text:
\section{Introduction}

It is well known that viscosity solutions were conceived by Crandall and Lions (1982) in the framework of optimal control theory. The goal was to show well posedness of Hamilton--Jacobi--Bellman equations in the whole space, and to prove, via dynamic programming principle, the value function of a suitable optimal control problem being the unique solution.

When trying to extend viscosity methods to the analysis of second order \emph{parabolic partial differential equations}, PDEs for short, and getting representation formulae, it appeared clear that some stochastic dynamics must be brought into play. Not surprisingly, this has been first done for stochastic control models. The hard work of a generation of mathematicians, \cite{hjlions2,hjlions1,bensoussa82,krylovbook,flemingrishel75,nisio15} among the others, allowed making effective dynamic programming approach to stochastic control problems.

Prompted by this body of investigations, a stream of research arose in the probabilistic community ultimately leading to the theory of \emph{backward stochastic differential equations}, BSDEs for short, which was introduced by Pardoux and Peng in \cite{Pardoux_Peng_1990} (1990). Since then, it has attracted a great interest due to its connections with mathematical finance and PDEs, as well as with stochastic control. This theory has been in particular used to extend the classical Feynman--Kac formula, which establishes a link between linear parabolic PDEs and \emph{stochastic differential equations}, SDEs for short, to semilinear and quasilinear equations, see for example \cite{Delarue_2002,Delarue_Guatteri_2006,Ma_Yong_2007}. See also \cite{pardouxbook} for a rather complete overview of the semilinear case.

For sake of clarity, let us consider the following semilinear parabolic PDE problem coupled with final conditions,
\begin{equation}\label{exparsemlin}
\begin{cases}
\begin{aligned}
\frac12\left\langle\sigma\sigma^\dag,D^2_xu\right\rangle(t,x)+(\nabla_xub)(t,x)&\\
+\partial_tu(t,x)+f(t,x,u,\nabla_xu\sigma)&=0
\end{aligned}
&t\in[0,T],x\in\mathds R^N,\\
u(T,x)=g(x),&x\in\mathds R^N,
\end{cases}
\end{equation}
then its viscosity solution can be written as $u(t,x)=\mathds E\left(Y^{t,x}_t\right)$, where $Y$ is given by the following system, called \emph{forward backward stochastic differential equation} or FBSDE in short, which is made in turn of two equations, the first one is a SDE, and the second one a BSDE depending on the first one
\[
\left\{\begin{aligned}
\!X^{t,x}_s\!=&x+\int_t^s\sigma\left(r,X^{t,x}_r\right)dW_r+\int_t^sb\left(r,X^{t,x}_r\right)dr,\\
\!Y^{t,x}_s\!=&g\left(X^{t,x}_T\right)\!+\!\!\int_s^T\!\!f\!\left(r,X_r^{t,x},Y_r^{t,x},Z_r^{t,x}\right)\!dr\!-\!\!\int_s^T\!\!Z_r^{t,x}dW_r,
\end{aligned}\right.s\!\in\![t,T],x\!\in\!\mathds R^N.
\]
As can be intuitively seen, the SDE takes care of the linear operator defined by $\sigma$ and $b$, also called the \emph{infinitesimal generators} of the SDE, while the BSDE depends on $f$ and $g$. In other words, this extension of the Feynman--Kac formula basically does not modify the treatment of the second order linear operator with respect to the completely linear case.

Subsequently, Peng introduced in \cite{math/0601035} (2006) the notion of \emph{$G$--expectation}, a nonlinear expectation generated by a fully nonlinear second order operator $G$ via its viscosity solutions. This work has originated an active research field, with relevant applications to Mathematical Finance.

Peng has improved this theory in several papers and has given a comprehensive account of it in the book \cite{book:peng}, where he highlights the role of the so--called \emph{sublinear expectations}, namely $G$--expectations generated by sublinear operators. Finally in \cite{art:pathpeng}, Peng provides representation formulae for viscosity solutions using these expectations. More precisely, given a sublinear operators $G$ and the $G$--heat equation
\begin{equation}\label{gheat}
\begin{cases}
\partial_tu(t,x)+G\left(D_x^2u(t,x)\right)=0,&t\in[0,T],x\in\mathds R^N,\\
u(T,x)=g(x),&x\in\mathds R^N.
\end{cases}
\end{equation}
he represents the viscosity solution as
\[
u(t,x)=\sup_{\sigma\in\mathcal A}\mathds E\left(g\left(x+\int_t^T\sigma_sdW_s\right)\right),
\]
where $\mathcal A$ is a family of stochastic process associated to $G$ and $W$ is a Brownian motion.
The key to prove it is a \emph{dynamic programming principle} that we will illustrate in the paper, see \cref{dynprincsec}. We point out that here the novelty with respect to the Feynman--Kac formula is essentially given by the sublinearity of the operator $G$.

The purpose of this article is to apply a generalized version of the dynamic programming principle of \cite{art:pathpeng} in order to give representation formulae of solutions to PDE problems of the type
\begin{equation}\label{parsubopgen}
\begin{cases}
\begin{aligned}
\partial_tu(t,x)+F\left(t,x,\nabla_xu,D^2_xu\right)&\\
+f(t,x,u,\nabla_xu)&=0,
\end{aligned}
&t\in[0,T],x\in\mathds R^N,\\
u(T,x)=g(x),&x\in\mathds R^N,
\end{cases}
\end{equation}
where $F$ is a sublinear operator, with respect the third and the fourth argument. This problem is clearly a blend between \eqref{exparsemlin} and \eqref{gheat}, where the additional difficulty with respect to \eqref{exparsemlin} is given by the sublinearity of the operator, while the generalization with respect to \eqref{gheat} is the dependence of $F$ on $(t,x)$ of and the presence of the term $f$.

This is hopefully just a first step to further extend the Feynman--Kac formula to problems with sublinear operators using a the BSDE theory in order to deal with general quasilinear problems. We also point out that there is close connection between this method and second order BSDEs, 2BSDEs for short. 2BSDEs were introduced by Cheridito, Soner, Touzi and Victoir in \cite{int2bsde} (2007). Then, in 2011, Soner, Touzi and Zhang \cite{sonertouzizhang11} provided a complete theory of existence and uniqueness for 2BSDEs under Lipschitz conditions. In those papers is also analyzed the connection between 2BSDEs and fully nonlinear PDEs. Among the subsequent developments of this theory we cite \cite{possamaitanzhou18,sonertouzizhang11,ekrentouzizhang16,ekrentouzizhang16_2,matoussipossamaizhou13,kazi-tanipossamaizhou15} and in particular \cite{possamai13}, which performs its analysis replacing the Lipschitz condition on $y$ with monotonicity as we do here.

This paper is organized as follows: in \cref{prelim} we make a preliminary study of the problem, analyzing the structure of the sublinear operator $F$ and developing a dynamic programming principle which is the core of our theory. Then, in \cref{parprob}, we perform the essential part of our analysis, and obtain in this way our main results. In \cref{2bsdesec} we summarily analyze the connection between 2BSDEs and our representation formula. The appendix at the end briefly gives some probability results we need, with a focus on the BSDE theory. We proceed setting the notation used in the paper.

\subsubsection*{Notation}
\pdfbookmark[3]{Notation}{Notation}

We will work on the filtered probability space $\left(\Omega,\mathcal F,\{\mathcal F_t\}_{t\in[0,\infty)},\mathds P\right)$,
\begin{itemize}
\item $\mathcal F$ is a complete $\sigma$--algebra on $\Omega$;
\item the stochastic process $\{W_t\}_{t\in[0,\infty)}$ will denote the $N$ dimensional Brownian motion under $\mathds P$;
\item $\{\mathcal F_t\}_{t\in[0,\infty)}$ is the filtration defined by $\{W_t\}_{t\in[0,\infty)}$ which respects the usual condition of completeness and right continuity;
\item $\{W_s^t\}_{s\in[t,\infty)}:=\{W_s-W_t\}_{s\in[t,\infty)}$ is a Brownian motion independent from $\{W_s\}_{s\in[0,t]}$ by the \emph{strong Markov property};
\item $\{\mathcal F^t_s\}_{s\in[t,\infty)}$ is the filtration generated by $\{W_s^t\}_{s\in[t,\infty)}$ which we assume respect the usual condition and is independent from $\mathcal F_t$;
\item we will say that a stochastic process $\{H_t\}_{t\in[0,\infty)}$ is adapted if $H_t$ is $\mathcal F_t$--measurable for any $t\in[0,\infty)$;
\item we will say that a stochastic process $\{H_t\}_{t\in[0,\infty)}$ is progressively measurable, or simply progressive, if, for any $T\in[0,\infty)$, the application that to any $(t,\omega)\in[0,T]\times\Omega$ associate $H_t(\omega)$ is measurable for the $\sigma$--algebra $\mathcal B([0,T])\times\mathcal F_T$;
\item a function on $\mathds R$ is called \emph{cadlag} if is right continuous and has left limit everywhere;
\item a cadlag (in time) process is progressive if and only if is adapted;
\item $B_\delta(x)$ will denote an open ball centered in $x$ with radius $\delta$;
\item for any Lipschitz continuous function $f$ we will denote its Lipschitz constant as $\Lip(f)$;
\item if $A\in\mathds R^{N\times M}$ then $A^\dag$ will denote its transpose and $\sigma_A$ its spectrum;
\item (Frobenius product) if $A,B\in\mathds R^{N\times M}$ then $\langle A,B\rangle:=\tr\left(AB^\dag\right)$ and $|A|$ is the norm $\sqrt{\langle A,A\rangle}=\sqrt{\sum\limits_{i=1}^N\sum\limits_{j=1}^MA_{i,j}^2}$;
\item $\mathds S^N$ is the space of all $N\times N$ real valued symmetric matrices and $\mathds S^N_+$ is the subset of $\mathds S^N$ made up by the definite positive matrices.
\end{itemize}

\section{Preliminaries}\label{prelim}

\subsection{Sublinear Operators}

We consider the space $\mathds R^N\times\mathds S^N$ with the inner product
\[
((p,S),(p',S')):=\frac12\langle S,S'\rangle+p^\dag p'
\]
and the norm $\|(p,S)\|:=\sqrt{((p,S),(p,S))}$.

\begin{assum}\label{subopassum}
In this subsection we will concentrate on the study of continuous operators of the form
\[
F:[0,T]\times\mathds R^N\times\mathds R^N\times\mathds S^N\to\mathds R
\]
such that the following properties hold true for any $(t,x)\in[0,T]\times\mathds R^N$, $(p,S)$ and $(p',S')$ in $\mathds R^N\times\mathds S^N$:
\begin{enumerate}[label=(\roman*)]
\item\label{subopassumsubad}\emph{(Subadditivity)} $F(t,x,p+p',S+S')\le F(t,x,p,S)+F(t,x,p',S')$;
\item\label{subopassumphom}\emph{(Positive Homogeneity)} If $\delta\ge0$ then $F(t,x,\delta p,\delta S)=\delta F(t,x,p,S)$;
\item\emph{(Uniform Ellipticity)} Exists a constant $\lambda>0$ such that, if $S'\ge0$,
\[
F(t,x,p,S+S')-F(t,x,p,S)\ge\lambda|S'|;
\]
\item\emph{(Lipschitz Continuity)} Exists a positive $\ell$ such that, for any $y\in\mathds R^N$,
\[
|F(t,x,p,S)-F(t,y,p,S)|\le\ell|x-y|\|(p,S)\|.
\]
\end{enumerate}
We will usually refer to $T$ as the terminal time of $F$, since it will play the role of terminal time in the parabolic problems which we will deal with later.
\end{assum}

The operators satisfying conditions \ref{subopassumsubad} and \ref{subopassumphom} are commonly known as \emph{sublinear operators}. Notice that \cref{subopassumphom,subopassumsubad} imply convexity in the third and fourth arguments and, vice versa, convexity and \ref{subopassumphom} imply \ref{subopassumsubad}.

