text:
\section{Introduction}
Stochastic dynamic programming incorporates uncertain events into a suitable framework to help the decision-maker to design an optimal plan of action.
A fruitful approach for showing the existence of optimal stationary plans is to prove that the dynamic programming equation admits a unique solution ---the value function--- in a suitable space of functions. See \cite{Blackwell}, \cite{Maitra}, \cite{Furukawa}, \cite{Bertsekas}, \cite {SLP} or \cite{HLLbook}, where this problem is analyzed in detail. There is a huge literature that applies stochastic dynamic programming to economics. \cite{BrockMirman}, \cite{MZ}, \cite{DMehra}, \cite{DD}, \cite{Nyarko}, \cite{HP} or \cite{Mitra} are only a few of the many relevant papers that have contributed to developing this field of research. \cite{OR} makes a review of the contributions to the stochastic optimal growth model.

However, only partial results have been developed so far about the existence and uniqueness of solutions to the dynamic programming equation with arbitrary utility functions and arbitrary shock spaces.
A satisfactory theory of stochastic dynamic programming should include these cases.
\cite{RZRP, RZRP2} developed a method to deal with unbounded utility functions in deterministic problems, the so--called local contraction approach, based on an extension of the Banach contraction principle for function spaces whose topology is defined by a countable family of seminorms.\footnote{\cite{Hadzic} is one of the first papers dealing with this extension. \cite{RZRP}, independently, introduced different hypotheses and applied the results to the deterministic Bellman equation. Recent contributions of the local contraction concept to dynamic programming are \cite{Vailakis}, \cite{MN} and \cite{BRW}.}\
The application of the method to the stochastic case is not straightforward if one wants to dispense with artificial bounds on the exogenous shocks. Usually, the stochastic dynamic programming theory imposes those bounds in the form of a compact shock space. This is for analytical convenience only.\footnote{In general, this assumption is incompatible with modelling the Markov chain as a first order stochastic difference equation, even if the underlying i.i.d. shocks takes only finitely many values. Think, for instance, of the simple random walk. It takes every integer number with positive probability. Thus, its range can only be limited by imposing exogenous constraints, alien to the economic model, a limitation which will modify the ``natural" solution and may distort its real implications.}

There are at least two problems when dealing both with unbounded utility functions and with an unbounded shock space. One of them is technical, related to the integrability of the functions involved and thus, to the correctness in the definition of the Bellman operator. A second problem concerns the choice of a suitable family of seminorms (or pseudodistances) that preserves the monotonicity of the Bellman operator. The local contraction approach in the deterministic Bellman equation constructs a family of compact sets in the endogenous state space to define the seminorms, see \cite{RZRP}. When trying to imitate this in the stochastic Bellman equation for the exogenous state space of shocks, one faces the difficulty that the Bellman equation requires the computation of a conditional expectation. This is an averaging of a random variable on the whole exogenous space, breaking down the monotonicity properties of the Bellman operator.
To overcome the difficulties, we work with an extended concept of contraction parameter(s) in the local contraction definition, to consider an operator that works on the whole family of seminorms. Thus, in this framework, in addition to the selfmap for which we are interested in finding fixed points, there is a companion operator that, acting on the seminorms, plays the role of the contraction parameter of the former selfmap. See Definition \ref{LCOP} below. We state a fixed point theorem, Theorem \ref{Theorem_main}, that applies to this more general framework and we show how it covers previous fixed point results, including the classical Banach ContractionTheorem --- and henceforth, the weighted norm approach in \cite{BoydIII}, or \cite{BB}
--- as well as those based on local contractions, see e.g., \cite{RZRP} and \cite{Vailakis}.
This idea is not new. \cite{KozlovThimTuresson} developed a fixed point theorem in locally convex spaces whose topology is given by a family of seminorms. However, the results obtained depend on the companion contraction parameter operator being linear. The proof in \cite{KozlovThimTuresson}, of the existence and uniqueness of fixed points, heavily exploits the linearity of the companion operator. This precludes the application of the results of this paper to the dynamic programming equation, as it genuinely demands a nonlinear companion contraction parameter operator, due to the presence of a maximization operation\footnote{It is for this reason that we have to develop our own fixed point theorem, departing from the approach of \cite{KozlovThimTuresson}, as we cannot make use of an equality as its formula (4). Instead, we provide an alternative condition, summarized in our assumption (VI) below.} On the other hand, we simplify somewhat the hypotheses made on the companion operator and extend the result to the consideration of arbitrary pseudodistances ---and hence the topological space is not a locally convex space anymore---, which could be useful for analyzing the unbounded from below case, and more importantly, we show how the dynamic programming equation fits well into this framework, attaining, to our knowledge, new existence and uniqueness results.

The present paper devotes a good deal of efforts to isolating a suitable space of functions and a suitable family of seminorms where the approach explained above is successful.
We find a suitable framework, where the averaging needed to compute the conditional expectation does not break down the monotonicity of the Bellman operator. The seminorms that we define combine the usual supremum norm in the endogenous variables, with an $L^1$ norm in the exogenous variables, and construct a complete space of functions for dealing with the Bellman operator --- a Carath\'eodory function space ---.

The paper is organized as follows. Section \ref{Sect:COP} develops a fixed point theorem for operators acting on topological spaces whose topology is given by a family of pseudodistances that makes it Hausdorff and sequentially complete. The operators enjoy a contraction property, materialized in an associated operator acting on the family of pseudodistances, which plays the role of the contraction parameter in Banach's Contraction Theorem. Thus, the result generalizes Banach's Contraction Theorem and shows that the local contraction approach used in previous papers is a particular case of this more general framework. Section \ref{Sect:BO} applies the theorem to the stochastic dynamic programming equation for models with shocks driven by an exogenous Markov chain and with an unbounded shock space. We carefully choose the set of functions where the Bellman operator is defined and we provide a way to methodically construct the objects needed to apply the fixed point theorem developed in Section \ref{Sect:COP}. We also show that the solution of the Bellman equation is the value function.
 In Section \ref{Sect:Appl}, we study a model of endogenous growth ---which encompasses the one sector optimal growth model---,
and the Lucas asset pricing model in an exchange economy, in all cases allowing for correlated and unbounded shocks.\
Section \ref{Sect:Conclusions} establishes the conclusions of the paper, and give some tips for further research. Appendix \ref{App:Proofs} contains the proofs that are not in the main text, with the exception of the proofs regarding the completeness of the function space and seminorms we consider to analyze the dynamic programming equation, which are developed in Appendix \ref{App:FS}.
Appendix \ref{App:Con} discusses the issue of continuity of the Markov operator appearing in the dynamic programming equation, giving sufficient conditions to establish continuity, and providing an example of non--continuity.

\section{Local Contractions}\label{Sect:COP}

Let $(E,{\mathcal D})$ be a topological space, where $E$ is a set whose topology is generated by a saturated family of pseudometrics ${\mathcal D}=\{d_a\}_{a\in A}$, with $A$ an arbitrary index set. Since the family ${\mathcal D}$ is saturated, the topology it generates is Hausdorff\footnote{A pseudometric $d:E\times E\rightarrow\mathbb{R}_+$ is a function satisfying $d(x,y)\ge 0$, $d(x,x)=0$, $d(x,y)=d(y,x)$ and $d(x,z)\le d(x,y)+d(y,z)$ for any $x,y,z\in E$, but $d(x,y)=0$ does not imply $x=y$. The family ${\mathcal D}$ of pseudometrics is saturated if $d_a(x,y)=0$ for all $a\in A$, implies $x=y$. Sometimes, the pseudometrics are defined through seminorms $p_a$, $a\in A$, by $d_a(x,y) = p_a(x-y)$, where now $E$ is a real vector space. A seminorm is a function $p:E\rightarrow \mathbb{R}_+$ that satisfies all the axioms to be a norm, except that $p(x)=0$ does not imply that $x$ is the null vector of $E$. If the family of seminorms is saturated, then the topology defined by the family is Hausdorff and the space and $E$ is a locally convex space. See \cite{Willard} for further details.}. We suppose that $(E,{\mathcal D})$ is sequentially complete: if $\{x_n\}$ is a sequence in $E$ which is Cauchy with respect to all $d_a\in {\mathcal D}$, that is, if $d_a(x_n,x_m)\to 0$ as $n,m\to \infty$, then there is $x\in E$ such that $d_a(x_n,x)\to 0$ as $n\to \infty$ for all $a\in A$.

Given a sequentially complete subset $F\subseteq E$, we study the existence and uniqueness of a fixed point of a mapping $T:F\rightarrow E$.

Let $\mathbb{R}^A$ be the set of functions $d:A\rightarrow \mathbb{R}_+$ and let $\mathbb{R}_+^A$ be the non--negative cone of $\mathbb{R}^A$. On this set we consider the order it generates, that is, for two elements $d,d'\in \mathbb{R}_+^A$, we say that $d\le d'$ if and only if $d(a)\le d'(a)$ for all $a\in A$. The family ${\mathcal D}$ can be embedded into $\mathbb{R}_+^A$, since that, for $x,y\in E$ given, the mapping $a\mapsto d_a(x,y)$ defines a function in $\mathbb{R}_+^A$, that we denote $d^{x,y}(a):=d_a(x,y)$. In general, for a given subset $F\subseteq E$, we let $D(F)$ be the set of functions in $\mathbb{R}^A_+$ which are generated by pairs $x,y\in F$, that is
\[
D(F):=\{d:A\rightarrow \mathbb{R}_+\,:\, d=d^{x,y}\mbox{ for some $x,y\in F$}\}.
\]

\begin{definition}\label{LCOP}
Let $F\subseteq E$. The mapping $T:F\rightarrow E$ is an $L$-local contraction on $F$ with contraction operator parameter $L$ (COP, for short), if there are a set $C\subseteq \mathbb{R}^A_+$ such that $D(F)\subseteq C$, and an operator $L:C\rightarrow \mathbb{R}_+^A$, such that
\[
d_a(T x, T y) \le (L d^{x,y}) (a),
\]
for all $x,y\in F$ and for all $a\in A$.
\end{definition}

Note that the inequality above can be rewritten $d^{Tx,Ty}\le L d^{x,y}$, that is, as an order relation in the space $\mathbb{R}^A_+$.  The definition of $L$--contractions for mappings $T:F\longrightarrow E$, not imposing $T:F\longrightarrow F$, will facilitate the definition of the COP parameter $L$ of the Bellman operator in Section \ref{Sect:BO}. Of course, the property $T:F\longrightarrow F$ is fundamental for Theorem \ref{GeneralDynPro} below, and will checked carefully in Section \ref{Sect:BO}.

The following two examples show that the operator $L$ is a generalization of the concept of contraction parameter of a (local) contraction mapping.
\begin{example}[Banach contractions]
In the classical Banach's Theorem, $E$ is dotted with a complete metric $d$, so the index set $A$ is a singleton, $\mathcal{D}=\{d\}$, and $T$ is a contraction of constant parameter $\beta$, with $0<\beta<1$: $d(Tx,Ty) \le \beta d(x,y)$, for any $x,y\in E$. The COP is $L=\beta I$, where $I$ is the identity map in $\mathbb{R}_+$.

A generalization of the Banach contraction concept is provided in \cite{Wong}, where it is considered  $T:E\longrightarrow E$ for which there is a function $L :\mathbb{R}_+ \longrightarrow \mathbb{R}_+$ satisfying
\begin{equation}\label{Wong}
d(Tx,Ty) \le L  (d(x,y)),
\end{equation}
for all $x,y\in E$. Note that our definition is an extension of this concept to topological spaces whose topology is given by a family of semidistances.
\end{example}
\begin{example}[$k$--local contractions]
Suppose that $A=\mathbb{N}$ is countable. In \cite{RZRP, RZRP1}, we introduced the concept of $k$--local contraction in the study of the deterministic Bellman and Koopmans equations, respectively. A $k$--local contraction on $F$, $k=0,1,2,\ldots$, is a mapping $T:F\subseteq E \longrightarrow E$ satisfying
\[
d_j(T x,T y)\le \beta_j d_{j+k}(x,y)
\]
for some fixed sequence of numbers $\{\beta_j\}_{j\in \mathbb{N}}$ with $0<\beta_j<1$, and for all $x,y\in F$.
If we let $s=\mathbb{R}^{\mathbb{N}}$ be the set of real sequences and $s^+$ be the subset of $s$ of nonnegative sequences, then the COP associated with $T$ is the linear operator $L:s^+ \longrightarrow s^+$ acting on sequences given by
\[
L(d_1,d_2,\ldots,d_j,\ldots)=(\beta_1 d_{1+k},\beta_2 d_{2+k},\ldots,\beta_j d_{j+k},\ldots),
\]
where $k\ge 0$ is fixed.

