text:
\section{Introduction and notation} \subsection{The existing literature}
The study of utility maximization in con\-ti\-nu\-ous-time stochastic
models of financial markets dates back to the seminal contributions of
Robert Merton \cite{Mer69, Mer71}.  General complete Brownian models were
considered by Karatzas, Lehoczky and Shreve \cite{KarLehShr87} and Cox and
Huang \cite{CoxHua89},  where the authors used convex-analytic (duality)
techniques to characterize the optimizer. Duality techniques for
incomplete It\^ o-process models were first developed by Karatzas,
Lehoczky, Shreve and Xu \cite{KarLehShrXu91}, and in a general
semimartingale setting, by Kramkov and Schachermayer \cite{KraSch99,
KraSch03}.

Cvitani\' c and Karatzas \cite{CviKar92} extended the existence results of
Karatzas et al \cite{KarLehShrXu91} to incorporate convex constraints on
the fraction of wealth invested in the risky securities. In the same, It\^
o-process driven setting, Cuoco \cite{Cuo97} attacked the primal problem
directly and established the existence of optimizers when investors face
convex constraints either on the number of shares or on the amount
invested.

Relying on a version of the Optional Decomposition Theorem of
F\"ollmer and Kramkov \cite{FolKra97}, Pham \cite{Pha02} and
Mnif and Pham \cite{MniPha01} studied constrained optimization in the
general semimartingale setting. In \cite{Pha02}, the author
generalized the shortfall objective considered in F\"ollmer and
Leukert \cite{FolLeu00}, while, in \cite{MniPha01},
investors subject to either convex constraints on the number of risky
securities, or American-type constraints on the wealth process have
been considered.
As in \cite{KraSch03}, both \cite{Pha02} and \cite{MniPha01} used
``Koml\'os-type'' arguments to establish the existence of primal
optimizers. The question of dual existence was, however, left open (see
the discussions on p.~154 in \cite{Pha02} and p.~167 in
\cite{MniPha01}). Constraints on the fractions of wealth
invested in the risky securities were investigated in \cite{Lon04} by
Long who established the existence of optimizers under a number of
strong additional assumptions.

Among several authors who studied  the existence of optimizers for
non-smooth utility functions, we mention Bouchard, Touzi and Zeghal
\cite{BouTouZeg04}, and we direct the reader to consult their references.
 The recent counterexample of Westray and Zheng \cite{WesZhe11}
illustrates some of the counterintuitive phenomena non-smooth dual
objectives can produce.

\subsection{Our contributions} The analysis in most of the papers mentioned
above requires that the investor be allowed to choose not to invest in the
risky securities at all,  with \cite{MniPha01} serving as a notable
exception.   In the present paper, no such condition is imposed:
one might be forced to invest in risky assets some or all of the time;
 the idea to apply constraints not containing the origin to
utility-maximization problems goes back at least to the work \cite{Kal99}
of Kallsen (see also \cite{Kal02}).  The study of
such a general class of constraints is interesting from both
mathematical and economical points of view. Mathematically, this setup produces an interesting convex-analytic situation where
the support
function is no longer necessarily nonnegative. Economically, such
constraints
correspond to the case when some of the available
assets are not perfectly liquid and the investor is effectively forced to
hold them.  The case of a terminal random endowment, studied by Cvitani\'
c, Schachermayer and Wang \cite{CviSchWan01} and Hugonnier and Kramkov
\cite{HugKra04} among others, can be embedded in our setting - it
corresponds to a constraint which forces the investor to hold one unit of a
specific asset to maturity.  Finally, a number of classical constraints,
including the prohibition or restriction of short selling, can be
interpreted as convex portfolio constraints and fit into our framework.

There are two main results in this paper and they both apply to a
general semimartingale model of a financial market.  The first
one
establishes the existence of the primal and dual optimizers in the
constrained utility-maximization problem, with the dual
problem defined over a class of finitely-additive measures. The
conjugacy of the primal and the dual value functions is an integral part of
our result. The only assumption imposed on the utility function, besides the defining properties of concavity, monotonicity, and the Inada condition at zero, is the reasonable asymptotic elasticity of \cite{KraSch99}.

Our second result is that the finitely-additive relaxation is, up to
attainment, in fact, not necessary, and that the dual problem can be
posed over a class of countably-additive measures. This result
generalizes Theorem 2.2 (iv) of \cite{KraSch99} to our constrained
case;  in particular, it subsumes the case of an unspanned endowment
considered in \cite{CviSchWan01}. The main technical difficulty we had
to  overcome is the absence of semicontinuity in the appropriate
direction of the dual objective function (in general, this objective
is not upper semicontinuous). Our solution is based on Theorem 2.2
(iv) of \cite{KraSch99} and methods of locally-convex convex analysis.
This countably-additive relaxation has several practical implications.
First of all, the classical stochastic-optimal-control framework and
the corresponding tools and notions, such as the dynamic programming
principle and the associated Hamilton-Jacobi-Bellman equation, rely on
having stochastic processes (in our case, densities of countably-additive measures)
as controls. These tools are not immediately available or applicable
in more general settings (such as the finitely-additive one).
Furthermore, the existence of $\varepsilon$-optimal countably-additive
measures
serves as a first step towards an efficient numerical treatment of the
problem.

As far as no-arbitrage-type assumptions are concerned, our main
existence and  conjugacy results are provided under the abstract assumption of closedness and boundedness in probability (convex compactness in the language of \v Zitkovi\' c \cite{Zit09}) of the $\el^0_+$-solid hull ${\mathcal C}(x)$ of the set of terminal wealths of admissible portfolios with
initial wealth $x$.  This condition is weaker than the celebrated
No Free Lunch with Vanishing Risk (NFLVR) of Delbaen and Schachermayer \cite{DelSch94} and is reminiscent of the No Unbounded Profit with Bounded Risk (NUPBR) condition of Karatzas and Kardaras \cite{KarKar07}.  Indeed, given the presence of constraints, the classical NFLVR can be too strong, as the
constraints will often prevent the investor from making riskless profit,
even if the asset prices would admit arbitrage in the unconstrained market.
Using a new closedness result of Czichowsky
and Schweizer \cite{CziSch11} for sets of constrained stochastic integrals
in the semimartingale topology, we give a general and easy-to-check
sufficient condition for the convex compactness of ${\mathcal C}(x)$.

\subsection{Notation and function spaces}
All stochastic objects are defined on a filtered probability space
$(\Omega,\sF,(\sF_t)_{t\in [0,T]},{\mathbb P})$ where the $T\in (0,\infty)$ is the time
horizon and the underlying filtration ${\mathbb F} := (\sF_t)_{t\in[0,T]}$
satisfies the usual conditions. For $p\in (0,\infty]$, $\el^p$ denotes
the Lebesgue space $\el^p(\Omega,\sF,{\mathbb P})$ and $\el^0$ denotes the
collection of all ${\mathbb P}$-a.s.~equivalence classes of finite-valued
random variables on $(\Omega,{\mathcal F})$ (topologized by convergence in
probability). If not stated otherwise, all processes are assumed to be
\cd and ${\mathbb F}$-adapted, with the exception of processes which
serve as integrands in stochastic integrals; those  are always assumed
to be ${\mathbb F}$-predictable.

While none of our results require their mention  in the
statements, finitely-additive measures are used quite frequently in
proofs. We naturally identify finite-valued finitely-additive set functions on
$(\Omega,{\mathcal F})$ which vanish on ${\mathbb P}$-null events with the topological
dual ${\mathrm{ba}}:=\linfd$ of $\el^{\infty}$ (see \cite{RaoRao83} for further
details). The dual pairing of ${\mathrm{ba}}$ and $\el^{\infty}$ is denoted
by $\scl{\cdot}{\cdot}:{\mathrm{ba}}\times\el^{\infty}\to\R$ and the (dual) norm
$\norm{\cdot}$ on ${\mathrm{ba}}$ is given by
$\norm{{\mathbb Q}}:=\sup\sets{|\scl{{\mathbb Q}}{f}|}{f\in\el^{\infty},\norm{f}_{\el^{\infty}}\leq
  1}$.  We do not differentiate between the elements of $\el^1({\mathbb P})$ and
their images under the natural bidual embedding $\el^1 \hookrightarrow
{\mathrm{ba}}$. In other words, we identify a countably additive measure ${\mathbb Q}$
absolutely continuous with respect to ${\mathbb P}$ with its Radon-Nikodym
derivative $\frac{d{\mathbb Q}}{d{\mathbb P}}$.

All of the spaces above admit natural positive cones, denoted by
$\el^p_+$, for $p\in [0,\infty]$ or ${\mathrm{ba}}_{+}$ in the case of
${\mathrm{ba}}$. The domain of the pairing
$\scl{\cdot}{\cdot}:{\mathrm{ba}}\times\el^{\infty}\to\R$ can be replaced by
${\mathrm{ba}}_+\times \el^0_+$ by setting $\scl{{\mathbb Q}}{f}:=\lim_{n\to\infty}
\scl{{\mathbb Q}}{f\wedge n}\in [0,\infty],\text{ for }{\mathbb Q}\in{\mathrm{ba}}_+\text{ and }
f\in\el^0_+$.  Each element ${\mathbb Q}\in{\mathrm{ba}}_+$ admits the unique
decomposition (called the Yosida-Hewitt decomposition)
${\mathbb Q}={\mathbb Q}^r+{\mathbb Q}^s$ into a countably-additive measure ${\mathbb Q}^r\in\el^1_+$
and a singular part ${\mathbb Q}^s\in{\mathrm{ba}}_+$ uniquely characterized by the fact
that ${\mathbb Q}'\equiv 0$, whenever ${\mathbb Q}'\in\el^1_+$ and ${\mathbb Q}'(A)\leq
{\mathbb Q}^s(A)$ for all $A\in{\mathcal F}$.

For an ordered normed space $N$ with the closed positive
orthant $N_+$ and $y\geq 0$, we set $\bn(y):=\sets{x\in
  N}{\norm{x}\leq y}$, $\bn_+(y):=\bn(y)\cap N_+$, $\sn(y):=\sets{x\in
  N}{\norm{x}= y}$, and $\sn_+(y):=\sn(y)\cap N_+$. For a dual pair
$(X,X^*)$ of vector spaces (with the pairing denoted by
$\scl{\cdot}{\cdot}$) and a map $f:X\to (-\infty,\infty]$, $f^*$
denotes the $(X,X^*)$-convex conjugate of $f$, i.e.,
$f^*(y):=\sup_{x\in X} \big( \scl{x}{y}-f(x) \big)$, $y\in X^*$.
Finally, we remind the reader that the (convex-analytic) indicator
$\cind{B}$ of a subset $B$ of $X$ is defined by $\cind{B}(x):= 0$ for
$x\in B$ and $+\infty$ otherwise.

\section{Problem formulation and the main results}
\subsection{The asset-price model}
We consider a financial market with $d\in{\mathbb N}$ risky assets modeled by a
 $d$-dimensional \cd semimartingale \[S=
(S^{(1)}_t,...,S^{(d)}_t)_{t\in [0,T]}.\] The existence of a num\'
eraire asset $(S^{(0)}_t)_{t\in [0,T]}$, with $S^{(0)}_t:=1$ for
$t\in[0,T]$ - a zero-interest money-market account -  is also postulated.

A predictable $S$-integrable process
$H=(H^{(1)}_t,...,H^{(d)}_t)_{t\in [0,T]}$ is called a
\define{portfolio} and its value $H_t$  is
interpreted as the number of shares of each risky asset held by the
investor at time $t\in [0,T]$.  If a portfolio $H$ is used to implement a
dynamic trading strategy, the gains/losses accrued by time $t$ are given by
$X^H_t$, where
\begin{align}\label{def:wealth}
  X^{H}_t := (H \cdot S)_t :=  \int_0^t \sum_{k=1}^d
  H^{(k)}_udS^{(k)}_u,\quad t\in[0,T].
\end{align}
The sum on the right has to be understood in the sense of
vector stochastic integration (see \cite{CheShi02}, \cite{Jac80}, and  Chapter VII, \S 1a, in \cite{Shi99}).
\subsection{Convex constraints}
\label{sse:conv-const}
Let $2^{\R^d}_{c}$ denote the set of all non-empty closed and convex subsets
of $\R^d$.
\begin{definition}
  \label{def:predictable}
  A map $\kappa:[0,T]\times \Omega\to2^{\R^d}_{c}$ is said to be
  \define{predictable} if the set
  \[ \Bsets{(t,\omega)\in [0,T]\times\Omega}{ \kappa(t,\omega)\cap F\ne
    \emptyset}\] is predictable for each closed set $F\subseteq \R^d$.
\end{definition}
We fix a predictable \define{constraint map} $\kappa:[0,T]\times
\Omega\to 2^{\R^d}_{c}$; it  is
used as a specification of an exogenously-imposed constraint on the
possible values the portfolio $H$ can take. The set of all portfolios
$H$ such that $H_t\in \kappa_t$ for all $t\in [0,T]$, ${\mathbb P}$-a.s., will
be denoted by $\sA^\kappa$. The investment in the money market account
is not restricted.

In addition to the constraint imposed through $\kappa$, we consider a
different kind of a constraint known as the \define{admissibility
  constraint}. More precisely, a portfolio $H$ for which the process
$X^H:= H\cdot S$ there exists a constant $a\geq 0$ such that
$X^H_t\geq -a$ for all $t\in[0,T]$, ${\mathbb P}$-a.s., is called
\define{admissible}. Such a constraint is commonplace in mathematical
finance and is imposed to rule out doubling strategies. The set of all
admissible portfolio processes is denoted by $\sA^\text{low}$.

Combining the above the two constraints produces the class $\sA$ of
\define{constrained admissible portfolios}:
\[ \sA:=\sA^\text{low}\cap \sA^\kappa.\] Many classical constraint
structures can be expressed in terms of a well-chosen $\kappa$, see,
e.g., Section 3 in \cite{Cuo97} and Chapter 5 in \cite{KarShr98}. We do exhibit, however, in some
detail the construction that allows us to treat the presence of a
random endowment in  our framework:
\begin{example}[Random endowment as a special case of a portfolio constraint]\label{ex:ran_endow}
  As above, let the financial market consist of the risky assets $S=
  (S^{(1)}_t,...,S^{(d)}_t)_{t\in [0,T]}$ and the riskless asset
  $S^{(0)}_t:= 1$. Let us also assume that $S$ admits no arbitrage
  in the sense of the condition NFLVR.  Consequently, there exists an
  equivalent $\sigma$-martingale measure ${\mathbb Q}$ (see \cite{DelSch98}
  for the terminology).