The main result of this section is the following characterization theorem:

\begin{theo}\label{charsublin}
Let $F$ be as in \cref{subopassum} and $K_F$ be the set of the elements $(b,a)\in C^0\left([0,T]\times\mathds R^N;\mathds R^N\right)\times C^0\left([0,T]\times\mathds R^N;\mathds S_+^N\right)$ such that, for any $(t,x,p,S)\in[0,T]\times\mathds R^N\times\mathds R^N\times\mathds S^N$,
\[
\frac12\langle a(t,x),S\rangle+p^\dag b(t,x)\le F(t,x,p,S),
\]
$\Lip(b(t))\le2\ell$, $\Lip(a(t))\le2\sqrt2\ell$ and the eigenvalues of $a(t,x)$ are bigger than $2\lambda$. Then $K_F$ is a non empty and convex set, and
\[
F(t,x,p,S)=\max_{(b,a)\in K_F}\frac12\langle a(t,x),S\rangle+p^\dag b(t,x)
\]
for any $(t,x,p,S)\in[0,T]\times\mathds R^N\times\mathds R^N\times\mathds S^N$. Furthermore for each $(b,a)\in K_F$ the linear operator
\[
(t,x,p,S)\mapsto\frac12\langle a(t,x),S\rangle+p^\dag b(t,x)
\]
has the same ellipticity conditions of $F$.
\end{theo}

To prove this we preliminarily need the followings two lemmas. The first one is an adaptation of \cite[Lemma 1.8.14]{schneiderconvex}, which permit us to express the Hausdorff distance using support function, while the second one is just an adaptation of the Hahn--Banach theorem.

\begin{lem}\label{supfunhausdis}
Given, $A$ and $B$, two compact and convex subset of $\mathds R^N\times\mathds S^N$ we define the application
\[
h_A,\:h_B:\mathds R^N\times\mathds S^N\to\mathds R
\]
as the support functions of $A$ and $B$ respectively, that is to say
\[
h_A((p,S))=\sup_{(p',S')\in A}((p,S),(p',S')),\quad h_B((p,S))=\!\sup_{(p',S')\in B}((p,S),(p',S')),
\]
for any $(p,S)\in\mathds R^N\times\mathds S^N$. Then, for the Hausdorff distance
\begin{multline*}
d_H(A,B):=\\
\max\left\{\max_{(p,S)\in A}\min_{(p',S')\in B}\|(p,S)-(p',S')\|,\max_{(p',S')\in B}\min_{(p,S)\in A}\|(p,S)-(p',S')\|\right\},
\end{multline*}
we have that
\[
d_H(A,B)=\max_{\substack{(p,S)\in\mathds R^N\times\mathds S^N\\\|(p,S)\|=1}}|h_A((p,S))-h_B((p,S))|.
\]
\end{lem}

\begin{lem}\label{charsublinaux}
If $F$ is like in \cref{subopassum} then the set
\[
\mathcal L_{F(t,x)}:=\{L\in\left(\mathds R^N\times\mathds S^N\right)^*:L\le F(t,x)\}
\]
is non empty, compact and convex for any $(t,x)\in[0,T]\times\mathds R^N$. Furthermore $F(t,x)=\max\limits_{L\in\mathcal L_F(t,x)}L$.
\end{lem}

\proof[Proof of \cref{charsublin}]
By \cref{charsublinaux} and the Riesz representation theorem we have that, for any $(t,x)\in[0,T]\times\mathds R^N$, there exists the non empty convex and compact set
\[
K^x_t:=\left\{
\begin{gathered}
(b,a)\in\mathds R^N\times\mathds S^N:\frac12\langle a,S\rangle+p^\dag b\le F(t,x,p,S)\\
\text{for any }(p,S)\in\mathds R^N\times\mathds S^N
\end{gathered}
\right\}.
\]
Given a $(t,x)\in[0,T]\times\mathds R^N$ and $\left(\overline b,\overline a\right)\in K^x_t$ we define the function
\[
(b,a):[0,T]\times\mathds R^N\to\mathds R^N\times\mathds S^N
\]
such that
\[
(b,a)(s,y):=
\begin{cases}
\left(\overline b,\overline a\right),&\text{if }(s,y)=(t,x),\\
\argmin\limits_{(b,a)\in K_s^y}\left\|\left(\overline b-b,\overline a-a\right)\right\|,&\text{if }(s,y)\neq(t,x).
\end{cases}
\]
This function is well defined because is well known that the projection of a point onto a convex set, i.e. $\argmin\limits_{(b,a)\in K_s^y}\left\|\left(\overline b-b,\overline a-a\right)\right\|$, exists and is unique. We will show that $(b,a)\in K_F$, since this yields that $K_F$ is a non empty convex set (the convexity proof is trivial, hence we skip it) such that, thanks to the arbitrariness of the construction,
\[
F(t,x,p,S)=\max_{(b,a)\in K_F}\frac12\langle a(t,x),S\rangle+p^\dag b(t,x)
\]
for any $(t,x,p,S)\in[0,T]\times\mathds R^N\times\mathds R^N\times\mathds S^N$.\\
As a consequence of the definition and \cref{supfunhausdis} we have, for any $(s,y)$ in $[0,T]\times\mathds R^N$,
\begin{align*}
\|(b,a)(t,x)-(b,a)(s,y)\|=&\min_{(b,a)\in K_s^y}\|(b(t,x)-b,a(t,x)-a)\|\\
\le&\max_{(b_1,a_1)\in K_t^x}\min_{(b_2,a_2)\in K_s^y}\|(b_1-b_2,a_1-a_2)\|\\
\le&d_H(K^x_t,K^y_s)\\
=&\max_{\substack{(p,S)\in\mathds R^N\times\mathds S^N:\\\|(p,S)\|=1}}|F(t,x,p,S)-F(s,y,p,S)|.
\end{align*}
Since $|a|\le\sqrt2\|(b,a)\|$ for any $(b,a)\in\mathds R^N\times\mathds S^N$ and
\[
|a(r,z)-a(s,y)|\le|a(t,x)-a(s,y)|+|a(r,z)-a(t,x)|
\]
the previous inequality yields that $\Lip(a(s))\le2\sqrt2\ell$ for any $s\in[0,T]$, and similarly that $\Lip(b(s))\le2\ell$ for any $s\in[0,T]$.\\
We now prove the ellipticity part of the statement and, as a consequence, that $\sigma_{a(t,x)}\subset[2\lambda,\infty)^N$ for any $(t,x)\in[0,T]\times\mathds R^N$, thus that $(b,a)\in K_F$. Let $\lambda$ be the ellipticity constants of $F$ and
\begin{align*}
L:[0,T]\times\mathds R^N\times\mathds R^N\times\mathds S^N\to&\:\mathds R\\
(t,x,p,S)\mapsto&\:\frac12\langle a(t,x),S\rangle+p^\dag b(t,x),
\end{align*}
then, by its linearity, we only have to prove that for any $S\in\mathds S^N_+$ and $(t,x)$ in $[0,T]\times\mathds R^N$
\begin{equation}\label{eq:unelllineq}
L(t,x,0,S)\ge\lambda|S|.
\end{equation}
Obviously we have, for any $S\in\mathds S^N_+$ and $(t,x)\in[0,T]\times\mathds R^N$,
\[
\lambda|S|\le F(t,x,0,0)-F(t,x,0,-S)\le-L(t,x,0,-S)=L(t,x,0,S),
\]
hence \eqref{eq:unelllineq}. Finally, let $q$ an element of $\mathds R^N$ and define $Q:=qq^\dag$, which is an element of $\mathds S^N_+$ such that
\[
|Q|^2=\tr(qq^\dag qq^\dag)=\tr(q|q|^2q^\dag)=|q|^4.
\]
Therefore \eqref{eq:unelllineq} yields, for any $(t,x)\in[0,T]\times\mathds R^N$,
\[
\lambda|q|^2=\lambda|Q|\le\frac12\langle a(t,x),qq^\dag\rangle=\frac12q^\dag a(t,x)q
\]
and the Rayleigh quotient formula proves that $\sigma_{a(t,x)}\subset[2\lambda,\infty)^N$, concluding the proof.
\endproof

\begin{rem}
We point out that can be easily proved that for the previous theorem holds a converse.
\end{rem}

We have characterized $F$ as the support function of a set of linear operators. Usually, to obtain representation formulas for viscosity solutions to a second order PDE with linear operator like \eqref{exparsemlin}, is useful to study a function $\sigma$ such that $\sigma\sigma^\dag$ is the diffusion part of that operator, hence we will do something similar: if we define the application from $\mathds S^N_+$ to itself which associate via singular value decomposition the matrix $a$ with its square root $\sigma$ then it is well defined, as can be seen in \cite[Section 6.5]{bellmanmatrixbook}. Moreover we know from \cite[Lemma 2.1]{sqmatrix} that, on the space of matrices with eigenvalues equal or bigger than $2\lambda$, this application is Lipschitz continuous with Lipschitz constant $c_\lambda:=\frac1{2\sqrt\lambda}$, therefore the application $(b,\sigma)\mapsto(b,\sigma\sigma)$ that maps the set $\mathcal K_F$, which contains the $(b,\sigma)$ in $C^0\left([0,T]\times\mathds R^N;\mathds R^N\right)\times C^0\left([0,T]\times\mathds R^N;\mathds S^N_+\right)$ such that $(b,\sigma\sigma)\in K_F$ and $\Lip(\sigma(t))\le2\sqrt2c_\lambda\ell$ for any $t\in[0,T]$,
into $K_F$ is surjective and consequently
\[
F(t,x,p,S)=\max_{(b,\sigma)\in\mathcal K_F}\frac12\left\langle\sigma^2(t,x),S\right\rangle+p^\dag b(t,x).
\]

Our method to obtain representation formulas relies on a \emph{dynamic programming principle}, which will be presented later and is based on a construction on a broader set than $\mathcal K_F$. This set, which we call $\mathcal A_F$, is made up of the functions
\[
(b,\sigma):[0,T]\times\Omega\times\mathds R^N\to\mathds R^N\times\mathds R^{N\times N}
\]
which are cadlag, i.e. right continuous and left bounded, on $[0,T]$ and such that, for any $(t,x,p,S)\in[0,T]\times\mathds R^N\times\mathds R^N\times\mathds S^N$ and $\omega\in\Omega$,
\[
\frac12\left\langle\sigma^2(t,\omega,x),S\right\rangle+p^\dag b(t,\omega,x)\le F(t,x,p,S),
\]
$\Lip(b(t,\omega))\le2\ell$, $\Lip(\sigma(t,\omega))\le2\sqrt 2c_\lambda\ell$, the eigenvalues of $(\sigma\sigma)(t,\omega,x)$ belong to $[2\lambda,\infty)^N$ and $\{(b,\sigma)(t,0)\}_{t\in[0,T]}$ is a progressive process. $\mathcal A_F$ is obviously non empty, since it contains $\mathcal K_F$. For any stopping time $\tau$ with value in $[0,T]$, an useful subset of $\mathcal A_F$, which we will use later, is $\mathcal A^\tau_F$, which consists of the $(b,\sigma)$ belonging to $\mathcal A_F$ such that $\{(b,\sigma)(\tau+t,x)\}_{t\in[0,\infty)\!}$ is progressive with respect to the filtration $\!\left\{\mathcal F_t^\tau\right\}_{t\in[0,\infty)}$. Trivially $\mathcal A^0_F=\mathcal A_F$.

\begin{rem}\label{aflp}
It is easy to see that, for any $p>0$, the process $(b,\sigma)(s,0)$ belongs to $\mathds L_N^p(t)\times\mathds L^p_{N\times N}(t)$ for each $(b,\sigma)\in\mathcal A_F$ and $t\in[0,\infty)$, since the image of $[0,t]\times\Omega\times\{0\}$ under $(b,\sigma)$ is contained on a compact set for any $(b,\sigma)\in\mathcal A_F$ and $t\in[0,\infty)$.
\end{rem}

\subsection{Dynamic Programming Principle}\label{dynprincsec}

The scope of this section is to provide the necessary instruments to prove the \cref{cauchydpp}, which will be used to derive representation formulas for viscosity solutions to the parabolic problem which we will study later. The dynamic programming principle is, in this contest, an instrument that permit us to break a stochastic trajectory in two or more part. In particular for the problem \eqref{exparsemlin} it means that
\begin{multline*}
\mathds E\left(u\left(s,X^{t,x}_s\right)+\int_s^Tf\left(r,X_r^{s,X^{t,x}_s},Y_r^{s,X^{t,x}_s},Z_r^{s,X^{t,x}_s}\right)dr\right)\\
\begin{aligned}
=&\mathds E\left(Y^{s,X^{t,x}_s}_s+\int_s^Tf\left(r,X_r^{s,X^{t,x}_s},Y_r^{s,X^{t,x}_s},Z_r^{s,X^{t,x}_s}\right)dr\right)\\
=&\mathds E\left(Y^{t,x}_s+\int_s^Tf\left(r,X_r^{t,x},Y_r^{t,x},Z_r^{t,x}\right)dr\right)\\
=&\mathds E\left(Y^{t,x}_t\right)=u(t,x),
\end{aligned}
\end{multline*}
which is just a simple consequence of the uniqueness of the solutions to the FBSDE, while for the $G$--heat equation \eqref{gheat} this means that
\begin{multline}\label{gheatdpp}
\sup_{\sigma\in\mathcal A_G}\mathds E\left(u\left(s,x+\int_t^s\sigma(r)dW_r\right)\right)\\
\begin{aligned}
=&\sup_{\sigma\in\mathcal A_G}\sup_{\sigma'\in\mathcal A_G}\mathds E\left(g\left(x+\int_t^s\sigma(r)dW_r+\int_s^T\sigma'(r)dW_r\right)\right)\\
=&\sup_{\sigma\in\mathcal A_G}\mathds E\left(g\left(x+\int_t^T\sigma(r)dW_r\right)\right)\\
=&u(t,x).
\end{aligned}
\end{multline}
The proof of \eqref{gheatdpp} is contained in \cite[Subsection 3.1]{art:pathpeng}. This also intuitively explain why we ask to the elements of $\mathcal A_F$ to be cadlag in time. We point out that in \cite{art:pathpeng} the authors ask to the elements of $\mathcal A_F$ to only be measurable in time, but for the analysis of the more general problem \eqref{parsubopgen} in \cref{parprob} we will use right continuity.

The dynamic programming principle exposed in theorem \ref{cauchydpp} is a generalization of the one presented by Denis, Hu and Peng in \cite{art:pathpeng} and can be obtained using a similar method, slightly adapting the proofs of \cite[Lemmas 41--44]{art:pathpeng}. Hence we will just present the results that are relevant for our analysis skipping the proofs.

