughter, who had passed away at too young of an age. She had a really cute candid shot she had taken with her cell phone and every time she tried to print it out, the resolution was horrible, unless she kept it at the original 2" x 3" size. She had the photographer try to enlarge it, but to no avail. He thought of me and asked if I could possibly meet with her. I agreed, and told her I would see what I could do. She provided me with some other larger photos for reference, but wanted the pose from the photo she took with the cell phone. When I finished the piece and the mother saw it for the first time, tears came down her face and she thanked me for restoring a precious memory that was so vivid in her mind. I thanked the Lord for letting me be a part of that blessing.


text:
Q: Cross domain ajax json I'm on site.com trying to grab some json data from my node.js server serving on port 8080.
I get this Error message:
XMLHttpRequest cannot load http://site.com:8080/json/1. Origin http://site.com is not allowed by Access-Control-Allow-Origin.

my code:
    $.get('http://site.com:8080/1/', {}, function (Data) {
       console.log(Data);
    }, "json");

But it is the same domain though! :(
Also consider my backbone.js model:
model = Backbone.Model.extend({
    url: function() {
        return 'http://site.com:8080/' + this.id
    }
});

Is there any way to resolve this other than using jsonp?
Thanks.

A: If you're making the call to the same domain, why do you have the absolute path in your $.get request?
Try this:
$.get('/1/', {}, function (Data) {
   console.log(Data);
}, "json");

model = Backbone.Model.extend({
    url: function() {
        return '/' + this.id
    }
});

If you are truly making the call on the same domain, then the above code should work.

A: Alternatively, you could modify your node.js server to allow Cross-Origin Resource Sharing (CORS).  This only works in modern browsers.  The simplest (and least secure) thing to do would be for your node.js server to emit the header "Access-Control-Allow-Origin: http://site.com".  You can learn more about CORS here: http://www.w3.org/TR/cors/

A: You have to make the call to the same domain and port your script was loaded from. You should be able to use jmort's code.

redpajama_set_name: RedPajamaStackExchange


text:
<?xml version="1.0" encoding="utf-8"?>
<manifest xmlns:android="http://schemas.android.com/apk/res/android"
    package="cr.ac.itcr.shopadvisor">

    <application
        android:allowBackup="true"
        android:icon="@mipmap/ic_launcher"
        android:label="@string/app_name"
        android:supportsRtl="true"
        android:theme="@style/AppTheme">
        <activity
            android:name=".LoginActivity"
            android:label="@string/title_activity_login"
            android:theme="@style/AppTheme.NoActionBar">
            <intent-filter>
                <action android:name="android.intent.action.MAIN" />

                <category android:name="android.intent.category.LAUNCHER" />
            </intent-filter>
        </activity>
        <activity android:name=".DashboardTestActivity" />
        <activity
            android:name=".MainActivity"
            android:label="@string/app_name"
            android:theme="@style/AppTheme.NoActionBar" />
        <activity
            android:name=".DashboardActivity"
            android:label="@string/title_activity_dashboard"
            android:theme="@style/AppTheme.NoActionBar"></activity>
    </application>

</manifest>

redpajama_set_name: RedPajamaGithub


text:
\section{Introduction}

		The optimal investment problem of choosing the best way to allocate an investor's capital is often formulated as the problem of maximizing, over admissible investment strategies, the expected utility of terminal wealth. The formulation relies on the axiomatic foundation developed by von Neumann and Morgenstern \cite{neumann07} and Savage \cite{Savage:1954wo}. In continuous time optimal portfolio selection, the study dates back to the seminal contributions of Merton \cite{merton,merton71}. In order to formulate the expected utility criterion, the agent needs to specify, on the one hand, her preferences via the investment horizon and the utility function and, on the other, her views about the future in providing the probability measure to compute the expectation. The specification of the latter may, however, itself be subject to uncertainty. This is referred to as ambiguity, or Knightian uncertainty in reference to the original contribution of Knight \cite{Knight:21}. It has been brought to prominence via the Ellsberg paradox \cite{ellsberg1961risk}.

	From a decision theoretic point of view, the issue was addressed in the seminal work of Gilboa and Schmeidler \cite{gilboa89}. They formulated axioms on investors' preferences that should account for aversion against both ambiguity and risk. Specifically, within an Anscombe-Aumann model, the axioms of von Neumann and Morgenstern were relaxed in that the axiom of independence was replaced by that of certainty independence. That led to a numerical representation of preferences in terms of a \emph{coherent} monetary utility functional. The robust representation of the latter (cf. \cite{delbaen00} for the case of bounded random variables), then yield the following representation of preferences over objects $X$ within some set $\mathcal{X}$:
	\begin{equation}
		X\longmapsto\inf_{\mathbb{Q}\in\mathcal{Q}}\mathrm{E}^\mathbb{Q}\big[U(X)\big],\label{utmmula}
	\end{equation}
	for some von Neumann Morgenstern utility function $U$ and set $\mathcal{Q}$ of probability measures.

	This motivated the study of the risk- and ambiguity-averse investor who trades over a fixed finite horizon in a continuous-time market model and evaluates terminal wealth according to the criterion \eqref{utmmula}. The investment problem associated with such, so-called, multiple-priors preferences has been well-studied. Stochastic control methods have successfully been applied and explicit solutions obtained for the choice of specific market models and utility functions. Specifically, for stochastic factor models and for non-Markovian models, solutions have been obtained, respectively, in terms of PDE's in \cite{hernandez2,owari09} and BSDEs in \cite{muller05,quenez04} (see \cite{tre} for a full list of references). For general market models and utility functions, we mention in particular the work by Quenez \cite{quenez04} and Schied and Wu \cite{wu}. Relying on the results for the classical utility maximization problem in Kramkov and Schachermayer \cite{kramkov,kramkov03}, the authors in \cite{quenez04} and \cite{wu} established a dual formulation and proved existence of an optimizer (see also \cite{burgert05,wittmuss08} for the case including consumption).

	The axiomatic results of Gilboa and Schmeidler were later generalized in Maccheroni et al. \cite{maccheroni06}, where the independence axiom was further relaxed. This led to a numerical representation in terms of a \emph{concave} monetary utility functional. Combined with the generalisation of the representation of coherent utility functionals to concave ones, for the case of bounded random variables obtained in F\"ollmer and Schied \cite{follmer02} and Fritelli and Rosazza-Gianin \cite{fritelli02}, it implied the numerical representation
	\begin{equation}
		X\longmapsto\inf_{\mathbb{Q}\in\mathcal{Q}}\Big(\mathrm{E}^\mathbb{Q}\big[U(X)\big]+\gamma(\mathbb{Q})\Big),\label{utmvara}
	\end{equation}
	for some penalty function $\gamma$. While the multiple-priors setup in \eqref{utmmula} is a worst-case approach, the appearance of $\gamma$ enables the investor to weight the possible market models according to their plausibility. This renders the presentation intuitively appealing.

	The investment-problem associated with these so-called variational preferences has also been studied. Particular attention has been paid to the case when the penalty function is given by the relative entropy of $\mathbb{Q}$ with respect to the reference model. Such criteria were introduced already in the seminal work of Hansen and Sargent \cite{anderson03,hansen01}. For this choice, the problem is most naturally formulated with respect to utility from consumption (or stochastic differential utilities) and the natural tool is the theory of BSDEs. While a systematic study was initiated in \cite{skiadas03}, these results have been considerably extended in a number of articles \cite{bordigoni07,chen02,faidi11,jeanblanc12,lazrak03}. For the case of utility from terminal wealth and general variational criteria, stochastic control methods have been successfully applied for the choice of logarithmic utility. This for stochastic factor models as well as non-Markovian ones; see \cite{hernandez,hernandez07b} and \cite{laeven12}, respectively. General existence and duality results have also been established for the variational preferences. Specifically, by use of similar methods to the ones used in \cite{quenez04,wu}, Schied \cite{schied} generalised these results to the concave case.

	The decision theoretic results in \cite{gilboa89} and \cite{maccheroni06}, were recently yet further extended by developments in Cerreia-Vioglio et al. \cite{cerreia}. Observing that all ambiguity averse preferences are axiomatized by a weakening of the independence axiom (the coordinate independence axiom within the Savage setting), the authors in \cite{cerreia} take this to its extreme by essentially removing this axiom altogether. Thereby, a numerical representation in terms of \emph{quasiconcave} utility functionals emerges. Recent advances also yield robust representations of the latter; besides \cite{cerreiab,cerreia}, see Drapeau and Kupper \cite{drapeau10} and Fritelli and Maggis \cite{fritelli11,fritelli12}. This motivates representing preferences via
	\begin{equation}
		X\longmapsto\inf_{\mathbb{Q}\in\mathcal{Q}}G\Big(\mathbb{Q},\mathrm{E}^\mathbb{Q}\big[U(X)\big]\Big),\label{quasia}
	\end{equation}
	for some function $G$ which is jointly quasiconvex, lower semicontinuous in its first argument and non-decreasing and right-continuous in its second. Similarly to the multiple-priors and variational cases (cf. \eqref{utmmula} and \eqref{utmvara}), these advances motivate the study of the associated investment problem. The aim herein is to initiate such a study.

Specifically, within a dominated setup, we consider an investor who trades over a fixed finite horizon in a continuous-time market model, evaluates terminal wealth according to \eqref{quasia} and maximizes this quantity over admissible trading strategies. While the investor's risk-aversion is governed by a standard utility function, the ambiguity preferences are, thus, determined by a quasiconcave utility functional. To the best of our knowledge, this problem has not been studied before. Indeed, although the advances in \cite{cerreia} recently motivated the use of quasiconvex risk measures within the area of portfolio optimization (see \cite{mastrogiacomo13}) and quasiconcave utility functions have been previously studied, the risk- and ambiguity-averse utility maximization problem associated with \eqref{quasia} seems to not have been addressed. We note that the notion is \emph{unifying} in the sense that all ambiguity averse preferences, in particular the multiple-priors, entropic and variational ones, are included as special cases. The class of quasiconcave preferences also includes interesting examples which do not correspond to variational preferences. Among others, the so-called smooth preferences axiomatized in \cite{klibanoff05}, which amount to considering a distribution over possible distributions rather than a worst-case approach.

	Our results extend the ones in \cite{schied} (cf. also \cite{quenez04,wu}) in that we prove existence of an optimal strategy and establish certain duality results for the quasiconcave case. As holds for the classical utility maximization, the study of the problem within the dual domain offers various advantages. Most importantly, in the case of robust preferences the dual problem amounts to a search for an infimum whereas the primal problem features a saddle-point. Similarly to \cite{schied}, we prove our results by building on existing results for the classical utility maximization problem (cf. \cite{kramkov,kramkov03}). However, the quasi-concavity, as opposed to the concavity, of the utility functional, required a slightly different approach to the one used in \cite{schied}. In particular, we obtain alternative proofs of some of the results therein.

	The paper is organized as follows. In Section \ref{model4}, the market model and the investment criterion are specified. The main results are presented in Section \ref{main4} while the proofs and further remarks are given in Section \ref{proof4}. The proof of an auxiliary Lemma is deferred to the appendix.

\begin{comment}
	In this chapter we study risk- and ambiguity-averse investment criteria defined in terms of quasiconcave utility functionals. Terminal payoffs are, thus, evaluated according to robust utility functionals of the form
		\begin{equation}
			X\longmapsto \inf_{\mathbb{Q}\ll\mathbb{P}}G\Big(\mathbb{Q},\mathrm{E}^\mathbb{Q}\left[U(X)\right]\Big), \label{crit1}
		\end{equation}
	where $G$ is the dual function in the robust representation (to be specified below) of some quasiconcave utility functional. We refer to the Introduction in Chapter 1 for the motivation and an overview of the related literature; see pp. \pageref{introchap4}--\pageref{introfinal}.