  Let us also assume that the agent receives a lump-sum \define{random
    endowment} ${\mathcal E}\in \el^{\infty}({\mathcal F}_T)$ at time $T$. For an arbitrary
  equivalent $\sigma$--martingale measure ${\mathbb Q}$, the process
  $\hat{S_t}$, defined as a \cd version of the bounded martingale
  ${\mathbb E}^{{\mathbb Q}}[{\mathcal E}|{\mathcal F}_t]$, can be
  added to $S$ to form a larger financial market. The constraint set
  $\kappa$ is defined so as to mimic the behavior in the original
  market with the presence of the random endowment:
  \[ \kappa_t(\omega):= \R^d\times \set{1}.\] Indeed, any admissible
  constrained portfolio in the augmented market $(S,\hat{S})$ leads
  to a total wealth of the form $(H\cdot S)_T+\hat{S}_T-\hat{S}_0=
  x+(H\cdot S)_T + {\mathcal E}$, for $x:=-{\mathbb E}^{{\mathbb Q}}[{\mathcal E}]$ (under the
  assumption that ${\mathcal F}_0$ is ${\mathbb P}$-trivial). Thanks to the boundedness of
  ${\mathcal E}$, the notions of admissibility in the two markets are equivalent.

  It is possible to extend the domain of this example in various
  directions. For example, to treat an unbounded random endowment,
  one would need to use a more sophisticated
  version of the admissibility requirement or resort to a change of
  num\'eraire.  \end{example}

\subsection{No-arbitrage conditions on the financial market} Moving
on  towards our main result, we introduce notation for the set of
gains processes of admissible constrained portfolios, as well as for
certain related sets:
\begin{equation
\label{def:C}
    \begin{split}
      \XX^c&:=\sets{(X^H_t)_{t\in{[0,T]}}}{H\in\sA},\\
      \sK&:=\sets{X_T}{X\in\XX^c},\\
      \sC &:= (\sK-\el^0_+)\cap \el^{\infty},\\
      \sC(x)&:=(x+\sK-\el^0_+)\cap \el^0_+,\text{ for }x\in\R.
    \end{split}
  \end{equation}
  As far as technical conditions are concerned, we start with a
  succinct umbrella assumption under which our main
  theorem holds. Natural sufficient conditions on separate ingredients
  - the market and the constraint correspondence - will be briefly
  described below, and then in detail in Section
  \ref{sec:suff}.

Following \cite{Zit09}, we say that a subset of a topological vector
  space is \define{convexly compact} if any family of closed and
  convex sets with the finite-intersection property admits a non-empty
  intersection. In \cite{Zit09}, it is shown  that a subset of
  $\el^0_+$ is convexly compact if and only if it is bounded and
  closed in probability.
\begin{assumption}\label{ass:C}
  ${\mathcal C}(x)$ is convexly compact  for all $x\in\R$, and there exists
  $x\in\R$ such that ${\mathcal C}(x)\ne \emptyset$.
\end{assumption}
\begin{remark}
  Let us  comment on the interpretation of Assumption
  \ref{ass:C}. The non-emptiness condition is equivalent to assuming
  $\sA\neq\emptyset$, i.e., that it is possible to produce a
  bounded-from-below wealth process without violating the
  constraints. Boundedness in probability serves as a weak
  no-arbitrage requirement and can be deduced, in may cases, already
  from the finiteness of the expected-utility value function. Similar
  weakenings of the no-arbitrage condition have already been
  considered in the literature (see, e.g., Section 3 in
  \cite{KarKar07}). The closedness requirement is a natural
  condition for the existence of an expected-utility optimizer and is
  present in virtually all widely-used no-arbitrage concepts.
\end{remark}

Let us preview a sufficient condition for Assumption
\ref{ass:C}. The definitions of the map $\Pi^S$ (the \emph{projection onto
the predictable range map} of \cite{CziSch11}) and the support measure
${\mathbb P}^S$ of $S$ are postponed until Section \ref{thm:closed}. Let us mention that the below condition (2) is always
satisfied for any $S$ if $\kappa_t(\omega)$ is polyhedral, compact, or if
it admits a continuous support function, for each $t\in [0,T]$,
${\mathbb P}$-a.s., see \cite{CziWesZhe11} for details.
However, \cite{CziWesZhe11} and \cite{CziSch11} contain examples showing that (2) in the following proposition is not true in general.

\newcommand{\proptf}[1]{
Assumption \ref{ass:C} holds if the following three conditions are satisfied:
\begin{enumerate}
	\setlength{\itemindent}{-2ex}
  \item $\sA \ne\emptyset$,
  \item the projection $\Pi_t^S(\omega)\kappa_t(\omega)$ is closed, for
${\mathbb P}^S$-a.e.
  \item there exist
  \begin{enumerate}
	  \setlength{\itemindent}{-5ex}
  \item a probability measure ${\mathbb Q}\sim{\mathbb P}$,

\item $\hH\in\AA$ with
${\mathbb E}^{{\mathbb Q}}[ (\hH\cdot S)_T]<\infty$ and $\hH\cdot S$ locally bounded, and
\item a nondecreasing predictable \cd process $\prf{A_t}$, with $A_0=0$
\end{enumerate}
such that
\begin{equation}
#1
 \begin{split}
H\cdot S - (\hH\cdot S + A) \text{ is a
${\mathbb Q}$-supermartingale, for all $H\in\AA$.}
  \end{split}
\end{equation}

 \end{enumerate}}

\begin{proposition}
\label{pro:sufficient}
\proptf{\label{equ:sup-cond}}
\end{proposition}
\begin{remark}\
\label{rem:suff-pro-suff}
\begin{enumerate}
\item Conditions on the constraint set $\kappa$ under which property (2) in
Proposition \ref{pro:sufficient} holds are presented in \cite{CziSch11}.
\item The process $A$ in (3)(c) above is allowed to depend on the measure
${\mathbb Q}$ from (3)(a) and the process $\hH$ from (3)(b). It has to guarantee
the supermartingale property of $H\cdot S- (\hH\cdot S+A)$, however, for
all $H\in\AA$ simultaneously.
\item In the unconstrained case, the existence of a local-martingale
measure for $S$ suffices for property (3) in the above proposition with $A=0$ and
$\hH=0$. When the constraint set forms a convex cone, the process $A$
scales away (unlike in \cite{KarKar07} where the admissibility criterion is
different) and the existence of a local supermartingale measure  suffices.
\item  The supermartingale requirement in Proposition \ref{pro:sufficient}
(3)(c) can be weakened by imposing additional regularity
on $A$ and $\hH\cdot S$. More precisely, if $A_T$ is ${\mathbb Q}$-integrable
and $\hH\cdot S$ is a ${\mathbb Q}$-uniformly integrable martingale,
it is enough to assume that the process $H\cdot S-(\hH\cdot S+A)$
is a ${\mathbb Q}$-{\em local} supermartingale. Indeed, the (full)
${\mathbb Q}$-supermartingality will then immediately follow by the (DL) property
of its negative part.
\end{enumerate}
\end{remark}

We conclude this section with an example in a ``Brownian'' setting.

\begin{example}[It\^ o-process-driven models]
  Let us consider the standard It\^o-process setting used, e.g., in
  \cite{KarShr98}. We fix $d\in{\mathbb N}$ and let $(W_t)_{t\in [0,T]}$ be a
  $d$-dimensional Brownian motion  and $(\sF_t)_{t\in [0,T]}$ its augmented
  filtration.  The stock price dynamics are given by
 \begin{equation}
   \label{equ:ito-S}
   \begin{split}
    dS_t :=
  \mu_t dt + \sigma_t dW_t, \quad S_0:=1,\quad  t\in [0,T],
	  \end{split}
	\end{equation}
  with the
  $d$-dimensional column-vector process $(\mu_t)_{t\in [0,T]}$ and the
  $d\times d$-matrix process $(\sigma_t)_{t\in [0,T]}$ are progressively
  measurable, and such that the integrals in \eqref{equ:ito-S} are well defined.

 With no invertibility requirements imposed
  on it, $\sigma_t$ can be assumed to be a square matrix,
 i.e., that there are as many risky assets as there are independent
 Brownian motions, without loss of generality.  For later use, we define the linear-subspace-valued
 process $(I_t)_{t\in [0,T]}$ - called the \define{span process} - by
 \[ I_t := \sets{\sigma_t \nu}{\nu\in\R^d}.\]

 As far as the constraints are concerned, we fix a closed convex constraint
 map $(\kappa_t)_{t\in [0,T]}$ and  associate to it is the
 \define{recession-cone process} $(R_t)_{t\in [0,T]}$ defined by \[
 R_t:=\sets{\xi\in\R^d}{ \forall\, t>0,\ \exists\, y\in\kappa_t, \,\
 y+t\xi\in \kappa_t}.\] In words, $R_t$ contains all the directions in
 which $\kappa_t$ is unbounded. We will also need the \define{barrier-cone
 process} whose values are  the polar cones of the values of $R_t$, i.e.,
 \[ B_t:=\sets{\eta\in\R^d}{\eta^T \xi\leq 0,\text{ for all }\xi\in R_t}.\]

\medskip

\noindent  Consider now the following condition: \begin{equation} \label{equ:cond1}
 \begin{split} I_t \cap \Big( \mu_t - B_t \Big) \ne \emptyset,
 \text{ on } \Omega\times [0,T], \end{split} \end{equation} In words, either
 $\mu_t$ is contained in the image of $\sigma_t$ (the typical no-arbitrage
 requirement in the unconstrained case), or we can travel to
 $\mu_t$ from some point in the image of $\sigma_t$ using one of the
 elements of the barrier cone as a velocity vector. We note that, by choosing an appropriate constraint structure, one can, without loss of
 generality, assume that $(\sigma_t)_{t\in [0,T]}$ is everywhere invertible, and, thus, that $I_t=\R^d$. For flexibility's sake, we opt to keep both processes at the current level of generality.

 The correspondence $(t,\omega)\to I_t(\omega) \cap
 \Big (\mu_t(\omega)-B_t(\omega))\Big)$ takes values in the set of
 non-empty  closed subsets of $\R^d$. Moreover, it is
weakly measurable with
 respect to the progressive $\sigma$-algebra (see Definition
 18.1, p.~592 of \cite{AliBor06} for various measurability notions for
 correspondences). Indeed, this follows easily from
 the progressive measurability of the processes $\mu$ and $\sigma$.
Therefore, we can apply the Kuratowski-Ryll-Nardzewski Selection
Theorem (see Theorem 18.13, p.~600 in \cite{AliBor06}) which guarantees
the existence of a progressively measurable process
$(\hat{\mu}_t)_{t\in [0,T]}$ with $\hat{\mu}_t\in I_t \cap (\mu_t -
B_t)$. Then, we can pick a process $(\nu_t)_{t\in [0,T]}$ such
that $\sigma_t \nu_t = \hat{\mu}_t$. This can be done, for example, through the
(measurable) operation of choosing the unique minimal-norm solution of
a solvable linear system, i.e., by taking the Moore-Penrose inverse
(see p.~35 of \cite{BenGre03} for definitions, and Ex.~25 on p.~101 for
the statement and the proof of the so-called Tihonov-regularization
representation which can be used to deduce the aforementioned
measurability of the Moore-Penrose pseudoinversion).

Assuming that the stochastic exponential ${\mathcal E}(-\nu\cdot W)$ is
 a (true) martingale, we define the measure ${\mathbb Q}^{\nu}\sim{\mathbb P}$ by
 $\RN{{\mathbb Q}^{\nu}}{{\mathbb P}}:={\mathcal E}(-\nu\cdot W)_T$. For two processes $H,\hH\in
 \AA$, we note that finite-variation part in the semimartingale
 decomposition of the process $(H-H')\cdot S$ under the probability
 measure ${\mathbb Q}^{\nu}$ is absolutely continuous with the derivative
 given by \[ (H_t-H'_t)^T(\mu_t - \sigma_t \nu_t)=
 (H_t-H'_t)^T\beta_t.\] Since $\beta_t\in B_t$, one can find the
 ``farthest'' point in $\kappa_t$ in the direction $\beta_t$. More
 precisely, we set $\hH_t=\argmax_{h\in \kappa_t} h^T \beta_t$. Then,
 it follows that $(h-\hH_t)(\mu_t-\sigma_t \nu_t)\leq 0$, for all $h\in
 \kappa_t$.  If one could ensure that the so-constructed process
 $(\hH_t)_{t\in [0,T]}$ indeed belongs to the admissible set
 $\AA^\text{low}$ and that $\hH\cdot S$ is a ${\mathbb Q}^{\nu}$-martingale,
 part (4) of Remark \ref{rem:suff-pro-suff} would guarantee
 that the requirement (3) in
 Proposition \ref{pro:sufficient} is fulfilled in a very parsimonious way: we
 could simply take $A_t:= 0$.

 \bigskip

 Alternatively, one can exchange some of the unpleasant regularity needed for the
 above approach for the necessity of the use of a  non-trivial process $A$.
 Indeed, let the processes $\nu$ and $\beta$ be as above, and let
 $\hH\in\AA$ be such that $\hH\cdot S$ is a ${\mathbb Q}^{\nu}$-martingale;
 $\hH_t:= 0$ is always a possibility.

 We define the process $A$ as \[ A_t := \int_0^t
 \Big( \delta_{\kappa_u}(\beta_u)- \hH_u^T \beta_u \Big)\, dt,\]
 where $\delta_{\kappa_t}(\xi):=\sup_{h\in \kappa_t} h^T \xi$ is the support
 function of the constraint set $\kappa_t$. This way,
 we can fulfill the
 requirement (3) in Proposition \ref{pro:sufficient}, by checking that
 ${\mathbb E}^{{\mathbb Q}^{\nu}}[ A_T]<\infty$.

 \bigskip

 Finally, let us shortly describe a case in which no equivalent
 local-martingale measure can be found in the unconstrained version of the
 market, but one can still verify the conditions of Proposition
 \ref{pro:sufficient}. We take $d:=1$, $\sigma_t:= 1$, and
 a  progressively-measurable process $\mu_t$ such that
 \begin{enumerate}
 \item ${\mathbb E}[\int_0^T \mu_t^2\, dt]<\infty$, but
 \item ${\mathbb E}[ {\mathcal E}(-\mu\cdot
	 W)_T]<1$, i.e.,
 ${\mathcal E}(-\mu\cdot W)$ is not a true martingale.
 \end{enumerate}
An example of such a process $\mu_t$ can be based on the
 three-dimensional Bessel
process, see, e.g., Example 2.2 in \cite{Lar09} for details.
 Girsanov's theorem implies that no local-martingale measure can exist for
 $S$.
 Indeed, the only candidate fails to be a probability measure.