First of all we notice that for any $(b,\sigma)\in\mathcal A_F$ we can define an $X_{(b,\sigma)}$ solution to the SDE $(b,\sigma)$ as in \eqref{eq:sde}. However, to ease notations, we will usually omit the dependence of $X$ from $(b,\sigma)$.

Given a sublinear operator $F$ with a positive terminal time $T$ as in \cref{subopassum}, a stopping time $\tau$ with value in $[0,T]$ and a measurable application $\varphi$ from $[0,T]\times\mathds R^N\times\mathcal A_F$ into $\mathds R$, continuous in probability with respect to $\mathcal A_F$ and such that
\[
\varphi_\tau:\mathds R^N\times\mathcal A_F|_{[\tau,T]}\to L^1\left(\Omega,\mathcal F_T;\mathds R\right),
\]
we define, for any $\zeta\in L^2\left(\Omega,\mathcal F_\tau;\mathds R^N\right)$, the function
\[
\Phi_\tau(\zeta):=\esss_{(b,\sigma)\in\mathcal A_F}\mathds E(\varphi_\tau(\zeta,b,\sigma)|\mathcal F_\tau).
\]
We assume that $\varphi_t(x,b,\sigma)$ is $\mathcal F^t_\infty$--measurable for any $t\in[0,T]$, $x\in\mathds R^N$ and $(b,\sigma)\in\mathcal A_F^t$, and, for any stopping time $\tau$ with value in $[0,T]$ and $\zeta$ in $L^2\left(\Omega,\mathcal F_\tau;\mathds R^N\right)$,
\begin{equation}\label{conddpp}
\sup_{(b,\sigma)\in\mathcal A_F}\mathds E(|\varphi_\tau(\zeta,b,\sigma)|)<\infty.
\end{equation}

In this section the function $\Phi$ represents, roughly speaking, the viscosity solution, $\zeta$ is the first part of a stochastic trajectory broken off at $\tau$ (this is why we restrict $\varphi_\tau$ on $\mathcal A_F|_{[\tau,T]}$) and $\varphi$ the function which we will use to build the viscosity solution.

\begin{lem}\label{peng40}
For each $(b_1,\sigma_1),(b_2,\sigma_2)$ in $\mathcal A_F$ and stopping time $\tau$ with value in $[0,T]$ there exists an $(b,\sigma)\in\mathcal A_F$ such that
\[
\mathds E(\varphi_\tau(\zeta,b,\sigma)|\mathcal F_\tau)=\mathds E(\varphi_\tau(\zeta,b_1,\sigma_1)|\mathcal F_\tau)\vee\mathds E(\varphi_\tau(\zeta,b_2,\sigma_2)|\mathcal F_\tau).
\]
Therefore there exists a sequence $\{(b_i,\sigma_i)\}_{i\in\mathds N}$ in $\mathcal A_F$ such that a.e.
\[
\mathds E(\varphi_\tau(\zeta,b_i,\sigma_i)|\mathcal F_\tau)\uparrow\Phi_\tau(\zeta).
\]
We also have
\[
\mathds E(|\Phi_\tau(\zeta)|)\le\sup_{(b,\sigma)\in\mathcal A_F}\mathds E(|\varphi_\tau(\zeta,b,\sigma)|)<\infty,
\]
and, for any stopping time $\tau'\le\tau$,
\[
\mathds E\left(\esss_{(b,\sigma)\in\mathcal A_F}\mathds E(\varphi_\tau(\zeta,b,\sigma)|\mathcal F_\tau)\middle|\mathcal F_{\tau'}\right)=\esss_{(b,\sigma)\in\mathcal A_F}\mathds E(\varphi_\tau(\zeta,b,\sigma)|\mathcal F_{\tau'}).
\]
\end{lem}

\begin{rem}
To prove \cref{peng40} the randomness of the elements of $\mathcal A_F$ is crucial, this is the reason why we consider a set of stochastic process instead of a deterministic one.
\end{rem}

To continue we need a density result on $\mathcal A_F$ endowed with the topology of the $L^2$--convergence on compact set, which is to say that a sequence in $\mathcal A_F$ converges to an element of $\mathcal A_F$ if and only if it converges in $L^2([0,T]\times\Omega\times K)$ for any compact set $K\subset\mathds R^N$.

\begin{lem}
The set
\[
\mathcal J^t:=\left\{
\begin{aligned}
\alpha\in\mathcal A_F:\alpha|_{[t,T]}=\sum_{i=0}^n\chi_{A_i}\alpha_i|_{[t,T]},\text{ where }\{\alpha_i\}_{i=0}^n\subset\mathcal A_F^t\\
\text{and }\{A_i\}_{i=0}^n\text{ is a }\mathcal F_t\text{--partition of }\Omega
\end{aligned}
\right\}
\]
is dense in $\mathcal A_F$ for any $t\in[0,T]$.
\end{lem}
\proof
To prove this we will show that, fixed a $k\in\mathds N$, we can approximate, in $L^2([0,T]\times\Omega\times B_k(0))$, any element of $\mathcal A_F$ with an element of $\mathcal J^t$.\\
Preliminarily notice that by our assumption each element of $\mathcal A_F$ can be approximated in $L^2([0,T]\times\Omega\times B_k(0))$ by a sequence of simple functions. We will denote with $\mathcal B([0,T]\times B_k(0))$ the Borel $\sigma$--algebra of $[0,T]\times B_k(0)$.\\
Furthermore, since the collection $\mathcal I$ of the rectangles $A\times B$ where $A\in\mathcal F_T$ and $B\in\mathcal B([0,T]\times B_k(0))$ is a $\pi$--system which contains the complementary of its sets and generate $\sigma(\mathcal F_T\times\mathcal B([0,T]\times B_k(0)))$, by \mbox{\cite[Dynkin's lemma A1.3]{williams08}} each set in $\sigma(\mathcal F_T\times\mathcal B([0,T]\times B_k(0)))$, which is the smallest $d$--system containing $\mathcal I$, can be approximate by a finite union of sets in $\mathcal I$. Similarly, each set in $\mathcal F_T$ can be approximated by finite intersection and union of sets in $\mathcal F_t$ and $\mathcal F_T^t$, since $\mathcal F_T=\sigma\left(\mathcal F_t,\mathcal F_T^t\right)$.\\
Therefore, fixed $(b,\sigma)\in\mathcal A_F$, for any $\varepsilon>0$ there exists a simple function $s_\varepsilon$ such that $s_\varepsilon(t,\omega,x)=\sum\limits_{i=1}^n\sum\limits_{j=1}^ms^j_i(t,x)\chi_{A_i}(\omega)\chi_{A'_j}(\omega)$ where $\{A_i\}_{i=1}^n$ and $\{A'_j\}_{j=1}^m$ are respectively a $\mathcal F_t$--partition and a $\mathcal F^t_T$--partition of $\Omega$ and
\begin{equation}\label{eq:exJdenseA1}
\mathds E\left(\int_0^T\int_{B_k(0)}|(b,\sigma)(t,x)-s_\varepsilon(t,x)|^2dxdt\right)<\varepsilon.
\end{equation}
Then, for each $A_i$ and $A'_j$ with $\mathds P(A_i\cap A'_j)>0$, there exists a $\omega_i^j\in A_i\cap A'_j$ such that
\[
\int_0^T\int_{B_k(0)}\left|(b,\sigma)\left(t,\omega_i^j,x\right)-s_i^j(t,x)\right|^2dxdt<\frac\varepsilon{\mathds P(A_i\cap A'_j)},
\]
otherwise we would have that
\begin{multline*}
\mathds E\left(\int_0^T\int_{B_k(0)}|(b,\sigma)(t,x)-s_\varepsilon(t,x)|^2dxdt\right)\\
\begin{aligned}
=&\mathds E\left(\sum_{i=1}^n\sum_{j=1}^m\int_0^T\int_{B_k(0)}\left|(b,\sigma)(t,x)-s_i^j(t,x)\right|^2\chi_{A_i}\chi_{A'_j}dtdx\right)\\
\ge&\mathds E\left(\int_0^T\int_{B_k(0)}\left|(b,\sigma)(t,x)-s_i^j(t,x)\right|^2\chi_{A_i}\chi_{A'_j}dtdx\right)\\
\ge&\varepsilon,
\end{aligned}
\end{multline*}
in contradiction with \eqref{eq:exJdenseA1}. Finally, let $\omega_i^j$ be any elements of $A_i\cap A'_j$ if $\mathds P(A_i\cap A'_j)=0$ and $\left(b_i^k,\sigma_i^k\right):=\sum\limits_{j=1}^m(b,\sigma)\left(\omega_i^j\right)\chi_{A'_j}$. Then $\left(b_i^k,\sigma_i^k\right)\in\mathcal A^t_F$ and $\left(b^k_\varepsilon,\sigma^k_\varepsilon\right):=\sum\limits_{i=1}^n\left(b^k_i,\sigma^k_i\right)\chi_{A_i}$ is an element of $\mathcal J^t$ satisfying
\[
\mathds E\left(\int_0^T\int_{B_k(0)}\left|(b,\sigma)(t,x)-\left(b^k_\varepsilon,\sigma^k_\varepsilon\right)(t,x)\right|^2dxdt\right)<4\varepsilon.
\]
This proves that $\mathcal J^t$ is dense in $\mathcal A_F$.
\endproof

\begin{lem}
For each $t\in[0,T]$ and $x\in\mathds R^N$, $\Phi_t(x)$ is deterministic. Furthermore
\begin{equation}\label{eq:artpeng42}
\Phi_t(x)=\esss_{(b,\sigma)\in\mathcal A_F}\mathds E(\varphi_t(x,b,\sigma)|\mathcal F_t)=\esss_{(b,\sigma)\in\mathcal A_F^t}\mathds E(\varphi_t(x,b,\sigma)|\mathcal F_t).
\end{equation}
\end{lem}

\begin{lem}\label{artpeng43}
We define the function
\begin{align*}
u:[0,T]\times\mathds R^N&\xrightarrow{\qquad}\mathds R\\
(t,x)&\xmapsto{\qquad}\Phi_t(x)
\end{align*}
and assume that it is continuous. Then, for each stopping time $\tau$ with value in $[0,T]$ and $\zeta\in L^2_N\left(\Omega,\mathcal F_\tau;\mathds R^N\right)$, we have that $u_\tau(\zeta)=\Phi_\tau(\zeta)$ a.e..
\end{lem}

\begin{rem}\label{supeqrem}
This lemma says, as a consequence of \eqref{eq:artpeng42}, that
\[
\esss_{(b,\sigma)\in\mathcal A_F}\mathds E(\varphi_t(\zeta,b,\sigma)|\mathcal F_t)=\esss_{(b,\sigma)\in\mathcal A_F^t}\mathds E(\varphi_t(\zeta,b,\sigma)|\mathcal F_t)
\]
for any $t\in[0,T]$.
\end{rem}

\section{Parabolic PDEs with Sublinear Operators}\label{parprob}

We analyze now the following problem:
\begin{prob}\label{parsubopgenprob}
Let $T$ be a terminal time, $F$ a uniformly elliptic operator satisfying \cref{subopassum} and
\[
f:[0,T]\times\mathds R^N\times\mathds R\times\mathds R^N\to\mathds R\text{ and }g:\mathds R^N\to\mathds R
\]
two continuous functions for which there exist two constants $\mu\in\mathds R$ and $\ell\ge0$ such that, for any $t\in[0,T]$, $x,x'\in\mathds R^N$, $y,y'\in\mathds R$ and $z,z'\in\mathds R^N$,
\begin{enumerate}[label=(\roman*)]
\item $|g(x)-g(x')|\le\ell|x-x'|$;
\item $|g(x)|\le\ell(1+|x|)$;
\item $|f(t,x,y,z)-f(t,x',y,z')|\le\ell(|x-x'|+|z-z'|)$;
\item $|f(t,x,y,z)|\le\ell(1+|x|+|y|+|z|)$;
\item $(y-y')(f(t,x,y,z)-f(t,x,y',z))\le\mu|y-y'|^2$.
\end{enumerate}
Find the solution $u$ to the parabolic PDE
\[
\begin{cases}
\partial_tu(t,x)+F\left(t,x,\nabla_xu,D^2_xu\right)+f(t,x,u,\nabla_xu)=0,&t\in(0,T),x\in\mathds R^N,\\
u(T,x)=g(x),&x\in\mathds R^N.
\end{cases}
\]
\end{prob}

\begin{rem}
To ease notations, we can assume without loss of generality that the $\ell$ in \cref{parsubopgenprob} is the same as in \cref{subopassum}. Since $F$ is continuous, we can also assume that, for any $(b,\sigma)\in\mathcal A_F$, $|b(t,0)|\le\ell$ and $|\sigma(t,0)|\le\ell$ for any $t\in[0,T]$. We also define the sets $\mathcal L_F^\tau$ whose elements are the operators $L_{(b,\sigma)}$ with $(b,\sigma)\in\mathcal A^\tau_F$ such that
\[
L_{(b,\sigma)}(p,S):=\frac12\left\langle\sigma^2(t,x),S\right\rangle+p^\dag b(t,x).
\]
As previously done with $\mathcal A_F$, we also define the set $\mathcal L_F:=\mathcal L_F^0$.
\end{rem}

Let us define what we mean with viscosity solution to \cref{parsubopgenprob}. For a detailed overview of the viscosity solution theory we refer to \cite{userguide}.