Suppose that $A=\mathbb{N}$ is uncountable and let a mapping $\alpha: A\longrightarrow E$ such that for any $a\in A$, $d_a\le d_{\alpha(a)}$. \cite{Vailakis} worked with the following generalization of the countable class above: $T:E\longrightarrow E$ is an $\alpha$-local contraction if there exists a function $\beta: A\longrightarrow [0,1)$ such that
\[
d_a(T x,T y)\le \beta (a) d_{\alpha(a)} (x,y).
\]
The COP $L$ acts as follows: given a function $d:A\longrightarrow\mathbb{R}_+$, the image function is $(L d)(a) = \beta(a) d(\alpha(a))$, that is, a translation in the independent variable by $\alpha$, and a multiplication by $\beta$. It turns out that $L$ is also a linear mapping, as in the countable case above.
\end{example}

In what follows, we use the standard notation for successive iterations of the operators $T$ and $L$. For instance, $L^0$ is the identity operator on $C$, $L^1=L$, and for $t\ge 2$, $L^t=L\circ L^{t-1}$.
We impose to $C$, $L$ and $T$ the assumptions (I) to (VI) listed below. The assumptions (I) to (V) concern the behavior of $L$ on the set $C$. Assumption (VI) links directly the operators $T$ and $L$.
\strut

\begin{enumerate}[{(}I{)}]
\item \it $D(F)\subseteq C$. For all $d,d'\in C$, the sum $d+d'\in C$, and any bounded subset of $C$  is countably chain complete\footnote{A subset $S\subseteq C$ is bounded with respect to the order inherited from $\mathbb{R}^A$ if there is $d'\in C$ such that $d\le d'$ for all $d\in S$. The bounded subset $S$ is countably chain complete if for any countably chain $d_1\le d_2\le \cdots d_t\le\cdots$ in $S$, $\sup_{t\in \mathbb{N}} d_t\in S$.} Moreover, if $d'\in C$, $d\in \mathbb{R}_+^A$ and $d\le d'$, then $d\in C$.\rm

\item \it $L0= 0$.\rm

\item \it $L$ is monotone: for all $d,d'\in C$ with $d\le d'$, $L d \le L d'$.\rm

\item \it $L$ is subadditive: for any $d, d'\in C$\rm
\[
L (d+d') \le L d + L d'.
\]
\item \it $L$ is upper semicontinuous sup-preserving\footnote{For instance, the sup-preserving property, $L (\sup_t d_t) = \sup_t L  d_t$, plays a prominent role in the Fixed Point Theorem of Kantorovich-Tarski. In our context, it can be weakened to a kind of upper semicontinuity.}: for any bounded countably chain in $C$,
$d_1\le d_2\le \cdots\le d_t\le \cdots$,
\[
L \sup_t d_t \le  \sup_t L  d_t.
\]\rm
\end{enumerate}

\begin{enumerate}[{(}VI{)}]
\item \it There are $x_0\in F$ and $r:0\in C$ with $d_a(x_0,Tx_0)\le r_0(a)$ and
\[
R_0(a):=\sum_{t=0}^{\infty} L^t r_0(a)<\infty,
\]
for all $a\in A$.
\end{enumerate}

\strut

Since $L^tr_0\in C$, for all $t=0,1,\ldots$, and the countable chain $\{r_0, r_0+Lr_0,\dots,r_0+Lr_0+\cdots +L^t r_0,\cdots\}$ is bounded in $C$ by (VI), $R_0$ is in $C$ by assumption (I).

For $F\subseteq E$, $x_0\in F$, and $m\in \mathbb{R}_+^A$, let the set
\[
V_F(x_0,m)=\{x\in F\,:\, d_a(x_0,x) \le m(a), \text{ $\forall a\in A$}\}.
\]
When $E$ is a metric space, that is, when $A$ is a singleton, the pseudometric is a metric, and $V_F(x_0,m)$ is simply the intersection with $F$ of the closed ball centered at $x_0$ and radius $m$.

\begin{lemma}\label{B_defined}
Let $T:F\longrightarrow F$ be an $L$-local contraction on $F\subseteq E$ and let $x_0\in F$ be such that (I)--(VI) hold true for a suitable $r_0\in C$. Let $R_0$ be defined as in (VI). Then
\begin{enumerate}[{(a)}]
\item $T: V_F(x_0,R_0) \longrightarrow V_F(x_0,R_0)$.
\item For any $a\in A$, $\lim_{t\to \infty} (L^t R_0) (a) = 0$.
\end{enumerate}
\end{lemma}
The following result is a fixed point theorem for $L$-local contractions.
\begin{theorem}\label{Theorem_main}
Let $(E,{\mathcal D})$ be a Hausdorff and sequentially complete topological space. Let $T:F\rightarrow F$ be an $L$-local contraction on the sequentially complete subset $F\subseteq E$ and let $x_0\in F$ be such that (I)--(VI) hold true. Then there is a unique fixed point $x^*\in V_F(x_0, R_0)$ of $T$, which is the limit of any iterating sequence $y_{t+1} = T y_t$, $t=0,1,2,\ldots$, where $y_0=x\in V_F(x_0,R_0)$ is arbitrary.
\end{theorem}
\begin{proof}
Consider first the iterating sequence $x_{t+1} = T x_t$, $t=0,1,2,\ldots$ (that is, the initial seed is $x_0$). By Lemma \ref{B_defined}, $x_t$ is in $V_F(x_0,R_0)$ for any $t=0,1,2,\ldots$. Since $T$ is an $L$-local contraction
\[
d_a(x_{t},x_{t+1}) = d_a(T x_{t-1}, Tx_{t}) \le L d^{x_{t-1},x_{t}} (a)
\]
and by induction $d_a(x_{t},x_{t+1}) \le ( L^t d^{x_0,Tx_0} )(a)$.
Let $r>s\ge 1$. Then by the triangle inequality extended to finite sums
\begin{equation}\label{estimate}
d_a(x_{s},x_{r+1}) \le \sum_{t=s}^{r} d_a (x_t,x_{t+1}) \le  \sum_{t=s}^{r} L^t d^{x_0,Tx_0} (a)\le R_0(a)<\infty,
\end{equation}
for all $a\in A$. By the Cauchy criterion for series, $d_a(x_{s},x_{r+1})$ tends to 0 as $r,s\to \infty$. Since $a$ is arbitrary, the sequence $\{x_t\}_{t=0}^{\infty}$ is Cauchy, hence it converges to some $x^*\in F$, as $F$ is sequentially complete. In fact, $x^*\in V_F(x_0,R_0)$. To see this, note that for all $a\in A$, the mapping $x\mapsto d_a(x_0,x)$ is trivially sequentially continuous in the topology generated by ${\mathcal D}$, hence $d_a(x_0,x^*)=d_a(x_0,\lim_{t\to \infty} x_t)=\lim_{t\to\infty} d_a(x_0,x_t)\le R_0 (a)$ for any $a\in A$. Next, we prove that $x^*$ is a fixed point of $T$. By estimate \eqref{estimate}, letting $r\to \infty$, $d_a(x_{s},x^*) \le \sum_{t=s}^{\infty} L^t d^{x_0,Tx_0} (a) = \sup_N \sum_{t=s}^{N} L^t d^{x_0,Tx_0} (a) $. Hence
\begin{align*}
L d^{x_s,x^*}(a) &\le  L\Big(\sup_N \sum_{t=s}^{N} L^t d^{x_0,Tx_0}\Big) (a) \\
&\le \sup_N L \Big(\sum_{t=s}^{N} L^t d^{x_0,Tx_0}\Big) (a) \\
&\le \sup_N \sum_{t=s}^{N} L^{t+1} d^{x_0,Tx_0} (a),
\end{align*}
which tends to 0 as $s\to \infty$, since $R_0$ is finite. Hence $L d^{x_s,x^*}\to 0$ as $s\to \infty$, for all $a\in A$. The first line above is due to the monotonicity of $L$, the second line since $L$ is upper sup-preserving, and the third one, since $L$ is subadditive. Now, given that
\[
d_a(x^*,Tx^*) \le d_a(x_{s+1},x^*) + d_a(x_{s+1},Tx^*) \le d_a(x_{s+1},x^*) + Ld^{x_s,x^*}(a)
\]
and that both summands tend to 0 as $s\to \infty$, we conclude that $x^*=Tx^*$.
To prove uniqueness, we argue by contradiction, supposing the existence of another fixed point $x^{**}\in V_F(x_0,R_0)$. Then $d_a(x^*, x^{**})\le 2 R_0(a)$ for all $a\in A$, and hence
\[
d_a(x^*, x^{**}) = d_a(T^tx^*, T^tx^{**}) \le L^t d^{x^*, x^{**}} (a) \le 2 L^t R_0 (a),
\]
since $L$ is both monotone and subadditive\footnote{It is easy to prove that $L^t$ is monotone and subadditive for any $t$.}. Taking the limit as $t\to \infty$ and using Lemma \ref{B_defined}, $x^* = x^{**}$ is proven. Finally, let $x\in V_F(x_0,R_0)$ and let the iterating sequence $y_{t+1} = T y_t$, for $n=0,1,2,\ldots,$, with $y_0=x$. Observe that
\[
d_a(x_t,y_t) \le L^t d^{x_0,x}(a) \le 2 L^t R_0(a) \to 0 \text{ as $n\to \infty$,}
\]
by Lemma \ref{B_defined}, so $d_a(y_{t}, x^*)\le d_a(x_t,y_t) + d_a(x_t,x^*)$ tends to 0 as $t\to \infty$.
\end{proof}

The next corollary provides conditions for the uniqueness of the fixed point in $F$ and not only in $V_F(x_0,R_0)$.
When $T$ is indeed an $L$-local contraction on the whole $E$, this result provides global uniqueness of the fixed point on $E$.
\begin{theorem}\label{GlobalU}
Let $(E,{\mathcal D})$ be a Hausdorff, sequentially complete space. Let $T:F\rightarrow F$ be an $L$-local contraction on the sequentially complete subset $F\subseteq E$ and let $x_0\in F$ be such that (I)--(VI) hold true. Suppose that, for any $x\in F$, it is possible to choose $r_0\in C$ satisfying (VI), such that $x\in V_F(x_0,R_0)$. Then there is a unique fixed point of $T$ in $F$ and convergence to the fixed point of successive iterations of $T$ is attained from any $x\in F$.
\end{theorem}
\begin{proof}
By Theorem \ref{Theorem_main}, $T$ admits a unique fixed point $x^*$ in $V_F(x_0,R_0)$, where $R_0=\sum_{t=0}^{\infty}L^t r_0$, for any $r_0\in C$ for which $R_0$ is a convergent series. Suppose, by contradiction, that $T$ admits another fixed point $x^{**}\neq x^*$ in $F$. By assumption, there is $r'_0\in C$ such that $x^{**}\in V_F(x_0,R'_0)$, where $R_0' = \sum_{t=0}^{\infty} L^tr_0'$ is finite. Hence, $x^*=x^{**}$. The convergence of iterating sequences is also an immediate consequence of Theorem \ref{Theorem_main}.
\end{proof}

Next we establish a useful sufficient condition for (VI). Note that the Bellman operator satisfies the extra condition imposed on $L$.
\begin{proposition}\label{SufficientB6}
Let $(E,{\mathcal D})$ be a Hausdorff and sequentially complete topological space. Let $T:F\longrightarrow F$ be an $L$--local contraction on $F\subseteq E$, with COP $L$ satisfying (I) to (V) and
$
L(\alpha d) \le \alpha L d$, for all $d\in C$, for all $ \alpha \in[0,1]$.
Let $x_0\in F$, for which there is $t_0\in \{0,1,2,\ldots\}$, $s\in C$, and $\theta\in[0,1)$ such that
\begin{equation}\label{Lalphad}
L^{t_0} d_0 \le s\quad \mbox{and}\quad Ls\le \theta s,
\end{equation}
where $d_0(a)=d_a(x_0,Tx_0)$. Then (VI) holds with $r_0 = d_0$.
\end{proposition}
\begin{proof}
Note that $\sum_{t=t_0}L^td_0\le s + L s+ L^2s+\cdots\le (1+\theta+\theta^2+\cdots)s=\frac 1{1-\theta}s$. Hence, $\sum_{t=0}^{\infty} L^td_0 =\sum_{t=0}^{t_0-1}L^td_0 + \sum_{t=t_0}^{\infty} L^td_0\le \sum_{t=0}^{t_0-1}L^td_0 +\frac 1{1-\theta}s$ is finite for all $a\in A$.
\end{proof}

\section{Stochastic Dynamic Programming and Bellman Equation}\label{Sect:BO}
Consider a dynamic programming model $(X,Z,\Gamma,Q,U,\beta)$, where
$X\times Z$ is the set of possible states of the system,
$\Gamma$ is a correspondence that assigns a nonempty set $\Gamma(x,z)$ of feasible actions to each state $(x,z)$ and
 $Q$ is the transition function, which associates a conditional probability distribution $Q(z,\cdot)$ on $Z$ to each $z\in Z$. Hence, the law of motion is assumed to be a first-order Markov process, which could be degenerated, giving rise to a deterministic model. We will use indistinctly the notation $Q_z(\cdot)=Q(z,\cdot)$; the function
$U$ is the one--period return function, defined on the graph of $\Gamma$, $\Omega=\{(x,y,z)\,:\, (x,y)\in X\times Z, y\in \Gamma(x,z)\}$, and
$\beta$ is a discount factor.

Starting at some state $(x_0,z_0)$, the agent chooses an action $x_1\in \Gamma(x_0,z_0)$, obtaining a return of $U(x_0,x_1,z_0)$ and the system moves to the next state $(x_1,z_1)$, which is drawn according to the probability distribution $Q(\cdot|z_0)$. Iteration of this process yields a random sequence $(x_0,z_0,x_1,z_1,\ldots)$ and a
total discounted return $\sum_{t=0}^{\infty} \beta^t U(x_t, x_{t+1},z_t)$. A history of length $t$ is $z^t=(z_0,z_1,\ldots, z_t)$. Let $Z^t$ be the set of all histories of length $t$. A (feasible) plan $\pi$ is a constant value $\pi_0\in X$ and a sequence of measurable functions $\pi_t : Z^t \longrightarrow X$, such that $\pi_t(z^t)\in \Gamma(\pi_{t-1}(z^{t-1}),z_t)$, for all $t=1,2,\ldots$. Denote by $\Pi(x_0,z_0)$ the set of all feasible plans starting at the state $(x_0,z_0)$. Any feasible plan $\pi\in \Pi(x_0,z_0)$, along with the transition function $Q$, defines a distribution $\mathbb{P}^{\pi,(x_0,z_0)}$ on all possible futures of the system $\{(x_t,z_t)\}_{t=1}^{\infty}$, as well as the expected total discounted utility
\[
u(\pi,x_0,z_0) = \mathbb{E}^{\pi,(x_0,z_0)}\left(\sum_{t=0}^{\infty} \beta^t U(x_t,x_{t+1},z_t)\right).
\]
The expectation $\mathbb{E}^{\pi,(x_0,z_0)}$ is taken with respect to the distribution $\mathbb{P}^{\pi,(x_0,z_0)}$.
The problem is then to find a plan $\pi\in \Pi(x_0,z_0)$ such that $u(\pi,(x_0,z_0))\ge u(\widehat \pi,(x_0,z_0))$ for all $\widehat \pi \in \Pi(x_0,z_0)$, for all $(x_0,z_0)\in X\times Z$. The value function of the problem is
$v(x_0,z_0)=\sup_{\pi\in\Pi(x_0,z_0)} u(\pi, (x_0,z_0))$.