 On the other hand, let us choose a constant constraint set $\kappa_t:=
 [-1,1]$ and take ${\mathbb Q}:={\mathbb P}$, $\hH:= 0$ and $ A_t := \int_0^t
 \delta_{\kappa_t}(\mu_t)\, dt = \int_0^t \abs{\mu_u}\, du$. For any
 $H\in\AA$, we have \[ (H\cdot S)_t - A_t = \int_0^t \Big(H_u \mu_u -
	 \abs{\mu_u} \Big)\, du + \int_0^t H_u\, dW_u,\] a process which is clearly
 a supermartingale. Consequently, the conditions of Proposition (3) are
 satisfied.

 A more extreme version of the above can be constructed by simply taking
 $S_t:=t$ and $\kappa_t := (-\infty,1]$. The original, unconstrained,  market allows for
 (unbounded) arbitrage which cannot be implemented without violating the
 constraints.  Constraints still allow for a limited riskless gain, but
the conditions of Proposition \ref{pro:sufficient} (3) hold.

\end{example}

\subsection{The primal problem}
The investor's preferences are modeled by a function $U$ - called a
\define{utility function} - which will always be assumed to satisfy the following
assumption:
\begin{assumption}
\label{ass:U}
$U:(0,\infty)\to \R$ is a nondecreasing and concave function with the
following two properties:
\begin{equation}%
\label{equ:RAE}
    \begin{split}
\exists\,  x_0>0,\, c\in (1,2)\quad \forall\, x \geq x_0\quad
U(2x)\leq c\, U(x),\\
\lim_{x\searrow 0} U'^{+}(x)=\infty,\text{ where $U'^{+}$ denotes the right derivative.}
    \end{split}
\end{equation}
\end{assumption}
\begin{remark}
The first part of the condition \eqref{equ:RAE} is a derivative-free restatement of the
notion of the reasonable asymptotic elasticity of \cite{KraSch99} (for
details, see Lemma 6.3(i) in \cite{KraSch99}) and it restricts the rate
of growth of $U$ in the neighborhood of $+\infty$. In
particular, \eqref{equ:RAE} implies that the Inada condition at $+\infty$, namely, $\lim_{x\to\infty}
U'(x)=0$,  is satisfied if $U'$ is interpreted as either the left or the right
derivative.
\end{remark}
To simplify the notation later on, we extend the definition of $U$ by
semicontinuity to $[0,\infty)$ by setting $U(0)= \inf_{\xi>0} U(\xi)$
and, further, to $\R$, by $U(x)=-\infty$, for $x<0$.  The
\define{(primal) value function} $u:\R\to [-\infty,\infty]$ of the
utility-maximization problem, parametrized by  the investor's initial
wealth $x\in\R$, is then defined by
\begin{align}\label{def:u}
  u(x):= \sup_{X\in\sK}\; {\mathbb E}[U(x+X)],
\end{align}
where we use the convention that for $\xi\in\el^0$, one has ${\mathbb E}[\xi]
= -\infty$ whenever ${\mathbb E}[\xi^-]=\infty$, even if ${\mathbb E}[\xi^+]=\infty$.

The monotonicity of $U$ and the fact that $U(x)=-\infty$ for
$x<0$, imply  \[u(x)=\sup_{X\in (\sK-\el^0_+)} {\mathbb E}[
U(x+X)]=\sup_{f\in \sC(x)} {\mathbb E}[ U(f)],\] where
$\sup\,\emptyset:=-\infty$. The monotone convergence theorem guarantees
that
\[ u(x)=\sup_{f\in \sC} \mathbb{U}(x+f),\] where
the map $\mathbb{U}:\el^{\infty}\to [-\infty,\infty)$ is
a shorthand for $f\mapsto {\mathbb E}[U(f)]$ with $U$ regarded as defined on
$(-\infty,\infty)$.
\subsection{The dual problem}
To introduce the dual optimization problem we first need to recall the
notion of a support function. Let $\sC$ be as in (\ref{def:C}) above
and let $\sP$ denote the set of all (countably additive) probability measures  on
$(\Omega,{\mathcal F})$ which are absolutely continuous with respect to ${\mathbb P}$.
The \define{support function $\alpha_{\sC}$ of $\sC$} is defined by
\begin{equation}
   \label{equ:support}

   \begin{split}
\sP\ni {\mathbb Q}\to \alpha_{\sC}({\mathbb Q}):=\sup_{f\in \sC}\; {\mathbb E}^{{\mathbb Q}}[f]\in (-\infty,\infty].
   \end{split}
\end{equation}

The optimization problem (for now only formally) dual to the primal
utility-maximization problem \eqref{def:u} above is defined by
its value function $v: [0,\infty)\to [-\infty,\infty]$:
\begin{align}\label{def:dual}
  v(y):= \inf_{{\mathbb Q}\in\sP}\Big( {\mathbb E}[ V(y \tRN{{\mathbb Q}}{{\mathbb P}})] +
  y\alpha_{\sC}({\mathbb Q})\Big),
\end{align}
 where
$V(y) := \sup_{x\in\R} \big( U(x) - xy \big)\in
(-\infty,\infty]$,  $y\in\R$, is the Fenchel-Legendre
transform of $-U(-\cdot)$.

\subsection{Main result}
The following theorem extends  some of the  main existence results in
\cite{CviSchWan01}, \cite{KraSch99} and \cite{KraSch03} to the
constrained case and shows that countably-additive measures suffice to
describe the dual value function.

\begin{theorem}\label{thm:main} Let $u$ and $v$ be defined by
  (\ref{def:u}) and (\ref{def:dual}) respectively, and assume
  that $u(x)\in\R$ for some $x\in \R$. Under the
  Assumptions \ref{ass:C} and \ref{ass:U},
with $\underline{x}:=\inf\sets{x\in\R}{u(x)>-\infty}$,
the following assertions
  hold:
  \begin{enumerate}
  \item The function $u$ is concave, upper semicontinuous and
  nondecreasing, while  $v$ is  convex and lower semicontinuous.
  \item We have $\underline{x}=-\inf_{{\mathbb Q}\in\sP} \alpha_{\sC}({\mathbb Q})$, where $\alpha_{\sC}$ is defined in
  \eqref{equ:support}. Furthermore,  $u(x)\in\R$ for
  $x\in(\underline{x},\infty)$ and $u(x)=-\infty$ for $x\in(-\infty,\underline{x})$.
\item For each $x\in\R$ with $u(x)\in\R$ (and, in particular,
  for $x>\underline{x}$), there exists $H^{(x)}\in \sA$ such that
    \[u(x)={\mathbb E}\left[ U\left(x+\int_0^T H^{(x)}_u\, dS_u\right)\right].\]
\item The following conjugacy relations hold:
\begin{align}
     v(y) &= \sup_{x\in\R} \Big( u(x) - xy\Big),\quad y\in\R \label{eq:u_conjugate},\\
     u(x) &= \inf_{y\in [0,\infty)} \Big( v(y) + xy\Big),\quad
     x\in\R.\label{eq:v_conjugate}
\end{align}

\end{enumerate}
\end{theorem}

\pagebreak
\begin{remark}\
\begin{enumerate}
  \item Theorem
  \ref{thm:main} and Example \ref{ex:ran_endow} show that Theorem 2.2\,(iv) in
  \cite{KraSch99} indeed carries over to the random-endowment setting of
  \cite{CviSchWan01} also when the utility function $U$ is non-smooth.
  Theorem 3.1\,(ii) in \cite{CviSchWan01} provides a link between the
  primal and dual optimizers. As we discuss in the next section, we can
  only guarantee the existence of a finitely additive dual minimizer
  $\hat{{\mathbb Q}}_y\in{\mathrm{ba}}$ and, in general, we will not have
  $\hat{{\mathbb Q}}_y\in\sP$.  Under the additional assumption that $U$ is
  strictly concave, the dual function $V$ is differentiable by
  Theorem 26.3 in \cite{Roc70}.  We can then  extend Theorem 3.1\,(ii) in
  \cite{CviSchWan01} to our setting by using the Yosida-Hewitt
  decomposition of ${\mathbb Q}\in{\mathrm{ba}}$ into its regular part ${\mathbb Q}^r\in\el^1_+$
  and its purely singular part ${\mathbb Q}^s$ as follows. For
  $x>\sup_{{\mathbb Q}\in\sP} -\alpha_{\sC}({\mathbb Q})$ we have the relation
  \begin{align}\label{eq:primal_dual} x + \int_0^T H^{(x)}_u\, dS_u = -
  V'(\hat{y} \frac{d\hat{{\mathbb Q}}_{\hat{y}}^r}{d{\mathbb P}}), \quad {\mathbb P}\text{-a.s.,}
  \end{align} where
  $\hat{y}$ attains the infimum in \eqref{eq:u_conjugate}, and
  $\hat{{\mathbb Q}}_{\hat{y}}^r$ denotes the regular part of $\hat{{\mathbb Q}}_{\hat{y}}$,
  a minimizer in the generalized dual problem $v^\text{ba}$ (see Section
	  3.1 for details). By using the positive homogeneity of the support
  function $\alpha_\sC$, the proof of \eqref{eq:primal_dual} is a
  straightforward application of the ideas in \cite{CviSchWan01}.

	\item When $U$ is not necessarily strictly concave, \cite{BouTouZeg04}
	  and later \cite{WesZhe09} establish the validity of
	  \eqref{eq:primal_dual} in the setting of Example \ref{ex:ran_endow}
	  when $V'$ is replaced by the $V$'s subdifferential $\partial V$.
	  However, as discussed in their Remark 3.9.3, the authors of
	  \cite{BouTouZeg04} assume a specific relationship between the domain
	  of $U$ and the norm $||{\mathcal E}||_{\el^{\infty}(\sF_T)}$,
	  which makes it difficult to compare their setting to
	  ours.  Finally, we mention \cite{WesZhe11} who illustrate a possible pitfall related to using
	  $\partial V$ instead of $V'$ in \eqref{eq:primal_dual} when $U$ is not strictly
	  concave.
\end{enumerate}
\end{remark}

\section{Proofs}

\subsection{A relaxation of the dual problem}
We first note that $\alpha_{\sC}$ naturally extends from $\sP$ to the space
${\mathrm{ba}}$ by replacing the expectation ${\mathbb E}^{{\mathbb Q}}[f]$ by the value
$\scl{{\mathbb Q}}{f}$ of the dual pairing in (\ref{equ:support}).  With such
an extended domain, $\alpha_{\sC}$ coincides with the convex
$(\el^{\infty},{\mathrm{ba}})$-conjugate $(\cind{\sC})^*$ of the convex indicator $\cind{\sC}$.  It
follows, in particular, that $\alpha_{\sC}$ is convex and $\sigma(\ba,\linf)$-lower
semicontinuous.

\pagebreak
To extend the dual value function, we
follow \cite{Zit05} and define the  map
$\mathbb{V}:{\mathrm{ba}} \to (-\infty,\infty]$ of $\mathbb{U}$ by
\begin{equation}
\label{gen:V}

   \begin{split}
\mathbb{V}({\mathbb Q}) := \sup_{f\in\el^{\infty}}\Big(
 \mathbb{U}(f) - \scl{{\mathbb Q}}{f} \Big),\quad\text{ for }{\mathbb Q}\in{\mathrm{ba}}.
   \end{split}
\end{equation}
We note that $\mathbb{V}=\hat{\UU}^*$, for the $(\el^{\infty},{\mathrm{ba}})$-duality, where
$\hat{\UU}(f)=-\mathbb{U}(-f)$.  A minimal modification of Lemma 2.1, p.~138, in
\cite{OweZit09} produces the following representation
\begin{equation}
   \label{equ:V-reg}

   \begin{split}
\mathbb{V}({\mathbb Q})=\begin{cases}
 {\mathbb E}[V(\tRN{{\mathbb Q}^r}{{\mathbb P}})],& {\mathbb Q}\in{\mathrm{ba}}_+,\\
\infty, & {\mathbb Q}\not\in{\mathrm{ba}}_+.
\end{cases}
   \end{split}
\end{equation}
As mentioned in the introduction, ${\mathbb Q}^r$ denotes the regular
part in the Yosida-Hewitt decomposition ${\mathbb Q}= {\mathbb Q}^r + {\mathbb Q}^s$.

With $\mathbb{V}$ and $\alpha_{\sC}$ extended as above, a relaxed version of
the dual value function can be posed over the $y$-sphere in ${\mathrm{ba}}$:
\[ v^{\ba}(y) := \inf_{{\mathbb Q}\in \sba_+(y)}\Big( \mathbb{V}({\mathbb Q}) + \alpha_{\sC}({\mathbb Q})\Big),\quad
\text{ for }y\geq 0.\] Since $y\sP$ can be identified with $\sl_+(y)$,
which, in turn, admits a natural embedding into $\sba_+(y)$, it is clear
that $v^{\ba}(y)\leq v(y)$. It is the equality between the two functions
(as demonstrated in Proposition \ref{pro:v-vba} below) that will be
one of the major steps in the proof of our main Theorem
\ref{thm:main}. Unfortunately, it is not true in general that the
involved quantities are $\sigma(\ba,\linf)$-upper semicontinuous so this equality
cannot be deduced from the $\sigma(\ba,\linf)$-density of $\sl(y)$ in $S^{\ba}(y)$.