\begin{defin}
Given an upper semicontinuous function $u$ we say that a function $\varphi$ is a \emph{supertangent} to $u$ at $(t,x)$ if $(t,x)$ is a local maximizer of $u-\varphi$.\\
Similarly we say that a function $\psi$ is a \emph{subtangent} to a lower semicontinuous function $v$ at $(t,x)$ if $(t,x)$ is a local minimizer of $v-\psi$.
\end{defin}

\begin{defin}
An upper semicontinuous function $u$ is called a \emph{viscosity subsolution} to \cref{parsubopgenprob} if, for any suitable $(t,x)$ and $C^{1,2}$ supertangent $\varphi$ to $u$ at $(t,x)$,
\[
\partial_t\varphi(t,x)+F\left(t,x,\nabla_x\varphi,D^2\varphi\right)+f(t,x,u,\nabla_x\varphi)\ge0.
\]
Similarly a lower semicontinuous function $v$ is called a \emph{viscosity supersolution} to \cref{parsubopgenprob} if, for any suitable $(t,x)$ and $C^{1,2}$ subtangent $\psi$ to $v$ at $(t,x)$,
\[
\partial_t\psi(t,x)+F\left(t,x,\nabla_x\psi,D^2\psi\right)+f(t,x,u,\nabla_x\psi)\le0.
\]
Finally a continuous function $u$ is called a \emph{viscosity solution} to \cref{parsubopgenprob} if it is both a super and a subsolution to \cref{parsubopgenprob}.
\end{defin}

We derive from \cref{defviscomp} that a comparison result holds true.

\begin{theo}\label{parsubopgencomp}
Let $u$ and $v$ be respectively a subsolution and a supersolution to \cref{parsubopgenprob} satisfying polynomial growth condition. If $u|_{t=T}\le v|_{t=T}$, then $u\le v$ on $(0,T]\times\mathds R^N$.
\end{theo}

When $F$ is a linear operator it is known that the representation formula of its viscosity solution is built from a FBSDE, see \eqref{exparsemlin}. To adapt this method to our case we will use the \cref{cauchydpp}.

\begin{defin}\label{fbsde}
Consider the FBSDE
\begin{equation}\label{eq:fbsde}
\left\{\begin{aligned}
X^{t,\zeta}_s=&\zeta+\int_t^s\sigma\left(r,X^{t,\zeta}_r\right)dW_r+\int_t^sb\left(r,X^{t,\zeta}_r\right)dr,\\
Y^{t,\zeta}_s=&g\left(X^{t,\zeta}_T\right)+\int_s^Tf_\sigma\left(r,X_r^{t,\zeta},Y_r^{t,\zeta},Z_r^{t,\zeta}\right)dr\\
&-\int_s^TZ_r^{t,\zeta}dW_r,
\end{aligned}\right.\,s\in[t,T],
\end{equation}
where $\zeta\in L^2\left(\Omega,\mathcal F_t;\mathds R^N\right)$, $(b,\sigma)\in\mathcal A_F$, the function $f_\sigma$ is defined as
\[
f_\sigma(t,x,y,z):=f\left(t,x,y,z(\sigma(t,x))^{-1}\right),
\]
for any $(t,x,y,z)$ in $[0,T]\times\mathds R^N\times\mathds R\times\mathds R^N$ and the functions $f$ and $g$ are as in the assumptions of \cref{parsubopgenprob}. Thanks to the uniformly ellipticity condition we know that $f_\sigma$ is well defined and that the Lipschitz constant for its the fourth argument is $\ell\sqrt{\frac N{2\lambda}}$, but for simplicity we will assume that it is $\ell$, possibly increasing it.\\
Note that under this conditions the \cref{sdeassum,bsdeassum} hold for $X$ and $(Y,Z)$ respectively. We will call $(X,Y,Z)$ a solution to the FBSDE if $X$ is a solution to the SDE part of this system and $(Y^{t,\zeta},Z^{t,\zeta})$ is a solution to the BSDE part for any $(t,\zeta)\in[0,T]\times L^2\left(\Omega,\mathcal F_t;\mathds R^N\right)$. Notice that, under our assumptions, there exists a unique solution to \eqref{eq:fbsde}, thanks to \cref{sdeex,bsdeex}. Due to \cref{sdestopstart,stopbsde}, this is true even if $t$ is an a.e. finite stopping time.
\end{defin}

\begin{rem}
Notice that the uniqueness property of the FBSDE imply that, for any $0\le t\le r\le s\le T$,
\[
\left(X^{r,X^{t,\zeta}_r}_s,Y^{r,X^{t,\zeta}_r}_s,Z^{r,X^{t,\zeta}_r}_s\right)=\left(X^{t,\zeta}_s,Y^{t,\zeta}_s,Z^{t,\zeta}_s\right).
\]
This holds true even if $t$, $r$ and $s$ are stopping time.
\end{rem}

\begin{rem}\label{L_FtoI_F}
We point out that since the elements of $\mathcal L_F^\tau$ and the solutions to SDEs $(b,\sigma)$ can be uniquely determined, except for the initial data of the SDEs, by an element of $\mathcal A_F^\tau$, we can uniquely link to each operator $L\in\mathcal L_F^\tau$ an $X_{(b,\sigma)}$. Moreover, for each problem, we can uniquely associate in the same way a solution of the FBSDE \eqref{eq:fbsde}.
\end{rem}

For the remainder of this section, we will simply write $Y$ to denote the second term of the triplet $(X,Y,Z)$ solution to the FBSDE defined in \cref{fbsde}, for $(b,\sigma)$ that varies in $\mathcal A_F$. For simplicity we will omit the dependence of $Y$ and $Z$ from $X$ and $\sigma$ or, equivalently, from $(b,\sigma)$.

We will prove that $u(t,x):=\sup\limits_{(b,\sigma)\in\mathcal A_F}\mathds E\left(Y^{t,x}_t\right)$ is a viscosity solution to the \cref{parsubopgenprob} breaking the proof in several steps.

\begin{prop}\label{parsubopgenviscont}
The function $u(t,x):=\sup\limits_{(b,\sigma)\in\mathcal A_F}\mathds E\left(Y^{t,x}_t\right)$ is $\frac12$--Hölder continuous in the first variable and Lipschitz continuous in the second one. Furthermore we have that there exists a constant $c$, which depends only on $\ell$, $\mu$ and $T$, such that
\begin{equation}\label{eq:parvisbound}
\mathds E\left(|u(\tau,\zeta)|^2\right)\le\sup_{(b,\sigma)\in\mathcal A_F}\mathds E\left(\left|Y^{\tau,\zeta}_\tau\right|^2\right)\le c\left(1+\mathds E\left(|\zeta|^2\right)\right),
\end{equation}
for any $t\in[0,T]$ and $\zeta\in L^2\left(\Omega,\mathcal F_t;\mathds R^N\right)$.
\end{prop}

We point out that this proposition permits us to use the results of \cref{dynprincsec} on $u$. In particular $Y^\tau_\tau$, which is $\mathcal F^\tau_\tau$--measurable and therefore a.e. deterministic, has the same role of $\varphi$ in \cref{dynprincsec}. We already know that $Y^\tau_\tau$ is continuous in probability, thanks to our assumptions, \cref{sdeex,bsdeex}, furthermore we prove here that it satisfies \eqref{conddpp} and the continuity of $u$, which is needed for \cref{artpeng43}.

\proof
To prove our statement note that by the definition and the Jensen's inequality
\begin{align*}
|u(t,x)-u(s,y)|=&\left|\sup_{(b,\sigma)\in\mathcal A_F}\mathds E\left(Y^{t,x}_t\right)-\sup_{(b,\sigma)\in\mathcal A_F}\mathds E\left(Y^{s,y}_s\right)\right|\\
\le&\sup_{(b,\sigma)\in\mathcal A_F}\left(\mathds E\left(\left|Y^{t,x}_t-Y^{s,y}_s\right|^2\right)\right)^\frac12
\end{align*}
and
\[
|u(t,\zeta)|=\left|\sup_{(b,\sigma)\in\mathcal A_F}\mathds E\left(Y^{t,\zeta}_t\right)\right|\le\left(\sup_{(b,\sigma)\in\mathcal A_F}\mathds E\left(\left|Y^{t,\zeta}_t\right|^2\right)\right)^\frac12,
\]
for any $t,s\in[0,T]$, $x,y\in\mathds R^N$ and $\zeta\in L^2\left(\Omega,\mathcal F_t;\mathds R^N\right)$. The statement is then a consequence of \cref{bsdeex,sdeex}.
\endproof

We can now prove the dynamic programming principle for $u$.

\begin{theo}[Dynamic programming principle]\label[dpp]{cauchydpp}
For any $(b,\sigma)\in\mathcal A_F$ we let $\left(\overline Y,\overline Z\right)$ be the solution of the BSDE
\begin{equation}\label{eq:cauchydpp1}
\overline Y_s=u\left(\tau,X_\tau^{t,x}\right)+\int_{s\wedge\tau}^\tau f_\sigma\left(r,X^{t,x}_r,\overline Y_r,\overline Z_r\right)dr-\int_{s\wedge\tau}^\tau\overline Z_rdW_r,
\end{equation}
where $s\in[t,T]$ and $\tau$ is a stopping time with value in $[t,T]$. Then we have $\sup\limits_{(b,\sigma)\in\mathcal A_F}\mathds E\left(\overline Y_t\right)=u(t,x)$.
\end{theo}
\proof
Fix $(\overline b,\overline\sigma)\in\mathcal A_F$ in \eqref{eq:cauchydpp1} and define $\overline X:=X_{(b,\sigma)}$ and the subset of $\mathcal A_F$
\[
\overline{\mathcal A}_F:=\left\{(b,\sigma)\in\mathcal A_F:(b,\sigma)(s)=\left(\overline b,\overline\sigma\right)(s)\text{ for any }s\in[t,\tau]\right\}.
\]
From \cref{artpeng43} we know that
\[
\esss_{(b,\sigma)\in\overline{\mathcal A}_F}\mathds E\left(Y_\tau^{t,x}\middle|\mathcal F_\tau\right)=\esss_{(b,\sigma)\in\overline{\mathcal A}_F}\mathds E\left(Y_\tau^{\tau,\overline X^{t,x}_\tau}\middle|\mathcal F_\tau\right)=u\left(\tau,\overline X_\tau^{t,x}\right)
\]
and \cref{peng40} yields the existence of a sequence $\{(b_n,\sigma_n)\}_{n\in\mathds N}$ in $\overline{\mathcal A}_F$ and a corresponding sequence $\{Y_n\}_{n\in\mathds N}$ such that
\[
\lim_{n\to\infty}\mathds E\left(Y_{n,\tau}^{t,x}\middle|\mathcal F_\tau\right)=\esss_{(b,\sigma)\in\overline{\mathcal A}_F}\mathds E\left(Y_\tau^{t,x}\middle|\mathcal F_\tau\right)=u\left(\tau,\overline X_\tau^{t,x}\right).
\]
Then, by \cref{bsdeex} and the dominated convergence theorem, there exists a constant $c$ such that
\[
\lim_{n\to\infty}\mathds E\left(\left|\overline Y_t-Y^{t,x}_{n,t}\right|^2\right)\le\lim_{n\to\infty}c\mathds E\left(\left|u\left(\tau,\overline X_\tau^{t,x}\right)-Y^{t,x}_{n,\tau}\right|^2\right)=0,
\]
hence, up to subsequences,
\begin{equation}\label{eq:cauchydpp2}
\lim\limits_{n\to\infty}\mathds E\left(Y_{n,t}^{t,x}\right)=\mathds E\left(\overline Y_t\right).
\end{equation}
Furthermore, thanks to \cref{bsdeconf}, $Y^{t,x}_t\le\overline Y_t$ for any $(b,\sigma)\in\overline{\mathcal A}_F$, which together with \eqref{eq:cauchydpp2} implies that $\sup\limits_{(b,\sigma)\in\overline{\mathcal A}_F}\mathds E\left(Y^{t,x}_t\right)=\mathds E\left(\overline Y_t\right)$. Therefore we can use the arbitrariness of $\left(\overline b,\overline\sigma\right)$ to obtain our conclusion:
\[
\sup_{(b,\sigma)\in\mathcal A_F}\mathds E\left(\overline Y_t\right)=\sup_{(b,\sigma)\in\mathcal A_F}\mathds E\left(Y_t^{t,x}\right)=u(t,x).
\]
\endproof