Consider the functional equation corresponding to the above dynamic programming problem as stated in \cite{SLP}.
For $x\in X$, $z\in Z$
\begin{equation}\label{Bellmaneq}
v(x,z)=\max _{y\in \Gamma(x,z)} \left\{U(x,y,z) + \beta \int_Z v(y,z')Q(z,dz')\right\}.
\end{equation}

A solution of the Bellman equation satisfying additional assumptions is the value function of the infinite programming problem. This is the content of Theorem \ref{GeneralDynPro} below, whose proof needs
the notion of the probability measure $\mu^t$ defined on the sequence space of shocks $(Z^t, \mathcal{Z}^t)$ for finite $t=1,2,\ldots$, where
\[
(Z^t, \mathcal{Z}^t) = (Z\times \cdots\times Z, \mathcal{Z}\times \cdots \times\mathcal{Z})\quad (\mbox{$t$ times}).
\]
For any rectangle $B=A_1\times \cdots\times A_t\in \mathcal{Z}^t$, $\mu^t$ is defined by
\[
\mu^t(z_0,B) = \int_{A_1}\ldots\int_{A_{t-1}}\int_{A_t} Q_{z_{t-1}}(dz_t) Q_{z_{t-2}}(dz_{t-1}) \cdots Q_{z_{0}}(dz_1),
\]
and by the Hahn Extension Theorems, $\mu^t(z_0,\cdot)$ has a unique extension to a probability measure on all of $\mathcal{Z}^t$. We omit the details, that can be found in \cite{SLP}, Section 8.2, whose presentation we follow closely

Defining the Bellman operator in a suitable function space $E$, such that for $f\in E$
\[
(Tf)(x,z) = \max _{y\in \Gamma(x,z)} \left\{U(x,y,z) + \beta \int_Z f(y,z')Q(z,dz')\right\},
\]
the Bellman functional equation \eqref{Bellmaneq} is a fixed point problem for $T$. This fixed point problem is completely understood for the case where $U$ is bounded. There are now also different approaches for some special cases for unbounded $U$. It is worth mentioning the constant returns to scale model and the logarithmic and the quadratic parametric examples analyzed in \cite{SLP}, pp. 270--280, and the weighted norm approach in \cite{BoydIII} and \cite{HLLbook}. One feature of all these approaches is that they consider a bounded (or compact) space of shocks, an assumption that we want to dispense with.\footnote{The weighted norm approach presents some limitations, which are explained, for instance, in Remark 9 of \cite{MN}. This paper constitutes a first attempt to translate the approach initiated by \cite{RZRP} for deterministic programs to the stochastic case. However, the results obtained do not cover a general model where shocks are driven by an exogenous transition probability. In fact, in the class of dynamic programming models described here and in \cite{SLP}, assumption (A4) on the generation of shocks $\{z_t\}_{t=0}^{\infty}$ imposed in \cite{MN}, basically implies that the space of shocks $Z$ is compact, or that the underlying probability has compact support.} Allow for a non--compact shock space is important for a qualitative analysis of models, see for instance \cite{BinderPesaran} and \cite{Stach}, and more recently, \cite{MaStachurski}.

We now impose the standing hypotheses. Most are taken from \cite{SLP}, but there are essential differences, as we admit an unbounded utility $U$ and an unbounded shock space $Z$.
\begin{enumerate}[{(B}1{)}]
\item $X\subseteq \mathbb{R}^l$,  $Z\subseteq \mathbb{R}^k$ are Borel sets, with Borel $\sigma$-algebra $\mathcal{X}$ and $\mathcal{Z}$, respectively. The set $X$ is endowed with the Euclidean topology.
\item $0<\beta<1$.
\item $Q:Z\times \mathcal{Z}\rightarrow [0,1]$ satisfies
\begin{enumerate}
\item for each $z\in Z$, $Q(z,\cdot)$ is a probability measure on $(Z,\mathcal{Z})$; and
\item for each $B\in \mathcal{Z}$, $Q(\cdot,B)$ is a Borel measurable function.
\end{enumerate}
\item The correspondence $\Gamma : X \times Z \longrightarrow X$ is nonempty, compact-valued and continuous.
\item $U: \Omega \longrightarrow \mathbb{R}$ is a Carath\'eodory function, that is, it satisfies
\begin{enumerate}
\item for each $(x,y)\in D := \{(x,y)\in X\times Y\,:\, \exists z\in Z, \ y\in \Gamma (x,z)\}$, the function of $z$, $U(x,y,\cdot): Z \longrightarrow \mathbb{R}$ is Borel measurable;
\item for each $z\in Z$, the function of $(x,y)$, $U(\cdot, \cdot,z):D\longrightarrow \mathbb{R}$ is continuous.
\end{enumerate}
\end{enumerate}

The reason for working with Carath\'eodory functions instead of continuous functions in the three variables $(x,y,z)$ is twofold. On the one hand, the Markov operator
\begin{equation}\label{MarkovO}
(Mf) (x,z) := \int _Z f(x,z') Q(z,dz'),
\end{equation}
does not preserve continuity of $f$, if $f$ is continuous but not bounded, as the simple example in Appendix \ref{App:Con} shows

On the other hand, the Bellman operator is well defined for the class of Carath\'eodory functions in the unbounded case, while working with the supremum norm is not possible. A direct attack of the Bellman equation in the space of $(x,z)$--continuous functions does not work for unbounded functions and/or unbounded shock space: known theorems on local contractions---with a countable or uncountable index set ---are not suitable, due to the averaging operation involved in the computation of conditional expectations. For this reason
we are going to use $L^1$-type seminorms, whose precise definition is given below.

We now describe the function space, which details are given in Appendix \ref{App:FS}.
For each $z\in Z$, let $L^1(Z,\mathcal{Z},Q_z)$ be the space of Borel measurable functions\footnote{It is well known that $L^1(Z,\mathcal{Z},Q_z)$ consists of equivalence classes rather than functions, identifying functions that are equal $Q_z$--almost everywhere.} $g:Z\longrightarrow \mathbb{R}$ such that
$
 \int_Z |g(z')|Q_z(dz')<\infty
$.
In what follows, we let $\mathcal{K}$ be the family of all compact subsets of $X$.

Consider the space $E:= \mathcal{L}^1(Z;C(X))$, formed by Carath\'eodory functions $f:X\times Z\longrightarrow  \mathbb{R}$ such that the function $z'\mapsto \max_{x\in K} |f_x(z')|$ is in $L^1(Z,\mathcal{Z},Q_z)$, for all compact sets $K\in \mathcal{K}$, and all $z\in Z$.
See Appendix \ref{App:FS} for the definitions and the notation, where it is also proved the following fundamental result.

\begin{lemma}\label{lemma:function_space}
$E= \mathcal{L}^1(Z;C(X))$ is a complete locally convex space with the topology generated by the family of seminorms $\mathcal{P}:=\{p_{K,z}\}_{K\in \mathcal{K}, z\in Z}$, given by
\begin{equation}\label{pKz}
p_{K,z} (f) :=\int_Z  \max_{x\in K} |f(x,z')| Q_z(dz').
\end{equation}
\end{lemma}
In particular, the lemma states that $E$ is sequentially complete.
In the notation of Section \ref{Sect:COP}, the index set of the family of seminorms is $A=\mathcal{K}\times Z$.

Given a solution $f\in \mathcal{L}^1(Z;C(X))$ of \eqref{Bellmaneq}, define the policy correspondence $G^f:X\times Z \rightarrow 2^X$ by
\begin{equation}\label{Gamma*}
G^f(x,z)=\{y\in \Gamma(x,z)\,:\, f(x,z)=U(x,y,z) + \beta Mf(y,z)\}.
\end{equation}
This is the optimal policy correspondence, denoted simply by $\Gamma^*$, when $f$ is the value function, $v$.

Remember from Section \ref{Sect:COP}, that for a subset $F \subseteq E$, the set $D(F)$ is in this context
\[
D(F)=\{p:\mathcal{K}\times Z\rightarrow \mathbb{R}_+\,:\, p(K,z)=p_{K,z}(f) \mbox{ for some $f\in F$}\}.
\]
\begin{notation}\label{notation1}Along the paper, we will use the notation
\[
\psi(x,z) = \max_{y \in \Gamma(x,z)} U(x,y,z) = T0(x,z)
\]
while, for $p:\mathcal{K}\times Z \longmapsto \mathbb{R}_+$, the function  $p[\Gamma]: X\times Z \longmapsto \mathbb{R}_+$ is defined by $p[\Gamma](x,z)= p(\Gamma(x,z),z)$, that is, it is the function of $(x,z)$ obtained through $p$, when the compact sets $K$ equal $\Gamma(x,z)$, for $x\in X$, $z\in Z$.
\end{notation}

The next result shows that $T$ is an $L$-local contraction, and gives the expression of $L$: Given $p:\mathcal{K}\times Z \mapsto \mathbb{R}_+$ for which $p[\Gamma]\in \mathcal{L}^1(Z;C(X))$, the operator $L$ computes the seminorm of the function $p[\Gamma]$, that is, $(Lp) (K,z) = \beta p_{K,z} (p[\Gamma])$. Note that $L$ is \emph{nonlinear}. The expanded definition of the operator $L$ is the expression \eqref{eq:LCOP} below.

\begin{proposition}\label{prop:LCOP}
Let the Bellman operator $T:F\longrightarrow E$, where $F\subseteq \mathcal{L}^1(Z;C(X))$, such that for all $p\in D(F)$, $p[\Gamma]\in \mathcal{L}^1(Z;C(X))$. Then, $T$ is
an $L$-local contraction on $F$ with COP $L:D(F)\longrightarrow \mathbb{R}_+^{\mathcal{K}\times Z}$ given by
\begin{equation}\label{eq:LCOP}
(Lp)(K,z) = \beta \int _Z  \max_{x\in K} p(\Gamma(x,z'),z') Q_z(dz'),
\end{equation}
for all $K\in \mathcal{K}$ and $z\in Z$.
\end{proposition}
\begin{proof}
Following \cite{Blackwell}, we exploit the fact that $T$ is monotone, in conjunction with the properties of the seminorms $p_{K,z}$. Let $f,g\in E$ and let $x\in X$, $K\in \mathcal{K}$ and $z\in Z$. Let $y\in \Gamma (x,z)$ and $z'\in Z$ arbitrary. Then $f(y,z')\le g(y,z') + |f(y,z')-g(y,z')|$ implies
$
f(y,z')\le g(y,z') + \max_{y\in \Gamma(x,z)} |f(y,z')-g(y,z')|
$
and then, by monotonicity and linearity of the integral,
\begin{align*}
\int_Z f(y,z')  Q_z(dz') \le &\int_Z g(y,z')  Q_z(dz') \\
&\quad + \int_Z \max_{y\in \Gamma(x,z)} |f(y,z')-g(y,z')| Q_z(dz').
\end{align*}
We are allowed to take the integral by Lemma \ref{welldefined}.
The inequality is maintained after multiplying by $\beta$ and adding $U(x,y,z)$ to both sides. Then, by taking the maximum in $y\in \Gamma (x,z)$ to both sides, we have
\begin{align*}
(Tf)(x,z) &\le (Tg)(x,z) + \beta \max _{y\in \Gamma(x,z)} \int_Z \max_{y\in \Gamma(x,z)} |f(y,z')-g(y,z')| Q_z(dz')\\
& =  (Tg)(x,z) + \beta \int_Z \max_{y\in \Gamma(x,z)} |f(y,z')-g(y,z')| Q_z(dz')\\
& =  (Tg)(x,z) + \beta p_{\Gamma(x,z),z}(f-g).
\end{align*}
Exchanging the roles of $f$ and $g$, we have
\[
|(Tf)(x,z)-(Tg)(x,z)|\le \beta  p_{\Gamma(x,z),z}(f-g).
\]
It is convenient to write this inequality with the dummy variable $z'$ instead of $z$. Now, taking the maximum in $x\in K$ and averaging with respect to the measure $Q_z$, we obtain
\[
\int_Z \max_{x\in K} |(Tf)(x,z')-(Tg)(x,z')|Q_z(dz') \le \beta \int_Z \max_{x\in K} p^{f-g}(\Gamma(x,z'),z')  Q_z(dz').
\]
This inequality can be rewritten $
p_{K,z}(Tf-Tg) \le (L p^{f-g}) (K,z)
$,
for all $K\in \mathcal{K}$, $z\in Z$, where $L$ is the operator defined in \eqref{eq:LCOP}.
\end{proof}

One of the difficulties in applying contraction techniques to the dynamic programming equation, when the return function and/or the space of shocks is unbounded, is the selection of a suitable space of functions where the Bellman operator is a selfmap. Assumption (B6) below provides a scheme to construct such a space along the lines of assumption (VI) in Section \ref{Sect:COP}.
This is in the same spirit of Assumption 9.3 in \cite{SLP}, pp. 248-249.\
Our assumption is not about bounding the one-shot utility function $U$ along any policy path by a function that depends only on time and the initial state, but about bounding its expected value with respect to the initial state. This is an important difference, as it allows us to deal with an unbounded space of shocks.