Some of the advantages that working with $v^{\ba}$ affords over $v$ are evident from the
following result which follows directly from the $\sigma(\ba,\linf)$-com\-pact\-ness
of $\sba_+(y)$ (the Banach-Alaoglu Theorem) and the $\sigma(\ba,\linf)$-lower
semicontinuity of $\mathbb{V}+\alpha_{\sC}$.
\begin{proposition}
\label{pro:first-uv}

If $\sC\ne\emptyset$ and Assumption \ref{ass:U} holds,

$v^{\ba}(y)$ admits a minimizer for each $y> 0$. More precisely,
   there exists $\hat{\QQ}^{(y)}\in\sba_+(y)$ such that
  $v^{\ba}(y)=\mathbb{V}(\hat{\QQ}^{(y)})+\alpha_{\sC}(\hat{\QQ}^{(y)})$.
\end{proposition}

\subsection{Conjugacy of value functions}
\begin{proposition}
\label{prop:conjugates}

Suppose that Assumptions \ref{ass:C} and \ref{ass:U} hold and that
$u(x) \in\R$ for some
$x\in\R$. Then
\begin{enumerate}
\item $v^{\ba}(y)=\sup_{x\in\R} \big( u(x)-xy \big)$, for all $y\in\R$,
  and
\item there exists $y>0$ such
  that $v^{\ba}(y)<\infty$.
\end{enumerate}
\end{proposition}
\pagebreak
\begin{proof}\
\begin{enumerate}
\item By the Banach-Alaoglu theorem, $\sba_+(y)$ is $\sigma(\ba,\linf)$-compact for
  any $y\geq 0$. Moreover, the Lagrangian
  $$L({\mathbb Q},(f,g)):=\mathbb{U}(f)-\scl{{\mathbb Q}}{f-g}$$
\begin{enumerate}
\item is concave in $(f,g)$ on  $\el^{\infty}\times\el^{\infty}$, and
\item convex, and $\sigma(\ba,\linf)$-lower semicontinuous in ${\mathbb Q}$ on
  ${\mathrm{ba}}$.
\end{enumerate}
Therefore, the Minimax Theorem (see \cite{Sio58}) can be used
  to interchange $\inf$ and $\sup$ in (\ref{equ:chain}) below. Also,
  let us note that for $h\in\el^{\infty}$ and $y\geq 0$, we have
  $$\sup_{{\mathbb Q}\in\sba_+(y)} \scl{{\mathbb Q}}{h}=y\esssup h.$$
  It follows that
\begin{equation}%
\label{equ:chain}
    \begin{split}
      v^{\ba}(y) &=\inf_{{\mathbb Q}\in\sba_+(y)} \Big( \mathbb{V}({\mathbb Q})+\sup_{g\in\sC}\,
      \scl{{\mathbb Q}}{g} \Big)\\ & =\inf_{{\mathbb Q}\in\sba_+(y)}\sup_{f\in\el^{\infty}}
      \Big( \mathbb{U}(f)-\scl{{\mathbb Q}}{f}+\sup_{g\in\sC} \,\scl{{\mathbb Q}}{g} \Big) \\
      &=\inf_{{\mathbb Q}\in\sba_+(y)}\sup_{(f,g)\in\el^{\infty}\times\sC} \Big(
      \mathbb{U}(f)-\scl{{\mathbb Q}}{f-g} \Big)\\ & =\sup_{(f,g)\in\el^{\infty}\times\sC}
      \inf_{{\mathbb Q}\in\sba_+(y)} \Big( \mathbb{U}(f)-\scl{{\mathbb Q}}{f-g}\Big) \\ &=
      \sup_{(f,g)\in\el^{\infty}\times\sC} \Big( \mathbb{U}(f)-y \esssup(f-g)\Big)
\end{split}
\end{equation}
We can split the last supremum according to the value of $\esssup
(f-g)$ and use the monotonicity of $\mathbb{U}$ to obtain
\begin{equation}
\nonumber
   \begin{split}
 v^{\ba}(y)&=\sup_{x\in\R}\  \sup_{g\in\sC, f\in\el^{\infty} f\leq g+x}\
\Big(\mathbb{U}(f)-yx \Big)\\&=\sup_{x\in\R} \;\sup_{g\in\sC} \;\Big(
\mathbb{U}(x+g)-yx\Big)=\sup_{x\in\R} \big( u(x)-xy \big).
   \end{split}
\end{equation}
\item This is a direct consequence of the standing assumption that $u$
  is proper and fact that properness is preserved under conjugacy (see
  Theorem 12.2, p.~104 in \cite{Roc70}).
\end{enumerate}
\end{proof}
\subsection{Existence in the primal problem}
We start with a variant of the argument developed in the proof of
Theorem 4.2 in \cite{DelSch94}, adjusted to our case of convex
constraints.

\begin{lemma}
  Under Assumption \ref{ass:C},
  the set $\sC$ is nonempty
  and $\sigma(\el^{\infty},\el^1)$-closed.
\end{lemma}

\begin{proof}
  Let $x\in\R$ be such that $\sC(x)$ is nonempty. Then, there
  exists $X\in\sK$ such that $x+X\geq 0$, ${\mathbb P}$-a.s., and, so, the constant random
  variable $-x$ belongs to $\sC$, proving that $\sC$ is nonempty.

To prove closedness, for $M>0$ we define the closed $\el^{\infty}$-ball $B^{\el^{\infty}}(M)=
\{f\in\el^{\infty} : ||f||_\infty \le M\}$. By a version of Grothendieck's
lemma (see, e.g., Theorem 5.1 in \cite{FolKra97}) and the convexity of
$\sC$ the claim is equivalent to showing that $\sC \cap B^{\el^{\infty}}(M)$
is closed in probability for all $M>0$. So let $(f_n)_{n\in{\mathbb N}} \subset
\sC \cap B^{\el^{\infty}}(M)$ converge to $f_0$ in probability. It is clear
that $f_0 \in B^{\el^{\infty}}(M)$ and we only need to show that $f_0\in\sC$.  We
have $f_n + M\ge 0$, hence, $f_n +M\in \sC(M)$. By Assumption \ref{ass:C}, the set $\sC(M)$
is closed in probability which ensures that $f_0+M\in\sC(M)$, i.e.,
there exists $H\in\sA$ such that $f_0 +M\le M + (H \cdot
S)_T$. Therefore, $f_0\in\sC$.
\end{proof}

 By using the extended definition $\scl{{\mathbb Q}}{f} := \lim_{n\to\infty}
\;\scl{{\mathbb Q}}{f\land n}$ for ${\mathbb Q}\in\sP$ and $f\in\mathbb{L}^0_+$, we have
the following characterization of the sets $\sC$ and $\sC(x)$.

\begin{corollary}
\label{cor:hedge}
Under Assumption \ref{ass:C}:
\begin{enumerate}
\item
$f\in\el^{\infty}$ belongs to $\sC$
if and only if $\scl{{\mathbb Q}}{f}\leq \alpha_{\sC}({\mathbb Q})$, for all ${\mathbb Q}\in\sP$,

\item $f\in \mathbb{L}^0_+$ belongs to $\sC(x)$
if and only if $\scl{{\mathbb Q}}{f}\leq x+\alpha_{\sC}({\mathbb Q})$, for all ${\mathbb Q}\in\sP$.

\end{enumerate}
\end{corollary}
\begin{proof}
  Closedness of $\sC$ implies that the convex function $\chi_{\sC}$ is
  lower semicontinuous for the
  $\sigma(\el^{\infty},\el^1)$-topology. Therefore, $\chi_{\sC}$ is its own
  $\sigma(\el^{\infty},\el^1)$-biconjugate, and consequently,
$\chi_{\sC}(f)=\sup_{{\mathbb Q}\in\sP}\left( \scl{{\mathbb Q}}{f}-\alpha_{\sC}({\mathbb Q})\right)$ which proves (1).

 For (2) we pick $f\in\sC(x)$ and $n\in{\mathbb N}$, and note that for some
$H\in\sA$ we have
$$(f-x)\land
n\le (H\cdot S)_T\land n\in\sC.$$ Therefore, for ${\mathbb Q}\in\sP$, Fatou's
lemma implies that $$ \scl{{\mathbb Q}}{f-x}\le \liminf_{n\to\infty}
\;\scl{{\mathbb Q}}{(f-x)\land n} \le \alpha_{\sC}({\mathbb Q}).  $$

Conversely, let $f\in \mathbb{L}^0_+$ be such that
$\scl{{\mathbb Q}}{f-x}\le \alpha_{\sC}({\mathbb Q})$ for all ${\mathbb Q}\in\sP$. Then for $n\in{\mathbb N}$ we also
have $\scl{{\mathbb Q}}{(f-x)\land n} \le \alpha_{\sC}({\mathbb Q})$. Hence, by (1),  $(f-x)\land n\in\sC$
 and, so, $(f-x)\land n +
x\in\sC(x)$, for all $n\in{\mathbb N}$. The claim now follows directly from  the closedness of $\sC(x)$
in probability.
   \end{proof}

\begin{lemma}
\label{lem:bd-tU}
Under Assumptions \ref{ass:C} and \ref{ass:U},
we have
$\sup_{f\in \sC(x)} {\mathbb E}[U^+(f)]<\infty$, whenever
$u(x)\in\R$.\end{lemma}
\begin{proof}
We define the constant $x'=\inf\{x>0: U(x) \ge 0\}$. If $x' = \infty$ there
is nothing to prove and so, in what follows, we assume that
$x'\in[0,\infty)$.  By Proposition \ref{pro:first-uv}, part (2), there exist $y>0$ and
  ${\mathbb Q}\in\sba_+(y)$ such that $\mathbb{V}({\mathbb Q})<\infty$ and $\alpha_{\sC}({\mathbb Q})<\infty$.   Since $\mathbb{U}(f)\leq \mathbb{V}({\mathbb Q})+\scl{{\mathbb Q}}{f}$ for each
  $f\in\el^{\infty}$, in particular, for $f\in (x+\sC)\cap \el^{\infty}_+$, we have
  \begin{equation}
\nonumber
   \begin{split}
     {\mathbb E}[ U^+(f)]&\le{\mathbb E}[U(f \inds{f\geq x'}+x'\inds{f<x'})]\\
&\leq
     \mathbb{V}({\mathbb Q})+\scl{{\mathbb Q}}{f}+
\scl{{\mathbb Q}}{x'}\\
&\leq
     \mathbb{V}({\mathbb Q})+\alpha_{\sC}({\mathbb Q})+x'y,
   \end{split}
\end{equation}
which is finite and independent of the choice of $f$.
\end{proof}

Let us choose and fix  constants $x_0>0$ and $c \in (1,2)$ as in Assumption
\ref{ass:U}.  For $h\in \el^{\infty}$ with $h\ge x_0$, we then have
$U(2h)\leq c U(h)$; iterating this inequality produces \begin{equation}
\label{equ:aaa} \begin{split} {\mathbb E}[ U(2^mh)] \leq c^m{\mathbb E}[ U(h)],\quad
\text{ for all } m\in{\mathbb N},\, h\in x_0+\el^{\infty}_+.  \end{split} \end{equation}
\begin{proposition}
\label{prop:primal_existence}
   Under Assumptions \ref{ass:C} and \ref{ass:U},
for each $x\in\R$ with  $u(x)\in\R$
  there exists $f^{(x)}\in\sC(x)$ such that $u(x)=\mathbb{U}(f^{(x)})$
\end{proposition}

\begin{proof}
The function $u$ is clearly concave, so the existence of $x\in\R$ such that
$u(x)<\infty$ implies that it is proper, i.e., that $u(x)<\infty$, for all
$x$. We pick $x\in\R$ with $u(x)<\infty$ and  let
$\seq{f}\subset \sC(x)$ be a maximizing sequence, i.e., a sequence in
$\sC(x)$ such that $\mathbb{U}(f_n)\to u(x)$. Since $\sC(x)$ is bounded in
probability, we may find a sequence $\seq{g}$, of convex combinations
$g_n\in\operatorname{conv}(f_n,f_{n+1},\dots)$, which converges in probability to some
$f^{(x)}\in\el^0_+$. The concavity of $\mathbb{U}$ implies that $g_n$ is also
a maximizing sequence. Furthermore, $f^{(x)}\in\sC(x)$ since $\sC(x)$ is closed
in probability.

  To show that $f^{(x)}$ is indeed a maximizer, we use
  Fatou's lemma to conclude that ${\mathbb E}[ - U^-(f^{(x)}) ] \geq \limsup_n
  {\mathbb E}[ - U^-(g_n)]$, so that it is enough to show that ${\mathbb E}[
  U^+(g_n)]\to {\mathbb E}[ U^+(f^{(x)})]$. This will follow once we show that the
  sequence
  $\sq{U^+(g_n)}$ is uniformly integrable.

  We start by defining the nonnegative constant $$ x' := \inf\big\{ x>x_0:
  U(x) >0\big\}.  $$ If $x'=\infty$ there is nothing to prove and so we
  assume that $x'\in[0,\infty)$. We argue by contradiction and assume that
  $\sq{U^+(g_n)}$ is not uniformly integrable. Lemma \ref{lem:bd-tU}
  ensures that $\sq{U^+(g_n)}$ is bounded in $\el^1$. Therefore, Corollary
  A.1.1 in \cite{Pha09} produces a subsequence, still labeled
  $\sq{U^+(g_n)}$, $\varepsilon>0$, and a pairwise disjoint sequence of events
  $\seq{A}$ such that \[ {\mathbb E}[ U^+(g_n) \ind{A_n}]\geq 2 \varepsilon>0\text{
	  for all }n\in{\mathbb N}.\]
  The monotone convergence theorem allows us to exchange $\varepsilon$
  in utility for boundedness and obtain the existence of a sequence
  $\seq{r}\subseteq \el^{\infty}_+\cap \sC(x)$ such that $r_n\leq g_n$ and ${\mathbb E}[
  U^+(r_n)\ind{A_n}]\geq \varepsilon$, for all $n\in{\mathbb N}$.   Let the sequence
  $\seq{h}$ of bounded random variables be defined by \[h_n:=x' +
  \sum_{k=1}^n r_k \ind{A_k}\in x'+\el^{\infty}_+\subseteq x_0+\el^{\infty}_+.\] For
  ${\mathbb Q}\in \sP$, we have $\scl{{\mathbb Q}}{h_n-x'-nx}=\sum_{k=1}^n
  \scl{{\mathbb Q}}{r_k\ind{A_k}-x} \leq n \alpha_{\sC}({\mathbb Q})$, so that $\tfrac{1}{n}h_n\in
  \sC(x+\frac1n x')\subseteq \sC(x+ x')$ for all $n\in{\mathbb N}$.  On the other
  hand, since $U(h_n)=U^+(h_n)$, we have \[{\mathbb E}[U(h_n)]\ge\sum_{k=1}^n {\mathbb E}[
  U^+(r_k)\ind{A_k}]\geq n \varepsilon. \] Using \eqref{equ:aaa} with $n=2^m$ for
  $m\in{\mathbb N}$ produces \begin{equation} \nonumber \begin{split} 2^m  \varepsilon &
  \leq {\mathbb E}[ U(h_{2^m})] \leq {\mathbb E}[ U(2^m x' \vee h_{2^m} )] \leq c^m {\mathbb E}[
  U(x' \vee \tfrac{1}{2^m} h_{2^m})]\\ &\leq c^m {\mathbb E}[U(x'+\tfrac{1}{2^m}
  h_{2^m})]\leq c^m u(x+2x'), \end{split} \end{equation} which, thanks
  to the fact that $c<2$, implies that $u(x+2x')\geq \varepsilon \lim_m
  (2/c)^m=\infty$, a statement in contradiction with the fact that $u$ is
  $[-\infty,\infty)$-valued everywhere.
  \end{proof}