	The chapter is organized as follows. In Section \ref{model4}, the market model and the investment criterion are specified. The main results are presented in Section \ref{main4} while the proofs and further remarks are given in Section \ref{proof4}.
\end{comment}

\section{The market model and the investment criterion}\label{model4}

	We consider a fixed finite horizon $T>0$ and a given filtered probability space $(\Omega,\mathbb{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P})$, where $\mathbb{P}$ is the so-called reference measure. We first precise the investor's preferences. Then, in turn, the market model and the investment problem are specified.

	The investor's \emph{risk} preferences are quantified via a utility function $U:(0,\infty)\to\mathbb{R}$, which is strictly increasing, strictly concave and satisfies the Inada conditions,
		\begin{equation}
			\lim_{x\to 0}U'(x)=\infty\quad\textrm{and}\quad\lim_{x\to\infty}U'(x)=0.\label{inadaq}
		\end{equation}
	As argued in the Introduction, following \cite{cerreia}, the investor's \emph{ambiguity} aversion is specified via a quasiconcave utility functional. The latter is a mapping $\phi:L^\infty\to[-\infty,\infty]$ that satisfies quasiconcavity and monotonicity. Recall that a function is quasiconcave if it is the negative of a quasiconvex function and that a function is quasiconvex if its level-sets are convex. For a function $f:\mathcal{X}\to\mathbb{R}$, that is equivalent to

			$f\big(\lambda x+(1-\lambda)y\big)\le \max\big\{f(x),f(y)\big\}$, $x,y\in\mathcal{X}$.

	Hence, a quasiconcave utility functional (the negative of a quasiconvex risk measure) satisfies
		\begin{equation*}
			\phi\big(\lambda X+(1-\lambda) Y)\big)\ge \min\big\{\phi(X),\phi(Y)\big\},\quad\textrm{and}\quad
			\phi(X)\ge \phi(Y), \;\;\textrm{if $X\ge Y$, $\mathbb{P}$-a.s.}
		\eq
	However, it need satisfy neither cash-invariance nor positive homogeneity.

	The specification of our criterion relies on the robust representation result (Theorem 3.2 in \cite{drapeau10}; see also Theorem 7 in \cite{cerreia}) stating that every $\sigma(L^\infty,L^1)$-upper semicontinuous quasiconcave utility functional $\phi(X)$ admits the (robust) representation
		\begin{equation*}
			\phi(X)=\inf_{\mathbb{Q}\in\mathcal{M}_1(\mathbb{P})}G\Big(\mathbb{Q},\mathrm{E}^\mathbb{Q}\big[X\big]\Big),
		\eq
	for some function $G\in\mathcal{G}$, where the set of functions $\mathcal{G}$ is specified next and $\mathcal{M}_1(\mathbb{P})$ is the set of $\sigma$-additive probability measures absolutely continuous with respect to $\mathbb{P}$. Conversely, for any $G\in\mathcal{G}$, the function $\phi$ defined in this way is a upper semicontinuous quasiconcave utility functional.  \\

	\begin{defa}\label{defg}
	Let $\mathcal{G}$ denote the set of functions $G:\mathcal{M}_1(\mathbb{P})\times\mathbb{R}\to[-\infty,\infty]$ satisfying the following conditions:

			\begin{itemize}
				\item[i)]{$G(\mathbb{Q},\cdot)$ is non-decreasing and right-continuous;}
				\item[ii)]{$G$ is jointly quasiconvex;}
				\item[iii)]{$G^-(\cdot,s)$ is weakly lower semicontinuous, where
					\begin{equation*}
						G^-(\mathbb{Q},s)=\sup_{t<s}G(\mathbb{Q},t),\;\; \mathbb{Q}\in\mathcal{M}_1(\mathbb{P}).
					\eq}
				\item[iv)]{$G$ has an asymptotic maximum in the sense that
					\begin{equation*}
						AM(G):=\lim_{s\to\infty}G(\mathbb{Q},s)=\lim_{s\to\infty}G(\bar\mathbb{Q},s),\quad\textrm{for all $\mathbb{Q},\bar\mathbb{Q}\in\mathcal{M}_1(\mathbb{P})$}.
					\eq}
			\end{itemize}
		In particular, $G(\mathbb{Q},\cdot)$ is quasi-linear and upper semicontinuous.
	\end{defa}

	Combined, the above implies that we consider an investor assessing the utility of terminal payoffs, modelled as random variables on $(\Omega,\mathcal{F},\mathbb{P})$, in terms of a robust utility functional of the form \eqref{quasia} with $G\in\mathcal{G}$. For the specific cases when $G$ corresponds to a coherent and concave utility functional, the criterion reduces, respectively, to the multiple priors and variational preferences studied in \cite{quenez04,wu} and \cite{schied}. In the same way as the study of the latter problems were motivated by the axiomatic results in \cite{gilboa89} and \cite{maccheroni06}, the study of the more general quasiconcave case relies on the recent axiomatic extensions in \cite{cerreia}.

	Without loss of generality, we may precise \eqref{quasia} by writing
		\begin{equation}
			X\to \inf_{\mathbb{Q}\in\mathcal{Q}}G\Big(\mathbb{Q},\mathrm{E}^\mathbb{Q}\big[U(X)\big]\Big), \label{utmquasi2}
		\end{equation}
	where, for given $G\in\mathcal{G}$,
		\begin{equation*}
			\mathcal{Q}:=\{\mathbb{Q}\in\mathcal{M}_1(\mathbb{P}): G(\mathbb{Q},t)<\infty, \;\textrm{for some $t>0$}\}.
		\eq
	We stress that while $U(X)\in L^0$ for the type of payoffs we are to consider (cf. below), the utility functionals defined with respect to $G\in\mathcal{G}$ are defined only for bounded random variables. Indeed, while the theory of quasiconcave risk measures has been extended to more general spaces, we do not restrict to functions $G\in\mathcal{G}$ for which that holds. In doing so, we follow \cite{schied} who in a similar manner studied the risk- and ambiguity-averse investment problem formulated with respect to penalty functions associated with convex risk measures on $L^\infty$. In order to ensure that \eqref{utmquasi2} is well defined, we let $\mathrm{E}^\mathbb{Q}[F]:=\infty$ if $\mathrm{E}^\mathbb{Q}[F^+]=\mathrm{E}^\mathbb{Q}[F^-]=\infty$. Moreover, we extend the domain of $G(\mathbb{Q},\cdot)$ to the extended real line, in that we define $G(\mathbb{Q},-\infty):=-\infty$ and $G(\mathbb{Q},\infty):=AM(G)$.\\ \label{expoperator}

	\begin{ex}
	An example of preferences defined in terms of a quasiconcave utility functional are the so-called smooth criteria:
		\begin{equation}
			X\longmapsto \phi^{(-1)}\Big(\int_{\mathcal{M}_1(\mathbb{P})}\phi\Big(\mathrm{E}^\mathbb{Q}\left[U(X)\right]\Big)d\mu(\mathbb{Q})\Big),\label{smoothfirst}
		\end{equation}
	where $\phi$ is an increasing and concave function modelling the ambiguity aversion. That is, rather than taking the infimum over possible models, the agent considers a distribution $\mu$ over them. Such criteria were axiomatized in Klibanoff et al. \cite{klibanoff05} (see also \cite{strzalecki11}) under axioms stronger than those used in \cite{cerreia}. Hence, (for bounded payoffs) these criteria constitute a particular case of the quasiconcave preferences. The specific form of the associated function $G$ is given in \cite{cerreia} and yields further intuition for the criteria.

	\end{ex}

	Next, we specify the continuous-time market model and the set of admissible trading strategies. The market model is defined by a $\mathbb{R}^d$-valued price process $S_t$, which is assumed to be a semi-martingale on $(\Omega,\mathbb{F},(\mathcal{F}_t),\mathbb{P})$. The price-process $S_t$ is assumed to satisfy the condition of NFLVR with respect to the reference-measure $\mathbb{P}$. For given initial capital $x>0$, and a predictable $S$-integrable trading strategy $\pi_t$, the associated wealth-process is given by
		\begin{equation*}
			X_t^\pi=x+\int_0^t\pi_sdS_s,\quad t\ge 0.
		\eq
	We denote a trading strategy admissible if $X_t^\pi\ge 0$, $\mathbb{P}$-a.s. for $t\in [0,T]$ and denote the associated set of wealth-processes by $\mathcal{X}(x)$.

	In consequence, we aim at studying the following investment problem. \\

	\begin{prob}\label{problem4}
		For given $G\in\mathcal{G}$, we consider the risk- and ambiguity-averse investment problem of maximizing the functional \eqref{utmquasi2} over admissible terminal payoffs $X_T^\pi$, $X\in\mathcal{X}(x)$. The associated value-function $u:\mathbb{R}_+\to\mathbb{R}$ is defined by
	\begin{equation}
		u(x):=\sup_{X\in\mathcal{X}(x)}\inf_{\mathbb{Q}\in\mathcal{Q}}G\Big(\mathbb{Q}, \mathrm{E}^\mathbb{Q}\big[U\big(X_T^\pi\big)\big]\Big).\label{u}
	\end{equation}
	\end{prob}

	We stress that the set of admissible strategies $\mathcal{X}$ is defined with respect to $\mathbb{P}$, the role of which is to specify the null-sets rather than representing the most likely model. In particular, we consider a dominated setting in which all measures $\mathbb{Q}\in\mathcal{Q}$ are absolutely continuous with respect to the reference measure (for ambiguity averse portfolio optimization with mutually singular measures, see \cite{nutz13} and the references therein). Also note that while $u(x)$ is non-decreasing, it need be neither concave nor continuous (cf. Remark \ref{tcrm}).

	We conclude this section by defining an auxiliary optimization problem which will be crucial for the analysis that follows. For $\mathbb{Q}\ll\mathbb{P}$, let
		\begin{equation}
			u_\mathbb{Q}(x):=\sup_{X\in\mathcal{X}(x)}\mathrm{E}^\mathbb{Q}\big[U(X_T)\big],\label{primal}
		\end{equation}
	where $\mathcal{X}(x)$ is as defined above. We also introduce the (dual) auxiliary problem
		\begin{equation}
			v_\mathbb{Q}(y):=\inf_{Y\in\mathcal{Y}(y)}\mathrm{E}\big[ZV(Y_T/Z)\big],\label{dual}
		\end{equation}
	where $Z=\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}$, $V(y)=\sup_{x\ge 0} (U(x)-xy)$ and $\mathcal{Y}(y)$ is the set of all positive $\mathbb{P}$-supermartingales such that $Y_0=y$ and $XY$ is a $\mathbb{P}$-supermartingale for all $X\in\mathcal{X}(1)$. Without further notice, these functions will also be denoted by $u_Z(x)$ and $u_Z(y)$, respectively. While the objective functions in \eqref{primal} and \eqref{dual} are defined with respect to the measure $\mathbb{Q}$, the set of admissible strategies and dual objects are defined with respect to the reference measure $\mathbb{P}$. Hence, while it holds for $\mathbb{Q}\sim\mathbb{P}$, that these auxiliary problems are, respectively, the standard investment problem and its dual counterpart for the market defined with respect to the measure $\mathbb{Q}$, this need not be the case for $\mathbb{Q}\ll\mathbb{P}$. In particular, $S_t$ need not satisfy the condition of NFLVR with respect to $\mathbb{Q}$ (see further discussion in Section \ref{aux} and Remarks \ref{normaldual1} and \ref{normaldual} below).