\strut

\begin{enumerate}[{(B}6{)}]
\item
There is a collection of nonnegative functions $\{l_t\}_{t=0}^{\infty}\in \mathcal{L}^1(Z;C(X))$, such that for all $x\in X$, for all $z\in Z$
\[
\begin{array}{l}
 l_0(x,z) \ge |\psi(x,z)|;\\[1ex]
l_{t+1}(x,z)\ge
\beta {\displaystyle \int_Z} \max_{y\in \Gamma(x,z)} l_t(y,z')Q_z(dz'),\quad \mbox{for all $t=0,1,\ldots$,}
\end{array}
\]
and the series $w:= \sum_{t=0}^{\infty}  l_t$
is unconditionally convergent, that is,
\[
R(K,z):=\sum_{t=0}^{\infty}  p_{K,z} (l_t) <\infty,
\]
for all $K\in \mathcal{K}$, for all $z\in Z$.
\end{enumerate}

\strut

Now we consider a suitable set $C$ where $L$ is defined.
\begin{equation}\label{SetC}
\begin{aligned}
C=\Big\{p:\mathcal{K}\times Z \longmapsto \mathbb{R}_+\,: \,p(K,z) \le c R_0(K,z) \mbox{ for some $c>0$,}&\\
\mbox{ and } p[\Gamma]\in \mathcal{L}^1(Z,C(X))\Big\}.&
\end{aligned}
\end{equation}
As it is proved in Lemma \ref{lemma:C}, $C$ is not trivial, as it contains the images of $V(0,R_0)$ by the family of seminorms $\mathcal{P}$.

Theorem \ref{GeneralDynPro} below is a fixed point theorem for the Bellman operator with unbounded utility and unbounded space of shocks. We state a previous lemma.
\begin{lemma}\label{Preparation_Th}
Let assumptions (B1) to (B6) hold. Then $T$ and $L$ with $C$ defined in \eqref{SetC}, satisfy (I) to (VI).
\end{lemma}

\begin{theorem}\label{GeneralDynPro}
Let assumptions (B1) to (B6) hold. The following is true.
\begin{enumerate}[{(a)}]
\item The Bellman equation admits a unique solution $v^*$ in $V(0,R_0)$
\item If the correspondence $G^{v*}$ defined in \eqref{Gamma*} admits a measurable selection, then $v^*$ coincides with the value function, $v=v^*$, and for all $v_0\in V(0,R_0)$, $T^nv_0\to v$ as $n\to \infty$, that is, $p_{K,z}(T^nv_0-v) \to 0$, for all $K\in \mathcal{K}$ and $z\in Z$. Moreover, for all $z\in Z$, the optimal policy correspondence $\Gamma^*(\cdot,z): X\rightarrow X$ is non-empty, compact valued and upper hemicontinuous.
\end{enumerate}
\end{theorem}
\begin{proof} (a) $T$ is an $L$--contraction by Proposition \ref{prop:LCOP} and all the assumptions of Theorem \ref{Theorem_main} hold true by Lemma \ref{Preparation_Th}. Hence $T$ admits a unique fixed point $v^*$ is $V(0,R_0)$ and the rest of conclusions of Theorem \ref{GeneralDynPro} hold true.

(b)
To see that $v^*$ is the value function of the problem, we invoke Theorem 9.2 in \cite{SLP}.
Recall that, for any function $F$ that is $\mu^t(z_0,\cdot)$-integrable, its conditional expectation can be expressed as
\begin{align*}
{\mathrm E}_{z_0}(F) &:= \int_{Z^t} F(z^t) \mu^t(z_0,dz^t)\\
& = \int_{Z^{t-1}}\left[ \int_Z F(z^{t-1},z_t) Q_{z_{t-1}}(dz_t)\right]\mu^{t-1}(z_0,dz^{t-1})\\
& =  \int_Z\left[ \int_{Z^{t-1}}F(z_1,z_2^t) \mu^{t-1}(z_1,dz_2^t)\right]Q_{z_0}(dz_1).
\end{align*}
The assumptions of Theorem 9.2 in \cite{SLP} are: (i) $\Gamma$ is non-empty valued, with a measurable graph and admits a measurable selection; (ii) for each $(x_0,z_0)$ and each feasible plan $\pi$ from $(x_0,z_0)$, $U(\pi_{t-1}(z^{t-1}),\pi_t(z^t),z_t)$ is $\mu^t(z_0,\cdot)$-integrable, $t=1,2,\ldots$, and the limit
\begin{equation}\label{Limii}
U(x_0,\pi_0,z_0) + \lim_{n\to \infty} \sum_{t=1}^{n} \int_{Z^t} \beta ^t U(\pi_{t-1}(z^{t-1}),\pi_t(z^t),z_t)\mu^t(z_0,dz^t)
\end{equation}
exists; and (iii) $\lim_{t\to \infty}  \int_{Z^t} \beta ^t v^*(\pi_{t-1}(z^{t-1}), z_t)\mu^t(z_0,dz^t) = 0$.

(i) is implied by (B5) and (ii) is implied by (B6), since $|U(\pi_{t-1}(z^{t-1}),\pi_t(z^t),z_t)|$ is clearly measurable, given that $U$ is a Carath\'eodory function. Moreover, since $\l_0$ in (B6) is in $\ca{X\times Z}$, we can apply Fubini's Theorem so that $l_0(\pi_{1}(z^1),z_2)$ is $\mu^2(z_0,\cdot)$-integrable and
\begin{align*}
\int_{Z^2} l_0(\pi_{1}(z^1),z_2)\mu^2(z_0,dz^2) &= \int_{Z^1}\left(\int_Zl_0 (\pi_{1}(z^1),z_2)Q_{z_1}(dz_2)\right)\mu^1(z_0,dz^1)\\
&\le \int_{Z^1}\frac 1{\beta} l_1 (\pi_{0}(z_0),z_1)\mu^1(z_0,dz^1)\\
&\le \frac 1{\beta^2} l_2(x_0,z_0).
\end{align*}
Both inequalities are due to assumption (B6). By induction, we get that $l(\pi_{t-1}(z^{t-1}),z_t)$ is $\mu^t(z_0,\cdot)$--integrable and
\[
\int_{Z^t} l_0(\pi_{t-1}(z^{t-1}),z_t)\mu^t(z_0,dz^t) \le \frac 1{\beta^t} l_t(x_0,z_0).
\]
Since
$ |U(\pi_{t-1}(z^{t-1}),\pi_t(z^t),z_t)|\le  l_0(\pi_{t-1}(z^{t-1}),z_t)$, the first part of (ii) is proved. Indeed, this estimate provides the bound
\begin{align*}
&|U(x_0,\pi_0(z_0),z_0)| + \sum_{t=1}^{n}  \int_{Z^t} \beta ^t |U(\pi_{t-1}(z^{t-1}),\pi_t(z^t),z_t)|\mu^t(z_0,dz^t)\\
&\quad  \le |U(x_0,\pi_0(z_0),z_0) | + \sum_{t=1}^{n} l_t(x_0,z_0)\le w_0(x_0,z_0),
\end{align*}
hence the second part of (ii) also holds, that is, the limit \eqref{Limii} is finite. Moreover, since the above inequality holds for any $\pi\in \Pi(x_0,z_0)$, it shows that the $n$--th iteration of $T$ on the null function as the initial seed satisfies $|T^n0(x_0,z_0)|\le w_0(x_0,z_0)$. Hence, since $\int_Z |T^n0(x_0,z_1) - v^*(x_0,z_1)|Q_{z_0}(dz_1)$ tends to 0 as $n\to \infty$, by part (a) above, we obtain the bound
\begin{equation}\label{v*}
\int_Z |v^*(x_0,z_1)|Q_{z_0}(dz_1) \le \int _Z w_0(x_0,z_1)Q_{z_0}(dz_1).
\end{equation}
This inequality will be used to show (iii). First, we claim that for any $t$, for any $\pi\in \Pi(x_0,z_0)$,
\[
\int_{Z^t} \beta^t w_0(\pi_{t-1}(z^{t-1}),z_t) \mu^t(z_0,dz^t)\le \sum_{s=t}^{\infty} l_s(x_0,z_0).
\]
To prove it, we employ mathematical induction. Let $t=1$. Then, by assumption (B6)
\begin{align*}
\int_{Z} \beta w_0(\pi_{0}(z_0),z_1) \mu^1(z_0,dz^1) &=\int_Z  \beta  \sum_{t=0}^{\infty}  l_t(\pi_{0}(z_0),z_1) Q_{z_0}(dz_1) \\
&= \sum_{t=0}^{\infty} \beta   \int_Z l_t(\pi_{0}(z_0),z_1) Q_{z_0}(dz_1)\\
&\le  \sum_{t=0}^{\infty} l_{t+1}(x_0,z_0).
\end{align*}
The exchange of the integral and infinite sum is possible by the Monotone Convergence Theorem. Suppose that the property is true for $t$ and let us prove it for $t+1$. Then it will hold for any $t$. Note
\begin{align*}
\int_{Z^{t+1}} \beta^{t+1} &w_0(\pi_{t}(z^{t}),z_{t+1}) \mu^{t+1}(z_0,dz^{t+1})\\
 &=
\int_Z \left(\beta \int_{Z^t} \beta^t w_0(\pi_{t-1}(z^{t-1}),z_t) \mu^t(z_0,dz^t)\right) Q_{z_0}(dz_1)
\\
&\le \int_Z \beta \sum_{s=t}^{\infty} l_s(\pi_{0}(z_0),z_1) Q_{z_0}(dz_1)\\
&\le \sum_{s=t+1}^{\infty} l_s(x_0,z_0),
\end{align*}
again by the Monotone Convergence Theorem, and where we have used Fubini's Theorem and the induction hypothesis. This and \eqref{v*} imply (iii), since the series $w_0$ converges. Thus, $v^*$ is the value function.
The claims about $\Gamma^*$ are
immediate from the Theorem of the Maximum of Berg\'e.
\end{proof}

The following result provides a sufficient condition for (B6).
\begin{proposition}\label{prop:l0r0}
Let assumptions (B1) to (B5) to hold. Suppose that there is $l_0\in \mathcal{L}^1(Z;C(X))$ with $|\psi|\le l_0$, $\alpha\ge 0$ such that $\alpha \beta <1$, and
\[
\int_Z \max_{y\in \Gamma(x,z)} l_0(y,z')Q_z(dz') \le \alpha l_0(x,z),
\]
for all $x\in X$, $z\in Z$.
Then (B6) holds, with $R_0(K,z)= \frac 1{1-\alpha\beta} p_{K,z}(l_{0})$.
\end{proposition}
\begin{proof}
Choose $l_{t} = (\alpha\beta)^t l_0$, for $t=0,1,\ldots$. Then
\begin{align*}
\beta \int_Z \max_{y\in \Gamma(x,z)} l_t(y,z')Q_z(dz')  &=\beta  (\alpha\beta)^t\int_Z  \max_{y\in \Gamma(x,z)}l_0(y,z')Q_z(dz')\\
& \le  (\alpha\beta)^{t+1}l_0(x,z) = l_{t+1}(x_0,z_0).
\end{align*}
Hence, $w(x_0,z_0)=\frac 1{1-\alpha\beta} l_0(x_0,z_0)$ and $R_0(K,z)=\frac 1{1-\alpha\beta}p_{K,z}(l_0)$, for $K\in\mathcal{K}$ and $z\in Z$.
\end{proof}