  \begin{proposition} \label{pro:usc} Under Assumptions \ref{ass:C} and
  \ref{ass:U}, the primal value function $u$ is upper-semicontinuous.\end{proposition}

\begin{proof} Thanks to $u$'s concavity and monotonicity, it will be enough to
show that $u(\underline{x})\geq \lim_n u(x_n)$ for each sequence $x_n\searrow
\underline{x}=\inf\sets{x\in\R}{u(x)>-\infty}$ with $x_n>\underline{x}$. We pick such a
sequence $\seq{x}$ and use Proposition \ref{prop:primal_existence} to
construct a sequence $\seq{f}$ of random variables such that $f_n\in
\sC(x_n)$ and $u(x_n)=\mathbb{U}(f_n)$. By the same argument as in the first
paragraph of the proof of Proposition \ref{prop:primal_existence}, we can
construct a limit $g\in \cap_n \sC(x_n)$ of a sequence of forward convex
combinations, i.e., $g_n := \sum_k \alpha_k^n f_k$ for positive constants $\alpha_k^n$ summing (over $k$) to one. By Fatou's lemma and Corollary \ref{cor:hedge}(2) we have for ${\mathbb Q}\in\sP$
$$
\langle{\mathbb Q},g\rangle \le \liminf_{n\to\infty} \;\sum_k \alpha_k^n \langle{\mathbb Q},f_k\rangle \le \liminf_{n\to\infty}\;\sum_k \alpha_k^n \Big( x_k +\alpha_\sC({\mathbb Q})\Big) = \underline{x} +\alpha_\sC({\mathbb Q}),
$$
since $x_n \searrow \underline{x}$. Corollary \ref{cor:hedge}(2) implies that $g\in \sC(\underline{x})$ and so
$u(\underline{x})\geq \mathbb{U}(g)$.  Using the ideas of the second paragraph of the proof
of Proposition \ref{prop:primal_existence}, we can establish the uniform
integrability of the sequence $\sq{U^+(g_n)}$, and conclude that
$u(\underline{x})\geq \mathbb{U}(g) \geq \lim_n u(x_n)$.  \end{proof}

\begin{remark}
The upper-semicontinuity of the value function of a utility maximization
problem has been established in the
dissertation \cite{Sio10} of Siorpaes, in the setting of
 utility maximization with random endowment of \cite{HugKra04} and
applies jointly to the initial wealth $x$ and the initial quantity of the
random endowment. The proof of   Proposition \ref{pro:usc} uses similar ideas
and generalizes the results of Siorpaes to constrained markets, but
considers only
the initial-wealth variable $x$.
\end{remark}

\subsection{No need to relax $v$}
We start with an observation about continuity  of
the upper-hedging-price map.
\begin{lemma}
\label{lem:uhp-lsc}
 Under the Assumption \ref{ass:C},
the upper-hedging-price map
\[ \el^{\infty}\ni f\mapsto \rho(f):=\inf\sets{x\in\R}{f\in x+\sC},\] is
convex, proper, and lower $\sigma(\el^{\infty},\el^1)$-semicontinuous. Moreover,
 there exist a
constant $M>0$ such that
\begin{equation}
   \label{equ:bd-rho}

   \begin{split}
\abs{\rho(f)} \leq M+\norm{f},\text{ for all
}f\in\el^{\infty}.
   \end{split}
\end{equation}

\end{lemma} \begin{proof}
  Thanks to Assumption
  \ref{ass:C}, there exists a constant $M>0$ such that  $\sC$ contains
  the set $-M-\el^{\infty}_+$.  Therefore, $\rho(f)\leq
  \norm{f}+M$, for any $f\in\el^{\infty}$.  To obtain the full bound
  (\ref{equ:bd-rho}), we assume, to the contrary, that there exists a
  sequence $\seq{f}$ in $\el^{\infty}$ such that \[ \rho(f_n) < -
  \norm{f_n}-n,\text{ for all }n\in{\mathbb N}.\] Therefore, $
  f_n+\norm{f_n}+n\in{\mathcal C}$ for each $n\in{\mathbb N}$, and, consequently, $n\in\sC$,
  for each $n\in{\mathbb N}$. This is, however, in contradiction with the Assumption
  \ref{ass:C}.

  Since properness of $\rho$ follows from the bounds in
  (\ref{equ:bd-rho}), and convexity follows directly from the
  definition, it remains to show that $\rho$ is
  $\sigma(\el^{\infty},\el^1)$-lower semicontinuous, i.e., that its epigraph
  \[ \epi \rho=\sets{(f,x)\in \el^{\infty}\times \R}{ \rho(f)\leq x}\] is
  closed.   This follows from the fact that $\epi\rho=\sets{ (f,x) }{
    f-x\in \sC}$ is the inverse image of the closed set $\sC$ under the
  continuous map $(f,x)\mapsto f-x$ from $\el^{\infty}\times \R$ to $\el^{\infty}$.
\end{proof}
\begin{lemma} \label{lem:Zalinescu}  Under the Assumption \ref{ass:C},  for each $y\geq 0$,
we have
$$
\inf_{{\mathbb Q}\in\sba_+(y)} \alpha_{\sC}({\mathbb Q})=\inf_{{\mathbb Q}\in\sl_+(y)} \alpha_{\sC}({\mathbb Q}).
$$
\end{lemma}
\begin{proof}
  For simplicity, we assume that $y=1$.
 The set $\sba_+(1)$ is $\sigma(\ba,\linf)$-compact by the
  Banach-Alaoglu theorem, so we can use the Minimax Theorem to conclude that
  \begin{equation}%
    \label{equ:mmax}
    \begin{split}
      \inf_{{\mathbb Q}\in\sba_+(1)} \alpha_{\sC}({\mathbb Q})&= \inf_{{\mathbb Q}\in\sba_+(1)} \sup_{f\in \sC}\,
      \scl{{\mathbb Q}}{f} = \sup_{f\in \sC} \inf_{{\mathbb Q}\in\sba_+(1)} \scl{{\mathbb Q}}{f} =
      \sup_{f\in\sC} \essinf f.
    \end{split}
\end{equation}
Now we focus on $\inf_{{\mathbb Q}\in\sl_+(1)} \alpha_{\sC}({\mathbb Q})$.  Since
$\alpha_{\sC}({\mathbb Q})=\infty$, for  ${\mathbb Q}\not\in\el^1\setminus\el^1_+$,
we have
\[ \inf_{{\mathbb Q}\in\sl_+(1)} \alpha_{\sC}({\mathbb Q})=\inf_{{\mathbb Q}\in\sS} \alpha_{\sC}({\mathbb Q}),\] where
$\sS:=\sets{{\mathbb Q}\in\el^1}{\scl{{\mathbb Q}}{1}=1}$.  Throughout the rest of this
proof, we work with the duality between the spaces $\el^{\infty}$ and
$\el^1$, and all notions of continuity and conjugation should be
understood with respect to this duality and the corresponding weak-$\ast$ and weak topologies.

We define the map $\gamma:\el^{\infty}\to \R\cup\set{+\infty}$ by
\[ \gamma(f):=
\begin{cases}
x, & f=x,\text{ a.s., for }x\in\R,\\
+\infty, & \text{otherwise}.
\end{cases}
\]
The convex conjugate $\gamma^*$ of $\gamma$ is the indicator $\cind{\sS}$ of $\sS$:
\[ \gamma^*({\mathbb Q}):=\sup_{f\in\el^{\infty}} \Big( \scl{{\mathbb Q}}{f} - \gamma(f) \Big) =
\sup_{x\in\R} \,x\Big( \scl{{\mathbb Q}}{1}-1 \Big) = \cind{\sS}({\mathbb Q}),\quad {\mathbb Q}\in\el^1.\]
Next, we define the infimal convolution $\cind{\sC}\Box \gamma$ of $\cind{\sC}$ and
$\gamma$ by
\[(\cind{\sC} \Box \gamma)(f) := \inf_{ g\in\el^{\infty}} \Big(
\cind{\sC}(f-g)+\gamma(g) \Big),\quad f\in\el^{\infty}.\]
Since $\gamma$ is only finite on constants, we have
\[ (\cind{\sC}\Box\gamma)(f)=\inf_{x\in\R}\, \big( \cind{\sC}(f-x)+x
\big)=\inf\sets{x\in\R}{ f\in x+{\mathcal C}}=\rho(f),\quad f\in\el^{\infty}.
\]
It follows from Lemma \ref{lem:uhp-lsc} that $\cind{\sC}\Box
\gamma$ is convex, proper, and lsc. Consequently, we have
\[ (\cind{\sC}\Box \gamma)^{**}(0)= (\cind{\sC}\Box\gamma)(0)=\rho(0)=-\sup_{h\in\sC}\,\essinf h.
\]
On the other hand, by Theorem 2.3.1(ix), p.~76 in \cite{Zal02} we have
$(\cind{\sC}\Box\gamma)^* = \cind{\sC}^*+\gamma^*= \alpha_{\sC}+\cind{\sS}$ and so
\begin{equation}
\nonumber
   \begin{split}
     (\cind{\sC}\Box\gamma)^{**}(0) &=\sup_{{\mathbb Q}\in\el^1} \Big(
     \scl{{\mathbb Q}}{0}-(\cind{\sC}\Box\gamma)^*({\mathbb Q}) \Big)\\ &=\sup_{{\mathbb Q}\in\el^1}
     -\Big(\alpha_{\sC}({\mathbb Q}) +\cind{\sS}({\mathbb Q}) \Big)=-\inf_{{\mathbb Q}\in\sS} \alpha_{\sC}({\mathbb Q}).
   \end{split}
\end{equation}

A comparison with (\ref{equ:mmax}) yields the statement.
\end{proof}
To prove Lemma \ref{lem:infs-equal}, we need a result from
\cite{KraSch99}. We state a rephrased version whose proof can be read
off the proof of Proposition 3.2, p.~924 of \cite{KraSch99} (in
particular, no additional smoothness assumptions on $V$ are required).
\begin{lemma}[Kramkov and Schachermayer] \label{lab:KraSch}
 Let ${\mathcal M}\subseteq{\mathcal D}$ be
   bounded subsets of $\el^1_+$ such that:
   \begin{enumerate}
   \item the mapping ${\mathcal D}\ni h \to {\mathbb E}[V(h)]$
     attains its minimum at some $\hat{h}\in{\mathcal D}$,
   \item ${\mathcal M}$ is closed under countable convex
     combinations, and
   \item there exists a sequence $\seq{h}\subseteq {\mathcal M}$
     which converges to $\hat{h}$ in probability.
   \end{enumerate}
   Then,  under Assumption
  \ref{ass:U}, we have
$\inf_{h\in{\mathcal D}} {\mathbb E}[V(h)]=\inf_{h\in{\mathcal M}} {\mathbb E}[V(h)]$.
 \end{lemma}

\begin{lemma}
\label{lem:infs-equal}
 Under Assumptions \ref{ass:C} and
  \ref{ass:U},  let $S\subseteq\sba_+(y)$ be of the form
$S=\sets{{\mathbb Q}\in\sba_+(y)}{\alpha_{\sC}({\mathbb Q})\leq M}$, for some constant
$M\in\R$. Then, provided that $S\cap\el^1\ne\emptyset$, we have
\begin{equation}
   \label{equ:infs-equal}

   \begin{split}
  \inf_{{\mathbb Q}\in S} \mathbb{V}({\mathbb Q})= \inf_{{\mathbb Q}\in S\cap\el^1} \mathbb{V}({\mathbb Q}).
   \end{split}
\end{equation}
\end{lemma}
\begin{proof}
  To simplify the notation, we assume that $y=1$ - the general case is
  completely analogous.  Let ${\mathcal D}$ denote the set of all
  (Radon-Nikodym derivatives of) regular parts of the elements in $S$,
  and let ${\mathcal M}\subseteq{\mathcal D}$ denote the set of all (Radon-Nikodym
  derivatives of) elements of $S\cap\el^1$. Since the passage to the
  regular part does not increase the total mass, ${\mathcal D}$ is clearly
  bounded in $\el^1$.

  The statement will follow from Lemma \ref{lab:KraSch}, once its
  assumptions are verified:
  \par{(1)} The set $S$ is a weak-$\ast$ closed (and therefore
  compact) subset of $\sba_+(1)$, and $\mathbb{V}$ is lower semicontinuous, so
  there exists $\hat{\QQ}\in\sba_+(1)$ at which the infimum on the
  left-hand-side expression of \eqref{equ:infs-equal} is achieved.  It
  follows from the representation (\ref{equ:V-reg}) that
  $\hh\in\argmin_{h\in{\mathcal D}} {\mathbb E}[V(h)]$, where $\hh:=\tRN{\hat{\QQ}^r}{{\mathbb P}}$.

\par{(2)} Let $\seq{{\mathbb Q}}$ be a sequence of (countably additive)
  probability measures in ${\mathcal M}$, and let $\seq{\alpha}$ be a sequence
  of positive constants with $\sum_n \alpha_n=1$. To show that the
  probability measure ${\mathbb Q}=\sum_n \alpha_n {\mathbb Q}_n$ belongs to ${\mathcal M}$, we
  need to show that $\alpha_{\sC}({\mathbb Q})\leq M$, i.e., that $\scl{{\mathbb Q}}{f}\leq M$,
  for all $f\in{\mathcal C}$. This follows by aggregation (combined with monotone convergence) of the
  inequalities $\scl{\alpha_n{\mathbb Q}_n}{f}\leq \alpha_n M$ over  $n\in{\mathbb N}$.

  \par{(3)} We first establish an auxiliary claim. We remind the
  reader that for $A\subseteq \el^0_+$, $A^{\circ}$ denotes the polar
  of $A$, i.e., $A^{\circ}:=\sets{g\in\el^0_+}{{\mathbb E}[fg]\leq 1,\text{ for
      all }f\in A}$.
\begin{claim}
\label{cla:one}
 For ${\mathbb Q}\in S$, we have
  ${\mathbb Q}^r\in {\mathcal M}^{\circ\circ}$.
\end{claim}
\begin{proof}[Proof of Claim \ref{cla:one}] Let us first note that
 \begin{equation}
   \label{equ:Seq}

   \begin{split}
  S= \{{\mathbb Q}\in\sba_+(1):\alpha_{\mathcal{C}'}({\mathbb Q}) \le 0\},
   \end{split}
\end{equation}
 where
  $\mathcal{C}'\subset \mathbb{L}^\infty$ denotes the weak-$\ast$
  closed convex cone generated by $\mathcal{C}-M-
  \mathbb{L}^\infty_+$. The inclusion $\supseteq$ clearly holds, and
  for the opposite one it suffices to note that $\scl{{\mathbb Q}}{\gamma(f-M-k)}\leq
  0$, for all ${\mathbb Q}\in S$ and all $\gamma\geq 0$, $f\in \mathcal{C}$
  and $k\in\el^{\infty}_+$.