	\section{The main results}\label{main4}

	In this section, we present the main results. The proofs are given in Section \ref{proof4}. The following standing assumption is imposed throughout. \\

	\begin{ass}\label{standing}
		There exists $\mathbb{Q}_0,\mathbb{Q}_1\in\mathcal{Q}$ such that
		\begin{equation}
			u_{\mathbb{Q}_0}(x_0)<\infty,\textrm{ for some $x_0>0$,}\label{ass0}
		\end{equation}
		and
		\begin{equation}
			G\big(\mathbb{Q}_1,u_{\mathbb{Q}_1}(x)\big)<\infty,\textrm{ for all $x>0$.}\label{ass1}
		\end{equation}
	\end{ass}

	Note that assumptions \eqref{ass0} and \eqref{ass1} are disjoint in the sense that none of them implies the other (although, they might both be satisfied for the same $\mathbb{Q}\in\mathcal{Q}$). Also note that \eqref{ass0} implies $u_{\mathbb{Q}_0}(x)<\infty$ for all $x>0$ and, furthermore, that Assumption \ref{standing} yields
		\begin{equation}
			u(x)=\sup_{g\in\mathcal{X}(x)}\inf_{\mathbb{Q}\in\mathcal{Q}}G\big(\mathbb{Q},\mathrm{E}^\mathbb{Q}\big[U\left(f\right)\big]\big)
			\le \inf_{\mathbb{Q}\in\mathcal{Q}}G\big(\mathbb{Q},u_\mathbb{Q}(x)\big)\;<\;\infty,\quad x>0.\label{finiteness}
		\end{equation}

	First we establish existence of a solution to Problem \ref{problem4}. The question of uniqueness is discussed below. \\

		\begin{thm}\label{existence}
		Let $G\in\mathcal{G}$ such that Assumption \ref{standing} holds and assume either that
			\begin{equation}
				v_\mathbb{Q}(y)<\infty,\;\; y>0, \label{aspt}
			\end{equation}
		for each $\mathbb{Q}\in\mathcal{Q}$, or that $G(\cdot,t)$, $t\in\mathbb{R}$, is convex and \eqref{aspt} holds for each $\mathbb{Q}\in\mathcal{Q}_e$. Then, there exists an optimal terminal payoff $\hat X\in\mathcal{X}(x)$ for which the supremum in \eqref{u} is attained.
		\end{thm}

		We make two general remarks on the assumptions of the above result. \\

		\begin{rem}
	That $G(\cdot,t)$, $t\in\mathbb{R}$, is convex is a necessary but not a sufficient condition for the associated utility functional to be concave. This class of preferences therefore includes but is not limited to the variational ones. Nevertheless, the very same methods as introduced in \cite{schied} can be used to establish existence for this case, cf. the discussion in Section \ref{aux}, and in consequence the weaker condition that \eqref{aspt} only holds for $\mathbb{Q}\in\mathcal{Q}_e$ suffices. \\
		\end{rem}

		\begin{rem} The assumption that \eqref{aspt} holds for $\mathbb{Q}\in\mathcal{Q}$ implies (cf. Lemma \ref{key} below) that $\mathcal{Q}=\mathcal{Q}^f$, where
		\begin{equation*}
			\mathcal{Q}^f:=\{\mathbb{Q}\in\mathcal{Q}:u_\mathbb{Q}(x)<\infty\}.
		\eq
	In fact, a sufficient condition for Theorem \ref{existence} to hold, is that $\mathcal{Q}=\mathcal{Q}^f$ and $AE_{+\infty}(U)<1$. Indeed, similarly to the standard case (cf. Note 2 in \cite{kramkov03}), this yields the finiteness of $v_\mathbb{Q}(y)$, $\mathbb{Q}\in\mathcal{Q}$. Yet another sufficient condition is that $v_\mathbb{Q}(y)<\infty$, $\mathbb{Q}\in\mathcal{Q}_e$, and that for each $\mathbb{Q}_0\in\mathcal{Q}$, there exists $\mathbb{Q}_1\in\mathcal{Q}_e$ such that
		\begin{equation}
			\mathrm{E}^{\mathbb{Q}_0}[U(g)]\le\mathrm{E}^{\mathbb{Q}_1}[U(g)]<\infty,\;\textrm{ for all }\;g\in\mathcal{C}(x).\label{limitasapm}
		\end{equation}
	Indeed, \eqref{limitasapm} implies that $u_{\mathbb{Q}_0}(y)\le u_{\mathbb{Q}_1}(y)<\infty$, where the last inequality follows from the assumption. Hence, the conjugacy relations of Lemma \ref{key} below holds with respect to $\mathbb{Q}_0$. This yields,
				\begin{equation*}
					v_{\mathbb{Q}_0}(y)=\sup_{x>0}\Big(u_{\mathbb{Q}_0}(y)-xy\Big)
					\le \sup_{x>0}\Big(u_{\mathbb{Q}_1}(y)-xy\Big)
					=v_{\mathbb{Q}_1}(y)<\infty.
				\eq
	In consequence, $v_{\mathbb{Q}_0}(y)<\infty$ for all $\mathbb{Q}_0\in\mathcal{Q}$ and the assumptions of Theorem \ref{existence} hold.
		\end{rem}

	We stress that the assumption that $v_\mathbb{Q}(y)<\infty$, $\mathbb{Q}\in\mathcal{Q}$, is rather natural. First, under some additional assumptions, it is established below (cf. Theorem \ref{interchange}) that the infimum and supremum in \eqref{u} can be interchanged. In effect, the set $\mathcal{Q}$ can be replaced by the set $\mathcal{Q}^f$ without affecting the indirect utility $u(x)$. More importantly, this assumption implies that the auxiliary investment problem, itself, is solvable for each individual measure $\mathbb{Q}\in\mathcal{Q}$ (cf. Lemma \ref{key} below). As pointed out above, the auxiliary problem is not a standard one. Nevertheless this might be understood as an arbitrage condition put on each individual model (cf. \cite{kardaras} where it is shown that the classical utility maximization problem admits a solution if and only if the market satisfies the no arbitrage condition NA$_1$).\label{kardaras} The assumption therefore relates to the interpretation of the ambiguity averse criterion. On the one hand, the criterion \eqref{utmquasi2} emerges due to the axioms posed on the preferences; this via the robust representation of quasiconcave utility functionals. This motivation - per se - does not imply that the measures $\mathbb{Q}\in\mathcal{Q}$ satisfy any market related conditions. However, it is common in the literature to also motivate ambiguity averse criteria by the fact that they, effectively, amount to taking the expectation with respect to different possible market models $\mathbb{Q}$ weighted according to their plausibility. While the weighting is determined by the penalty function $\gamma$ for the variational case, the function $G$ allows for more flexibility in the quasiconcave case. Given the latter interpretation, it is natural to assume that each market model $\mathbb{Q}\in\mathcal{Q}$, itself constitutes a sensible market model which excludes arbitrage. This is exactly what \eqref{aspt} guarantees. The important question of how to ensure that this condition holds for specific choices of $G$ remains to be addressed.

	Under the assumptions of Theorem \ref{existence}, it is not clear whether the infimum and supremum in \eqref{u} can be interchanged (neither whether the infimum is attained). Hence, there need \emph{not} exist an auxiliary problem (cf. \eqref{primal})\label{equivaux} producing the same investment behaviour as the ambiguity averse criterion \eqref{utmquasi2}. This is, however, the case under some additional assumptions; we will come back to this (see page \pageref{auxreplace}).

	Next, we turn to the study of the dual version of the risk and ambiguity averse investment problem \eqref{utmquasi2}. Specifically, we establish relations between the primal and dual problems and their respective solutions. As holds for the variational case (cf. \cite{schied}), the study of the dual problem is, in fact, not needed for proving existence of a solution to the primal problem. Indeed, the proof of Theorem \ref{existence} relies on properties of the auxiliary problem \eqref{primal} and, thus, rather on the study of the dual version thereof. Nevertheless, the study of its dual counterpart is of interest as it gives a further understanding of the problem and of the optimal strategy. Similarly to the standard case, the study of the dual problem rather than the primal one offers various advantages. This is particularly evident for ambiguity averse preferences where the dual problem amounts to the search for a pure infimum, whereas the primal problem features a saddle-point. In particular, most articles providing explicit solutions for specific choices of utility and penalty functions, notably focus on the dual rather than the primal problem. Hence, there is reason to believe that the dual formulation is helpful in obtaining more explicit results also for the quasiconcave case. The results also enable us to draw important conclusions about the optimal strategy. Specifically, regarding the existence of an equivalent auxiliary problem and uniqueness of the optimal strategy (cf. the discussion after Theorem \ref{relation}).

	We impose the following additional assumption.\\

		\begin{ass}\label{dualass}
			The function $G$ is jointly lower semicontinuous and the level sets $\mathcal{Q}_t(c)$ are relatively weakly compact, where
				\begin{equation*}
					\mathcal{Q}_t(c):=\big\{\mathbb{Q}\in\mathcal{Q}:G(\mathbb{Q},t)\le c\big\},\quad t\in\mathbb{R}, c\ge 0.
				\eq
			Furthermore, either $U:\mathbb{R}_+\to\mathbb{R}_+$ or $G(\mathbb{Q},\cdot)$, $\mathbb{Q}\in\mathcal{Q}$, is concave.
		\end{ass}

		Note that since $\mathbb{Q}\to G(\mathbb{Q},t)$ is weakly lower semicontinuous, $\mathcal{Q}_t(c)$ is weakly closed and, thus, due to Dunford-Pettis theorem, relative weak compactness is equivalent to uniform integrability. Moreover, recall that for functions $G\in\mathcal{G}$, $G^-(\cdot,t)$, $t\in\mathbb{R}$, is lower semicontinuous and $G(\mathbb{Q},\cdot)$, $\mathbb{Q}\in\mathcal{Q}$, is upper semicontinuous. Hence, the assumption that $G$ is jointly lower semicontinuous is, in fact, equivalent to the assumption that $G(\mathbb{Q},\cdot)$ is continuous. Both these properties are closely related to the fact that one considers (evenly) quasiconcave utility functionals which are continuous not only from above but also from below and, thus, weakly continuous (as opposed to weakly upper semicontinuous). We refer to \cite{cerreiab,bok,fritelli11,fritelli12} for further details but note that this class of quasiconcave utility functionals is natural within the present context. Indeed, in \cite{cerreia} the axioms were formulated (including a continuity axiom) so as to render a numerical representation in terms of such weakly continuous utility functionals.