Now we proceed to show that $u$ is a viscosity subsolution. In order to do that, we need the following lemma:

\begin{lem}\label{parsemlinvis}
For any $t\in(0,T)$, let $L$ be an element of $\mathcal L^t_F$ and $(X,Y,Z)$ the solution to the FBSDE \eqref{eq:fbsde} associated to $L$ as in \cref{L_FtoI_F}. If we define, for any $x\in\mathds R^N$ and $s\in[t,T]$, $u_L(s,x):=\mathds E\left(Y^{s,x}_s\right)$ we have that, for any supertangent $\varphi$ to $u_L$ at $(t,x)$,
\[
L\left(t,x,\nabla_x\varphi,D^2_x\varphi\right)\ge-\partial_t\varphi(t,x)-f\left(t,x,u_L,\nabla_x\varphi\right).
\]
\end{lem}
\proof
We preliminarily denote by $(b,\sigma)$ the element of $\mathcal A_F^t$ associated to $L$ and point out that since $(b,\sigma)$, restricted in $[t,T]$, is progressive with respect to the $\sigma$--algebra $\{\mathcal F^t_s\}_{s\in[t,T]}$, so are $L$, $X^t$ and $Y^t$. They are therefore constants a.e. in $t$. As a consequence $u_L(t,x)=Y^{t,x}_t$ a.e. for any $x\in\mathds R^N$.\
Given $x\in\mathds R^N$ and a supertangent $\varphi$ to $u_L$ at $(t,x)$ we can assume without loss of generality that $u_L(t,x)=\varphi(t,x)$, so we suppose that, a.e.,
\begin{equation}\label{eq:semlinvis0}
\partial_t\varphi(t,x)+L\left(t,x,\nabla_x\varphi,D^2_x\varphi\right)+f_\sigma(t,x,u_L,\nabla_x\varphi\sigma)<0
\end{equation}
and we will find a contradiction. Note that, as a consequence of the Blumenthal's 0--1 law, \eqref{eq:semlinvis0} is a deterministic inequality a.e.. By the definition of supertangent, there exists a $\delta\in(0,T-t)$ such that, for any $s\in[t,t+\delta]$ and $y\in B_\delta(x)$,
\begin{equation}\label{eq:semlinvis1}
u_L(s,y)\le\varphi(s,y),
\end{equation}
hence we define the stopping time
\[
\tau:=(t+\delta)\wedge\inf\left\{s\in[t,\infty):\left|X_s^{t,x}-x\right|\ge\delta\right\}
\]
and assume, possibly taking a smaller $\tau$, that
\begin{equation}\label{eq:semlinvis2}
\begin{aligned}
\partial_t\varphi\left(s\wedge\tau,X^{t,x}_{s\wedge\tau}\right)+L\left(s\wedge\tau,X^{t,x}_{s\wedge\tau},\nabla_x\varphi,D^2_x\varphi\right)&\\
+f_\sigma\left(s\wedge\tau,X^{t,x}_{s\wedge\tau},\varphi,\nabla_x\varphi\sigma\right)&<0.
\end{aligned}
\end{equation}
We point out that, by \eqref{eq:semlinvis0} and \cref{cadlagcont}, the previous inequality holds true on a set of positive measure for the $\chi_{[t,t+\delta)}dt\times d\mathds P$ measure, thus $\tau>t$ on a set of positive measure.\\
Let $\left(\overline Y_s,\overline Z_s\right):=\left(Y^{t,x}_{s\wedge\tau},Z^{t,x}_{s\wedge\tau}\right)$, which solve the BSDE
\[
\overline Y_s=Y_\tau^{t,x}+\int_{s\wedge\tau}^\tau f_\sigma\left(r,X^{t,x}_r,\overline Y_r,\overline Z_r\right)dr-\int_{s\wedge\tau}^\tau\overline Z_rdW_r,\quad s\in[t,T],
\]
and $\left(\hat Y_s,\hat Z_s\right):=\left(\varphi\left(s,X^{t,x}_{s\wedge\tau}\right),(\nabla_x\varphi\sigma)\left(s,X^{t,x}_{s\wedge\tau}\right)\right)$ which, by Itô's formula, is solution to
\[
\begin{aligned}
\hat Y_s=&\varphi\left(\tau,X^{t,x}_\tau\right)-\int_{s\wedge\tau}^\tau\hat Z_rdW_r\\
&-\int_{s\wedge\tau}^\tau\left(\partial_t\varphi\left(r,X^{t,x}_r\right)+L\left(r,X^{t,x}_r,\nabla_x\varphi,D^2_x\varphi\right)\right)dr,
\end{aligned}
\quad s\in[t,T].
\]
By \eqref{eq:semlinvis1} we have that
\[
u_L\left(\tau,X_\tau^{t,x}\right)-\varphi\left(\tau,X_\tau^{t,x}\right)=Y_\tau^{\tau,X_\tau^{t,x}}-\varphi\left(\tau,X_\tau^{t,x}\right)\le0
\]
and \eqref{eq:semlinvis2} imply, thanks to \cref{bsdeconf}, that $Y_t^{t,x}<\varphi(t,x)$ a.e., but this lead to a contradiction since we know that, by our assumptions, $\varphi(t,x)=Y^{t,x}_t$ a.e.. This conclude the proof.
\endproof

\begin{prop}\label{parsubopgensubvis}
The function $u(t,x)$ is a continuous viscosity subsolution to the \cref{parsubopgenprob}.
\end{prop}
\proof
We know from \cref{parsubopgenviscont} that $u$ is continuous, thus we just have to prove the subsolution property to conclude the proof.\\
Let $L$ be an element of $\mathcal L_F^t$ and $u_L$ as defined in \cref{parsemlinvis}, then if $\varphi$ is a supertangent to $u_L$ in $(t,x)$ we have that, by the definition of $\mathcal L_F^t$,
\[
F\left(t,x,\nabla_x\varphi,D^2_x\varphi\right)\ge L\left(t,x,\nabla_x\varphi,D^2_x\varphi\right)\ge-\partial_t\varphi(t,x)-f(t,x,u_L,\nabla_x\varphi),
\]
therefore $u_L$ is a viscosity subsolution to the \cref{parsubopgenprob} at $(t,x)$. Thanks to the arbitrariness of $t$, $L$ and $x$ we then have that $u_L$ is a viscosity subsolution in $(t,x)$ for any $L\in\mathcal L_F^t$, $x\in\mathds R^N$ and $t\in(0,T)$. From \cref{supeqrem} we have that
\[
\sup_{L\in\mathcal L_F^t}u_L(t,x)=\sup_{(b,\sigma)\in\mathcal A_F^t}\mathds E\left(Y^{t,x}_t\right)=\sup_{(b,\sigma)\in\mathcal A_F}\mathds E\left(Y^{t,x}_t\right)=u(t,x),
\]
therefore the family of functions $\{u_L\}$ is locally equibounded, thanks to \cref{parsubopgenviscont}. Well known properties of viscosity solutions hence yield that
\[
\sup_{L\in\mathcal L_F^t}u_L(t,x)=u(t,x)
\]
is a viscosity subsolution for any $(t,x)\in(0,T)\times\mathds R^N$.
\endproof

We conclude this \namecref{parprob} with our main statement.

\begin{theo}
The function $u(t,x):=\sup\limits_{(b,\sigma)\in\mathcal A_F}\mathds E\left(Y^{t,x}_t\right)$ is the only viscosity solution to the \cref{parsubopgenprob} satisfying polynomial growth condition such that $u(T,x)=g(x)$ for any $x$ in $\mathds R^N$.
\end{theo}
\proof
The uniqueness is a consequence of \cref{parsubopgencomp} and \eqref{eq:parvisbound}, hence we only have to show that $u$ is a viscosity solution.\\
From \cref{parsubopgensubvis} we know that $u$ is a continuous viscosity subsolution and it is easy to see that $u(T,x)=g(x)$ for any $x\in\mathds R^N$, so we only need to prove the supersolution property of $u$. Fixed $(t,x)$ in $(0,T)\times\mathds R^N$, let $\psi$ be a subtangent to $u$ in $(t,x)$ which we assume, without loss of generality, equal to $u$ at $(t,x)$ and $\delta$ a positive constant such that
\begin{equation}\label{eq:subtandis}
\psi(s,y)\le u(s,y)\qquad\text{for any }(s,y)\in[t,t+\delta]\times B_\delta(x).
\end{equation}
We know, thanks to \cref{charsublin}, that there exists a continuous and deterministic $L\in\mathcal L_F$ for which
\[
F\left(t,x,\nabla_x\psi,D^2_x\psi\right)=L\left(t,x,\nabla_x\psi,D^2_x\psi\right)
\]
and assume by contradiction
\begin{align*}
F\left(t,x,\nabla_x\psi,D^2_x\psi\right)=L\left(t,x,\nabla_x\psi,D^2_x\psi\right)>-\partial_t\psi(t,x)-f_\sigma(t,x,u,\nabla_x\psi\sigma).
\end{align*}
Then, by continuity,
\begin{equation}\label{eq:supsoldisass}
\partial_t\psi(s,y)+L\left(s,y,\nabla_x\psi,D^2_x\psi\right)>-f_\sigma(s,y,\psi,\nabla_x\psi\sigma)
\end{equation}
for any $(s,y)\in[t,t+\delta]\times B_\delta(x)$, possibly taking a smaller $\delta$.\\
We denote with $(b,\sigma)$ and $(X,Y,Z)$, respectively, the element of $\mathcal A$, which, to repeat, is continuous and deterministic, and the solution to the FBSDE \eqref{eq:fbsde} associated to $L$. We define the stopping time
\[
\tau:=(t+\delta)\wedge\inf\left\{s\in[t,\infty):\left|X_s^{t,x}-x\right|\ge\delta\right\},
\]
let $\left(\overline Y_s,\overline Z_s\right)$ be the solution to the BSDE
\[
\overline Y_s=u\left(\tau,X_\tau^{t,x}\right)+\int_{s\wedge\tau}^\tau f_\sigma\left(r,X^{t,x}_r,\overline Y_r,\overline Z_r\right)dr-\int_{s\wedge\tau}^\tau\overline Z_rdW_r,\quad s\in[t,T]
\]
and $\left(\hat Y_s,\hat Z_s\right):=\left(\psi\left(s,X^{t,x}_{s\wedge\tau}\right),(\nabla_x\psi\sigma)\left(s,X^{t,x}_{s\wedge\tau}\right)\right)$ which, by Itô's formula, is solution to
\[
\begin{aligned}
\hat Y_s=&\psi\left(\tau,X^{t,x}_\tau\right)-\int_{s\wedge\tau}^\tau\hat Z_rdW_r\\
&-\int_{s\wedge\tau}^\tau\left(\partial_t\psi\left(r,X^{t,x}_r\right)+L\left(r,X^{t,x}_r,\nabla_x\psi,D^2_x\psi\right)\right)dr,
\end{aligned}
\quad s\in[t,T].
\]
We know from the \cref{cauchydpp} that
\begin{equation}\label{eq:eqassum}
\sup_{(b,\sigma)\in\mathcal A_F}\mathds E\left(\overline Y_t\right)=u(t,x)=\psi(t,x),
\end{equation}
but by \eqref{eq:subtandis} we have $u\left(\tau,X^{t,x}_\tau\right)\ge\psi\left(\tau,X^{t,x}_\tau\right)$, which together with \eqref{eq:supsoldisass} imply, thanks to \cref{bsdeconf}, that $\overline Y_t>\psi(t,x)$ a.e., contradicting \eqref{eq:eqassum}.
\endproof

\section{Connection with 2BSDEs}\label{2bsdesec}

In this section we will briefly present a different approach to second order BSDEs, 2BSDEs for short, using our dynamic programming principle. This is intended as a short survey and not as a complete analysis of the subject.

We start giving the classical formulation of 2BSDE. Following \cite{sonertouzizhang11,possamai13} assume that $\Omega:=\left\{\omega\in C\left([0,T];\mathds R^N\right):\omega_0=0\right\}$ and let $\mathds P_0$ be the Wiener measure. Note that in this space the Brownian motion $W$ is a projection, i.e. $W_t(\omega)=\omega_t$. Denote with $[W]_t$ the quadratic variation of the projection and with
\[
\hat a_t:=\lim_{\varepsilon\downarrow0}\frac{[W]_t-[W]_{t-\varepsilon}}{\varepsilon}
\]
its variation. We will then denote with $\mathcal P_W$ the set of the probability measures $\mathds P$ such that $[W]$ is absolutely continuous in $t$ and $\hat a\in\mathds S^N_+$, $\mathds P$--a.e.. In particular $\mathds P_0\in\mathcal P_W$ because $[W]_t=tI_N$ and $\hat a_t=I_N$, $\mathds P_0$--a.e., where $I_N$ is the $N\times N$ identity matrix. Moreover, let $\mathcal P_S$ be the subset of $\mathcal P_W$ composed by the probability measures $\mathds P^\alpha:=\mathds P_0\circ(X^\alpha)^{-1}$, where
\[
X^\alpha_t:=\int_0^t\alpha^{1/2}_sdW_s,\qquad\mathds P_0\text{--a.e.},
\]
and $\alpha$ is a progressive process in $\mathds S^N_+$ such that, for two fixed $\underline a,\overline a$ in $\mathds S^N_+$, $\underline a\le\alpha\le\overline a$, $\mathds P_0$--a.e.. It is then apparent a link between $\mathcal P_S$ and the control set $\mathcal A_F$.\\
Now, given a function
\[
h:[0,T]\times\Omega\times\mathds R\times\mathds R^N\times D_h\to\mathds R,
\]
where $D_h$ is a subset of $\mathds R^{N\times N}$ containing 0, define for any $a\in\mathds S^N_+$
\[
f(t,\omega,y,z,a):=\sup_{\gamma\in D_h}\left(\frac12\langle\gamma,a\rangle-h(t,\omega,y,z,\gamma)\right).
\]
Furthermore let $\hat f(t,y,z):=f(t,y,z,\hat a_t)$,
\[
\mathcal P_h^2:=\left\{\mathds P\in\mathcal P_S:\mathds E^{\mathds P}\left(\int_0^T\left|\hat f(t,0,0)\right|^2dt\right)<\infty\right\}
\]
and $\mathcal P_h^2(t,\mathds P):=\left\{\mathds P'\in\mathcal P_h^2:\mathds P'=\mathds P\text{ on }\mathcal F_t\right\}$.