\section{Applications}\label{Sect:Appl}

\subsection{Endogenous growth}\label{Sect:Endgrowth}
Endogenous growth models have become fundamental to understand economic growth. From the huge literature studying this field, few contributions consider an unbounded shock space. Some exceptions are \cite{Stach} and \cite{Kami}, but with uncorrelated shocks. I consider here the stochastic endogenous growth model studied in \cite{JMSS}, which is described as follows. The preferences of the agent over random consumptions sequences are given by
\begin{equation}\label{end0}
\max\  {\mathrm E}\sum_{t=0}^{\infty} \beta^t\frac{c_ t^{1-\sigma} \upsilon (\ell_t)}{1-\sigma},
\end{equation}
subject to
\begin{align}
\label{end1}&c_t + k_{t+1} + h_{t+1} \le z_t A k_t^{\alpha}(n_th_t)^{1-\alpha} + (1-\delta_k) k_t +  (1-\delta_h) h_t,\\
\label{end2}&\ell_t + n_t \le 1,\\
\label{end3}&c_t,k_t, h_t,\ell_t, n_t \ge 0
\end{align}
for all $t=0,1,\ldots$, with $k_0$ and $h_0$ given. Here, $\{z_t\}$ is a Markov stochastic process with transition probability $Q_z(\cdot)$ and $Z=[1,\infty)$; $c_t$ is consumption; $\ell_t$ is leisure; $n_t$ is hours spent working; $k_t$ and $h_t$ are the stock of physical and human capital, respectively; $\delta_k$ and $\delta_h$ are the depreciation rates on physical and human capital, respectively; and $\upsilon$ is a continuous function on $(0,1]$, strictly increasing. The usual non-negativity constraints on consumption, investment, leisure and hours worked apply. The feasible correspondence is thus
\begin{align*}
\Gamma(k,h,z)=\Big\{(k',h',c,n,\ell)\,:\, \mbox{\eqref{end1}--\eqref{end3} hold with $x'=x_{t+1}$, $x=x_t$}&\\
\mbox{ for $x=k,h,c,n,\ell,z$}\Big\}&,
\end{align*}
and the utility function is $U(c,\ell)= \frac{c^{1-\sigma} \upsilon (\ell)}{1-\sigma}$. Regarding the function $\upsilon$, we consider $\upsilon(\ell) = \ell ^{\psi (1-\sigma)}$.
The endogenous state space is $X=\mathbb{R}_+\times \mathbb{R}_+$ and the family of compact sets $\mathcal K$ is formed by compact sets in the product space $\mathbb{R}_+\times \mathbb{R}_+$.
The Markov chain is given by the log--log process
\begin{equation}\label{lnz}
\ln{z_{t+1}} = \rho \ln{z_t} + \ln{w_{t+1}},
\end{equation}
with $\rho\ge 0$ and where the $w$'s are i.i.d., with support in $W\subseteq [1,\infty)$. Let $\mu$ be the distribution measure\footnote{With correlated shocks, the method developed in \cite{MN} would require $\mu(z'\in Z_{j+1}|Z_j=z)=1$ for a suitable increasing family $\{Z_j\}_{j=1}^{\infty}$ of compact sets that fills $Z$. We do not impose this strong constraint on $\mu$. In fact, \eqref{lnz} do nos satisfy it if $W$ is unbounded and $\mu$ has not compact support.} of the $w$'s. Note that $\rho = 0$ corresponds to shocks $z_t$ that are i.i.d.. \cite{JMSS} suppose that $z_t=\exp{\left(\zeta_t-\frac{\sigma_{\epsilon}^2}{2(1-\rho^ 2)}\right)}$, where $\zeta_{t+1} = \rho \zeta_t + \epsilon_{t+1}$ and the $\epsilon$'s are i.i.d., normal with mean 0 and variance $\sigma^2_{\epsilon}$. This corresponds to \eqref{lnz} with\footnote{Since, from \eqref{lnz}, $z_{t+1} = z_t ^{\rho} w_{t+1}$, it is clear that, to keep $z\ge 1$, it is necessary (and sufficient) to have $w\ge 1$. Thus, the assumption that the random variable $\epsilon $ is normally distributed with mean 0 should be modified to fulfill the requirement that the random variable $w$ has support $W$ in $[1,\infty)$. The assumption $z\ge 1$ is usually imposed in growth models with a multiplicative structure, see \cite{SLP}.} $w_{t+1} = \exp{\left(\epsilon_{t+1} - \frac {\sigma_{\epsilon}^2}{2(1+\rho)}\right)}$.
We do not need to restrict $\epsilon$ to be normally distributed.
To shorten notation, let us define $\delta = \min\{\delta_k,\delta_h\}$, $\gamma = A \alpha^{\alpha} (1-\alpha)^{1-\alpha} + (1-\delta)$, and $g(k,h) = A k^{\alpha}(nh)^{1-\alpha} + (1-\delta) (k + h)$. Also, let $\Theta = {\mathrm E}{(w^{\frac{1-\sigma}{1-\rho}})} = \int_W w^{\frac{1-\sigma}{1-\rho}} \mu(dw)$.
\begin{theorem}\label{prop:end_bounded}
Consider the endogenous growth model described in \eqref{end0}--\eqref{lnz} with $0\le \sigma <1$ and $0\le \rho <1$.
If
\begin{equation}\label{OptG_condition}
\beta \gamma^{1-\sigma} \Theta  <1,
\end{equation}
then the associated Bellman equation admits a unique solution, $v^*$, in the set $V(0,R_0)$, where, for $K\in \mathcal K$ and $z\in Z$
\[
R_0(K,z) = \left(\frac{\Theta}{1-\beta\gamma^{1-\sigma} \Theta }\right)  z^{\frac{\rho(1-\sigma)}{1-\rho}} \,\max_{(k,h)\in K} g(k,h)^{1-\sigma}.
\]
Moreover, $v^*$ is the value function $v$ and
$p_{K,z}(T^nv_0 - v)$ converges to 0 as $n\to \infty$, for all $K\in \mathcal K$, $z\in Z$ and all initial guess $v_0\in V(0,R_0)$.
\end{theorem}
\begin{proof} We check all the hypotheses of Theorem \ref{GeneralDynPro}. It is clear that (B1)--(B5) are fulfilled. Regarding (B6), we will use Proposition \ref{prop:l0r0} for a suitable function $l_0$. Since $0\le \sigma <1$, both $U$ and $\upsilon$ are bounded from below by zero, and $\upsilon$ is bounded above by 1.
Since $z\ge 1$ and by the definition of $\delta$, we have
$
 zA k^{\alpha}(nh)^{1-\alpha} + (1-\delta_k) k + (1-\delta_h)h\le zg(h,k)
$.
Then
\[
\psi(k,h,z) \le \frac 1{1-\sigma}z^{1-\sigma}g(k,h)^{1-\sigma}\le  \frac 1{1-\sigma} z^{\frac{1-\sigma}{1-\rho}} g(k,h)^{1-\sigma} = l_0(k,h,z).
\]
Let us prove that $\beta \int_Z \widehat l_0(k,h,z,z')Q_z(dz')\le \alpha l_0(k,h,z)$, for all $(k,h)\in K$, for all $z\in Z$, and for all $K\in \mathcal{K}$, where $\alpha = \gamma^{1-\sigma} \Theta $. Here, to simplify notation in what follows, we have defined
\[
\widehat l_0(k,h,z,z')= \max_{(k',h',c,n,\ell)\in \Gamma(k,h,z)} l_0(k,h,z).
\]
First, we determine a bound for $\widehat l_0$.
To this end, consider the Lagrange problem
\begin{equation}\label{Lagrange}
\begin{aligned}
&\max \ g(k',h')^{1-\sigma},\\
&\mbox{s. t.: } k'+h' \le z g(k,h),\\
&\qquad k',h'\ge 0,
\end{aligned}
\end{equation}
and notice that its feasible set is larger than $\Gamma(k,h,z)$. The constraint is binding at the optimal solution, which is $k' = \alpha z g(k,h)$, $h'=(1-\alpha) z g(k,h)$. Substituting this into the objective function of \eqref{Lagrange}, we find its optimal value, $\gamma^{1-\sigma}  z^{1-\sigma} g(k,h)^{1-\sigma}$. Thus, $\widehat l_0(k,h,z,z') \le \frac 1{1-\sigma} (z')^{\frac{1-\sigma}{1-\rho}}\gamma^{1-\sigma}z^{1-\sigma} g(k,h)^{1-\sigma}$. Second, we use the conditional expectation $\int_Z (z')^{\frac{1-\sigma}{1-\rho}}Q_z(dz') = z^{\rho \frac{1-\sigma}{1-\rho}}\Theta $ to estimate
\begin{align*}
\int_Z \widehat l_0(k,h,z')Q_z(dz')& \le \gamma^{1-\sigma} z^{1-\sigma} z^{\rho \frac{1-\sigma}{1-\rho}}\Theta  g(k,h)^{1-\sigma}\\
& = \gamma^{1-\sigma}\Theta  l_0(k,h,z).
\end{align*}
Since $\beta \gamma^{1-\sigma} \Theta <1$, Proposition \ref{prop:l0r0} applies. The expression for $R_0$ requires a simple computation.
\end{proof}

\subsection{Asset Prices in an Exchange Economy}\label{sect:Exchange}
\cite{Lucas} studied the determination of equilibrium asset prices in a pure exchange economy in a framework that has become classical in the economics and financial literature. Boundedness of the utility function, as well as compactness of the space of shocks, are important hypotheses in the development of this model. In this section, we show that these hypotheses can be dispensed with by using the results of Theorem \ref{GeneralDynPro}. In this way, we significantly extend the model's range of applicability.

We closely follow \cite{SLP} in the exposition of the problem.
The preferences of the representative consumer over random consumption sequences are
\begin{equation}\label{ULucas}
{\mathrm E}\ \sum_{t=0}^{\infty} \beta ^t u(c_t),
\end{equation}
where $u:\mathbb{R}_+\longrightarrow \mathbb{R}_+$ is continuous, not necessarily bounded, with\footnote{If $u$ does not satisfy $u\ge 0$ and $u(0)=0$, but is bounded from below, it may be modified to $u(c)-u(0)$ to fulfill our hypotheses.} $u(0)=0$ and where $\beta\in (0,1)$. There are $i=1,\ldots,k$ productive assets taking values on a set $Z\subseteq \mathbb{R}^k_+$, not necessarily compact, with Borel sets $\mathcal{Z}$. The components $z_i$ of the vector $z=(z_1,\ldots,z_k)^{\top}$ in $Z$ represents the dividend paid by one unit of asset $i$. In the description of the model, all vectors are considered column vectors, and the symbol $^{\top}$ denotes transposition. We assume that the dividends follow a Markov process, with stationary transition function $Q$ on $(Z,\mathcal{Z})$. Assets are traded on a competitive stock market at an equilibrium price given by a stationary continuous price function $p:Z\longrightarrow \mathbb{R}^k_+$, where $p(z) = (p_1(z),\ldots,p_k(z))^{\top}$ is the vector of asset prices if the current state of the economy is $z$ (the notation for prices should not be confused with the notation for seminorms, which always carry a subindex). The goal is to characterize equilibrium asset prices. Let $x=(x_1,\ldots, x_k)^{\top}\in \mathbb{R}^k_+$ be the vector of the consumer's asset holdings. Given the price function $p$, the initial state of the economy $z_0$ and initial asset holdings $x_0$, the consumer chooses a sequence of plans for consumption and end-of-period asset holdings that maximizes discounted expected utility \eqref{ULucas} subject to
\begin{align}
&c_t + x^{\top} _{t+1}p(z_t) \le  x_t^{\top} (z_t + p(z_t))  \quad \mbox{for all $z^t$, for all $t$,}\label{Lconst1}\\
&c_t, x_{t+1} \ge 0\quad \mbox{for all $z^t$, for all $t$.}\label{Lconst2}
\end{align}
The consumer holds exactly one unit of each asset in equilibrium, hence we can restrict the state space to $X=[0,\overline x]^k$, with $\overline x >1$, with its Borel subsets $\mathcal{X}$. The correspondence $\Gamma:X\times Z \longrightarrow 2^X$ is
\[
\Gamma(x,z) =\{y\in X\,:\, y^{\top} p(z)  \le x^{\top} (z+p(z))\}.
\]
Assuming that $p$ is continuous, $\Gamma$ is nonempty, compact valued and continuous.
Given the price $p$, the dynamic programming equation is
\begin{equation}\label{LucasDP}
v(x,z) = \max_{y\in \Gamma(x,z)} \Big\{u(x^{\top} z  + (x-y)^{\top}p(z)) + \beta \int_Z v(y,z') Q_z(dz')\Big\}.
\end{equation}
We will look for solutions to this functional equation in the class $\mathcal{L}^1(Z;C(X))$. Since the state space is compact, and the utility function $u$ is bounded from below, we take the trivial family of compact sets $\mathcal{K}=\{X\}$ in this model, and not the whole family of compact subsets of $X$.

We impose the following assumptions.
\begin{align}
&u \mbox{ is nondecreasing and concave}\label{uLucas};\\
&p_i(z) \le a_i^{\top}  z + b_i \mbox{ for some vectors  $a_i\ge 0$ and scalars $b_i> 0$},\label{pLucas}\\
& \mbox{$i=1,\ldots,k$}\nonumber.
\end{align}
Hence, we look for equilibrium prices in the class of functions that are bounded by an affine function. Other possibilities could obviously be explored. We state two results about the existence of equilibrium in a Lucas asset pricing model satisfying \eqref{uLucas} and \eqref{pLucas} under two different regimes for the Markov chain

\begin{enumerate}[{(}{M}1{)}]
\item
The Markov chain is given by $z_{t+1} =  Bz_t + w_t$, where $B$ a matrix of order $k$ with non--negative entries and norm\footnote{The norm of a matrix $B$ is defined by $\|B\| = \sup\left\{\frac{\|Bx\|}{\|x\|}\,:\, x\in \mathbb{R}^l\mbox{ with $x\neq 0$}\right\}$. The condition $\|B\|<1$ is equivalent to saying that the spectral radius of $B$---the maximum of the module of the eigenvalues of $B$---is less than one.} $\|B\|<1$, and where $\{w_t\}_{t=1}^{\infty}$ are i.i.d. random vectors with support in a Borel subset $W\subseteq \mathbb{R}_+^k$ with finite expectation, $0\le {\mathrm E}{w}<\infty$.
\item
The Markov chain is given by $z_{i,t+1} = z_{i,t} ^{\rho_i} w_{i,t+1}$, for all $t=0,1,2,\ldots$, where $0\le \rho_i \le 1$ for all $i=1,\ldots,k$, and where $\{w_t\}_{t=1}^{\infty}$ are i.i.d. random vectors\footnote{Hence, we are now considering a linear log--log system of uncoupled equations for the evolution of dividends.} with support in a Borel subset $W\subseteq [1,\infty)^k$ such that
\begin{equation}\label{mulLucas}
\rho_i \le 1 \mbox{ for all } i=1,\ldots,k \mbox{ and, if $\rho_i=1$, then }{\mathrm E}{w_i} < 1/\beta.
\end{equation}
\end{enumerate}

In the proof that follows, as well as in the rest of the paper, we use the same notation for inequalities between scalars and inequalities between vectors, which have to be understood in a pointwise manner.

\begin{theorem}\label{th:Lucas1}
Consider the Lucas Asset Pricing model described above in \eqref{ULucas}-\eqref{Lconst2}, for which \eqref{uLucas} and \eqref{pLucas} hold and the Markov chain satisfies either (M1) or (M2). Then there is a unique solution of \eqref{LucasDP} in $V(0, R_0)$ for a suitable $R_0$, which is the value function $v$ of the problem, and the conclusions of Theorem \ref{GeneralDynPro} hold.
\end{theorem}
\begin{proof} Let $A$ be the matrix whose columns are the vectors $a_1,\ldots,a_k$ in \eqref{pLucas}, and let $b=(b_1,\ldots,b_k)^{\top}$, hence we can write $0\le p(z)\le Az + b$.
Let us construct a family of functions $\{l_t\}_{t=0}^{\infty}$ satisfying assumption (B6). Obviously
\[
x^{\top} (z+p(z)) \le \overline x ^{\top} z(I_k + A)  + \overline x ^{\top} b,
\]
where $I_k$ is the indentity matrix. Since $u$ is increasing and concave, for a supergradient $\overline u$ of $u$ at $\overline x ^{\top} b>0$, we have
\[
\psi (x,z) = u(x^{\top} (z+p(z)) \le u(\overline x^{\top} b) + \overline u\; \overline x^{\top} (I_k+A)z.
\]
We define $l_0(z) =  u(\overline x^{\top} b) + \overline u\; \overline x^{\top} (I_k+A) z$ and,  recursively, $l_{t+1}(z) = \beta \int_Z l_t(z')Q(z,dz')$, for $t=0,1,\ldots$.