		While this (natural) additional continuity assumption is crucial for the duality results, the restriction to positive utility functions is needed for technical reasons. The possibility of relaxing this assumption is left for future study. Note, however, that the results in, among others, \cite{quenez04} are established only for positive utility functions. The restriction to $G(\mathbb{Q},\cdot)$ concave yields, essentially, the variational case treated in \cite{schied}. It is included as a specific case in order to compare our methods and relate our results to the ones therein. \\

	\begin{thm}\label{interchange}
		Let $G\in\mathcal{G}$ such that Assumptions \ref{standing} and \ref{dualass} hold. Then,
			\begin{itemize}
				\item[i)]{the robust value-function satisfies
					\begin{equation}
					u(x):=\sup_{X\in\mathcal{X}(x)}\inf_{\mathbb{Q}\in\mathcal{Q}}G\Big(\mathbb{Q},\mathrm{E}^\mathbb{Q}\big[U(X_T)\big]\Big)
					=\inf_{\mathbb{Q}\in\mathcal{Q}}\sup_{X\in\mathcal{X}(x)}G\Big(\mathbb{Q},\mathrm{E}^\mathbb{Q}\big[U(X_T)\big]\Big),\label{hoppas}
					\end{equation}
				and, furthermore,
					\begin{equation}
						u(x)=\inf_{y>0}v(y;x),\label{udual}
					\end{equation}
				where $v(y;x)$ is the dual value function given by
					\begin{equation}
						v(y;x):=\inf_{\mathbb{Q}\in\mathcal{Q}}G\Big(\mathbb{Q},v_\mathbb{Q}(y)+xy\Big).\label{vyx}
					\end{equation}}
				\item[ii)]{Moreover, if $v(y;x)<\infty$, then the dual problem \eqref{vyx} admits a solution $(\widehat\mathbb{Q},\widehat Y)$ that is maximal in the sense that any other solution $(\mathbb{Q},Y)$ satisfies $\mathbb{Q}\ll\widehat\mathbb{Q}$ and $Y_T/Z=\widehat Y_T/\widehat Z$, $\mathbb{Q}$-a.s. }
			\end{itemize}
		\end{thm}

			While the above result imposes stronger conditions on the structural properties of $G$ (and $U$) than needed for Theorem \ref{existence}, finiteness assumptions of the type needed for the existence are not required here. This should be related to the results in \cite{kramkov,kramkov03}, where the conjugacy relations are proven under much weaker conditions than needed for the existence.\label{discinterchange}

			For the variational case, the relation between the primal and dual value function (cf. \eqref{udual}) was established in \cite{schied} under the additional assumptions that $u_{\mathbb{Q}_0}(x)<\infty$ for some $\mathbb{Q}_0\in\mathcal{Q}_e$ and
			\begin{equation}
				\textrm{$v(y;x)<\infty$ implies $v_{\mathbb{Q}_1}(y)<\infty$ for some $\mathbb{Q}_1\in\mathcal{Q}_e$.}\label{additional}
			\end{equation}
			Although very natural, this assumption is, in fact, not needed. However, for the variational case, a slightly stronger result can be proven under this additional assumption; cf. Remark \ref{normaldual} below.

	The next result relates the respective solutions to the primal and dual problems. In particular, the result yields the existence of an equivalent auxiliary problem and results on the uniqueness of the optimal strategy.\\

		\begin{thm}\label{relation}
			Let $G\in\mathcal{G}$ and assume Assumption \ref{dualass} and the assumptions of Theorem \ref{existence} to hold. Let $\bar X$ be a solution to the primal problem. Then, the primal problem admits a saddle point $(\bar X,\bar \mathbb{Q})$ and there exists $y^*>0$ for which the infimum in \eqref{udual} is attained.

			Assume, furthermore, that $G(\mathbb{Q},\cdot)$, $\mathbb{Q}\in\mathcal{Q}$, is strictly increasing and let $(\widehat Y,\widehat\mathbb{Q})$ be any solution to the dual problem at level $y^*$. Then, $(\bar X, \widehat\mathbb{Q})$ is a saddle point for the primal problem and, moreover,
				\begin{equation}
					\bar X=I(\widehat Y_T/\widehat Z),\quad \textrm{$\widehat\mathbb{Q}$-a.s.}\label{relpp}
				\end{equation}
		\end{thm}

		The above result is of particular interest as it gives a further understanding of the optimal strategy. Let $\widehat\mathbb{Q}$ be the measure in the maximal solution to the dual (or, equivalently, primal) problem. Following the related literature, we refer to it as the least favourable measure. Relation \eqref{relpp} then implies that the optimal solution $\widehat X_T$, in fact, is $\widehat\mathbb{Q}$-a.s. unique. Consequently, if the least favourable measure is equivalent to $\mathbb{P}$, the solution is $\mathbb{P}$-a.s. unique\label{uniqueness}. In particular, it can then be recovered from the dual solution. In general, the least favourable measure need however not be equivalent to $\mathbb{P}$. Nevertheless, an optimal strategy can still be constructed from a given solution of the dual problem by superhedging of an appropriate claim (cf. Corollary 2.7 in \cite{schied}).

		The existence of a saddle point also implies that, \emph{a posteriori}, there exists an auxiliary investment problem \eqref{primal} producing the same optimal behaviour as the original criterion (in particular, this always holds for the variational criteria). Specifically, the auxiliary problem defined with respect to the least favourable measure $\widehat\mathbb{Q}$ admits a solution (cf. Lemma \ref{key} below) which $\widehat\mathbb{Q}$-a.s. coincides with the solution to the original problem. Since the least-favourable measure is part of the solution to the original problem, the equivalence between the ambiguity averse and the auxiliary problem is an a posteriori result. Given the existence of an auxiliary problem (for which the objective function is concave), the $\widehat\mathbb{Q}$-a.s. uniqueness of the optimal strategy is natural. Under the weaker assumptions of Theorem \ref{existence}, it is not clear though whether an equivalent auxiliary problem exists (cf. the discussion on page\label{auxreplace} \pageref{equivaux}). The question of to which extent uniqueness holds under more general assumptions (given $G(\mathbb{Q},\cdot)$ strictly increasing) is left for future study.  \\

	\begin{rem}[Time consistency]\label{tcrm}
		In general, Problem \ref{problem4} is not a time-consistent investment problem (see \cite{schied} for counter-examples in the variational case). A natural question is therefore under what assumptions it is. Indeed, while time-consistency is of interest in its own right, it is also of essential importance as it enables the use of stochastic control methods. In consequence, it is a prerequisite for extending to the quasiconcave case the explicit results obtained in terms of PDEs and BSDEs for the variational preferences (cf. the references in the Introduction). Time consistency of quasiconcave utility functionals (quasiconvex risk measures) remains, however, an open problem and, in particular, feasible explicit examples are few. Indeed, while necessary and sufficient conditions for temporal consistency of convex risk measures were established in \cite{delbaen10} (see also \cite{bionnadal06} and \cite{irina}), such results are, yet, lacking for the quasiconvex
		case. In recent work, \cite{fritelli11,fritelli12} initiated such a programme via the study of conditional quasiconvex risk measures. Questions of temporal consistency and the relation to $g$-expectations have also been studied within the more general framework of non-linear expectations in \cite{peng05}.

		We also note that in \eqref{u}, the risk preferences of the investor are modelled via a standard continuous and concave utility function. While this assumption makes perfect sense for the static problem, the value function $u(x)$ will only satisfy weaker properties (a more precise study of the properties of the value function is left for future study). Hence, a study of time-consistent problems of this type also requires a study of the risk and ambiguity averse investment problem \eqref{u} under weaker assumptions on $U(x)$. In summary, while questions of time-consistency are of great interest, they impose challenging additional problems which are left for future study.

	\end{rem}

	\section{Proofs}\label{proof4}

		The above theorems extend results established for the variational preferences in \cite{schied} (cf. \cite{quenez04,wu} for the coherent case) to the quasiconcave risk- and ambiguity-averse investment criteria. Naturally, our proofs are therefore inspired by and in many ways similar to those in the former articles. Specifically, also here the idea is to establish results for the risk- and ambiguity-averse problem by relying on existing results for the auxiliary problem \eqref{primal}. As pointed out above, for $\mathbb{Q}\sim\mathbb{P}$, this is the classical utility maximization problem as studied in \cite{kramkov,kramkov03}. For $\mathbb{Q}\ll\mathbb{P}$, this need not be the case. In \cite{schied}, this is dealt with by use of certain limiting arguments. However, for our case, the weaker properties of $G$ imply that this approach does not apply. Hence the need for somewhat different arguments.

		In Section \ref{aux}, we explain why we were not able to directly apply the arguments developed in \cite{schied} and describe the approach we use in further detail. This illustrates the differences in the required assumptions. The existence and duality results are proven, respectively, in Sections \ref{existproof} and \ref{dualityproof}. The proof of Lemma \ref{key} below is deferred to the Appendix.

	\subsection{The auxiliary problem and its significance}\label{aux}

	For the variational case studied in \cite{schied}, the appearance of measures in $\mathcal{Q}$ which are absolutely continuous but not necessarily equivalent to $\mathbb{P}$, is addressed in the following way. For $\mathbb{Q}_0\in\mathcal{Q}\setminus\mathcal{Q}_e$, the authors let $\mathbb{Q}_1\in\mathcal{Q}_e$ and define the measures $\mathbb{Q}_t$, $t\in[0,1]$, via the Radon-Nikodym derivative $Z_t:=(1-t)Z_0+tZ_1$, where $Z_0$ and $Z_1$ correspond, respectively, to $\mathbb{Q}_0$ and $\mathbb{Q}_1$. Then, $\mathbb{Q}_t\in\mathcal{Q}_e$ for $t\in(0,1]$. Moreover, under suitable finiteness-assumptions, the functions
	\begin{equation}
		t\to G\big(\mathbb{Q}_t,u_{\mathbb{Q}_t}(x)\big)\;\textrm{ and }\;  t\to G\big(\mathbb{Q}_t,v_{\mathbb{Q}_t}(y)+xy\big),\label{schiedcase}
	\end{equation}
	are continuous and upper semicontinuous, respectively. Combined, this implies that the results can be established by relying on results for auxiliary problems (cf. \eqref{primal}) defined with respect to measures equivalent to $\mathbb{P}$ only. To the latter, the results in \cite{kramkov,kramkov03} can, in turn, be directly applied. In consequence, certain assumptions need only be posed on the set of measures $\mathcal{Q}_e$ as opposed to $\mathcal{Q}$ (cf. Theorem \ref{existence}). This approach is closely related to the fact that the set of absolutely continuous measures in the representation of a concave utility functional, under the assumption that there exists an equivalent measure for which the penalty is finite, can be replaced by the equivalent ones; cf. \cite{kloppel05}.

	The continuity properties in \eqref{schiedcase} rely, however, on the fact that for the variational case $G(\cdot,t)=\gamma(\cdot)+t$ and, thus, the mapping is convex. Indeed, this implies that the mapping
		\begin{equation}
			t\to G(Z_t,s), \quad t\in [0,1], s\in\mathbb{R},\label{whynot}
		\end{equation}
	is convex as well. If it is finite, it is therefore upper-semicontinuous. According to Lemma 3.3 in \cite{wu}, $t\to u_{Z_t}(x)$ is continuous and, under rather weak assumptions, also $t\to v_{Z_t}(y)$ is upper semicontinuous. Hence, (under suitable finiteness assumptions) the required continuity properties in \eqref{schiedcase} follow. The crucial point is that for the quasiconcave case we consider, the function $G(\cdot,t)$ might not be convex. Therefore, the continuity properties in \eqref{schiedcase} might not hold and, in consequence, the arguments developed in \cite{schied} do not apply.

	Our methods are rather based on a closer study of the auxiliary problem \eqref{primal}. Although not a standard problem, it is, in fact, only a minor modification thereof and can therefore be solved by use of the same methods as developed in \cite{kramkov} (cf. Lemma \ref{key} below). By use of this observation, we then address the risk and ambiguity averse investment problem. In particular, we obtain alternative proofs of some results established in \cite{schied}. Indeed, once the relevant properties have been obtained for auxiliary problems with $\mathbb{Q}\ll\mathbb{P}$, the proofs can be simplified. In particular, our approach enables establishing the relation between the primal and dual value functions (cf. \eqref{udual} below) under slightly weaker assumptions than in \cite{schied}. However, for the existence of an optimizer, stronger assumptions are required. These assumptions, which are a consequence of the more general type of criteria we consider, are still economically feasible; see the discussion after Theorem \ref{existence} above.