A pair of progressive processes $(Y,Z)$ is solution to the 2BSDE
\begin{equation}\label{eq:2bsde}
Y_t=\xi+\int_t^T\hat f(s,Y_s,Z_s)ds-\int^T_tZ_sdW_s+K_T-K_t
\end{equation}
if, for any $\mathds P\in\mathcal P_h^2$,
\begin{enumerate}[label=(\roman*)]
\item $Y_T=\xi$, $\mathds P$--a.e.;
\item the process $K^{\mathds P}$ defined below has non decreasing path $\mathds P$--a.e.,
\[
K_t^{\mathds P}=Y_0-Y_t-\int_0^t\hat f(s,Y_s,Z_s)ds+\int^t_0Z_sdW_s,\quad t\in[0,T],\,\mathds P\text{--a.e.};
\]
\item the family $\left\{K^{\mathds P},\mathds P\in\mathcal P^2_h\right\}$ satisfies the minimum condition
\begin{equation}\label{eq:kmincond}
K_t^{\mathds P}=\essi_{\mathds P'\in\mathcal P_h^2(t,\mathds P)}\mathds E^{\mathds P}\left(K_T^{\mathds P'}\middle|\mathcal F_t\right),\qquad\mathds P\text{--a.e.}.
\end{equation}
\end{enumerate}
Under suitable conditions the 2BSDE \eqref{eq:2bsde} admits a unique solution and, if we denote with $\left(Y^{\mathds P}(r,\xi),Z^{\mathds P}(r,\xi)\right)$ the solution to the BSDE
\[
Y_t=\xi+\int_t^r\hat f(s,Y_s,Z_s)ds-\int^r_tZ_sdW_s,\qquad t\in[0,r],\,\mathds P\text{--a.e.},
\]
it can be proved that $Y_t=\esss\limits_{\mathds P'\in\mathcal P_h^2(t,\mathds P)}Y_t^{\mathds P'}(r,Y_r)$ for any $r\in[t,T]$ and $\mathds P\in\mathcal P_h^2$. The last identity is a dynamic programming principle and can be seen as the connection between 2BSDE and our method.

Now we will show a different formulation of 2BSDEs, using controls instead of probability measures. Let $\mathcal A$ be our control set, made up by the progressive processes in $L^2([0,T]\times\Omega;B)$, where $B$ is a Banach space, and, for any $t\in[0,T]$,
\[
\mathcal A(t,\alpha):=\left\{\alpha'\in\mathcal A:\alpha'_s=\alpha_s\text{ for any }s\in[0,t]\right\}.
\]
Then define the function
\[
f:[0,T]\times\Omega\times\mathds R\times\mathds R^N\times B\to\mathds R
\]
and assume that there exists a $C>0$ such that
\[
|f(t,y,z,\alpha)-f(t,y,z,\alpha')|\le C(1+|y|+|z|)|\alpha-\alpha'|.
\]
We will also assume that, for each $\alpha\in\mathcal A$, $f_\alpha(t,y,z):=f(t,y,z,\alpha)$ satisfies \cref{bsdeassum} uniformly with respect to $\alpha$. We point out that these conditions are not intended to be minimal. We then have that the BSDEs
\[
Y^\alpha_t=\xi_\alpha+\int_t^Tf_\alpha(s,Y_s^\alpha,Z_s^\alpha)ds-\int_t^TZ^\alpha_sdW_s,\qquad t\in[0,T],
\]
admit a unique solution for any $\alpha\in\mathcal A$. If we moreover require that, for any $t\in[0,T]$, $\sup\limits_{\alpha\in\mathcal A}\mathds E\left(|Y^\alpha_t|^2\right)<\infty$ (which can be achieved if, for example, $B$ is compact), then by the \cref{cauchydpp},
\begin{equation}\label{eq:poordpp}
\overline Y_t^\alpha:=\esss_{\alpha'\in\mathcal A(t,\alpha)}Y^{\alpha'}_t=\esss_{\alpha'\in\mathcal A(t,\alpha)}Y^{\alpha'}_t\left(r,\overline Y_r^{\alpha'}\right),\quad\text{for any }0\le t\le r\le T,
\end{equation}
where $Y^{\alpha'}\left(r,\overline Y_r^{\alpha'}\right)$ is solution to the BSDE $\left(\overline Y^{\alpha'}_r,f_\alpha,r\right)$. It is easy to see that, for any $\alpha\in\mathcal A$, $\overline Y^\alpha$ is a continuous progressive process in $L^2$ and $\overline Y^\alpha_T=\xi_\alpha$ a.e..\\
Using the same arguments of \cite{possamai13} we that, for each $\alpha\in\mathcal A$, there exist two progressive processes in $L^2$, $Z^\alpha$ and $K^\alpha$, such that $K^\alpha$ is a continuous and increasing process in $t$ with $K_0^\alpha=0$ and
\[
\overline Y_t^\alpha=\xi_\alpha+\int_t^Tf_\alpha\left(s,\overline Y^\alpha_s,\overline Z_s^\alpha\right)ds-\int^T_t\overline Z_s^\alpha dW_s+K^\alpha_T-K^\alpha_t,\qquad t\in[0,T].
\]
We also have that, as in \eqref{eq:kmincond},
\[
K_t^\alpha=\essi_{\alpha'\in\mathcal A(t,\alpha)}\mathds E\left(K_T^{\alpha'}\middle|\mathcal F_t\right),\qquad\text{a.e. for any }t\in[0,T],\,\alpha\in\mathcal A.
\]

If we let $X$ be as in \eqref{eq:fbsde} and $\left(\overline Y,\overline Z,K\right)$ be the solution to the 2BSDE (we omit the dependence on the control set $\mathcal A_F$)
\[
\overline Y_s^{t,x}=g\left(X^{t,x}_T\right)+\int_s^Tf_\sigma\left(r,X^{t,x}_T,\overline Y_r^{t,x},\overline Z_r^{t,x}\right)ds-\int^T_s\overline Z_r^{t,x}dW_r+K^{t,x}_T-K^{t,x}_s
\]
for any $x\in\mathds R^N$ and $0\le t\le s\le T$, we then have that the function $u(t,x):=\overline Y^{t,x}_t$ is the viscosity solution to \cref{parsubopgenprob}.

\begin{appendices}

\section{Some Probability Results}

Here we give some probability results we use in this paper.

\begin{lem}\label{cadlagcont}
Let $\{U_t\}_{t\in[0,\infty)}$ be a cadlag process, then for any $\varepsilon>0$ there exists a $\delta>0$ such that
\[
\mathds P(\{|U_t-U_s|<\varepsilon,\text{ for any }s\in[t,t+\delta)\})>0.
\]
\end{lem}
\proof
Our argument is by contradiction. Assume that there exists an $\varepsilon>0$ such that for any $\delta>0$
\[
\mathds P(\{|U_t-U_s|<\varepsilon,\text{ for any }s\in[t,t+\delta)\})=0,
\]
which is equivalent to
\[
\mathds P(\{|U_t-U_s|\ge\varepsilon,\text{ for any }s\in[t,t+\delta)\})=1.
\]
Let, for any positive integer $n$,
\[
A_n:=\left\{|U_t-U_s|\ge\varepsilon,\text{ for any }s\in\left[t,t+\frac1n\right)\right\},
\]
then $A_{n}\subseteq A_{k}$ if $k\le n$ and
\[
A:=\bigcap_{n=1}^\infty A_n=\left\{\lim_{s\downarrow t}|U_t-U_s|\ge\varepsilon\right\}.
\]
Since $U$ is right continuous we know that $\mathds P(A)=0$ which contradicts our assumption, since $\mathds P(A)=\lim\limits_{n\to\infty}\mathds P(A_n)=1$.
\endproof

Consider the SDE
\begin{equation}\label{eq:sde}
X^{t,\zeta}_s=\zeta+\int_t^s\sigma\left(r,X^{t,\zeta}_r\right)dW_r+\int_t^sb\left(r,X^{t,\zeta}_r\right)dr,\qquad s\in[t,\infty),
\end{equation}
under the following assumptions:
\begin{assum}\label{sdeassum}
$t\in[0,\infty)$, $\zeta\in L^2\left(\Omega,\mathcal F_t;\mathds R^N\right)$ and for the functions
\[
b:[0,\infty)\times\Omega\times\mathds R^N\to\mathds R^N\text{ and }\sigma:[0,\infty)\times \Omega\times\mathds R^N\to\mathds R^{N\times M}
\]
there exists a positive constant $\ell$ such that a.e., for any $r\in[0,\infty)$, $x,y\in\mathds R^N$,
\begin{enumerate}[label=(\roman*)]
\item $|b(r,x)-b(r,y)|+|\sigma(r,x)-\sigma(r,y)|\le\ell|x-y|$;
\item $\{(b,\sigma)(t,0)\}_{t\in[0,\infty)}$ is a progressive process belonging to $L^2([0,T]\times\Omega)$ for any $T\in[0,\infty)$.
\end{enumerate}
\end{assum}

A solution to this SDE is a continuous progressive process $X$ as in \eqref{eq:sde} such that $X\in L^2([0,T]\times\Omega)$ for any $T\in[0,\infty)$. The next theorem summarizes some SDE result given in \cite{krylovbook}.

\begin{theo}\label{sdeex}
Under \cref{sdeassum} there exists a unique solution to the SDE \eqref{eq:sde}. Moreover, for any $T\in[t,\infty)$, there exists a constant $c$, depending only on $\ell$ and $T$ such that
\begin{gather*}
\mathds E\left(\sup_{s\in[t,T]}\left|X^{t,\zeta}_s-\zeta\right|^2\right)\le c\mathds E\left(\int_t^T\left(|b(s,0)|^2+|\sigma(s,0)|^2\right)ds\right),\\
\mathds E\left(\sup_{s\in[t,T]}\left|X^{t,\zeta}_s\right|^2\right)\le c\mathds E\left(|\zeta|^2+\int_t^T\left(|b(s,0)|^2+|\sigma(s,0)|^2\right)ds\right),\\
\mathds E\left(\sup_{s\in[t,T]}\left|X^{t,\zeta}_s-X^{t,\zeta'}_s\right|^2\right)\le c\mathds E\left(|\zeta-\zeta'|^2\right).
\end{gather*}
\end{theo}

\begin{rem}\label{sdestopstart}
The results obtained in this section hold even for SDEs with an a.e. finite stopping time $\tau$ as starting time. In fact if for any $\zeta$ in $L^2\left(\Omega,\mathcal F_\tau;\mathds R^N\right)$ we define
\[
\overline b(t,x):=b(t,x+\zeta)\chi_{\{\tau\le t\}}\text{ and }\,\overline\sigma(t,x):=\sigma(t,x+\zeta)\chi_{\{\tau\le t\}},
\]
then $X^{\tau,\zeta}$ is solution of the SDE $(b,\sigma)$ if and only if $\overline X^{0,0}:=X^{\tau,\zeta}-\zeta$ is solution of the SDE $\left(\overline b,\overline\sigma\right)$. The claim can be easily obtained from this.
\end{rem}

\begin{rem}
By the strong Markov property, for any a.e. finite stopping time $\tau$, the process $\{W_t^\tau\}_{t\in[0,\infty)}:=\{W_{\tau+t}-W_\tau\}_{t\in[0,\infty)}$ is a Brownian motion. Thus if $b$ and $\sigma$ are are progressive with respect to the filtration $\{\mathcal F^\tau_t\}_{t\in[0,\infty)}$ then any solution to the SDE $(b,\sigma)$ with initial data $\tau+t$ and $\zeta\in L^2\left(\Omega,\mathcal F_t^\tau;\mathds R^N\right)$ is also progressive with respect to that filtration. In fact, in this case, the stochastic integral with respect to $W_t^\tau$ is the same as the one with respect to $W_{\tau+t}$.
\end{rem}

\subsection{Backward Stochastic Differential Equations}

In this subsection we give some results on BSDEs used in our investigation. Most of them are well known and actually hold under more general assumptions. We refer to \cite{syspardoux,phamsdebook,pardouxbook,brianddelyonhupardouxstoica03} for their proofs.