Suppose first that $Q$ satisfies (M1). In this case, ${\mathrm E}_{z}{z'} = Bz + {\mathrm E} {w}$, hence
\begin{align*}
l_1(z) &= \beta \int_Z l_0(z')  Q(z,dz')\\
& =   \beta  u(\overline x^{\top} b) + \beta \overline u\; \overline x^{\top}  (I_k+A) {\mathrm E}_{z}{z'}\\
& = \beta (  u(\overline x^{\top} b) + \overline u\; \overline x^{\top} (I_k+A) (Bz+{\mathrm E}{w}).
\end{align*}
We will prove by induction that
\begin{align*}
l_{t}(z) &= \beta^{t} u(\overline x^{\top} b) + \beta^{t}  \overline u\; \overline x^{\top} (I_k+A) \big(B^tz + (B^{t-1}+\cdots +  I_k ){\mathrm E}{w}\big),
\end{align*}
for all $t=1,2,\ldots$. For $t=1$ it has been just proved. Suppose it is true for $t$. Then
\begin{align*}
l_{t+1} &= \beta^{t+1} u(\overline x^{\top} b) +  \beta^{t+1}  \overline u\; \overline x^{\top} (I_k+A) \big(B^t{\mathrm E}_z{z'} + (B^{t-1}+\cdots + I_k ){\mathrm E}{w}\big)\\
&=\beta^{t+1} u(\overline x^{\top} b) \\
&\quad + \beta^{t+1}  \overline u\; \overline x^{\top} (I_k+A) \big(B^{t}(Bz+{\mathrm E}{w})
+ (B^{t-1}+\cdots + I_k ){\mathrm E}{w}\big),
\end{align*}
and we are done. On the other hand, $(B^{t-1}+\cdots +  I_k ){\mathrm E}{w}\le  (I_k -B)^{-1}{\mathrm E}{w}$, since $B$ has nonnegative entries, $\|B\|<1$ and ${\mathrm E}{w}>0$. Hence, the series
$w_0(z) = \sum_{t=0}^{\infty}l_t(z)$ is unconditionally convergent, since it is bounded by the function of  $L^1(Z)$ defined by
\[
\overline w_0(z) := \frac 1{1-\beta} (u(\overline x^{\top} b)  + \beta \overline u\; \overline x^{\top}(I_k+A)  \big((I_k - \beta B)^{-1} z + (I_k - B) ^{-1} {\mathrm E}{w} \big),
\]
where we have used $\sum_{t=0}^{\infty} (\beta B)^t = ( I_k - \beta B)^{-1}$.
Hence (B6) holds with $R_0(z)=\sum_{t=0}^{\infty}p_z(l_t)$.

If $Q$ satisfies (M2), then ${\mathrm E}_z{z'}= (z_1^{\rho_1}{\mathrm E}{w_1},\ldots,z_k^{\rho_k}{\mathrm E}{w_k})^{\top}$. Define, as above, $l_0(z) =  u(\overline x\cdot b) + \overline u\; \overline x\cdot z (I_k+A)$ and let $l_{t+1}(z) = \beta \int_Z l_t(z')Q(z,dz')$, for $t=0,1,\ldots$. Then it is easy to prove by induction that
\[
l_{t} (z) = \beta^{t} u(\overline x^{\top} b) + \beta^{t}  \overline u\; \overline x^{\top} (I_k+A) \left(z_1^{\rho_1^t}\Pi_{s=0}^{t-1}{\mathrm E}{\left(w_1^{\rho_1^s}\right)},\ldots,z_k^{\rho_k^t}\Pi_{s=0}^{t-1}{\mathrm E}{\left(w_k^{\rho_k^s}\right)}\right)^{\top}.
\]
By Jensen's inequality, ${\mathrm E}{\left(w_i^{\rho_i^s}\right)} \le ({\mathrm E}{w_i})^{\rho_i^s}$ and thus
\[
l_{t}(z) \le \beta^{t} u(\overline x^{\top} b) + \beta^{t}  \overline u\; \overline x^{\top} (I_k+A) \left(z_1^{\rho_1^t}\left({\mathrm E}{w_1}\right)^{1/(1-\rho_1)},\ldots, z_k^{\rho_k^t}\left({\mathrm E}{w_k}\right)^{1/(1-\rho_k)}\right)^{\top}.
\]
In the case that $\rho_i<1$ for all $i=1,\ldots, k$, the series $w_0(z) = \sum_{t=0}^{\infty} l_t(z)$ is clearly (unconditionally) convergent since $\beta <1$ and $\rho_i<1$ for all $i=1,\ldots,k$. The ratio test can be used to prove this claim. In the case in which $\rho_j=1$ for some $j$, then the bound above no longer applies, as a term $z_j ({\mathrm E}{w_j})^t$ appears in position $j$ of the vector $\left(z_1^{\rho_1^t}\Pi_{s=0}^{t-1}{\mathrm E}{w_1^{\rho_1^s}},\ldots,z_k^{\rho_k^t}\Pi_{s=0}^{t-1}{\mathrm E}{w_k^{\rho_k^s}}\right)^{\top}$. However, the assumption $\beta {\mathrm E}{w_j}<1$ assures convergence of the series $w_0(z) = \sum_{t=0}^{\infty} l_t(z)$.

Hence, in both cases considered, (M1) and (M2), the condition \eqref{mulLucas} guarantees that (B6) holds with $R_0(z)=\sum_{t=0}^{\infty}p_z(l_t)$. Thus, Theorem \ref{GeneralDynPro} applies.
\end{proof}

To complete the circle, we have to prove that our conjecture \eqref{pLucas} about the equilibrium price holds. Following \cite{Lucas} or \cite{SLP}, we now assume the further conditions:
\begin{align}
\label{eqLucas1}
&\mbox{$u(0)=0$, $u$ is continuously differentiable, with $u'(c)>0$ for all $c\ge 0$,}\\
&\mbox{and strictly concave;}\nonumber
\end{align}
we also impose
\begin{align}
\label{eqLucas3}
&\mbox{there are constants $\gamma,\delta \ge 0$ such that } cu'(c) \le \gamma c + \delta,  \mbox{ for all $c\ge 0$;}\\
& \mbox{there exists $a>0$ such that }u'(\mathbf{1}^{\top}z) \ge a \mbox{ for all $z\in Z$, where $\mathbf{1}=(1,\ldots,1)^{\top}$}.\label{eqLucas4}
\end{align}

A function like $u(c) = c^{1-\sigma}/(1-\sigma) + c$, with $0\le \sigma <1$, satisfies \eqref{eqLucas1}-\eqref{eqLucas4}. Also, if $Z$ is bounded, then \eqref{eqLucas1} implies \eqref{eqLucas4}.

Finding an equilibrium price function $p(z)=(p_1(z),\ldots,p_k(z))^{\top}$ is equivalent to finding functions $\phi_1(z),\ldots, \phi_k(z)$ that satisfy the $k$ independent functional equations
\begin{equation}\label{Lucasphi}
\phi_i(z) = h_i(z) + \beta \int _Z \phi_i(z') Q(z,dz'), \quad i=1,\ldots,k,
\end{equation}
where $h_i(z)=\beta \int_Z z'_i u'(\mathbf{1}^{\top}z')  Q(z,dz')$, for all $i =1,\ldots,k$.
\cite{Lucas} shows that a solution to \eqref{Lucasphi} provides an equilibrium price $p$ given by
\begin{equation}\label{priceLucas}
p_i(z) = \frac{\phi_i(z)}{u'(\mathbf{1}^{\top} z )},\quad \mbox{for $i=1,\ldots,k$}.
\end{equation}
Let, as in \cite{Lucas}, the operator $T_i$ be
\[
T_if (z) = h_i(z) + \beta \int _Z f(z') Q(z,dz'),\quad \mbox{for all } f\in L^1(Z,\mathcal{Z},Q_z),\quad  i=1,\ldots,k.
\]
Note that, in this context, the seminorms are simply defined by $p_z(f) = \int_Z |f(z')|Q_z(dz')|$. It is pretty clear that the COP associated to $T_i$ is given by
\[
Lp (z) = \beta \int_Z p(z')Q_z(dz'),
\]
where $p$ belongs to a suitable set $C$ as defined in \eqref{SetC}.

\begin{theorem}\label{prop:Lucas1}
Consider the Lucas Asset Pricing model described above in \eqref{ULucas}-\eqref{Lconst2}, for which \eqref{eqLucas1}--\eqref{eqLucas4} hold and the Markov chain satisfies either (M1) or (M2). Then there is an equilibrium price $p$ satisfying \eqref{pLucas}.
\end{theorem}
\begin{proof}
Note that $z\ge 0$ and \eqref{eqLucas3} imply
\begin{equation}\label{hineq}
z_i u'(\mathbf{1}^{\top} z)  \le (\mathbf{1}^{\top} z) u'(\mathbf{1}^{\top} z)   \le \gamma (\mathbf{1}^{\top} z) + \delta.
\end{equation}
Suppose that $Q$ satisfies (M1). Then
\begin{align*}
h_i(z) &\le \beta  \left(\gamma \left(\mathbf{1}^{\top} {\mathrm E}_z{z'}\right) + \delta\right)
\le
\beta  \left(\gamma \left(\mathbf{1}^{\top} (Bz+{\mathrm E}{w})\right) + \delta\right).
\end{align*}
This implies that the operator $T_i$ is a self--map in $L^1(Z)$, for all $i=1,\ldots,k$.  We want to apply Theorem \ref{GeneralDynPro} to each of the operators $T_i$, where the COP associated to $T_i$ is given just above the theorem.
Let $l_0(z)=h_i(z)$ and define
\[
l_t(z) = \beta^t  \left(\gamma \left(\mathbf{1}^{\top} (B^t z+{\mathrm E}{w})\right) + \delta\right),
\]
for $ t=1,2,\ldots$.
It is immediate to check that $l_{t+1}\ge \beta \int_Z l_t(z')Q(z,dz')$ and that the series $w_0(z)=\sum_{t=0}^{\infty} l_t(z)$ is (unconditionally) convergent, since $\beta \|B\| <1$. The sum of this series is $w_0(z)=\gamma \mathbf{1}^{\top} (I_k-\beta B)^{-1}z  + \frac 1{1-\beta}(\gamma \mathbf{1}^{\top} {\mathrm E} w + \delta)$. Hence, (B6) holds. Moreover, following analogous reasonings as in the proof of part (b) of Theorem \ref{GeneralDynPro}, the fixed point of $T_i$, $\phi_i$, satisfies $\phi_i\le w_0$, and thus, by \eqref{eqLucas4}
\[
p_i(z) = \frac{\phi_i(z)}{ u'(\mathbf{1}^{\top} z)} \le \frac 1a \gamma \mathbf{1} ^{\top} (I_k-\beta B)^{-1} z + \frac 1{a(1-\beta)}(\gamma \mathbf{1}^{\top} {\mathrm E} w  + \delta),
\]
for all $i=1,\ldots,k$, where the right hand side is an affine function of $z$. Then, $p=(p_1,\ldots,p_k)^{\top}$ satisfies \eqref{pLucas} with
\[
a_i^{\top} = \frac 1a \gamma \mathbf{1} ^{\top} (I_k-\beta B)^{-1},\quad b_i = \frac 1{a(1-\beta)}(\gamma \mathbf{1}^{\top} {\mathrm E} w  + \delta),\mbox{ for all $i=1,\ldots,k$}.
\]

Suppose that $Q$ satisfies (M2). Now ${\mathrm E}_z{z'}= (z_1^{\rho_1}{\mathrm E}{w_1},\ldots,z_k^{\rho_k}{\mathrm E}{w_k})^{\top}$. From \eqref{hineq}, we have
\[
h_i(z)\le \beta(\gamma (\mathbf{1}^{\top} (z_1^{\rho_1}{\mathrm E}{w_1},\ldots,z_k^{\rho_k}{\mathrm E}{w_k}))+\delta).
\]
Let $l_0(z)=h_i(z)$ and $l_{t+1}(z) = \beta \int_Z l_t(z')Q(z,dz')$, for $t=0,1,\ldots$.
When $0\le \rho_i<1$ for all $i=1,\ldots,k$, using similar arguments as in the proof of Theorem \ref{th:Lucas1}, we have
\begin{align*}
l_t(z) &\le \beta^t \gamma \mathbf{1}^{\top} \left(z_1^{\rho_1^t} ({\mathrm E}{w_1})^{1/(1-\rho_1)},\ldots, z_k^{\rho_k^t}({\mathrm E}{w_k})^{1/(1-\rho_k)}\right)+\beta^t\delta \\
&\le \beta^t (\gamma \mu \mathbf{1}^{\top} z + \delta),
\end{align*}
where $\mu:=\max\left\{z_1({\mathrm E}{w_1})^{1/(1-\rho_1)},\ldots, z_k({\mathrm E}{w_k})^{1/(1-\rho_k)}\right\}$.
Hence, the infinite series $w_0(z)=\sum_{t=0}^{\infty} l_t(z)=\frac {\gamma \mu \mathbf{1}^{\top} z + \delta}{1-\beta} $ is (unconditionally) convergent, (B6) holds and the fixed point of $T_i$, $\phi_i$, satisfies $\phi_i\le w_0$. It is clear then that the price $p=(p_1,\ldots,p_k)^{\top}$ defined in \eqref{priceLucas} satisfies \eqref{pLucas} with  $a_i = \frac {\gamma \mu}{a(1-\beta)}$ and $b_i = \frac{\delta}{a(1-\beta)}$, for all $i=1,\ldots,k$. In the case in which some $\rho_j=1$, the coordinate $j$ on the vector  $\left(z_1^{\rho_1^t}\Pi_{s=0}^{t-1}{\mathrm E}{(w_1^{\rho_1^s})},\ldots,z_k^{\rho_k^t}\Pi_{s=0}^{t-1}{\mathrm E}{(w_k^{\rho_k^s})}\right)$ is equal to $z_j {\mathrm E}{w_j}$, and then $\beta {\mathrm E}{w_j}<1$ is required to have convergence of the series $\sum_{t=0}^{\infty} l_t(z)$, which is then bounded by an affine expression in $z$; hence, as in the previous case, \eqref{pLucas} holds.
\end{proof}