	The properties of the auxiliary problem that will be made use of are presented next.\\

		\begin{lem}\label{key}
			Let $U$ satisfy the Inada conditions (cf. \eqref{inadaq}) and assume $u_Z(x_0)<\infty$ for some $x_0>0$. Then, the function $u_Z(x)$ and $v_Z(y)$ defined, respectively, in \eqref{primal} and \eqref{dual} satisfy the following:
			\begin{itemize}
				\item[i)]{It holds that
				\begin{equation*}
					v_Z(y)=\sup_{x>0}\big(u_Z(x)-xy\big)
					\;\;\textrm{and}\;\;
					u_Z(y)=\inf_{y>0}\big(v_Z(x)+xy\big),
				\eq
			and, furthermore,
				\begin{equation*}
					u_Z'(0)=\infty\;\;\textrm{and}\;\; v_Z'(\infty)=0.
				\eq}
				\item[ii)]{Under the additional assumption that $v_Z(y)<\infty$, $y>0$, it also holds that
				\begin{equation*}
					u_Z'(\infty)=0\;\;\textrm{and}\;\; v_Z'(0)=-\infty.
				\eq
			Moreover, the set $\{ZU^+(g):g\in\mathcal{C}(x)\}$ is $\mathbb{P}$-UI and the primal problem admits a solution.}
			\end{itemize}
		\end{lem}

	For $\mathbb{Q}\sim\mathbb{P}$, the above result was established in \cite{kramkov,kramkov03}. For $\mathbb{Q}\ll\mathbb{P}$, note that while the objective in the auxiliary problem is defined with respect to the measure $\mathbb{Q}$, the set of strategies is still defined in the standard way with respect to the reference measure $\mathbb{P}$. This is the key reason for the above result to hold also for auxiliary problems defined with respect to $\mathbb{Q}\ll\mathbb{P}$ (cf. also Remark \ref{normaldual}). Indeed, the proofs in \cite{kramkov,kramkov03} make use of the assumption of NFLVR in order to treat the strategies (via the characterization in \eqref{iff} below), not the objective function. In consequence, Lemma \ref{key} follows by minor modifications the respective proofs in \cite{kramkov,kramkov03}. For completeness, the details are presented in the Appendix. We note that the auxiliary problem also might be viewed as utility maximization under $\mathbb{P}$ with respect to the stochastic utility function $\tilde U(x)=Z_TU(x)$ and refer to, among others, \cite{follmer2000efficient} for alternative arguments. For an economic interpretation of the condition that $u_Z(x_0)<\infty$, see the discussion on p. \pageref{kardaras}.

	\subsection{Proof of the existence of an optimal investment strategy}\label{existproof}

		The proof first establishes upper semicontinuity and quasi-concavity of the objective function. The existence of an optimizer then follows by use of a Komlos-type argument.

		\begin{proof}[Proof of Theorem \ref{existence}]
			For the case when $G(\cdot,t)$ is convex, we refer to Lemma 4.7 in \cite{schied}. Note that the optimization over $X\in\mathcal{X}(x)$ can be replaced by optimization over the set $\mathcal{C}(x):=\{g\in L^0_+(\mathcal{F}_T):g\le X_T, X\in\mathcal{X}(x)\}$ of random variables. That is to say,
			\begin{equation}
				u(x)=\sup_{g\in\mathcal{C}(x)}\inf_{\mathbb{Q}\in\mathcal{Q}}G\Big(\mathbb{Q}, \mathrm{E}^\mathbb{Q}\big[U\big(g\big)\big]\Big).\label{ug}
			\end{equation}

			Due to assumption \eqref{aspt}, $u_\mathbb{Q}(x)<\infty$, $\mathbb{Q}\in\mathbb{Q}$, (cf. Lemma 3.5 in \cite{wu}) and Lemma \ref{key} applies. Hence, $\{U^+(f)\}$, $f\in\mathcal{C}(x)$ is $\mathbb{Q}$-UI, $\mathbb{Q}\in\mathcal{Q}$. Combined with an application of Fatou's Lemma to the negative part $U^-(f)$, this yields that $g\to\mathrm{E}^\mathbb{Q}[U(g)]$, $g\in\mathcal{C}(x)$, is upper semicontinuous for $\mathbb{Q}\in\mathcal{Q}$. Since also $G(Z,\cdot)$ is upper semicontinuous and the pointwise infimum of u.s.c. functionals is again u.s.c., it follows that the mapping
			\begin{equation}
				g\to V(g):=\inf_{\mathbb{Q}\in\mathcal{Q}}G\Big(\mathbb{Q},\mathrm{E}^\mathbb{Q}[U(g)]\Big), \quad g\in \mathcal{C}(x),
			\end{equation}
			is upper semicontinuous as well. Moreover, as $g\to\mathrm{E}[ZU(f)]$ is concave, $G(Z,\cdot)$ is quasi-linear and the point-wise infimum of quasiconcave functionals is again quasiconcave, it follows that $g\to V(g)$ is quasiconcave. More precisely, the semicontinuity and quasi-concavity can be argued as follows. Since $t\to G(q,t)$ is non-decreasing and right-continuous, it holds according to Proposition B.2 in \cite{drapeau10} that
				\begin{equation*}
					G(q,t)\ge m\quad\textrm{if and only if}\quad t\ge G^{(-1,l)}(q,m),
				\eq
			where the left inverse is given by $G^{(-1,l)}(q,m)=\inf\{n\in\mathbb{R}:G(q,m)\ge n\}$. Consequently, the following holds:
				\begin{equation}
					\Big\{g\in\mathcal{C}(x):G\big(q,\langle U(g),q\rangle\big)\ge m\Big\}
					=\Big\{g\in\mathcal{C}(x):\langle q,U(g)\rangle\ge G^{(-1,l)}(q,m)\Big\}.\label{cf}
				\end{equation}
			This level set is then closed and convex since $g\to\langle q,U(g)\rangle$, $g\in\mathcal{C}(x)$, $q\in\mathcal{Q}$, is concave and upper semicontinuous according to the above. Then, in turn, it follows that the set
				\begin{equation*}
					\Big\{g\in\mathcal{C}(x):V(g)\ge m\Big\}
					=\cap_{q\in\mathcal{Q}}\Big\{g\in\mathcal{C}(x):G\big(q,\langle U(g),q\rangle\big)\ge m\Big\},
				\eq
			is closed and convex and, thus, that $g\to V(g)$ is upper semicontinuous and quasiconcave.

			The existence of an optimizer now follows by a standard Komlos-type argument. Indeed, let $(g_n)$ be a sequence in $\mathcal{C}(x)$ such that $V(g_n)\nearrow u(x)$. Since each $g_n\ge 0$, Komlos Lemma gives $\tilde g_n\in\textrm{conv}(g_n,g_{n+1},...)$ converging $\mathbb{P}$-a.s. to some $g$. Since $\mathcal{C}(x)$ is convex, $\tilde g_n\in \mathcal{C}(x)$. Furthermore, according to Proposition 3.1 in \cite{kramkov}, $\mathcal{C}(x)$ is closed under convergence in $L^0$ and, thus, $g\in\mathcal{C}(x)$. Next, due to the quasi-concavity of $g\to V(g)$ it holds that
				\begin{equation*}
					V\left(\tilde g_n\right)=V\bigg(\sum_{k\ge n}\lambda^kg_k\bigg)\ge \inf_{k\ge n} V(g_k)=V(g_n).
				\eq
			Since $V(\tilde g_n)\le u(x)$ and $V(g_n)\nearrow u(x)$ it, thus, follows that $\lim_{n\to\infty}V(\tilde g_n)=u(x)$; that is to say, $\tilde g_n$ is an optimizing sequence. Then, in turn, the upper semi-continuity of $g\to V(g)$ yields that
				\begin{equation*}
					V(g)\ge \limsup_{n\to\infty} V(\tilde g_n)=u(x),
				\eq
			which concludes the proof.
		\end{proof}

	\subsection{Proof of the duality results}\label{dualityproof}

		For the proof of Theorem \ref{interchange}, note that since $G(Z,\cdot)$ is non-decreasing, it immediately follows that
			\begin{equation}
				u(x)\le \inf_{\mathbb{Q}\in\mathcal{Q}}G\big(\mathbb{Q},u_{\mathbb{Q}}(x)\big)
				\le \inf_{\mathbb{Q}\in\mathcal{Q}}G\big(\mathbb{Q},v_{\mathbb{Q}}(y)+xy\big)=v(y;x), \quad \textrm{ for all}\; y>0. \label{proo}
			\end{equation}
		The proof verifies that the inequalities, in fact, are equalities by use of, respectively, Sion's minimax theorem and Lemma \ref{key}.

			\begin{proof}[Proof of Theorem \ref{interchange}]
			\emph{Part i)} For the case when $G(Z,\cdot)$ is concave and $U:\mathbb{R}_+\to\mathbb{R}_+$ let, respectively, $\varepsilon>0$ and $\varepsilon=0$. According to \eqref{finiteness}, $u(x)<\infty$, $x>0$. Hence, for each $g\in\mathcal{C}(x)$,
			\begin{equation*}
				\inf_{Z\in\mathcal{Q}}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big)
				\le
				\sup_{g\in\mathcal{C}(x)}\inf_{Z\in\mathcal{Q}}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big)
				\le u(x+\varepsilon)<\infty.
			\eq
		Consequently,
			\begin{equation}
				\inf_{Z\in\mathcal{Q}}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big)
				=\inf_{Z\in\tilde \mathcal{Q}}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big),\label{framtid}
			\end{equation}
		where $\tilde\mathcal{Q}$ is the set of measures in $\mathcal{Q}$ for which $G\big(Z,\mathrm{E}[ZU(\varepsilon+g)]\big)\le u(x+\varepsilon)+1$. Since $\mathrm{E}[ZU(\varepsilon+g)]\ge U(\varepsilon)\wedge 0$, for $Z\in\mathcal{Q}$, it holds for each $Z\in\tilde\mathcal{Q}$ that $G(Z,t)\le c$ for $t:=U(\varepsilon)\wedge 0$ and $c:=u(x+\varepsilon)+1$. Hence, $\tilde\mathcal{Q}$ in \eqref{framtid} might be replaced by $\mathcal{Q}_t(c)$. Since this holds for each $g\in\mathcal{C}(x)$, it thus follows that
			\begin{equation}
				\sup_{g\in\mathcal{C}(x)}\inf_{Z\in\mathcal{Q}}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big)
				=\sup_{g\in\mathcal{C}(x)}\inf_{Z\in\mathcal{Q}_t(c)}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big).\label{gard}
			\end{equation}