We will work under the followings assumptions:

\begin{assum}\label{bsdeassum}
Let $T\in[0,\infty)$, $\xi\in L^2\left(\Omega,\mathcal F_T,\mathds P;\mathds R^M\right)$ and
\[
f:[0,T]\times \Omega\times\mathds R^M\times\mathds R^{M\times N}\to\mathds R^M
\]
a function which admits a positive constant $\ell$ and a real number $\mu$ such that a.e., for any $t\in[0,T]$, $y,y'\in\mathds R^M$ and $z,z'\in\mathds R^{M\times N}$,
\begin{enumerate}[label=(\roman*)]
\item $\{f(s,0,0)\}_{s\in[0,T]}$ is a progressive process belonging to $L^2([0,T]\times\Omega)$;
\item $|f(t,y,z)|\le|f(t,0,z)|+\ell(1+|y|)$;
\item $|f(t,y,z)-f(t,y,z')|\le\ell|z-z'|$;
\item $(y-y')(f(t,y,z)-f(t,y',z))\le\mu|y-y'|^2$;
\item $v\mapsto f(t,v,z)$ is continuous.
\end{enumerate}
\end{assum}

A solution to the BSDE $(\xi,f,T)$, where $\xi$ and $T$ have respectively the role of a final condition and a terminal time, is a pair $(Y,Z)$ of progressive processes belonging to $L^2([0,T]\times\Omega)$ such that
\begin{equation}\label{eq:bsde}
Y_t=\xi+\int_t^Tf(s,Y_s,Z_s)ds-\int_t^TZ_sdW_s,\quad\text{for any }t\in[0,T].
\end{equation}

The followings are classical results of BSDE theory.

\begin{theo}\label{bsdeex}
Under the \cref{bsdeassum} the BSDE \eqref{eq:bsde} has a unique solution $(Y,Z)$. Furthermore there exists a constant $c$, which depends on $T$, $\mu$ and $\ell$, such that
\[
\mathds E\left(\sup_{t\in[0,T]}|Y_t|^2+\int_0^T|Z_t|^2dt\right)\le c\mathds E\left(|\xi|^2+\int_0^T|f(t,0,0)|^2dt\right)
\]
and, if $(Y',Z')$ is the solution to the BSDE $(\xi',f',T)$,
\begin{multline*}
\mathds E\left(\sup_{t\in[0,T]}|Y_t-Y'_t|^2+\int_0^T|Z_t-Z'_t|^2dt\right)\\
\le c\mathds E\left(|\xi-\xi'|^2+\int_0^T|f(t,Y'_t,Z'_t)-f'(t,Y'_t,Z'_t)|^2dt\right).
\end{multline*}
\end{theo}
\begin{theo}\label{bsdeconf}
Assuming $M=1$, let $(Y,Z)$ be the solution to the BSDE $(\xi,f,T)$ under the \cref{bsdeassum} and
\[
Y'_t=\xi'+\int_t^TV_sds-\int_t^TZ'_sdW_s,\qquad t\in[0,T],
\]
where $\xi'\in L^2\left(\Omega,\mathcal F_T,\mathds P;\mathds R\right)$, $Y',V\in L^2([0,T]\times\Omega)$ and $Z'\in L^2([0,T]\times\Omega)$. Suppose that $\xi\le\xi'$ a.e. and $f(t,Y'_t,Z'_t)\le V_t$ a.e. for the $dt\times d\mathds P$ measure. Then, for any $t\in[0,T]$, $Y_t\le Y'_t$ a.e..\\
If moreover $Y_0=Y'_0$ a.e., then $Y_t=Y'_t$ a.e. for any $t\in[0,T]$, or in other words, whenever either $\mathds P(\{\xi<\xi'\})>0$ or $f(s,Y'_s,Z'_s)<V_s$, for any $(y,z)$ in $\mathds R\times\mathds R^N$ on a set of positive $dt\times d\mathds P$ measure, then $Y_0<Y'_0$.
\end{theo}
\begin{prop}\label{stopbsde}
Let $(Y,Z)$ be the solution to the BSDE \eqref{eq:bsde} and assume that there exists a stopping time $\tau$ such that $\tau\le T$, $\xi$ is $\mathcal F_\tau$--measurable and $f(t,y,z)=0$ on the set $\{\tau\le t\}$. Then $Y_t=Y_{\tau\wedge t}$ a.e. and $Z_t=0$ a.e. on the set $\{\tau\le t\}$.
\end{prop}

\section{Comparison Theorem}

Consider the parabolic problem
\begin{equation}\label{eq:defparpde}
\partial_tu(t,x)+F\left(t,x,u,\nabla u,D^2u\right)=0,\quad t\in(0,T),x\in\mathds R^N,
\end{equation}
where $F$ is a continuous elliptic operator which admits, for any $t\in[0,T]$, $(x,r,p,S)$ and $(y,r',p',S')$ in $\mathds R^N\times\mathds R\times\mathds R^N\times\mathds S^N$, a $\mu\in\mathds R$ and a positive constant $\ell$ such that
\begin{enumerate}[label=(\roman*)]
\item $|F(t,x,r,p,S)-F(t,x,r,p,S')|\le\ell\left(1+|x|^2\right)|S-S'|$;
\item $|F(t,x,r,p,S)-F(t,x,r,p',S)|\le\ell(1+|x|)|p-p'|$;
\item $|F(t,x,r,p,S)-F(t,y,r,p,S)|\le\ell(1+|x|+|y|)|x-y|\|(p,S)\|$;
\item\emph{(Monotonicity)} $(F(t,x,r,p,S)-F(t,x,r',p,S))(r-r')\le\mu|r-r'|^2$;
\item\label{domteocond} the continuity of the function $r\mapsto F(t,x,r,p,S)$ is independent from the fourth variable.
\end{enumerate}
Notice that, given a compact set $K\subset\mathds R^N\times\mathds R\times\mathds S^N$, \cref{domteocond} and the Heine--Cantor theorem yield the existence of a modulus of continuity $\omega_K$ such that, if $(x,r,S),(x,r',S)\in K$,
\begin{equation}\label{eq:domteocond}
|F(t,x,r,p,S)-F(t,x,r',p,S)|\le\omega_K(|r-r'|)
\end{equation}
for any $t\in[0,T]$ and $p\in\mathds R^N$.

Here we give a comparison result, which is an adaptation of \cite[Theorem C.2.3]{book:peng} for problem \eqref{eq:defparpde}. To prove it we will need the following \namecref{domteo}, adaptation of \cite[Theorem C.2.2]{book:peng} which can be proved similarly. Notice that condition \textbf{(G)} in \cite{book:peng} is replaced by \eqref{eq:domteo.1}.

\begin{theo}\label{domteo}
Let $\{F_i\}_{i=1}^k$ be a collection of continuous functions from $[0,T]\times\mathds R^N\times\mathds R\times\mathds R^N\times\mathds S^N$ to $\mathds R^N$ and assume that, if $(x,r,S),(y,r,S)$ belong to a compact set $K\subset\mathds R^N\times\mathds R\times\mathds S^N$, there exist a constant $C_K$ and a modulus of continuity $\omega_K$ such that, for any $i\in\{1,\cdots,k\}$,
\begin{equation}\label{eq:domteo.1}
|F_i(t,x,r,p,S)-F_i(t,y,r,p,S)|\le C_K(1+|p|)|x-y|+\omega_K(|x-y|).
\end{equation}
Furthermore assume the following domination condition: there exists a collection of positive constants $\{\beta_i\}_{i=1}^k$ satisfying
\[
\sum_{i=1}^k\beta_iF_i(t,x,r_i,p_i,S_i)\le0
\]
for each $(t,x)\in[0,T]\times\mathds R^N$ and $(r_i,p_i,S_i)$ such that $\sum\limits_{i=1}^k\beta_ir_i\ge0$, $\sum\limits_{i=1}^k\beta_ip_i=0$ and $\sum\limits_{i=1}^k\beta_iX_i\le0$.\\
For any $i\in\{1,\cdots,k\}$, let $u_i$ be a viscosity subsolution of
\[
\partial_tu(t,x)+F_i\left(t,x,u,\nabla u,D^2u\right)=0,\quad t\in(0,T),x\in\mathds R^N,
\]
and assume that $\sum\limits_{i=1}^k\beta_iu_i(T,\cdot)\le0$ and $\left(\sum\limits_{i=1}^k\beta_iu_i(\cdot,x)\right)^+\to0$ uniformly as $|x|\to\infty$. Then $\sum\limits_{i=1}^k\beta_iu_i(t,\cdot)\le0$ for any $t\in(0,T)$.
\end{theo}

\begin{theo}\label{defviscomp}
Let $u$ and $v$ be, respectively, a viscosity subsolution and a viscosity supersolution to \eqref{eq:defparpde} satisfying polynomial growth condition. Then, if $u|_{t=T}\le v|_{t=T}$, $u\le v$ on $(0,T]\times\mathds R^N$.
\end{theo}
\proof
We set $\phi(x):=(1+|x|^2)^\frac c2$,
\[
\gamma>\mu+\ell\sup_{x\in\mathds R^N}\left((1+|x|)\frac{|\nabla\phi(x)|}{\phi(x)}+\left(1+|x|^2\right)\frac{|D^2\phi(x)|}{\phi(x)}\right),
\]
$u_1(t,x):=\dfrac{e^{-\gamma t}u(t,x)}{\phi(x)}$ and $u_2(t,x):=-\dfrac{e^{-\gamma t}v(t,x)}{\phi(x)}$, where $c$ is such that both $|u_1|$ and $|u_2|$ converge uniformly to 0 as $|x|\to\infty$. Notice that
\[
\nabla\phi(x)=c\frac{\phi(x)x^\dag}{1+|x|^2}\quad\text{and}\quad D^2\phi(x)=\phi(x)\left(\frac c{1+|x|^2}I+\frac{c(c-2)}{(1+|x|^2)^2}x\otimes x\right),
\]
therefore $\gamma$ is well defined. We also set the operators $F_1(t,x,r,p,S)$, given by
\[
\frac{e^{-\gamma t}}{\phi(x)}F\!\left(t,x,e^{\gamma t}r\phi,e^{\gamma t}\left(r\nabla\phi+\phi p^\dag\right),e^{\gamma t}\left(rD^2\phi+\nabla\phi\otimes p+p\otimes\nabla\phi+\phi S\right)\right)\!,
\]
and $F_2(t,x,r,p,S)$, given by
\[
-\frac{e^{-\gamma t}\!}{\phi(x)}F\!\left(\!t,x,\!-e^{\gamma t}r\phi,\!-e^{\gamma t}\!\left(r\nabla\phi\!+\!\phi p^\dag\right)\!,\!-e^{\gamma t}\!\left(rD^2\phi\!+\!\!\nabla\phi\!\otimes\!p+p\!\otimes\!\nabla\phi\!+\!\phi S\right)\right)\!.
\]
It is easy to check that, for $i\in\{1,2\}$, $F_i$ is still continuous, elliptic, Lipschitz continuous in $p$ and $S$, that its monotonicity constant is
\[
\mu+\ell\sup_{x\in\mathds R^N}\left((1+|x|)\frac{|\nabla\phi(x)|}{\phi(x)}+\left(1+|x|^2\right)\frac{|D^2\phi(x)|}{\phi(x)}\right),
\]
i.e. is lower than $\gamma$, and $u_i$ is a viscosity subsolution to
\[
\partial_tu(t,x)-\gamma u(t,x)+F_i\left(t,x,u,\nabla u,D^2u\right)=0,\quad t\in(0,T),x\in\mathds R^N.
\]
It can also be checked that, if $(x,r,S),(y,r,S)$ belong to a compact set $K\subset\mathds R^N\times\mathds R\times\mathds S^N$, there exist a constant $C_K$ and a modulus of continuity $\widetilde\omega_K$ bigger than $\omega_K$ in \eqref{eq:domteocond} such that
\[
|F_i(t,x,r,p,S)-F_i(t,y,r,p,S)|\le C_K(1+|p|)|x-y|+\widetilde\omega_K(|x-y|)
\]
for any $i\in{1,2}$, $t\in[0,T]$ and $p\in\mathds R^N$.\\
Furthermore $(u_1+u_2)|_{t=T}\le0$ and $F_1(t,x,r,p,S)+F_2(t,x,-r,-p,-S)=0$. From these properties we have, for any $r_i\in\mathds R$, $p_i\in\mathds R^N$ and $S_i\in\mathds S^N$ such that $r=r_1+r_2\ge0$, $p_1=-p_2$ and $S=S_1+S_2\le0$,
\begin{multline*}
-\gamma r+F_1(t,x,r_1,p_1,S_1)+F_2(t,x,r_2,p_2,S_2)\\
\begin{aligned}
=&-\gamma r+F_1(t,x,r_1,p_1,S_1)+F_2(t,x,-r_1,-p_1,-S_1)\\
&+(F_2(t,x,r_2,p_2,S_2)-F_2(t,x,r_2-r,p_2,S_2-S))\frac rr\\
\le&-\gamma r+(F_2(t,x,r_2,p_2,S_2)-F_2(t,x,r_2-r,p_2,S_2))\frac rr\\
\le&-\gamma r+\gamma r=0.
\end{aligned}
\end{multline*}
As a consequence we have that all the conditions of \cref{domteo} are satisfied, thus $u_1+u_2\le0$, or equivalently, $u\le v$ in $(0,T]\times\mathds R^N$.
\endproof