\section{Conclusions}\label{Sect:Conclusions}
In this paper, we develop a general framework to analyze stochastic dynamic problems with unbounded utility functions and unbounded shock space. We obtain new results concerning the existence and uniqueness of solutions to the Bellman equation through a fixed point theorem that generalizes the results known for Banach contractions and local contractions. This generalization is possible by considering seminorms that give a different treatment to the endogenous state variable and the exogenous one. While a supremum norm on arbitrary compact sets is considered in the former variable, an $L^1$ type norm is in the latter variable. Putting together this definition with the aforementioned generalization of the local contraction concept, we are able to maintain the monotonicity (in a mild sense) of the Bellman operator, thus proving that it is essentially a contractive operator. The usefulness of the approach and the applicability of the results are clearly revealed in the analysis of two fundamental models of economic analysis: an endogenous growth model with a multiplicative structure in the shocks and the Lucas model of an exchange economy. The combination of unbounded rewards and unbounded shocks makes it hard to prove the existence of a unique fixed point of the Bellman equation. In this sense, another benefit of the paper is to provide a secure method to check the hypotheses needed to apply the approach, based on assumption (B6), and one that can be used straightforwardly to analyze other models. A challenging problem is to extend the theorems to deal with the unbounded from below case in a more satisfactory way, as done in \cite{RZRP} or \cite{Vailakis} for the deterministic case, by introducing a suitable family of pseudodistances.

\newpage

redpajama_set_name: RedPajamaArXiv


text:
Brookfield Shooting Raises Concerns About Restraining Orders
The tragic shooting deaths of four women in Brookfield on Sunday raise troubling questions about how to enforce restraining orders in domestic violence cases.
There were some warning signs before 45-year-old Radcliffe Haughton took his gun to the Azana Salon and spa in Brookfield and allegedly killed his wife Zina and...
Kewaunee Nuke Plant to Close
Owners of the Kewaunee nuclear power plant say the plant will permanently close next year, and the site will begin a decommissioning process.
The Virginia-based owner of the Kewaunee plant, Dominion Resources has been trying to sell the facility for months. But there have been no buyers. Dominion managers say...
Early Voting Underway in Wisconsin
Early, in-person absentee voting begins today throughout Wisconsin. Municipal clerks are expecting a lot of traffic.
In the last few presidential elections Wisconsin voters had three full weeks and three weekends to vote early at their local clerk's office. The legislature changed that and this year the window is just two...
Money Pouring into 7th Congressional District Race
Political candidates in northern and central Wisconsin's 7th Congressional District have raised more money than any other district in the state except one. But both candidates say they can't control the role money plays in politics.
Republican freshman Congressman Sean Duffy has raised twice as much money as his Democratic challenger...
Thompson Not Sure His Medicare Plan Would Save Money
Republican U.S. Senate Candidate Tommy Thompson says he doesn't know whether his plan to rework Medicare will save the government any money.
While health care has dominated the discussion in the race for U.S. Senate, Thompson didn't start talking about this plan until just a few weeks ago. The way...
Nuke Plant Closure to Cut Kewaunee Jobs
A small northeastern Wisconsin county is losing its largest private employer. The owners of the Kewaunee nuclear power plant announced it will go dark in six months.
The Kewaunee plant employs 400 people, plus around 200 contractors. The plant sits on the shores of Lake Michigan and is owned by...
Senate Candidates Step Up Campaigning in Final Two Weeks
With two weeks to go until Election Day, Wisconsin's U.S. Senate candidates were on the campaign trail Monday to kick off early voting in the state. They each spent the time criticizing their opponent.
Republican Tommy Thompson made two stops in Green Bay, one as part of a National Federation of...
Studying Why Some Shy Away from Math and Science
Researchers from the UW-Madison and University of Colorado-Boulder will use a $4 million grant from the National Science Foundation to try to figure out why students shy away from math or science.
For decades there's been concern that the U.S is losing its edge as a global leader in science and...
Medical Examiner Says Gunman's Wife Among Those Killed
The Waukesha County Medical Examiner's Office says 42-year-old Zina Haughton of Brown Deer was among those killed in a shooting rampage at a day spa in Brookfield yesterday. It says 45-year old Radcliffe Haughton was also among the dead. Police say he shot seven people before taking his own life.
Report Card Day for Wisconsin Public Schools
The Department of Public Instruction will unveil new reports cards for Wisconsin public schools Monday.
The report cards will rate each school on a scale of 0 to 100. Schools will be classified in one of five categories based on their numerical rating ranging from significantly exceeding expectations to failing...
Hoping to Fight Violent Crime Through Brain Research
A researcher at UW-Madison hopes the evidence from the brain scans of convicted psychopaths at a Wisconsin prison may someday lead to effective treatment for a disorder that often leads to violent crimes.
In a bus equipped with an MRI machine parked at the Fox Lake prison psychiatry professor Mike Koenigs...
Younger Sturgeon Spearers Permitted
Twelve year-olds will be able to take part in next winter's sturgeon spearing season on Lake Winnebago and up-river lakes. A citizens' council approved the change.
The deadline to buy a $20 spearing permit is October 31. As the DNR says, 2013 will be the first year "in modern history"...
Gasoline Prices Fall; Diesel Stays Put
While the average price of gas in Wisconsin has dropped to $3.67 a gallon, the price of diesel remains above $4.00.
Industry analysts at Gasbuddy.com say the cost of diesel is as high as $4.54 per gallon in western Wisconsin. Senior Petroleum Analyst Patrick DeHaan says it's a sign of the...
Group Releases Concerns About WEDC
The Wisconsin Public Interest Research Group weighed in Wednesday on oversight problems at the Wisconsin Economic Development Corporation. The group released a report documenting a decline in public accountability for the agency's loans and grants programs.
The WISPIRG report cites the groups numerous efforts to get a full accounting of the...
Dems Want More Transparency from WEDC
The Democratic co-chair of the legislature's Audit Committee says she'll introduce legislation next year that calls for more transparency in Wisconsin's new economic development agency.
When the legislature and governor created the quasi-private Wisconsin Economic Development Corporation, or WEDC, they billed it as an agency with a singular focus on...
First Lady Stumps for President in Wisconsin
First Lady Michelle Obama is in Wisconsin today. She's in Racine and Wausau, campaigning for her husband's reelection, and encouraging people to take advantage of early voting.
Firing up supporters in Racine, First Lady Michelle Obama said the difference between winning and losing in a battleground state like Wisconsin could be...
Millions Flow into Northern Wisconsin Congressional Race
The race to unseat Wisconsin's two freshmen Republican congressman is drawing millions of dollars, according to the latest campaign finance reports. Although the incumbents have raised twice as much as their challengers, the Democrats are racking up impressive amounts too.
First-term Congressman Reid Ribble has raised two-million dollars and some...
Wisconsin ROTC Cadet in Vietnam
Forty years ago, an American military officer visiting Hanoi might have been thrown in prison. But this summer, a Wisconsin cadet from the Army's Reserve Officer Training Program spent a month in the Vietnamese capitol teaching English to her counterparts.
Nineteen year old Rachel Liskar of New Berlin traveled to Vietnam...
In Lac du Flambeau: Most Who Will Vote, Will Vote for Obama
Voters in Lac du Flambeau who live on the reservation plan to vote for President Obama, if they are going to vote at all. That wasn't true for those at the Lake of the Torches casino who were either on vacation visiting the area or gambling.
These photos are part...
Amid Partisan Rancor, People Remain Lovely
WPR reporters Maureen McCollum and Lindsey Moon have spent the last week talking with voters along Route 51, hearing what they think about the presidential election. Here are some closing thoughts from the road.
One family-sized bag of M and M's, 245 photos, 6-hours, 39-minutes and 44 miliseconds of audio, and...

redpajama_set_name: RedPajamaCommonCrawl


text:
Yale-NUS saga: Academic freedom can't be carte blanche for misusing academic institutions for political advocacy, says Ong Ye Kung
A proposed Yale-NUS College module titled Dissent and Resistance was cancelled by the college about two weeks before it was scheduled to begin.PHOTO: ST FILE
Rei Kurohi
Oct 8, 2019, 1:15 am
Oct 7, 2019, 1:12 pm SGT
https://str.sg/J3Je
SINGAPORE - The worry that institutes of higher learning (IHLs) may be used to conduct partisan political activities to sow dissent against the Government is not unfounded, said Minister for Education Ong Ye Kung.
The Ministry of Education (MOE) had this concern when it looked at the itinerary of a proposed Yale-NUS College module titled Dissent and Resistance, Mr Ong told Parliament on Monday (Oct 7). He said the ministry was informed of the college's decision to cancel the module and fully supported the decision.
The module, which would have been run by Singaporean poet and playwright Alfian Sa'at, was cancelled by the college on Sept 13, about two weeks before it was scheduled to begin.
Mr Ong said the programme would have involved making protest placards and visiting the Speaker's Corner in Hong Lim Park.
It would also have included dialogues with personalities like Mr Jolovan Wham and Mr Seelan Palay - who have both previously been convicted of public order-related offences - and Ms Kirsten Han and Dr Thum Ping Tjin of the New Naratif media platform, which receives "significant foreign funding", he said.
Mr Ong quoted from a 1998 poem by Mr Alfian titled "Singapore You Are Not My Country", as well as other statements he had made in the past that compared Malaysia favourably against Singapore.
Mr Alfian had also spoken about "a revival of student activism in Singapore, especially in areas such as political conscientisation" in a Facebook post over the weekend, Mr Ong said.
"The term 'political conscientisation' comes from radical left-wing thought. It means agitation aimed at making people conscious of the oppression in their lives, so that they will take action against these oppressive elements," he said. "I think this is how Mr Alfian saw his project."
The minister told Parliament that those responsible for the programme are entitled to their views and feelings about Singapore.
"They can write about them, even vent them on social media," he said.
"But we have to decide whether we allow such forms of political resistance free rein in our educational institutions, and even taught as compulsory, credit-bearing programmes."
Guiding Principles for Educational Institutions
Why Yale-NUS course on dissent was scrapped: Ong Ye Kung
Parliament: Educational institutions should not expose students to risk of breaking law, says Ong Ye Kung
Parliament: Liberal education can make Singaporeans better, more active citizens, says Nominated MP Walter Theseira
Responding to questions filed by Dr Intan Azura Moktar (Ang Mo Kio GRC), Mr Seah Kian Peng (Marine Parade GRC) and Nominated MP Walter Theseira, he said the cancellation has drawn many comments.
While a few disagreed, most academics, including those from Yale-NUS, supported the cancellation.
Many members of the public also supported the decision, but for a "simpler and more fundamental" reason. "They did not see why inciting and teaching students to protest should be condoned in our educational institutions," Mr Ong said. "This is a valid view that we cannot ignore."
The minister said some may argue that academic freedom grants universities the licence to run such programmes, in the spirit of critically engaging the minds of their undergraduates. A few may go even further to claim that dissent is good for democracies, and hence, so is teaching students to become dissidents.
"I much prefer the test of an ordinary Singaporean exercising his common sense," he said. "He would readily conclude that taking into consideration all the elements and all the personalities involved, this is a programme that was filled with motives and objectives other than learning and education."
"MOE's stand is we cannot have such activity in our schools or institutes of higher learning. Political conscientisation is not the taxpayer's idea of what education means."
Mr Ong reassured the House that the ministry values academic freedom.
"We believe in this fundamental value. Modern-day Galileos would not exist without our academics and researchers being free to pursue the truth, wherever it may lead," he said.
"But let us also be aware that, given the state of the world today, there will be people who want to misuse it as a cloak to advance their hidden agendas," he said.
"To preserve what we cherish, we must be ready to protect it when the situation calls for it. Academic freedom cannot be carte blanche for anyone to misuse an academic institution for political advocacy, for this would undermine the institution's academic standards and public standing."
Alfian Sa'at responds to report on cancelled dissent module at Yale-NUS College
Decision to cancel Yale-NUS module on dissent made without government interference: Yale president
Singapore does not need a 'colour revolution'
He added that Singapore's autonomous universities (AUs) have always been places where different ideas are explored and debated, and where public discourse is carried out vigorously and rigorously. This is why a liberal arts school like Yale-NUS will have a place in the Singapore education landscape.
In fact, there is increasing focus on inter-disciplinary learning and development of critical thinking skills in students in all the AUs, Mr Ong said.
"But thinking critically is quite different from being unthinkingly critical, and any course offered by our AUs must be up to mark. Otherwise it does not deserve to be part of a liberal arts programme."
Political dissent is also a legitimate topic of academic inquiry, Mr Ong said. Students here read and assess works by revolutionary figures such as Karl Marx, Friedrich Engels, Vladimir Lenin, Sun Yat Sen and Mao Zedong.
He added that it would also be valuable for students to critically examine present-day issues such as the causes and implications of protests against climate change or globalisation, or the demonstrations currently happening in Hong Kong.
"Students can and should also discuss the implications of such political developments for a small country like Singapore. Such open academic inquiry will continue."
In a Facebook post on Monday, Mr Alfian said Mr Ong had not quoted a line in his poem in full, saying it continues with "...although the lyrics are still bleeding from the bark of my sapless heart".
Mr Alfian wrote: "Just stopping on the word 'anthem' might suggest that I am somehow rejecting symbols of the state. But the whole line makes clear that I have grown up with the anthem as a Singaporean, that it bleeds from my heart."
YALE-NUS


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How To Migrate A Blog From Blogger To WordPress (Self-Host)?