By \eqref{equ:Seq} we have
\[\scl{{\mathbb Q}^r}{g+1}\le \scl{{\mathbb Q}}{g+1} =\scl{{\mathbb Q}}{g} +1\le 1,\]
for all ${\mathbb Q}\in S$ and $g\in {\mathcal C}'$ with $1+g\in \el^{\infty}_+$. Therefore,
${\mathbb Q}^r\in A^{\circ}$, for all ${\mathbb Q}\in S$, where
$A=(\mathcal{C}'+1)\cap \mathbb{L}^0_+$. Consequently, Claim
\ref{cla:one} will be proven once we show that $$ A^{\circ} \subset
\sets{S^{\mathbb{L}^1}_+(1)}{\alpha_{\mathcal{C'}}({\mathbb Q}) \le
  0}^{\circ\circ}.
$$
To this end we argue by
contradiction and assume that there exists
\[ \hat{{\mathbb Q}}\in A^{\circ} \setminus \{{\mathbb Q}\in
S^{\mathbb{L}^1}_+(1):\alpha_{\mathcal{C'}}({\mathbb Q}) \le
0\}^{\circ\circ}.\] In other words, we assume that there exist
$\hat{{\mathbb Q}}\in A^{\circ}$ and $\hat{h}\in \{{\mathbb Q}\in
S^{\mathbb{L}^1}_+(1):\alpha_{\mathcal{C'}}({\mathbb Q}) \le 0\}^{\circ}$ such
that
\begin{align}\label{eq:bipolar1}
  \scl{\hat{{\mathbb Q}}}{\hat{h}}>1,\quad \scl{\hat{{\mathbb Q}}}{f}\le 1\quad
  \text{for all}\quad f\in A.
\end{align}
General solidity of polars and the monotone convergence theorem imply
that for all $n\in{\mathbb N}$, we have $\hat{h}\wedge n\in \{{\mathbb Q}\in
S^{\mathbb{L}^1}_+(1):\alpha_{\mathcal{C'}}({\mathbb Q}) \le 0\}^{\circ}$ and,
for large enough $n\in{\mathbb N}$, it additionally holds that
$\scl{\hat{{\mathbb Q}}}{\hat{h}\wedge n}>1$.  Therefore, we may assume that
already $\hat{h}\in \el^{\infty}_+$.

 Trivially, \eqref{eq:bipolar1} shows $\hat{h}\notin
A$, and, equivalently, $\hat{h}-1\notin \mathcal{C}'$. By the Hahn-Banach
separation theorem there exists $\tilde{{\mathbb Q}}\in \el^1$ and $\beta\in\R$
such that
\begin{align}\label{eq:bipolar2}
\scl{\tilde{{\mathbb Q}}}{\hat{h}-1} >\beta \ge \scl{\tilde{{\mathbb Q}}}{g} \quad \text{for all}\quad g\in \mathcal{C}'.
\end{align}
Given that $\mathcal{C}'$ contains $M'-\el^{\infty}_+$ for some $M'\in\R$, we
must have $\tilde{{\mathbb Q}}\in\el^1_+$ and, since $\mathcal{C}'$ is a cone,
we must also have $\beta= 0$. Nontriviality of $\tilde{{\mathbb Q}}$ allows
us safely to assume - by scaling, if necessary - that $||\tilde{{\mathbb Q}}||_1 = 1$.
The second inequality in \eqref{eq:bipolar2} shows $\tilde{{\mathbb Q}} \in
\{{\mathbb Q}\in S^{\mathbb{L}^1}_+(1):\alpha_{\mathcal{C'}}({\mathbb Q}) \le
0\}$. However, we have assumed that $\hat{h} \in \{{\mathbb Q}\in
S^{\mathbb{L}^1}_+(1):\alpha_{\mathcal{C'}}({\mathbb Q}) \le 0\}^{\circ}$ which
implies $\scl{\tilde{{\mathbb Q}}}{\hat{h}}\le 1$ and thereby contradicts the
first inequality in \eqref{eq:bipolar2}.
\end{proof}

Returning to the proof of (3), we note that the weak-$\ast$
compactness of $S$ (via the Banach-Alaoglu Theorem) guarantees the
existence of a minimizer $\hat{{\mathbb Q}} \in S$ for the
left-hand side of \eqref{equ:infs-equal}. Thanks to  representation
\eqref{equ:V-reg}, all we need to do is construct a sequence
$\sq{\frac{d{\mathbb Q}_n}{d{\mathbb P}}}\subset {\mathcal M}$ which converges almost surely
to the regular part $\frac{d\hat{{\mathbb Q}}^r}{d{\mathbb P}}\in{\mathcal D}$, and for that  we will
use a variant of an argument in \cite{CviSchWan01}. By the Bipolar
Theorem (see \cite{BraSch99}), ${\mathcal M}^{\circ\circ}$ is the
closure in probability of the solid  hull of ${\mathcal M}$.
Therefore, there exist  sequences $\seq{f}\subseteq \el^0_+$ and
$\seq{{\mathbb Q}}\subseteq {\mathcal M}$ such that ${\mathbb P}$-a.s.
$$
0\le f_n \le \frac{d{\mathbb Q}_n}{d{\mathbb P}},\quad f_n \to \frac{d\hat{{\mathbb Q}}^r}{d{\mathbb P}}\quad \text{in probability as} \quad n\to\infty.
$$
Furthermore, by passing to a subsequence, ${\mathbb P}$-a.s.~convergence can
be substituted for the convergence in probability. Koml\'os' lemma can
be used to justify the existence of a nonnegative random variable $Y$
and a double array $\sets{\beta_n^k}{n\in{\mathbb N}, k=1,...,K(n)}$ with
$0\leq \beta_n^k\leq 1$ such that
$$
\sum_{k=n}^{K(n)}\beta_n^k =1,\, n\in{\mathbb N},\quad
\frac{d\tilde{{\mathbb Q}}_n}{d{\mathbb P}} = \sum_{k=n}^{K(n)}\beta_n^k
\frac{d{\mathbb Q}_k}{d{\mathbb P}} \to Y,\quad {\mathbb P}\text{-a.s. as }n\to\infty.
$$
It follows from the
convergence $f_n \to \frac{d\hat{{\mathbb Q}}^r}{d{\mathbb P}}$  that
$$
\frac{d\hat{{\mathbb Q}}^r}{d{\mathbb P}} = \lim_n\sum_{k=n}^{K(n)}\beta_n^k f_k \le
\lim_n\sum_{k=n}^{K(n)}\beta_n^k \frac{d{\mathbb Q}_k}{d{\mathbb P}} = Y.
$$
By the convexity of ${\mathcal M}$, we have $\frac{d\tilde{{\mathbb Q}}_n}{d{\mathbb P}} \in {\mathcal M}$,
so it suffices to verify the equality $Y= \frac{d\hat{{\mathbb Q}}^r}{d{\mathbb P}}$,
a.s.

Since $S$ is weak-$\ast$ compact, the sequence
$(\tilde{{\mathbb Q}}_n)_{n\in{\mathbb N}}\subset S$ must have an accumulation point
$\tilde{{\mathbb Q}}\in S$, which, by
Proposition A.1, p.~271 in
\cite{CviSchWan01}, must satisfy
$\frac{d\tilde{{\mathbb Q}}^r}{d{\mathbb P}}=Y$. Assuming that ${\mathbb P}[
\frac{d\hat{{\mathbb Q}}^r}{d{\mathbb P}} < Y]>0$, the representation
  \eqref{equ:V-reg} produces the contradiction
$$
\inf_{{\mathbb Q}\in S} \mathbb{V}({\mathbb Q})= \mathbb{V}(\hat{{\mathbb Q}}) =
{\mathbb E}[V(\tfrac{d\hat{{\mathbb Q}}^r}{d{\mathbb P}})] >{\mathbb E}[V(Y)] =
{\mathbb E}[V(\tfrac{d\tilde{{\mathbb Q}}^r}{d{\mathbb P}})] = \mathbb{V}(\tilde{{\mathbb Q}}^*),
$$
where the strict inequality is the consequence of the strict decrease
of $V$ which, in turn, follows from the second part of
\eqref{equ:RAE}.  Therefore, we have $Y=\frac{d\hat{{\mathbb Q}}^r}{d{\mathbb P}}$,
${\mathbb P}$-a.s., and the proof is complete.
\end{proof}
\begin{proposition}  Under Assumptions \ref{ass:C} and
  \ref{ass:U}, $v^{\ba}=v$.
\label{pro:v-vba}
\end{proposition}
\begin{proof} As we already commented in the paragraph following
  \eqref{equ:V-reg}, the inequality $v^{\ba}\leq v$ is immediate. It is, therefore,
  enough to prove that $v^{\ba}(y)\geq v(y)$ for all $y>0$ with
  $v^{\ba}(y)<\infty$. We fix such $y>0$, pick $\varepsilon>0$, and choose a
  minimizer $\hat{\QQ}^{(y)}$ for $v^{\ba}(y)$. By Lemma \ref{lem:Zalinescu}, the
  family
\[ \sl_{\eps}(y):=\sba_{\eps}(y)\cap\el^1\text{ where }\sba_{\eps}(y):=\sets{{\mathbb Q}\in\sba_+(y)}{\alpha_{\sC}({\mathbb Q})\leq \alpha_{\sC}(\hat{\QQ}^{(y)})+\varepsilon},\]
is nonempty. Then, by Lemma \ref{lem:infs-equal}, we have
\begin{equation}
\nonumber
   \begin{split}
 v^{\ba}(y) &=\alpha_{\sC}(\hat{\QQ}^{(y)}) + \mathbb{V}(\hat{\QQ}^{(y)})
\geq \alpha_{\sC}(\hat{\QQ}^{(y)})+ \inf_{{\mathbb Q}\in\sba_{\eps}(y)} \mathbb{V}({\mathbb Q})
\\ &= \inf_{{\mathbb Q}\in\sl_{\eps}(y)} \Big( \mathbb{V}({\mathbb Q})+\alpha_{\sC}(\hat{\QQ}^{(y)}) \Big)
\geq \inf_{{\mathbb Q}\in\sl_{\eps}(y)} \Big( \mathbb{V}({\mathbb Q})+\alpha_{\sC}({\mathbb Q}) \Big) - \varepsilon\\
& \geq \inf_{{\mathbb Q}\in\sl_+(y)} \Big( \mathbb{V}({\mathbb Q})+\alpha_{\sC}({\mathbb Q}) \Big)-\varepsilon=v(y)-\varepsilon.
   \end{split}
\end{equation}
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:main}] \
\begin{enumerate}
\item The properties of the function $u$ follow either directly from the
definition, or, in the case of upper semicontinuity, from Proposition
\ref{pro:usc}. Convexity and lower semicontinuity of $v^{\ba}$ follow from
the representation in part (1) of Proposition \ref{prop:conjugates}.
Finally, $v$ and $v^{\ba}$ are identical, by Proposition \ref{pro:v-vba}.
\item

For $x\in\R$ with $u(x)\in\R$,  there clearly exists $f\in\sC$ such that $x+f \ge
0$, and, so, $x + \langle {\mathbb Q},f\rangle \ge0$ for all ${\mathbb Q}\in\sP$. If we
take the
supremum over $f\in\sC$ followed by the infumum over
${\mathbb Q}\in\sP$ in this inequality, we get  $x\ge \sup_{{\mathbb Q}\in\sP} -\alpha_{\sC}
({\mathbb Q})$, and consequently,
$ \underline{x}\geq \sup_{{\mathbb Q}\in\sP} -\alpha_{\sC}({\mathbb Q}) = -\inf_{{\mathbb Q}\in\sP} \alpha_{\sC}({\mathbb Q}).$

On the other hand, by Corollary \ref{cor:hedge}, for $x>
\sup_{{\mathbb Q}\in\sP} -\alpha_{\sC}({\mathbb Q})$, we can find $\varepsilon>0$
such that $\varepsilon-x \in\sC$. Therefore, $u(x)\geq U(\varepsilon)>-\infty$, and, so
$x \geq \underline{x}$. The second statement follows from the fact that $u$ is proper
and nondecreasing.

\item The existence of primal optimizers is proven in  Proposition \ref{prop:primal_existence}.

\item The relation  \eqref{eq:u_conjugate} is proven in Proposition
\ref{prop:conjugates}, part (1) and Proposition \ref{pro:v-vba}. The
symmetric relation \eqref{eq:v_conjugate} follows directly from
\eqref{eq:u_conjugate} and the upper semicontinuity of $u$.
 \end{enumerate}
\end{proof}

\section{A sufficient condition for Assumption \ref{ass:C}}
\label{sec:suff}

The closedness in probability of the sets ${\mathcal C}(x)$, $x\in\R$, is the
central condition of our main results. It is, however, not immediately
obvious how to test its validity in a given model. Thanks to a recent
result of \cite{CziSch11}, a much more workable sufficient condition can be given. We start by recalling that
each ($\R^d$-valued) semimartingale $S$ can be represented in terms
of its {\em predictable characteristics}:
\[ S=S^c+F+ (x\inds{\abs{x}\leq 1})\ast (\mu-\tilde{\mu})+
(x\inds{\abs{x}> 1})\ast \mu,\] where $S^c$ is a continuous
semimartingale, $F$ is a predictable process of finite variation,
$\mu$ is the jump measure of $S$, and $\tilde{\mu}$ is its
compensator. Instead of explaining these terms we refer the
reader to the standard reference \cite{JacShi03}. Furthermore, it is well-known that there exists a  nondecreasing
process $B$, a  $\R^d$-valued process $b$, a
nonnegative-definite $\R^{d\times d}$-matrix valued process $c$, and a
 L\' evy-measure-valued process $\Gamma$, all predictable, such that
\[ F=b\cdot B,\ [S^c,S^c]=c\cdot B,\text{ and }\tilde{\mu}= \Gamma
\cdot B.\] The triplet $(b,c,\Gamma)$ is usually referred to as the
\emph{triplet of semimartingale characteristics} of $S$.