			Next, as $U(\varepsilon+\cdot)$ is bounded from below and $\mathcal{Q}_t(c)$ is UI due to Assumption \ref{dualass}, an application of Fatou's Lemma yields $Z\to\mathrm{E}[ZU(\varepsilon+g)]$, $g\in\mathcal{C}(x)$, lower semicontinuous with respect to a.s. convergence on $\mathcal{Q}_t(c)$. As $\mathcal{Q}_{t,T}$ is UI, that is equivalent to lower semicontinuity with respect to convergence in $L^1$. As the functional is convex (affine) that, in turn, implies weak lower semicontinuity. Since $G$ is jointly lower semicontinuous and quasiconvex it, thus, follows that
				\begin{equation}
					Z\to G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big),\quad g\in\mathcal{C}(x),
  					     \label{semi}
				\end{equation}
			is weakly lower semicontinuous and quasiconvex. Furthermore, as established above, it holds that
				\begin{equation}
					g\to G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big),\quad Z\in\mathcal{Q},\label{quasiconcave}
				\end{equation}
			is quasiconcave. Recall that $\mathcal{C}(x)$ is convex. Moreover,
				\begin{equation*}
					G(\lambda Z+(1-\lambda)\bar Z,t)\le \max\{G(Z,t),G(\bar Z,t)\}\le c,
				\eq
			for $Z,\bar Z\in\mathcal{Q}_t(c)$. Hence, $\mathcal{Q}_t(c)$ is also convex. The latter set is also weakly compact due to Assumption \ref{dualass}. Given the properties of the mappings defined in \eqref{semi} and \eqref{quasiconcave}, respectively, we might thus apply Sion's minimax theorem (cf. \cite{sion58}). This yields,
				\begin{equation}
					\sup_{g\in\mathcal{C}(x)}\inf_{Z\in\mathcal{Q}_t(c)}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big)
					=\inf_{Z\in\mathcal{Q}_t(c)}\sup_{g\in\mathcal{C}(x)}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big).\label{sion}
				\end{equation}
		Note that
			\begin{equation*}
				\inf_{Z\in\mathcal{Q}}\sup_{g\in\mathcal{C}(x)}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big)
				\le
				\inf_{Z\in\mathcal{Q}_t(c)}\sup_{g\in\mathcal{C}(x)}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big)\le u(x+\varepsilon),
			\eq
		where the first inequality is trivial and the second follows from \eqref{sion} combined with \eqref{gard}. Hence, by use of the same argument as above, the set $\mathcal{Q}_t(c)$ in the right hand side of \eqref{sion} can be substituted for the set $\mathcal{Q}$. That is to say,
				\begin{equation}
					\sup_{g\in\mathcal{C}(x)}\inf_{Z\in\mathcal{Q}}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big)
					=\inf_{Z\in\mathcal{Q}}\sup_{g\in\mathcal{C}(x)}G\big(Z,\mathrm{E}\big[ZU\left(\varepsilon+g\right)\big]\big).\label{sion2}
				\end{equation}
			For the case when $U:\mathbb{R}_+\to\mathbb{R}_+$, this completes the proof of part i).

			For the case when $G(Z,\cdot)$ is concave, a straight-forward argument yields $u(x)$ concave. According to \eqref{finiteness} it is also finite. Consequently, it is continuous as a concave function is continuous on the interior of the set where it is finite (cf. Theorem 10.1 in \cite{rock70}). On the other hand, since the expression on the left-hand-side in \eqref{sion2} clearly is smaller than $u(x+\varepsilon)$, it follows that
				\begin{eqnarray}
					u(x+\varepsilon)
					&\ge &\inf_{Z\in\mathcal{Q}}\sup_{g\in\mathcal{C}(x)}G\big(Z,\mathrm{E}\big[ZU\left(g\right)\big]\big)\nonumber\\
					&\ge &\sup_{g\in\mathcal{C}(x)}\inf_{Z\in\mathcal{Q}}G\big(Z,\mathrm{E}\big[ZU\left(g\right)\big]\big)\label{magic}
					\;=\;u(x).
				\end{eqnarray}
			Since $u(x)$ is (upper semi-) continuous, the result then follows by letting $\varepsilon\searrow 0$.

			Next, according to Assumption \ref{standing}, $\mathcal{Q}^f\neq\emptyset$. Since $G$ has an asymptotic maximum in the sense of Definition \ref{defg}, this implies that the set $\mathcal{Q}$ on the right-hand side in \eqref{hoppas} can be replaced by the set $\mathcal{Q}^f$. For $\mathbb{Q}\in\mathcal{Q}^f$, Lemma \ref{key} applies. Hence, use of part i), the fact that $G(Z,\cdot)$ is non-decreasing and the duality relations between $u_Z(x)$ and $v_Z(y)$ given in Lemma \ref{key}, yields
				\begin{eqnarray*}
					u(x)&=&
					\inf_{Z\in\mathcal{Q}^f}G\Big(Z,u_Z(x)\Big)\\
					&=&\inf_{Z\in\mathcal{Q}^f}G\Big(Z,\inf_{y>0}\left(v_Z(y)+xy\right)\Big)
					\;=\;\inf_{y>0}\inf_{Z\in\mathcal{Q}^f}G\Big(Z,v_Z(y)+xy\Big).
				\end{eqnarray*}
			Given the definition of $v(y;x)$, it thus only remains to show that
				\begin{equation}
					\inf_{Z\in\mathcal{Q}^f}G\Big(Z,v_Z(y)+xy\Big)=\inf_{Z\in\mathcal{Q}}G\Big(Z,v_Z(y)+xy\Big)=:v(y;x).\label{prokl}
				\end{equation}
			The inequality "$\ge$" follows as $\mathcal{Q}^f\subseteq\mathcal{Q}$. Without loss of generality, assume $v(y;x)<\infty$ and let $\tilde\mathcal{Q}:=\{Z\in\mathcal{Q}:v_Z(y)<\infty)\}$. Clearly the set $\mathcal{Q}$ on the right-hand side of \eqref{prokl} can then be replaced by $\tilde \mathcal{Q}$. On the other hand, $v_Z(y)<\infty$ implies that $u_Z(x)<\infty$. Hence, $\tilde \mathcal{Q}\subseteq \mathcal{Q}^f$ which yields the inequality "$\le$".

			\emph{Part ii)} Let
			\begin{equation*}
				H(Z,h):=G(Z,\mathrm{E}[ZV(h/Z)]+xy).
			\eq
			According to Lemma 3.7 in \cite{wu}, $(Z,h)\to\mathrm{E}[ZV(h/Z)]$ is lower semicontinuous. Hence, since $G$ is jointly lower semicontinuous, it follows that so is $(Z,h)\to H(Z,h)$. Furthermore, since $(z,y)\to zV(y/z)$ is convex, $G(Z,\cdot)$ non-decreasing and $G$ jointly quasiconvex, it follows that $(Z,h)\to H(Z,h)$ is jointly quasiconvex. Indeed, let $Z_t=tZ_0+(1-t)Z_1$ and $h_t=th_0+(1-t)h_1$. Then,
				\begin{eqnarray*}
				 H(Z_t,h_t)&=&
				 G(Z_t,\mathrm{E}[Z_tV(h_t/Z_t)]+xy)\\
				 &\le & G(Z_t,t\mathrm{E}[Z_0V(h_0/Z_0)]+(1-t)\mathrm{E}[Z_1V(h_1/Z_1)]+xy)
				 \;\le \; H(Z_0,h_0)\vee H(Z_1,h_1).
				\end{eqnarray*}
			Let $(Z_n,h_n)\in\mathcal{Q}\times\mathcal{D}(y)$ be an optimizing sequence such that
			\begin{equation}
				G\Big(Z^n,\mathrm{E}\left[Z^nV\left(h^n/Z^n\right)\right]+xy\Big)\sumpport_{n\to\infty}v(y;x)<\infty.\label{mot1}
			\end{equation}
		Note that
			\begin{equation}
				\mathrm{E}[ZV(h/Z)]+xy \ge U(x), \label{mot2}
			\end{equation}
		for all $Z\in\mathcal{Q}$ and $h\in\mathcal{D}(y)$. Indeed, as $V$ is convex, use of Jensen's inequality yields
			\begin{equation*}
				\mathrm{E}\big[ZV\left(h/Z\right)\big]
				=\mathrm{E}\big[ZV\left(h/Z1\!\!1_{Z>0}\right)\big]
				\ge V\big(\mathrm{E}\left[Zh/Z1\!\!1_{Z>0}\right]\big)
				=V\big(\mathrm{E}\left[h1\!\!1_{Z>0}\right]\big)
				\ge V(y),
			\eq
		as $V$ is decreasing and $h\in\mathcal{D}(y)$ which implies that $\mathrm{E}[h1\!\!1_{Z>0}]\le \mathrm{E}[h]\le y$. The fact that $G(Z,\cdot)$ is non-decreasing combined with \eqref{mot1} and \eqref{mot2}, then yields
			\begin{equation*}
				\tilde c:=\limsup_{n\to\infty}G\Big(Z^n,U(x)\Big)<\infty
			\eq
		and w.l.o.g., we can assume that $Z^n\in\{Z\in\mathcal{Q}:G(Z,U(x))\le \tilde c+1\}$. That is to say, that $Z^n\in\mathcal{Q}_t(c)$, for $t:=U(x)$ and $c:=\tilde c+1$.

		Applying twice the Komlos Lemma, yields a sequence $(\tilde Z_n,\tilde h_n)\in\textrm{conv}\{(Z_n,h_n), (Z_{n+1},h_{n+1}),...\}$ which converges $\mathbb{P}$-a.s. to some $(Z_0,h_0)$. Since $\mathcal{Q}_t(c)$ and $\mathcal{D}(y)$ both are convex, $(\tilde Z_n,\tilde h_n)\in\mathcal{Q}_t(c)\times\mathcal{D}(y)$. Furthermore, since $\mathcal{Q}_t(c)$ is uniformly integrable (as it is weakly compact) and $\mathcal{D}(y)$ is closed in $L^0$ according to Proposition 3.1 in \cite{kramkov}, it follows that $(Z_0,h_0)\in\mathcal{Q}_t(c)\times\mathcal{D}(y)$. Moreover, use of the quasiconvexity yields
				\begin{equation*}
					H(\tilde Z_n,\tilde h_n) = H\bigg(\sum_{k\ge n}\lambda^kZ_k,\sum_{k\ge n}\lambda^kh_k\bigg)
					\le \max_{k\ge n} H(Z_k,h_k)=H(Z_n,h_n)\searrow v(y).
				\eq
			Hence, $(\tilde Z_n,\tilde h_n)$ is also an optimizing sequence. Consequently, by use of the lower semicontinuity, it follows that
				\begin{equation*}
					H(Z_0,h_0)\le \liminf_{n\to\infty} H(\tilde Z_n,\tilde h_n)=v(y),
				\eq
			which proves that the optimum is attained for $(Z_0,h_0)$.