Concrete Cylindrical sample are taken from concrete for checking the compressive strength regularly. Concrete temperature & slump test also made before pouring concrete. These tests are made by in plant testing and third party testing in order to assure the concrete quality.
For reinforcing steel and accessories, test certificate is provided from the supplier periodically.
When mold is ready for pouring and reinforcing steel is placed with necessary installations, QC/QA representative makes the inspection and permits to pour the concrete.
Post pouring inspections are made by QC/QA representative after the final finishing of wet concrete for each product.
An inspection checklist is performed to provide a method of identifying and communicating the quality of our products and recognize problems as they occur. It serves as a method of gathering data or identification of any trends that may be evident. After a utility structure is stripped from the form, it is inspected for conformance with the fabrication drawings.
Final inspection through the final checklist is made after the product has been loaded in order to assure that all accessories including covers have been installed as per required specification.


Dinosaur fear has a name and a date.
Steven Spielberg brings his long-awaited अगला installment of the Jurassic Park series to theaters June 12, 2015. It also has a name -- Jurassic World -- Universal Pictures announced on Tuesday.
Colin Trevorrow has been tapped to direct the epic action-adventure from a draft of the screenplay he wrote with Derek Connolly. It will be shot in 3-D.
The filmmakers have already promised new dinosaur villains for the fourth installment of the franchise. The first three films in the series have taken in $1.9 billion at the box office.
Earlier this year, a 3-D re-release of the original 1993 Jurassic Park inspired a return to the dinosaur madness of decades past and brought in $114 million globally.
The news comes the same दिन that डिज़्नी studios announced that Pirates of the Caribbean 5 would हटाइए off the summer 2015 calendar and be released in 2016.
The excitement about the Jurassic film immediately made its way across Hollywood Tuesday to the तारा, स्टार Trek Into Darkness party to celebrate the DVD and Blu-Ray release.
"Jurassic World? Sign me up. Obviously, I'll be there," कहा तारा, स्टार Trek director J.J. Abrams. "I don't know what the story is but it sounds like fun."


text:
(3) Application of Biomimetic Dehalogenation in Drug Design
Chemistry of the Thyroid Hormone Deiodination
Thyroid hormones, produced by the thyroid gland, are iodine-containing compounds that regulate gene expression in every vertebrate tissue and control the metabolism in the body. The iodide ions are transported into the follicular cells of the thyroid gland by the sodium/iodide symporter (NIS) and for each iodide anion (I–), NIS transports two sodium cations (Na+) into the cell. The synthesis of the prohormone thyroxine (T4) on thyroglobulin is catalyzed by thyroid peroxidase (TPO), a heme enzyme that utilizes iodide and hydrogen peroxide to perform two distinct reactions: iodination of tyrosyl residues and phenolic coupling of the resulting iodinated tyrosyl residues of thyroglobulin. Subsequently, the thyroxyl residues are cleaved by proteolysis to produce free T4 (Figure 7).
Figure 7. Biosynthesis of thyroid hormones from thyroglobulin by thyroid peroxidase and regioselective deiodination of thyroxine (T4) (inset). The figure is adapted from Manna, D.; Roy, G.; Mugesh, G. Acc. Chem. Res. 2013, 11, 2706.
Although a small amount of T3 (3,3',5-triiodothyronine), the active thyroid hormone, is also produced during synthesis, most of the circulating T3 is generated via 5'-deiodination of T4, catalyzed by the selenoenzymes, type 1 and 2 iodothyronine deiodinases (ID-1 and ID-2). The deactivation of thyroid hormone is catalyzed by the type 3 deiodinase (ID-3), which removes iodine exclusively from the tyrosyl (inner) ring of T4 to produce 3,3',5'-triiodothyronine (rT3). The triiodo derivatives T3 and rT3 are further metabolized by the three selenoenzymes to produce 3,3'-diiodothyronine (3,3'-T2) (Figure 7, inset). It is known that the activation of thyroid stimulating hormone (TSH) receptor by auto-antibodies leads to an overproduction of thyroid hormones. In addition, these auto-antibodies also stimulate ID-1 and probably ID-2, which then together produce relatively more T3, leading to hyperthyroidism. The overproduction of T3 is controlled by specific inhibitors which either block the thyroid hormone biosynthesis (Figure 8) or reduce the conversion of T4 to T3. The most commonly used drug for hyperthyroidism are the thiourea drugs, 6-n-propyl-2-thiouracil (PTU), 6-methyl-2-thiouracil (MTU), methimazole (MMI) and carbimazole (CBZ).
Figure 8. (A) Mechanism of the antithyroid action of methimazole. (B) A simple and facile one-pot synthesis of carbimazole (12) and its methyl analogue (13) by using heterocyclic carbenes generated in situ from the corresponding imidazolium salts.
During last ten years, we have contributed significantly to the understanding of the mechanism of the inhibition of thyroid hormone synthesis by antithyroid drugs. The isosteric substitutions in the existing drugs led to compounds that can effectively and reversibly inhibit the iodination. In contrast to MMI, the chemically modified analogues inhibit the enzyme by reducing the H2O2 that is required for the oxidation of the iron center. These studies reveal that the degradation of the intracellular H2O2 by some of the anti-thyroid drugs may be beneficial to the thyroid gland as these compounds may act as antioxidants and protect thyroid cells from oxidative damage. As the inhibition is reversible, the drugs with a more controlled action could be of great importance in the treatment of hyperthyroidism. (Roy, G. and Mugesh, G. J. Am. Chem. Soc. 2004, 126, 2712; J. Am. Chem. Soc. 2005, 127, 15207, Das, D.; Roy, G.; Mugesh, G. J. Med. Chem. 2008, 51, 7313).
Recently, we have carried out extensive work on the regioselective deiodination of thyroid hormones as the regioselectivity of the reaction controls the thyroid hormone homeostasis in human body. When the work was started, nothing was known about the reaction mechanism. They demonstrated the first chemical model for the inner-ring deiodination of thyroxine (T4) and 3,5,3'-triiodothyronine (T3) by iodothyronine deiodinase (Figure 9). This study suggests that the nature of substituents around the selenol functionality may modulate its reactivity towards outer- or inner-ring iodines. The presence of an in-built thiol group in the close proximity to selenium is important as this group not only acts as thiol cofactor in the deiodination reactions, but may also assist the selenol in polarizing the C-I bond. The effective removal of iodine from the
Figure 9. The first example of a selective removal of 5-iodine by a synthetic compound.
inner-ring of T4 indicates that an enol-keto tautomerism is not required for the deiodination. Although the activation of selenocysteine by histidine is believed to be important for the ID-1-mediated deiodination, such activation is probably not essential for the ID-3-catalyzed reaction. (Manna, D.; Mugesh, G. Angew. Chem. Int. Ed. 2010, 49, 9246). The replacement of the thiol group in the deiodinase mimetic (Figure 9) by a selenol group lead to a dramatic increase in the activity, without affecting the regioselectivity of deiodination (Manna, D.; Mugesh, G. J. Am. Chem. Soc. 2011, 133, 9980).
Figure 10. Proposed mechanism for the deiodination of thyroxine (T4) involving chalcogen and halogen bonding.
Mechanistic investigations reveal that the formation of a halogen bond between the iodine and chalcogen (S or Se) and the peri-interaction between two chalcogen atoms (chalcogen bond) are important for the deiodination reactions. Although the formation of a halogen bond leads to elongation of the C-I bond, the chalcogen bond facilitates the transfer of more electron density to the C-I σ* orbitals, leading to a complete cleavage of the C-I bond (Figure 10). The higher activity of amino-substituted selenium compounds can be ascribed to the deprotonation of thiol/selenol moiety by the amino group, which not only increases the strength of halogen bond, but also facilitates the chalcogen-chalcogen interactions (Manna, D.; Mugesh, G. J. Am. Chem. Soc. 2012, 134, 4269). This work has been highlighted in Nature Chemistry (Metrangolo, P.; Resnati, G. Nature Chem. 2012, 4, 437).
Development of synthetic deiodinase mimetics is important not only to understand the mechanism of enzyme catalysis, but also to develop novel therapeutic agents. It is known that abnormal thyroid hormone levels have implications in different diseases, such as hypoxia, myocardial infarction, critical illness, neuronal ischemia, tissue injury and cancer. In the above work, we showed that the napthyl-based compounds shown in Figure 9 (thiol-selenol pair) and Figure 10 (selenol pair) always mediate the 5-deiodination of thyroxine (T4) to produce rT3 as the only product. However, very recently, we showed that the introduction of tellurium atoms in place of sulfur/selenium in deiodinase mimetics alters not only the reactivity but also the regioselectivity of deiodination. We showed for the first time that compounds having two tellurol moieties or a thiol/tellurol pair can mediate sequential deiodination of T4 to produce all the possible thyroid hormone derivatives under physiologically relevant conditions (Figure 11).
As shown in Figure 11, T4 is converted to the fully deiodinated derivative T0 through rT3-3,3'-T2-3-T1 pathway, involving two IRD and two ORD reactions. When compounds 16 and 17 were used in large excess (50 equiv.), a complete conversion of T4 to T0 was observed, indication that 3'-T1 also undergoes deiodination to give T0. The diiodo derivative 3,3′-T2 serves as a common intermediate for both 3-T1 and 3'-T1. These observations reveal that the reactivity of C-I bonds in T4 is remarkably altered upon deiodination. This study provides the first experimental evidence that the regioselectivity of thyroid hormone deiodination is controlled by the nucleophilicity and the strength of halogen bond between the iodine and chalcogen atoms. While the larger size and greater polarizability of Te supports more efficient deiodination than observed with Se compounds, the change in regiospecificity discovered here was not anticipated. This change will be crucial in the future for distinguishing between the various mechanisms proposed for the model reactions as well as those catalyzed by the natural deiodinases including the three isozymes of iodothyronine deiodinases (Raja, K.; Mugesh, G. Angew. Chem. Int. Ed. 2015, doi/10.1002/ anie.201502762).
Figure 11. (A) Chemical structures of compounds 14-19. (B) HPLC chromatograms for the reactions of T4 with 16 and 17. (C) Initial rates for the deiodination of T4 and its derivatives. (D) Sequential deiodination of T4 by 16 and 17. IRD: inner-ring deiodination, ORD: outer-ring deiodination, T4: L-thyroxine, T3: 3,5,3'-triiodothyronine, rT3: 3,3',5'-triiodothyronine, 3,3'-T2: 3,3'-diiodothyronine. Raja, K.; Mugesh, G. Angew. Chem. Int. Ed. 2015, doi/10.1002/ anie.201502762.
(4) Halogen Bond-Mediated Membrane Transport. Click Here to Continue


June 22, 2016, 12:51 a.m.
MSI acknowledged that cards for reviewers are different because that best mimics the experience most users get.
MSI and Asus have been accused of sending product reviewers graphics cards with higher clock speeds than consumers will find out of the box.
As the review team at TechPowerUp notes, MSI's GeForce GTX 1080 Gaming X graphics card has three settings: the standard "Gaming" mode, a faster "OC" mode, and slower "Silent" mode. The site, however, found that review units are set to OC mode by default but Gaming mode out of the box. The result, then, is better performance on the tests than what consumers might get out of the box.
"The 'OC Mode' is just there so consumers can overclock it a little further at the push of a button, without having any knowledge of overclocking," TechPowerUp says. "Now if the OC mode is enabled for review samples of one company and not for the others, this means that potential customers comparing reviews will think one card performs better than the other, even if it's just 1 percent, people do base their buying decision on such small differences."
Today on Random Access: Chinese cyber attacks, overclocked NVIDIA GTX 1070 and 1080s, and Mighty No. 9 plus a demo of the BlueSmart One.
After this discovery, the site looked at older reviews, and "were shocked to realize that this practice has been going on for years in MSI's case. It looks like ASUS has just adopted it, probably because their competitor does it, too, 'so it must be ok.'"
Asus did not immediately respond to a request for comment. But in a statement, MSI acknowledged that cards for reviewers are indeed set to OC mode because that setting best mimics the experience most users get.
"In order to enjoy the best performance and all features of MSI Gaming products, we highly recommend to use the MSI Gaming App, which is available for free on MSI.com and the driver CD. The MSI Gaming App allows you to apply one of three performance profiles with a single click, instantly giving you the desired performance," MSI said.
"As several reviewers have stated, software like the MSI Gaming App is often not used in reviews. This is why review samples of the MSI GeForce GTX 1080 and GTX 1070 Gaming X graphics cards are set to 'OC Mode' to ensure that reviews demonstrate the same performance available through the MSI Gaming App."
MSI has launched downloadable BIOS modes, allowing owners of the MSI GeForce GTX 1080 or 1070 to run in OC Mode without using its software.
See what PCMag's Sascha Segan and Tim Torres have to say about the controversy in this morning's episode of Random Access, embedded above.


Comments: A fantastic day for a great walk. Taking a route up the flanks of Seat Sandal was quite tough and made the climb onto Fairfield seem much easier. Views across to the Coniston fells were very clear making it possible to name a number of Wainwright summits.
Tongue Gill, before we leave the road . . .
to head up Great Tongue . . .
Taking a break on Seat Sandal . . .
Windermere comes into view . . .
as we take time out for some photos . . .
Time for more 'doggy' photos . . .
above Riggs Crags . . .
before heading down to Stone Arthur . . .
and eventually reaching . . .
A beautiful display . . .
of rhodedendron flowers . . .
demands our attention . . .
and a closer view . . .