It can be shown that the measure ${\mathbb P}\otimes dB$ is $\sigma$-finite
and can, therefore, be replaced by an equivalent probability measure
on the predictable sets of $\Omega\times [0,T]$, which we denote by
${\mathbb P}^S$. We refer the reader to \cite{CziSch11} for a discussion and
the interpretation of the probability measure ${\mathbb P}^S$ (this measure is denoted by ${\mathbb P}_B$ in \cite{CziSch11}), as well as for the proof of the
following proposition.
\begin{proposition}[Czichowsky and Schweizer, \cite{CziSch11}] There exists a
  predictable process $\prf{\Pi^S_t}$, with values in the orthogonal
  projections in $\R^d$ with the following property. For predictable
  processes $\theta,\varphi$ with $\theta$ being $S$-integrable, the following
  two statements are equivalent:
  \begin{enumerate}
  \item $\varphi$ is $S$-integrable with $\theta\cdot S$ and $\varphi\cdot S$
  indistinguishable, and
  \item $\Pi^S \theta=\Pi^S \varphi$, ${\mathbb P}^S$-a.e.
  \end{enumerate}
\end{proposition}
We fix a version of such a $\Pi^S$ and we call it the \emph{projection on
  the predictable range of $S$}. One can think of $\Pi^S \theta$ as
the ``relevant'' portion of $\theta$, as far as stochastic integration
with respect to $S$ is concerned.
It was shown in \cite{CziSch11} that closedness of the set of constrained
stochastic integrals is closely related to the interplay between
$\Pi^S$ and the constraint $\kappa$:
\begin{theorem}[Czichowsky and Schweizer, \cite{CziSch11}]
\label{thm:closed}
Let $\kappa$ and $\AA^\kappa$ be as in Subsection
\ref{sse:conv-const}. Then, the set of stochastic integrals
$\sets{H\cdot S}{H\in \AA^\kappa}$ is closed with respect to the
semimartingale topology if and only if $\Pi^S_t(\omega)
\kappa_t(\omega)$ is a closed subset of $\R^d$, ${\mathbb P}^S$-a.e.
\end{theorem}
\begin{remark}
  Since closedness of the set $\Pi^S_t(\omega) \kappa_t(\omega)$ is
  going to play a prominent role in the sequel, let us briefly comment
  on its financial interpretation. It states, essentially, that when
  the constraints are imposed, one should take into account those
  aspects of the portfolio that actually matter for the evolution of
  the wealth process,

  see Section 3 of \cite{CziSch11} for a detailed explanation.  In most
  models of interest, $\Pi^S$ is the identity, i.e., there are no
  ``redundant'' assets and the closedness condition is automatically
  satisfied. For other sufficient conditions, see \cite{CziSch11}.
  Let us mention that closedness is guaranteed for all semimartingales
  $S$  when, for example, with probability one, for each
  $t\in [0,T]$ one of the following three properties holds:
  \begin{enumerate}
\item $\kappa_t(\omega)$ compact,
\item $\kappa_t(\omega)$ is polyhedral (i.e., representable as an
  intersection of finitely many half-planes), or
\item the support map $\R^d\ni \bsx \mapsto \sup_{\bsy\in
    \kappa_t(\omega)} \bsx^T \bsy$ of $\kappa_t(\omega)$ is
  continuous.
    \end{enumerate}
\end{remark}
Using Theorem \ref{thm:closed} as one of the central ingredients, we
can prove the sufficiency of our conditions for convex compactness of
${\mathcal C}(x)$ in Proposition \ref{pro:sufficient}. For the reader's
convenience, we repeat its statement below:
\begin{propositiontwofive}
\proptf{\label{equ:super-N}}
\end{propositiontwofive}
\begin{proof}
  The condition $\AA\ne\emptyset$ implies that ${\mathcal C}(x)\ne \emptyset$ for
  some $x\in\R$, so it will be enough to show that ${\mathcal C}(x)$ is convexly
  compact.

  First, we show that ${\mathcal C}(x)$ is closed in probability. Let
  $\seq{f}$ be a sequence in ${\mathcal C}(x)$ with
  $$
  f_n = x+(H^n\cdot S)_T - g_n \to f,\text{ in probability, }
  $$
where $H^n
  \in \sA$ and $g_n\in\mathbb{L}^0_+$. for all $n\in{\mathbb N}$.
  By passing to a sequence of convex combinations (justified by
  Koml\'os' theorem and the fact that our constraints are convex) we can
  - and will - assume that $g_n =0$, ${\mathbb P}$-a.s., for all $n\in {\mathbb N}$. It
  therefore suffices to find $H\in \sA$ such that $(H\cdot S)_T\ge
  \lim_{n\to\infty} (H^n\cdot S)_T$.

  Let ${\mathcal N}$ denote the set of all pairs $({\mathbb Q},A)$ (with ${\mathbb Q}$
   as in (3)(a) and $A$ as in (3)(c)) for which there
   exists $\hH$ as in (3)(b) such that \eqref{equ:super-N} holds.
  We fix $({\mathbb Q},A)\in{\mathcal N}$ so that for each element $V^n$ in the
  sequence
  $$
  V^n= (H^n - \hH)\cdot S, \quad n\in{\mathbb N},
  $$
  the process $V^n-A$ is a ${\mathbb Q}$-supermartingale.  In particular, we have
  \begin{equation}
   \nonumber
   \begin{split}
    V^n_t-A_t &\geq {\mathbb E}^{{\mathbb Q}}[ V^n_T-A_T|{\mathcal F}_t]\\ &=
    {\mathbb E}^{{\mathbb Q}}[ (H^n\cdot S)_T|{\mathcal F}_t] -
   {\mathbb E}^{{\mathbb Q}}[ A_T+(\hH\cdot S)_T|{\mathcal F}_t] \geq - M_t,
   \end{split}
   \end{equation}
   where $M_t={\mathbb E}^{{\mathbb Q}}[x+(\hH\cdot S)_T+A_T|{\mathcal F}_t]$ is a
${\mathbb Q}$-martingale.
Indeed, $(\hH\cdot S)_T\in\el^1({\mathbb Q})$ by assumption and $A_T\in \el^1({\mathbb Q})$
because the process $-A= (\hH - \hH)\cdot S-A$ is a ${\mathbb Q}$-supermartingale.

From the above we conclude that the processes
$V^n-A+M-M_0$, $n\in{\mathbb N}$, are uniformly lower bounded
${\mathbb Q}$-supermartingales starting at zero. Therefore, we can use the
Koml\'os-type Lemma 5.2.(1), p.~14 in \cite{FolKra97}, to extract a
Fatou-convergent sequence of convex combinations. By the convexity of
our constraint sets, these convex combinations are still of the form
$(\tilde{H}^n-\hH)\cdot S - A + M-M_0$ and converge towards a lower
bounded ${\mathbb Q}$-supermartingale, which we write in the form $V-A+M-M_0$,
for some semimartingale $V$. Using the properties of Fatou-convergence
and the already assumed convergence of the terminal values $(H^n\cdot
S)_T$ we have
\[ V_0\leq 0 \text{ and } V_T = f - x - (\hH\cdot S)_T.\]

Since the processes $M$ and $A$ are independent of $n$ we also have
Fatou-convergence of $V^n$ towards $V$.  It is important to note that
Fatou-convergence is measure-independent (as long as we stay in the same
equivalence class), so that, for each pair $({\mathbb Q}',A')\in{\mathcal N}$, the
${\mathbb Q}'$-supermartingale $V^n-A'$ Fatou-converges towards $V-A'$. The
processes $\hH\cdot S$ and $A'$ are locally bounded ($\hH\cdot S$ is by
assumption whereas $A'$ is thanks to predictability and the \cd property)
so all
$V^n-A'$ are locally bounded from below, with the same localization
sequence. It follows that their Fatou limit $V-A'$ is a
locally-bounded-from-below local ${\mathbb Q}'$-supermartingale for each
$({\mathbb Q}',A')\in{\mathcal N}$.

  The next step is to apply a version of the Optional Decomposition Theorem
  developed in \cite{FolKra97}, namely Theorem 3.1 on p.~6.
  We need to check that all of its assumptions are
  satisfied, i.e., that the family $\sS$ of semimartingales
  $\sS=\sets{ (H-\hH)\cdot S}{ H\in \AA}$ satisfies
  \begin{enumerate}
  \item $\sS$ is predictably convex (in the language of \cite{FolKra97}),
  \item $\sS$ contains processes locally bounded from below,
  \item $\sS$ is closed in the semimartingale topology for uniformly-bounded from
    below sequences (Assumption 3.1~in \cite{FolKra97}), and
  \item $\sS$ contains the constant process $0$.
  \end{enumerate}
  Indeed, (1) follows from the convexity of $\kappa$, (2) holds thanks to the
  local boundedness of $\hH\cdot S$, (3) is the content of Theorem
  \ref{thm:closed}, and (4) is true by the construction of $\sS$.

  Therefore, the fact that $V-A$ is a ${\mathbb Q}$-local supermartingale
  for each $({\mathbb Q},A)\in{\mathcal N}$ and Theorem 3.1 in \cite{FolKra97} allow us
  to conclude that there exists $H\in \sA$ such that
  $$
  V=V_0 + (H-\hH)\cdot S - C,
  $$
  for some nondecreasing, nonnegative, c\' adl\' ag, and adapted process $C$.
 We then have the representation
  $$
  f = x+V_T + (\hH\cdot S)_T = x+ (H \cdot S)_T +
  V_0 - C_T \le x+ (H \cdot S)_T.
  $$

  To finish the proof we need to show that ${\mathcal C}(x)$ is bounded in
  probability. For $f\in{\mathcal C}(x)$ we let $H\in\AA$ be such that
  $x+H\cdot S \geq f$ and pick $({\mathbb Q},A)\in{\mathcal N}$. Then
  \[ {\mathbb E}^{{\mathbb Q}}[f]\leq x+ {\mathbb E}^{{\mathbb Q}}[ ((H-\hH)\cdot S)_T - A_T] +
  {\mathbb E}^{{\mathbb Q}}[ (\hH\cdot S)_T+ A_T]\leq M_0,\] where - as before -
  $M_0=x+{\mathbb E}^{{\mathbb Q}}[ (\hH\cdot S)_T+ A_T]<\infty$. This shows that
  ${\mathcal C}(x)$ is bounded in $\el^1({\mathbb Q})$, so, by Markov's inequality, it
  is bounded in probability under ${\mathbb Q}$ and, by equivalence, also under
  ${\mathbb P}$.
\end{proof}

\bibliographystyle{imsart-number}

redpajama_set_name: RedPajamaArXiv


The Lances West condos for sale are located at about 4 o'clock on the loop called Columbine Drive in the Four Seasons section of the Warriors Mark neighborhood. The base of Peak 9 of the Breckenridge Ski Resort is due west and is a quick walk in summer or winter.
The free town shuttle stops just outside the door and makes everything in Breckenridge easily accessible from any condo for sale in Lances West.
Lances West was built in the nineteen-seventies and has been recently repainted, but maintains its original Board & Batten exterior. This style is being used in current construction to give a nod to the architecture of the early mining era.
Find out more about these stellar residences, or browse the Lances West condominium listings below.
Maintained trails make it easy to ski over to the base of Peak 9, although this connection looks completely different – like a short forest hike across the Blue River – in the summer.
The floor plan of all the units is open and the living room section has floor to ceiling glass doors that lead onto a small balcony. Depending on which side of the building, any condo for sale in Lance's West will have a nice view of mountains, either Baldy Mountain or the Breckenridge Ski Resort. One space of off-street parking next to the building is assigned to each condo.
Size: 585 to 1285 sq. ft.
146 BROKEN LANCE DRLocated at 146 Broken Lance in Warriors Mark, across the Blue River from the base of Peak 9, on the inner circle of the Columbine Loop, just east of the Columbine neighborhood pool.
Drive south on Main Street to last stoplight at Boreas Pass Road, turn right on Broken Lance Drive. At the three-way stop make a left and remain on Broken Lance, before it intersects with Columbine, Lances West will be on the right, between the Snowdrop and Miners Candle condos.
Get started on your home ownership today! There are two ways you can contact us - by email or by phone at (970) 453-2200.


On Wednesday, Year 5 had a talk from John Atkinson about trees. We learned about how useful trees are for our environment and then John took us into the playground and talked about the trees we have here at school. Did you know that the Ginko tree, which is in the woodland playground, can survive the radiation from a nuclear bomb? Did you know that one of the trees by the wall is over 300 years old? We all learnt a lot and it was really lovely to hear from John again, after he had talked to us when we were in Year 4.


text:
TATA STEEL ADVENTURE FOUNDATION PLANS CLEANING DRIVE AT DIMNA
Jamshedpur, September 07, 2017
Inspired by the appeal made by Prime Minister Shri Narendar Modi in the Mann Ke Baat' programme of observing October 2nd as 'safai divas' the Tata Steel Adventure Foundation is organizing a cleaning drive on September 8 at Dimna Area from 8 am onwards till all the areas are covered. Starting out from Brick Kiln area, the cleaning drive would cover the open ground in front of the kiln , the Dam area, the Management Development Centre area, the Guest House area, the picnic area, the vehicle cleaning area by side of the lake and the area by the side of the road coming from lake back to the starting point.
A 35 member TSAF Team comprising of its staff together with few volunteers equipped with collection bags , tongs and face masks would collect all the strewn garbage of the non-biodegradable kind. A mini pick up is being arranged to bring back the garbage for proper disposal.
TSAF sources said that such efforts would be made periodically from time to time keeping in mind the PM's Swachh Bharat Abhiyan (SBA) – a movement that seeks to create a Clean India. The cleanliness drives by TSAF from time to time would be a tribute to Mahatma Gandhi as cleanliness was very close to his heart.
The Secretary of TSAF informed that Environment Education was an integral part of its Outdoor Leadership Courses conducted in Jamshedpur and in the Himalayas. TSAF had previously carried out cleaning exercises in the river Ganga, in the Himalayas and in other places in and around Jamshedpur.
About Tata Steel:
Tata Steel Group is among the top global steel companies with an annual crude steel capacity of 27.5 million tones per annum (MTPA) as on March 31, 2017. It is the world's second-most geographically-diversified steel producer, with operations in 26 countries and a commercial presence in over 50 countries. The Group recorded a consolidated turnover of US $18.12 billion (INR 117,420 crore) in FY17.
Tata Steel Group is spread across five continents with an employee base of nearly 74,000. Having bagged the Deming Application Prize and Deming Grand Prize for continuous improvement in 2008 and 2012 respectively, Tata Steel has now been recognized as the global 'Industry Leader' in 'Steel category' by Dow Jones Sustainability Index (2015). Besides being a member of the World Steel Climate Action Programmer, Tata Steel has also been felicitated with several awards including the Prime Minister's Trophy for the best performing integrated steel plant for 2013-14 (received in 2017), Best Risk Management by CNBC TV18 (2016), 'Best-in-class Manufacturing' award from TIME India (2016) and the 'Most Ethical Company' award from the Euthyphro Institute (2016), IIM Sustainability Award (2015), among several others.
Disclaimer: Statements in this press release describing the Company's performance may be "forward looking statements" within the meaning of applicable securities laws and regulations. Actual results may differ materially from those directly or indirectly expressed, inferred or implied. Important factors that could make a difference to the Company's operations include, among others, economic conditions affecting demand/ supply and price conditions in the domestic and overseas markets in which the Company operates, changes in or due to the environment, Government regulations, laws, statutes, judicial pronouncements and/ or other incidental factors.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: how to kill an app started by `quickly run` I am building an app and it hangs on some thread . how can i kill that app ?
tried ctrl+c but it didn't work (the icon stays in the notification area)
thanks