			Next, suppose $(\tilde Z_1,\tilde h_1)$ is another optimal pair. Let $h_t:=t\tilde h_1+(1-t)\tilde h_0$ and $Z_t:=t\tilde Z_1+(1-t)\tilde Z_0$, $t\in[0,1]$. As $(Z,h)\to\mathrm{E}[ZV(h/Z)]$ is convex and $G$ jointly quasiconvex, it thus follows that
		{\setlength{\arraycolsep}{-0.5cm}
				\begin{eqnarray*}
					G(Z_t,\mathrm{E}\left[Z_tV\left(h_t/Z_t\right)\right]+xy)\\
					&&\le G\left(Z_t,t\mathrm{E}\left[Z_1V\left(h_1/Z_1\right)\right]+(1-t)\mathrm{E}\left[Z_0V\left(h_0/Z_0\right)\right]+xy\right)\\
					&&\le  \max\big \{G\left(Z_1,\mathrm{E}\left[Z_1V\left(h_1/Z_1\right)\right]+xy\right),G\left(Z_0,\mathrm{E}\left[Z_0V\left(h_0/Z_0\right)\right]+xy\right)\big\}
					\;=\; v(y),
				\end{eqnarray*}}
			due to the optimality of $(\tilde Z_1,\tilde h_1)$ and $(\tilde Z_0,\tilde h_0)$, respectively. Hence, also $(h_t,Z_t)$ is optimal. We proceed as in the proof of Lemma 4.3 in \cite{schied}. Note that for $t\in(0,1)$, $\{Z_t>0\}=\{Z_0>0\}\cup\{Z_1>0\}$. Also note that according to (25) in \cite{schied}, the ration $h_t/Z_t$, does not depend on $t$. Hence, there exists a random variable $Y_T\ge 0$ and a sequence $\bar Z_1$, $\bar Z_2$, ... such that:
				\begin{itemize}
					\item[(a)]{$\mathbb{P}[\bar Z_n]$ tends to the maximum $\mathbb{P}$-probability for the support of any optimal $Z$;}
					\item[(b)]{$\{\bar Z_1>0\}\subseteq \{\bar Z_2>0\}\subseteq ...$;}
					\item[(c)]{for each $n$, $\bar h_n:=Y_T\bar Z_n\in\mathcal{D}(y)$ and $(\bar h_n,\bar Z_n)$ is optimal.}
				\end{itemize}
			By use of a Komlos-type argument, we may assume that $\bar Z_n$ converge $\mathbb{P}$-a.s. to some $\bar Z\in\mathcal{Q}$. Then $\bar h:=Y_T\bar Z\in\mathcal{D}(y)$ by Proposition 3.1 in \cite{kramkov} (cf. \eqref{iff} below). As above, it follows that $(\bar h,\bar Z)$ is optimal and, clearly, it is maximal.
		\end{proof}

	\begin{comment}
		\begin{rem}\label{usc}
			Note that if it could be proven that $u(x)$ is upper semicontinuous, then Theorem \ref{interchange} would hold for general utility functions from $\mathbb{R}_+$ to $\mathbb{R}$. Indeed, the result would follow by simply exploiting upper semicontinuity of $u(x)$ in \eqref{magic}. Alternatively, it would be enough if one could verify that
		\begin{equation*}
			\lim_{\varepsilon\searrow 0} \sup_{g\in\mathcal{C}(x)}\inf_{Z\in\mathcal{Q}} G\Big(Z,\mathrm{E}\big[ZU\big(\varepsilon+g\big)\big]\Big)
			\le \sup_{g\in\mathcal{C}(x)}\lim_{\varepsilon\searrow 0} \inf_{Z\in\mathcal{Q}} G\Big(Z,\mathrm{E}\big[ZU\big(\varepsilon+g\big)\big]\Big),
		\eq
			as the right-hand-side of the above expression trivially (by use of upper semicontinuity and, thus, under assumptions similar to those in Theorem \ref{existence}) is smaller than $u(x)$.
		\end{rem}
	\end{comment}

	Next, we prove Theorem \ref{relation} which establishes the existence of a saddle point and the link between the primal and dual solutions.

	\begin{proof}[Proof of Theorem \ref{relation}]
			\emph{Part i)} In order to verify the first statement, it remains to show that the infimum on the right hand side in \eqref{hoppas} is attained. To this end, note that since $U\ge 0$, $G$ is jointly lower semicontinuous and quasiconvex and the operation of point-wise supremum preserves lower semicontinuity and quasiconvexity, it follows that
			\begin{equation}
				Z\to \sup_{g\in\mathcal{C}(x)}G\Big(Z,\mathrm{E}\big[ZU(g)\big]\Big),\label{sist}
			\end{equation}
			is lower semicontinuous and quasiconvex. Next, let $Z^n\in\mathcal{Q}$ be a sequence such that
				\begin{equation}
					G\Big(Z^n,u_{Z^n}(x)\Big)\searrow u(x).
				\end{equation}
			As $u_{Z^n}(x)\ge U(x)$, for all $n$ and $G(Z,\cdot)$ is increasing, it follows that
				\begin{equation*}
					\tilde c:=\limsup_{n\to\infty}G\big(Z^n,U(x)\big)<\infty.
				\eq
			Hence, w.l.o.g., we may assume $Z^n\in\mathcal{Q}_t(c)$ with $t:=U(x)$ and $c=\tilde c+1$. Application of Komlos Lemma, yields a sequence $\tilde Z_n\in\textrm{conv}\{Z_n,Z_{n+1},...\}$ which converges $\mathbb{P}$-a.s. to some $Z_0$. Since $\mathcal{Q}_t(c)$ is convex, $\tilde Z_n\in\mathcal{Q}_t(c)$, $n=1,2,...$. Furthermore, since $\mathcal{Q}_t(c)$ is uniformly integrable (as it is weakly compact), it follows that $Z_0\in\mathcal{Q}_t(c)$. Moreover, as in the proof of Theorem \ref{interchange} part iii), the quasiconvexity implies that also $Z^n$ is an optimizing sequence. Use of the lower-semicontinuity of the mapping in \eqref{sist}, then yields that the infimum is attained for $Z_0$.

			\emph{Part ii)} Let $\tilde g$ and $\widetilde Z$ be a saddle-point for the primal problem. Let $y^*>0$ such that the infimum for the auxiliary conjugacy relations with respect to $\widetilde Z$ are attained for $y^*$. Then, it follows that
				\begin{equation*}
					u(x)=G\Big(\widetilde Z,u_{\widetilde Z}(x)\Big)=G\Big(\widetilde Z,v_{\widetilde Z}(y^*)+xy^*\Big).
				\eq
			Hence, the infimum in \eqref{udual} is attained for $y^*$. Next, let $(\widehat Z,\hat g)$ be any solution to the dual problem corresponding to $y^*$. Since $G(Z,\cdot)$ is non-decreasing, use of the assumptions and Lemma \ref{key} yields
				\begin{equation}
					u(x)=v(y^*;x)=G\left(\hat Z,v_{\hat Z}(y^*)+xy^*\right)\ge G\left(\hat Z,u_{\hat Z}(x)\right)\ge u(x).\label{sol}
				\end{equation}
			Hence, we have equality which, in turn, implies that
				\begin{equation*}
					u(x)=G\left(\hat Z,u_{\hat Z}(x)\right)
					\ge G\left(\hat Z,\mathrm{E}\Big[\hat ZU(\hat X)\Big]\right)
					\ge \inf_{Z\in\mathcal{Q}}G\left(Z,\mathrm{E}\Big[ZU(\hat X)\Big]\right)= u(x).
				\eq
			Consequently, $(\widehat\mathbb{Q},\widehat X)$ is a saddle point for the primal problem.

			Next, from the definition of $\mathcal{Y}(y)$, it follows that $\mathrm{E}[X_TY_T]\le xy$, for $X\in\mathcal{X}(x)$ and $Y\in\mathcal{Y}(y)$. Thus, it follows that
				\begin{eqnarray*}
					G\left(\hat Z,\mathrm{E}\Big[\hat ZV(\hat Y/\hat Z)+\hat X\hat Y\Big]\right)-G\left(\hat Z,\mathrm{E}\left[\hat ZU(\hat X)\right]\right)
					&=&G\left(\hat Z,\mathrm{E}\left[\hat ZV(\hat Y/\hat Z)\right]+\mathrm{E}\left[\hat X\hat Y\right]\right)-u(x)\\
					&\le &G\left(\hat Z,\mathrm{E}\left[\hat ZV(\hat Y/\hat Z)\right]+xy\right)-u(x)\\
					&=&v(y^*;x)-u(x)\;=\;0.
				\end{eqnarray*}

			Since $G(Z,\cdot)$ strictly increasing, this implies that
				\begin{equation*}
					\mathrm{E}^{\hat\mathbb{Q}}\Big[V(\hat Y/\hat Z)+\hat X\hat Y/\hat Z\Big]\le \mathrm{E}^{\hat\mathbb{Q}}\Big[U(\hat X)\Big].
				\eq
			On the other hand $V(\hat Y/\hat Z)+\hat X\hat Y/\hat Z\ge U(\hat X)$, $\hat\mathbb{Q}$-a.s. Consequently, $V(\hat Y/\hat Z)+\hat X\hat Y/\hat Z= U(\hat X)$ $\hat\mathbb{Q}$-a.s. and, thus, it follows that $\hat X=I(\hat Y/\hat Z)$, $\hat\mathbb{Q}$-a.s.
		\end{proof}

	\subsection{Further remarks on the auxiliary problem}

	We conclude with some further remarks on the auxiliary value functions.\\

	\begin{rem}\label{normaldual1}
		Let $\mathcal{\hat Y}(y)$ the set of all positive $\mathbb{Q}$-supermartingales such that $Y_0=y$ and $XY$ is a $\mathbb{Q}$-supermartingale for all $X\in\mathcal{X}(1)$. Then, as shown in Lemma 4.2 in \cite{schied}, it holds that
				\begin{equation*}
					v_\mathbb{Q}(y)=\inf_{Y\in\mathcal{\hat Y}(y)}\mathrm{E}^\mathbb{Q}\big[V(Y_T)\big].
				\eq
		Indeed, let $0\le s\le t\le T$. For $\hat Y\in\mathcal{\hat Y}(y)$ and $X\in\mathcal{X}(1)$,
			\begin{equation*}
				X_s\hat Y_s\ge \mathrm{E}^\mathbb{Q}\left[X_t\hat Y_t|\mathcal{F}_s\right]=\frac{1}{Z_s}\mathrm{E}\left[X_t\hat Y_tZ_t|\mathcal{F}_s\right],\quad \textrm{$\mathbb{P}$-a.s. on $\{Z_s>0\}$}.
			\eq
		On $\{Z_s=0\}$, it holds that $Z_t=0$ $\mathbb{P}$-a.s. It follows that $X\hat Y Z$ is a $\mathbb{P}$-supermartingale and hence that $\hat YZ\in\mathcal{Y}(y)$. Conversely, let $Y\in\mathcal{Y}(y)$. Then, $\mathbb{Q}$-a.s. for each $X\in\mathcal{X}(1)$,
			\begin{equation*}
				\mathrm{E}^\mathbb{Q}\left[X_t\frac{Y_t}{Z_t}|\mathcal{F}_s\right]
				=\frac{1}{Z_s}\mathrm{E}\left[Z_tX_t\frac{Y_t}{Z_t}1\!\!1_{Z_t>0}|\mathcal{F}_s\right]
				\le\frac{1}{Z_s}\mathrm{E}\big[X_tY_t1\!\!1_{Z_s>0}|\mathcal{F}_s\big]\le \frac{X_sY_s}{Z_s}1\!\!1_{Z_s>0}.
			\eq
		Hence, $XY/Z$ is a $\mathbb{Q}$-supermartingale, for all $x\in\mathcal{X}(x)$ and $Y/Z1\!\!1_{Z>0}\in\mathcal{\hat Y}(y)$. \\
		\end{rem}

		\begin{rem}\label{normaldual}
		For $\mathbb{Q}\ll\mathbb{P}$, let $\mathcal{X}_\mathbb{Q}(x)$ the set of wealth-processes such that $X_0\le x$ and $X_t\ge 0$, $\mathbb{Q}$-a.s. on $t\in[0,T]$ and $\mathcal{Y}_\mathbb{Q}(y)$ the set of all positive $\mathbb{Q}$-supermartingales such that $Y_0=y$ and $XY$ is a $\mathbb{Q}$-supermartingale for all $X\in\mathcal{X}_\mathbb{Q}(1)$. Then, consider the following two problems
			\begin{equation*}
				\tilde u_\mathbb{Q}(x)=\sup_{X\in \mathcal{X}_\mathbb{Q}(x)}\mathrm{E}^\mathbb{Q}\big[U(X_T^\pi)\big]\quad\textrm{and}\quad
				\tilde v_\mathbb{Q}(y)=\inf_{Y\in\mathcal{Y}_\mathbb{Q}(y)}\mathrm{E}^\mathbb{Q}\big[V(Y_T)\big],
			\eq
		where $\inf\emptyset:=\infty$. Note that $\tilde u(x)$ is the standard investment problem with respect to $\mathbb{Q}$ as it is normally defined. However, since it is not clear whether $S_t$ satisfies NFLVR with respect to $\mathbb{Q}$, it is (a priori) not clear whether $\tilde u$ and $\tilde v$ are each others conjugate. Also note that $\mathcal{X}(x)\subseteq \mathcal{X}_\mathbb{Q}(x)$, which, in turn, implies that $\mathcal{Y}_\mathbb{Q}(y)\subseteq\mathcal{\hat Y}(y)$. Hence,
				\begin{equation}
					u(x)\le \tilde u(x)\quad\textrm{and}\quad v_\mathbb{Q}(y)\le \tilde v_\mathbb{Q}(y).\label{remaining}
				\end{equation}
		In particular, the condition that $\tilde v_\mathbb{Q}(y)<\infty$, $\mathbb{Q}\in\mathcal{Q}$, is therefore a sufficient condition for Theorem \ref{existence} to hold. For $\mathbb{Q}\sim\mathbb{P}$, \eqref{remaining} holds with equality. The question whether there are models $\mathbb{Q}\ll\mathbb{P}$, for which the inequality is strict is left for future study. We limit ourselves to note that given that the market is continuous, equality may in fact hold under rather weak conditions. Regardless of this, under the additional assumption \eqref{additional}, it holds for the variational case (i.e. when $G(\mathbb{Q},t)=\gamma(\mathbb{Q})+t$) that
				\begin{equation*}
					v(y;x)=\inf_{\mathbb{Q}\in\mathcal{Q}}G\Big(\mathbb{Q},\tilde v_\mathbb{Q}(y)+xy\Big),
				\eq
		and, in consequence, Theorem \ref{interchange} holds also with $v_\mathbb{Q}(y)$ replaced by $\tilde v_\mathbb{Q}(y)$. Indeed, $v(y;x)=v(y)+xy$, where
			 	\begin{equation*}
					v(y)=\inf_{Z\in\mathcal{Q}^e}\Big(v_Z(y)+\gamma(Z)\Big)
					=\inf_{Z\in\mathcal{Q}^e}\Big(\tilde v_Z(y)+\gamma(Z)\Big)
					\ge \inf_{Z\in\mathcal{Q}}\Big(\tilde v_Z(y)+\gamma(Z)\Big).
				\eq
		Here the first equality follows from Lemma 4.4 in \cite{schied}. Since $v_\mathbb{Q}(y)\le \tilde v_\mathbb{Q}(y)$, $\mathbb{Q}\in\mathcal{Q}$, equality follows.
	\end{rem}

redpajama_set_name: RedPajamaArXiv


And the street far below.
It's much hotter this time.
And run from it all.
of our own kitchen sinks.
Where they still have some snow.
I'm scared half to death.
And we are so cold here!! I can't wait for sunshine to come back. You should take vacations to cool places during these times. People here go somewhere warm at least once every winter.
Beautiful poetry with a message.
Thank you, Isha! 🙂 Your comments mean so much to me!!!! 🙂 Glad you liked this one. We had a thunderstorm today. After the storm, we were blessed with a breathtaking, bold sunset. When I saw the sun, it looked like a huge ball of fire. I couldn't help but worry about the long-term effects of pollution. I kept thinking what it would be like, if there was no barrier between us and the intense, powerful rays of the sun.
It's all very scary! :O We've got to…..SAVE OUR PLANET!!!!!!!
Well written poem. 🙂 The whole world has had enough of us and extreme weather is becoming the norm – snow storms in the north, drought here in CT.
It's not good! I hate to think about….what we are leaving behind for generations to come. 😦 Thank you for reading and commenting. I really appreciate it!


CRL 6" Square End Caps for Sidelite Rails are used to finish the installation of the CRL Sidelite Rails.
Each End Cap matches the finish of the Sidelite and Door Rails perfectly. Attaches with provided matched finish machine screw.


text:
Catholics for Choice
Yes, People Of Faith Can Absolutely Support Planned Parenthood
Carol Kuruvilla
A Hindu, Muslim, Pagan, Christians, and Jews offer arguments in defense of a woman's right to choose.
How Hobby Lobby Killed Religious Liberty
The Supreme Court's Hobby Lobby decision is a disaster for women, and we can lay the blame squarely at the feet of the Obama administration.
Jon O'Brien
Polling Shows That Bishops' Campaign Failed
Catholics know what's at stake, and they're making up their own minds despite the influence the bishops claim, or the relative importance the hierarchy's influence and divisive campaigns garner in the media.
From Russia Without Love
There is no easy answer to demographic decline, and making abortion the scapegoat for decades of neglect to women's well-being will not help. Building a stronger society that supports women and children is the long, slow, sure road to a thriving Russia.
Protect Life Act: New Bill Would Allow Hospitals To Refuse To Perform Abortions
"Unfortunately in the Catholic system, someone who's a bishop, who has no medical qualifications whatsoever, can dictate
Laura Bassett
Condom Activists Push Debate At Catholic World Youth Day
By David Gibson c. 2011 Religion News Service (RNS) Sending a small band of young people to talk about condoms at World Youth
Steve King: Covering Birth Control Could Make Us A 'Dying Civilization' (VIDEO)
The socially-conservative King called the new guidelines "bizarre" and "Orwellian." WATCH: An estimated 98 percent of sexually
Catholic Women And Contraception: It's Beyond Don't Ask, Don't Tell
The divide between the faithful and the Catholic hierarchy on contraception has a long history. Now, surveys show that 98 percent of sexually active Catholic women have used family planning methods.
Birth Control? Really? (Yes, Really!)
Today the Institute of Medicine reported what most breathing humans know more or less reflexively: That contraception is a preventive health care service. Why is this even a question? And why must we answer it now?
Ellen R. Shaffer
The Church And Contraception
Recent research shows what everybody already knows to be true: Catholics use methods of contraception that are banned by the church hierarchy.
hide/seek exhibit
We at Catholics for Choice are very disappointed in your decision to remove David Wojnarowicz's "A Fire in My Belly." Censorship of the arts is the last thing that an art institution should be doing.
GOP Ploy to Curry Latino Favor Via Puerto Rican Statehood Is Misguided
The recent GOP intelligentsia strategy to curry favor for Latino votes via support for Puerto Rican Statehood while opposing immigration reform is misguided, misinformed, and probably disingenuous.
Maritza Stanchich
Catholics Call on Pope Benedict to Reconsider Vatican's Ban on Contraceptive Pill
It would be a lasting and wholly positive legacy if the current pope lifted the ban on the use of contraceptives to allow Catholics to plan their families.
Truth Is Indeed One of the First Casualties of War
Despite his carefully nuanced positions, we know that Jim Wallis is anti-choice and opposes access to comprehensive reproductive health services for women.
What Catholics Want in Healthcare Reform
American Catholics can picture themselves as patients, and want to be able to get birth control and condoms at their doctor. They also think they can speak for themselves.
Stepping Up the Pressure: The Archbishop Comes to the Abortion Clinic
Every woman has different life circumstances. Every pregnancy produces different challenges. Who is more able to assess the moral validity of every woman's abortion? I certainly don't think it's the Archbishop.
We Believe in Health Care Reform
When men and women have access to high-quality sexual and reproductive health care, our country is healthier and stronger.
"Don't Be Mad at Me for Wanting to Solve the Problem"
An extraordinary relationship between DeLauro (D-CT) and Ryan (D-OH) has been forged that will both help people avoid unplanned pregnancies and assist those who choose to continue a pregnancy.
Catholics for Choice Protest Appointment of Anti-choice Advocate to Senior Position at HHS
The appointment yesterday of Alexia Kelley as Director of Faith-based and Community Partnerships at the Department of Health and Human Services marks "a defeat for reason and logic."
Jodi Jacobson

redpajama_set_name: RedPajamaCommonCrawl


text:
This #LearnAnywhere Bus Gives A New Meaning to #MobileLearning #MLearning
By Lisa Nielsen 2015-07-09T05:40:00Z
One of the most exciting things I found at #ISTE2015 was a school bus transformed into a technology learning studio.
The bus is operated by Estella who at age 79 is a self-identified "Gadget Gal." She loves technology and gadgets so much, she wanted to bring them to others in her community who might not have access to computers and the internet.
So, she drew up her dream on a piece of paper and put the word out that she was looking for folks to help her bring this to life. She found volunteers, donors, and invested much of her own retirement funds on the project.
That was eight years ago. Below is a picture of Estella in her bus today.
Estella travels around Palm Beach County talking with schools, communities, and local agencies to see how various technology services she provides can help them. This might include digital citizenship, literacy, GED or test prep, tutoring, homework assistance, resume writing, and more. Estella understands that not everyone has access to technology and the internet, so she brings it to their door...literally. She says, "When kids get on this bus, they're able to learn about the world and connect with the world using Microsoft technology and I can empower them to make their lives better." Estella says she is grateful for the contributions she has received from Microsoft YouthSpark to help bring her dream to life. She said she feels blessed because she has been fortunate enough to have people step up to the plate and say, "Let me help you do what you're doing because I believe in your cause."
Estella explained that because of her dad she grew up around trucks and buses. Little did she know back then, this would help prepare her for most recent passion.
In addition to supporting the local community, Estella has also taken her show on the road. Last summer Estella's Brilliant Bus took a "Tech Innovation Bus Ride." They journeyed on the "Road to Yes We Code!" A multi-city bus caravan roadtrip traveling from West Palm Beach, Fla; Atlanta, Ga; Birmingham, Al; Selma, Al; Memphis, Tenn., Jackson, Miss, New Orleans, La. The Tech Innovation Bus Ride featured turnkey hands-on custom coding classes, robotics and tech entrepreneurship workshops.Most recently, Estella was featured in a Superbowl commercial which you can see here.
To innovative educators, the idea of a mobile technology learning studio is fascinating. What if we wired up more school buses like Estella's? Rather than sitting in a bus depot, how could they be utilized during school hours? What about in rural areas or others where students spend a significant amount of time on buses? This commuting time could be utilized for homework time, tutoring, reading, and more. What can you imagine doing if your community had a mobile technology learning studio? What kind of programs and services do you envision taking place?Want to follow what's new with Estella's Brilliant Bus? Like the Facebook page here.
Lisa Nielsen writes for and speaks to audiences across the globe about learning innovatively and is frequently covered by local and national media for her views on "Passion (not data) Driven Learning," "Thinking Outside the Ban" to harness the power of technology for learning, and using the power of social media to provide a voice to educators and students. Ms. Nielsen has worked for more than a decade in various capacities to support learning in real and innovative ways that will prepare students for success. In addition to her award-winning blog, The Innovative Educator, Ms. Nielsen's writing is featured in places such as Huffington Post, Tech & Learning, ISTE Connects, ASCD Wholechild, MindShift, Leading & Learning, The Unplugged Mom, and is the author the book Teaching Generation Text.
Disclaimer: The information shared here is strictly that of the author and does not reflect the opinions or endorsement of her employer.
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