A: Try to kill it via the process list:

Or - if you want to be more nerdy - kill it via the console, using $>top

A: If you know the name of your process, grep against ps to get the PID (process ID).  Example: If I wanted to stop shotwell, I would do this (from a terminal session):
$ ps -ef | grep shotwell
xaap001   1779  1139  8 08:08 ?        00:00:01 shotwell
xaap001   1792  1703  0 08:08 pts/1    00:00:00 grep --color=auto shotwell

The first process listed is the one I want (in this example), and the first number listed is the PID.  You can also see that the command I just ran is listed in there, too.
Now that I have my PID (1779), I can stop the process with this command:
$ kill -9 1779

A: I suggest that you enter
Cntl-z

That will suspend the process and tell you its job number.
then:
kill -9 %1

if its job number is 1. After you press enter the first time it will mention that [1] is stopped. The next time you press enter it will tell you it is killed or terminated.

redpajama_set_name: RedPajamaStackExchange


text:
Flowers A-F
Flowers G-M
Flowers N-S
Flowers T-Z
YodelOut! Flowers
You are here: Home / Flowers T-Z / Yellow Adder's Tongue, Trout Lily
Yellow Adder's Tongue, Trout Lily
(Erythronium Americanum) Lily family
Flower—Solitary, pale russet yellow, rarely tinged with purple, slightly fragrant, 1 to 2 in. long, nodding from the summit of a footstalk 6 to 12 in. high, or about as tall as the leaves. Perianth bell-shaped, of 6 petal-like, distinct segments, spreading at tips, dark spotted within; 6 stamens; the club-shaped style with 3 short, stigmatic ridges. Leaves : 2, unequal, grayish green, mottled and streaked with brown or all green, oblong, 3 to 8 in. long, narrowing into clasping petioles.
Preferred Habitat—Moist open woods and thickets, brooksides.
Flowering Season—March—May.
Distribution—Nova Scotia to Florida, westward to the Mississippi.
Colonies of these dainty Iittle lilies, that so often grow beside leaping brooks where and when the trout hide, justify at least one of their names ; but they have nothing in common with the violet or a dog's tooth. Their faint fragrance rather suggests a tulip; and as for the bulb, which in some of the lily-kin has tooth.. like scales, it is in this case a smooth, egg-shaped corm, producing little round offsets from its base. Much fault is also found with another name on the plea that the curiously mottled and delicately pencilled leaves bring to mind, not a snake's tongue, but its skin, as they surely do. Whoever sees the sharp purplish point of a young plant darting above ground in earliest spring, however, at once sees the fitting application of adder's tongue. But how few recognize their plant friends at all seasons of the year!
Every one must have noticed the abundance of low-growing spring flowers in deciduous woodlands, where, later in the year, after the leaves overhead cast a heavy shade, so few blossoms are to be found, because their light is seriously diminished. The thrifty adder's tongue, by laying up nourishment in its store-room underground through the winter, is ready to send its leaves and flower upward to take advantage of the sunlight the still naked trees do not intercept, just as soon as the ground thaws. But the spring beauty, the rue-anemone, bloodroot, toothwort, and the first blue violet (palmata) among other early spring flowers, have not been slow to take advantage of the light either. Fierce competition, therefore, rages among them to secure visits from the comparatively few insects then flying—a competition so severe that the adder's tongue often has to wait until afternoon for the spring beauty to close before receiving a single caller. Hive-bees, and others only about half their size, of the Andrena and Halictus clans, the first to fly, the Bombylius frauds, and common yellow butterflies, come in numbers then. Guided by the speckles to the nectaries at the base of the flower, they must either cling to the stamens and style while they suck, or fall out. Thus cross-fertilization is commonly effected; but in the absence of insects the lily can fertilize itself. Crawling pilferers rarely think it worth while to slip and slide up the smooth footstalk and risk a tumble where it curves to allow the flower to nod—the reason why this habit of growth is so popular. The adder's tongue, which is extremely sensitive to the sunlight, will turn on its stalk to follow it, and expand in its warmth. At night it nearly closes.
A similar adder's tongue, bearing a white flower, purplish tinged on the outside, yellow at the base within to guide insects to the nectaries, is the White Adder's Tongue (E. albidum), rare in the Eastern States, but quite common westward as far as Texas and Minnesota.


text:
FROM mcr.microsoft.com/dotnet/sdk:5.0 AS build
WORKDIR /app
COPY Benchmarks .
RUN dotnet publish -c Release -o out

FROM mcr.microsoft.com/dotnet/aspnet:5.0 AS runtime
ENV ASPNETCORE_URLS http://+:8080
WORKDIR /app
COPY --from=build /app/out ./

EXPOSE 8080

ENTRYPOINT ["dotnet", "Benchmarks.dll", "scenarios=mvcplaintext,mvcjson"]

redpajama_set_name: RedPajamaGithub


My take for this weeks photography challenge – living in the layers is inevitable when you're in the Scottish Highlands. I am here alot at the moment, and see Rannoch Moor in all its epic glory in every weather imaginable.
This entry was posted in photography, weekly wordpress challenge and tagged photography, Rannoch Moor, Scottish Highlands, stanley kunitz on September 23, 2017 by modestly.


Deborah McCollin, the Butterfly Queen and artiste, Mariposa Reina, is a woman who wears many hats.
By day she is a primary school teacher, but since July 2012 she has made a name for herself as a spoken word artiste, author, story teller and motivational speaker.
However, it is her role as an advocate for victims of domestic violence that has motivated her latest work, Journey of the Butterfly.
She recently launched the publication at the UWI Bookshop at the Cave Hill Campus. It is a collection of poems highlighting her personal experience as a victim of abuse.
That chapter of her life began in her pre-teen years when she suffered at the hands of a now deceased great-aunt who raised her.
Recounting that difficult period of her life, the author said it got worst because she was the only girl and the most vocal of the children left in the care of her great-aunt.
One poem entitled The Dark examines a time when the woman tortured all the children and left them alone in the house during a power outage.
Mariposa said while at school she followed the wrong company and that led in part to her downfall. During the book launch, the artiste recalled her worst experience in 1989 when she was physically attacked and raped by an abusive boyfriend.
The teacher described Journey of the Butterfly as complex.
"It takes the reader on a journey from abuse to the turmoil and soul-searching in between, and success at the end," she explained.
Host of the book launch, Troy Holder, commended Mariposa for her resilience, as did Robert Passion Poet Gibson, who edited Journey of the Butterfly.
Holder described the book as "inspirational" adding that it was "good to see people getting up and going again instead of taking on the role of victim". This was a point on which Mariposa expanded.
"When you are feeling overwhelmed, there are two ways in which you can respond. You can either say 'Why me?' or 'Try me!' The latter is empowering, as it challenges whoever or whatever is causing the problem to back off, or makes you rise higher to conquer the problem," she added.
Holder told those present, "Success is not success unless you are willing to encourage and nurture others", adding that the teacher and artiste had certainly taken that to heart.
In her role as a motivational speaker, Mariposa has addressed audiences at churches and summer camps around the island and was featured on an American radio show entitled Smooth Operator.
According to the author, Journey of the Butterfly started as a book but had become a special project.
A planned visit to Tortola last month was postponed due to Hurricane Irma, but come next March she heads to Trinidad and Tobago to promote her work.
Mariposa has also taken six young females under her wing. The women who are all victims of abuse will be guided on how to live their best lives and to love themselves.


text:
Holiday events at Churchill Museum kick off with Victorian Christmas Nov. 14
Dean Asher | The Fulton Sun | November 7, 2013
For three decades, the National Churchill Museum has been a key part of Fulton's Christmas celebration.
The tradition continues next Thursday, Nov. 14, with their 30th annual Victorian Christmas.
Festivities begin at 10 a.m. with shopping and the traditional English kettledrum tea, with spirits, food and an auction running from 5-7 p.m.
All proceeds benefit the Museum.
"For the original Friends (of the Churchill Museum) group, it was just another way to raise money and do something for the community," said Becky McCue, museum store and visitor services manager. "It's evolved very much from just a few tables and trees."
Jody Paschal, Mid-Missouri Friends of the National Churchill Museum president, said that at 5 p.m. the evening's wine and appetizer party kicks off. The Friends will be auctioning off gift baskets from Elliott's Custom Cabinets, Saults, Smockingbirds, Edward Jones, and Mary Martin Shelter.
But holiday festivities don't stop there. The museum will also host a Christmas tree lighting Dec. 2.
The museum will place a large tree in the neighboring garden, which Fulton Mayor LeRoy Benton will light at around 6:15 p.m. There will also be snacks and crafts for children.


All my life I've sacrificed.
I've focused on being a wife.
I'm going to focus on being a woman.
My son doesn't know. Neither does Jason's mom.
It would be one thing if Jason wasn't my son's best friend.
We have a secret love.
But secrets have a way of coming out when you least expect them.
And love has a way to make people envious.
All they'll want to do is destroy what you have.
They'll want to tear down your happiness.
Do Jason and I have a love that's strong enough to survive?
Or will we lose each other forever?


XENTAURIX is a cross-application platform for the transmission, recording and playback of� TV/ radio and video content via networks and the Internet. The platform combines encoding, streaming, recording, transcoding and repurposing.
Different types of solutions for TV/ radio compliance recording, competition monitoring, ratings analytics, training, education as well as for activity and evidence recoding, IPTV/ OTT purposes can be realized with XENTAURIX.
In addition to the content, all relevant metadata such as EPG, subtitles, bookmarks or third party data are synchronously recorded and can be replayed via search function exact to the second. The content is available as multibit or adaptive stream to all common end devices including smartphones and tablet computers. Users can continuously switch from live into time-shift mode. Numerous signals can be recorded for months or even years - depending on the storage system's size. XENTAURIX systems are truly scalable from one to thousands of channels and process different types of signal sources. Due to the available interfaces and the open database architecture different applications and customer-specific front ends may be realized. We offer a wide range of XENTAURIX products and solutions which cover all common applications.


MHK MarketProminence offers industry-leading enrollment processing, member maintenance, billing and reconciliation solutions that allow health plans to process and reconcile enrollment and payment at the member level, remediate differences to maximize revenue, and track expected revenue. With over 20 years in the industry, MHK MarketProminence is recognized as the leading platform for Medicare Advantage and Part D (MAPD) plans, helping plans meet the increasing complexities of CMS requirements. Collaborative, agile implementation and on-going support coupled with custom-configured workflow management design ensures a highly effective, compliant, and cost-efficient solution where care and knowledge converge.
The MHK MarketProminence Enrollment & Member Maintenance solution comes out-of-the-box compliant with CMS Medicare rules and regulations and can be configured to meet state-specific Medicaid rules and business requirements.
The MHK MarketProminence Financial Reconciliation solution allows plans to reconcile enrollment and payment at the member level. This helps to remediate differences and maximize revenue, as well as track expected revenue moving forward at multiple plan levels.
The MHK MarketProminence Premium Billing solution enables health plans to perform accurate and timely premium billing processing. This solution also serves as an effective audit tool to produce required reporting and ensure compliance.
We understand that health plans need a compliant online enrollment solution that's easy to use. That's why we developed MHK MarketProminence Web Portals to meet the needs of the various types of users who need to apply for benefits online.


I'm still on such a high from last night.
Special thank you to everyone who came out to support!
to my team (lol) ,Hannah and Tiff, who are just too amazing for words... truly.
and maybe, just maybe, our girlfriends are our soulmates.
Special shoutout to everyone who supported - whether you waited in that crazy line for over an hour, or didn't quite make it in... I appreciate that you showed up. I'm so grateful for the love that you all continue to show me and blessed to have such amazing supporters of my art and message. Words can't explain how it feels to have people connect with your art.
To everyone that approached me to tell me something resonated with them, purchased a piece, asked me questions to delve a little deeper into the meaning behind my art or just took a card; the people who have been to EVERY show, continue to believe in me, and those who "got it" way before I even knew what my message was... thank you.


A fact that will doubtless help the RAV4's phenomenal sales growth to continue.
It took just four years for Toyota to double sales of the RAV4. That's a phenomenal achievement and one the Japanese carmaker would love to maintain. Realistically, that rate of growth won't be sustained, but with the reveal of the in New York last week, plans to increase volume were brought to light by Bill Fay, Senior Vice President, Automotive Operations. "The RAV4 has become our sales leader, a very important vehicle for us," Fay said.
Viewing the three models together, it's clear from my mind at least the new XSE Hybrid has been positioned as the most premium of the trio. It looks the best, inside and out, and boasts the best performance. "The hybrid is at 10-12% of the mix now, give or take 40,000 RAV hybrids. We're going to push that to probably 20% of the mix," Fay claimed. When discussing the hybrid system with RAV4 chief engineer Yoshikazu Saeki later in the day, a telling fact was leaked that—once officially confirmed— will attract plenty of interest in the RAV4 Hybrid. "The first reason for choosing this specific setup was to gain the longest driving distance. In between, if you need to extend the range you can fill up with gasoline quite easily," Saeki said.
A Toyota spokesperson was quick to add the range is "not yet confirmed, these are preliminary figures," but the cat is out of the bag. The RAV4 Hybrid will have a range of 600 miles, or thereabouts, on a single fill up. Knowing that, doubling sales of the hybrid variant won't be a challenge at all.