\end{appendices}

\phantomsection

\pdfbookmark[1]{References}{References}
\emergencystretch=2em

redpajama_set_name: RedPajamaArXiv


text:
"We're in the dark": Questions surround Roma street death case
https://english.radio.cz/were-dark-questions-surround-roma-street-death-case-8721150#player=on
Length of audio 4:37
Photo: Ondřej Hájek, ČTK
The death of a Romany man in Teplice on Saturday has been making international headlines, after video emerged of officers restraining him shortly before he reportedly died in an ambulance. The police say, however, that a preliminary autopsy has found a drug overdose to be the cause of death. I discussed the case with Roma rights advocate Gwendolyn Albert, who is among those comparing the man's death to that of George Floyd in the US.
"The parallels are extraordinary. First of all there's the very fact that without a bystander filming the incident we would not know what the nature of the police intervention was.
"Then there's the fact that the police used the exact same tactic that the police used in the George Floyd case – kneeling on somebody's neck, which is a very, very risky tactic to use.
"The third parallel is that this person is a member of a community that is pretty much despised here sadly, and experiences a lot of racism.
"And lastly the last few days have shown that the other parallel is that the response of the authorities has been defensive, rushed, callous, and I would even say arrogant."
What do you say to the police's assertion that they had to use force because this man was out of control and was fighting and damaging a car?
"The police are allowed to use force – I don't think anybody questions that.
"But the question is what kind of force do they use and for how long.
"And in this case they chose a method that is life threatening, extremely risky and universally condemned after the George Floyd case.
"You know, it doesn't take a rocket scientist to understand that if you cut off somebody's air flow they risk either enormous injury or death.
"It's just very clear."
But isn't the issue then that the Czech police may have questionable procedures and is not about the officers in this case, who say at least that they followed the correct procedures?
"They say that they followed procedure and the question is – and this is something that an independent investigation of this incident would determine, and that's what needs to happen – whether it was genuinely proportionate.
"Is it proportionate to keep kneeling on the person's neck after he stops moving, for minutes? That's the question.
"That's something that hopefully an independent investigation will answer and the courts will answer in the future, but we're at the very, very beginning phases of this."
What about the preliminary autopsy? That reportedly says that the death of this man wasn't caused by the police's actions but was a drug overdose. Doesn't that kind of end the comparison between this case and that of George Floyd?
"Not at all. The police asserted the exact same thing in George Floyd's case in the United States.
"They pointed to substances in his body and a heart condition, which the autopsy also mentions."
Yes, but the autopsy here says that he specifically died of a drug overdose.
"Well that's why the family needs to ask for a second autopsy, as soon as possible.
"There needs to be an independent autopsy conducted.
"What we need to ask is, What is the medical reasoning?
"And when the coroner conducted that autopsy, was the information revealed to them of how the police had proceeded? Were they aware, did they see the video or not?
"That's a question."
But is it really a question? Because surely the coroner does a medical examination and then they see what the cause of death was – they don't have to look at a video.
"We haven't seen the report itself. So we don't know what all it says.
"What we know is what the police say it says, and the police may not have told us everything that it says.
"The autopsy may have found, for example, marks on the body consistent with what the police did, etcetera.
"We don't know… we don't know anything about anything.
"We don't know any details, we don't know about his blood oxygen level and all that kind of think.
"We're just simply in the dark really.
"The approach that the police are taking is to say, Trust us, these two things are unrelated.
"And that's not good enough. It's not good enough when somebody dies to say that."
Romský George Floyd? Muž zemřel po zásahu policie v Teplicích. Ta odmítá, že za smrt může zákrok
Video of Romský George Floyd? Muž zemřel po zásahu policie v Teplicích. Ta odmítá, že za smrt může zákrok
Author: Ian Willoughby
Security expert: More can be done to make hospitals safer
Gunman kills six patients in Ostrava hospital, two more fighting for their lives
Dutch nationals face charge of grievous bodily harm in attack on Czech waiter
Police welcome club's reward to find Petra Kvitová's attacker

redpajama_set_name: RedPajamaCommonCrawl


text:
:: The Newsboys to Tuck & Patti
The Newsboys Biography
Find The Newsboys on Amazon.com
If you would like to share The Newsboys lyrics with other users of this site, please see the bottom of this page on how to submit The Newsboys lyrics.
Members include Jody Davis, guitar; Jeff Frankenstein, keyboards; Peter Furler , drums, lead vocals; John James, (left group, 1997), lead vocals; Phil Joel, bass. Addresses: Record company--Virgin Records America, 338 N. Foothill Rd., Beverly Hills, CA 90210 Phone: (310)-278-1181.
The early days of The Newsboys were filled with rough and crtical barbs. Some critics felt the group relied too much on lyrical religious cliches. Undaunted by such attacks, they forged ahead in what they believed to be a mission. Each musician was a seasoned professional who could work in the secular rock market had they wanted to. Yet they each wanted to combine their professional talents with their personal spiritual beliefs and decided to start a Christian rock band. Well known for their wild, high-energy live-show performances and high-tech production numbers, The Newsboys are an example of a unique musical phenomenon. They are a Christian rock group that has managed to break into into the cross-over market of secular pop music, following such talents as Amy Grant, Audio Adrenaline, Matthew Sweet and others., The group itself formed in the late 1980s with an original lineup of lead vocalist John James, guitarists Roscoe Meeks and Phillip Urry, and drummer Peter Furler. They are considered to be one of the most popularly known Christian rock bands of the 1990s
Star Song/Virgin Records also believed in their potential and gave them a recording contract. The Newsboy's debut album Hell Is for Wimps was released in 1990 to a mild response. They released Turn in 1991 to much of the same. Undauanted, they continued with Not Ashamed, Boys Will Be Boyz, and Read All About It in 1992. They received a Grammy nomination for Rock Album of the Year in 1993. However, their first big success came with 1994's Going Public. It hit the charts like a bullet and became their first gold record. The next public accolade came from the Christian music sector when they received a Dove Nomination for Best Short Form Video in 1994 for "I Cannot Get You Out of My System." Their public expanded exponentially and since they were now approaching the crossover horizon, they wanted to maximize the impact of their deeply held spiritual beliefs. To do this in the most effective way possible, they teamed up with a Dallas-based evangelical organization called Teen Mania.Teen Mania regularly schedules weekend events and The Newsboys chose to use this venue and reach out to Christian teenagers. The following years reaped even sweeter harvests. They released Take Me To Your Leader in 1996. It went gold, making Take Me to Your Leader their second gold record in as many years.
On November 1, 1997, The Newsboys made contemporary Christian Music History. Their Houston Astrodome concert held over 33,000 people. Also, over 900,000 fans showed up to attend the Take Me To Your Leader Tour. The latter tour had the distinction of being named both Christian Contemporary Music's Release Magazine's favorite live concert of 1997.
It was at this time that vocalist John James decided to quit. According to Furler and Joel, he left the group in pursuit
"It&'s like a cheeky way of saying both `let the chips fall where they may'," Furler told the Newsboys Website, "and at the same time, for the fans and the youth group leaders and the like, showing them that on this record, we've gone through a lot of self-examination. We've been thinking about `why did Jesus die on the cross? And more important, `why do we believe what we believe? The phrase `Step Up To The Microphone' obviously means `make your statement." But to me, it also means `what are you going to say when you get there?"
After putting on the finishing touches, their latest album, Step Up To The Microphone was released in 1998. A week before it hit the streets, the band promoted the event by headlining the annual Creation Fest held in Mount Union, Pennsylvania. Audience reception was positive. The fans seemed to love Furler as the lead singer. Though the differences in style were apparent, it was a positive one. Because of the radical personnel changes, the Newsboys knew they had a lot to prove to their long-time fans. It was a given that the audience would make the inevitable comparisons between lead singers. Some would probably even judge the overall production tone of their album, the signature of a producer. Another change for the group was the collaborational effort of their new album, the whole band took part in writing all of the songs together. "We all stepped up and took on a whole lot more," said Joel to Price. "I can't say it was a shock that James moved on, so it didn't rock the boat immensely. Peter has a great voice. He's had to take on being a front man and he's doing a great job." "There were a lot of challenges going into this record." Furler added, "But I think that made us more creative. I'm really proud of the lyrics. It's really a very personal record."
Having faith in their potential for reaching an even wider secular cross-over audience, Virgin Records launched a massive promotional campaign to back Step Up To The Microphone. They also worked through EMI Christian Music Group Distribution to release the album into mainstream outlets. Special promotional events were launched in over twenty key market areas and, for the first time in their decade long career, the crossover became a great success. Along with having their songs played on the traditional alternate Christian radio stations, they also started appearing on the playlist of major Top 40 radio stations across the country. Their 120 city Step Up To The Microphone Fall Tour surpassed initial expectations. Everyone was pleased with the results. The tour has even surpassed the success of Amy Grant's Age to Age Tour. It looks like the Newsboys'mainstream crossover is here to stay.
by Timothy Kevin Perry
The Newsboys's Career
Group formed in late 1980s; released debut album Hell Is For Wimps, Star Song, 1990; released Not Ashamed, Boys Will Be Boyz, and Read All About It, 1992; Going Public , 1994; Take Me To Your Leader, Chordant, 1996; Step Up To The Microphone, 1998.
The Newsboys's Awards
Gold record for album Going Public, 1994; Gold record for album Take Me To Your Leader, 1996.
Hell Is For Wimps , Star Song, 1990.
Boys Will Be Boyz , Star Song, 1992.
Going Public , Star Song, 1994.
Take Me To Your Leader Chordant, 1996.
Step Up To The Microphone , Star Song/Virgin Records, 1998.
Billboard Magazine , June 6, 1998, pg. 14.
All Music Guide Web Site, http://www.allmusic.com/cg/x.exe
The Official Newsboys Web Site, http://www. newsboys.com
The Newsboys Lyrics
Feel free to share The Newsboys lyrics. Just click on "Add a comment…" below and paste the song name and the lyrics. However, please do not post The Newsboys lyrics unless you have received permission from the copyright owner. Make sure to include the name of the The Newsboys album along with the lyrics.
Phil Joel please come back you ARE A NEWSBOY.
it is YOU we adore....
it is you
this page has a lot of info.
this page is very helpful.


A Writer's Journey: Which way?
What do you do with hurt feelings? Why are they hurt? I'm going to be a little transparent tonight. I've got cotton candy so I may veer off course.
Prayer unanswered still? No, I know it's been answered. This is where the patience part comes in. The faith part. The bear all things, believe all things, hope all things, endure all things 1 Cor13. True love part comes in.
What I'm learning is that love is not that butterfly in your stomach feeling. It's the action part. The I forgive you when you are wrong and hurt me part. The I believe in you and support you part. That's love that lasts a lifetime. It's hard, but like anything in life if it's important to you you work at it. Like an artist, you don't stop. God is love. He loves me even during those times that I'm not very loving, not being obedient. That's the same for you or me. We are to follow his example. It hasn't been easy. I've been stubborn and wanted my way, but I see why God's way is perfect.


There has been something that's been bothering me in recent years, while taking my kids to birthday parties. It's the "present-opening" time of the event -- or lack thereof. Yes, there's a trend towards not opening presents and I'm trying to figure out why.
There's a new trend that I've noticed that makes me very uncomfortable. I say "new" because "back in the day," it wasn't done, but I realize that things are different now.
Nonetheless, it's something that's been bothering me in recent years, while taking my kids to birthday parties.
Yes, there's a trend towards not opening presents and I'm trying to figure out why.
When I was a kid, there was a pretty standard format for birthday parties.
It's this last standard of the classic children's birthday party that I miss -- that is because too often than not, "the opening of the presents" part of the birthday party no longer exists. What I'm trying to figure out is why.
Feelings: What if little Emma doesn't like the gift that she receives? What if she shows her distaste for the present and, in turn, shows up her parents for not teaching her how to handle herself with grace and diplomacy, even if she is only old enough to count her birthdays on one or maybe two hands?
There's a shift that's rapidly becoming the norm and I didn't get the memo. I assume it says "Don't let your kid open their birthday presents in front of their guests at the risk of offending/hurting feelings/being compared to others/wasting time." I'm glad I didn't get the memo because it would have been deleted.
I'm not a supporter of "helicopter parenting" and its extreme forms that have become the standard in today's society. Accordingly, the reasons noted above are not, in my opinion, good enough to eliminate what was once one of the high points of attending a birthday party, both for the attendees and the Birthday Girl.
Eliminating this important part of the event deprives the giver of the gift the joy of seeing their friend receive and appreciate the present that was especially chosen for them.
Let's take a look at each of the reasons given for not opening presents, and why they just don't hold up.
1) Time: It's often cited that for reasons of expediency and to save time, the opening of gifts at a birthday party is eliminated. OK. I would propose that any parents who are planning a birthday party for their child, regardless of size, factor in the time needed to leisurely open gifts and appreciate them, and extend the amount of time for the birthday party accordingly. It's not difficult. It's all part of the planning -- depending on the number of kids coming, adding the "opening presents" part of the event isn't a big deal.
2) Comparison: One of the most important lessons that a child can learn is that comparing oneself to others is often unnecessary as well as an exercise in futility. There will always be people who have bigger and better things than others; that's just a part of life. What's key to a child -- both the receiver of the gift and the gift giver -- is to convey a graceful response to the receipt or the giving of said gift. No need for comparison -- each gift was chosen for a reason and should be received and recognized as such.
3) Feelings: Teaching a child about grace and empathy will help them well beyond their younger years. In the case of a birthday party, the child receiving the present should be taught how to gracefully receive every gift, regardless of size, extravagance (or lack thereof) or cost. By showing their guests through their actions that each gift is appreciated, no feelings will be hurt and no child will feel slighted.
4) Culture: If a child is attending a party where culture is key and the adherence to certain cultural norms is important, then by all means, don't open the presents. That being said, it's safe to assume that if it's a traditionally-planned Western-style party in all other ways, e.g. party games, cake, the singing of "happy birthday" etc., then go ahead with that other tenet of these types of parties and let the Birthday Boy open his presents!
5) Jealousy: That green-eyed monster is one that we all feel from time to time and, as ugly as it is, it needs to be contained. Sometimes jealousy about not receiving gifts or the fact that a friend has received something really cool is bothersome to children and can't be helped. That being said, feelings of jealousy and envy are opportunities for parents to discuss why the child feels this way, whether the feelings are appropriate, and finally, how a child can manage such feelings when they're experiencing them.
Look at this opportunity or future opportunities (birthday party gift-giving) as yet another chance to teach your child about the realities of life and how, very often, life isn't always fair. Yes -- your child's friend got a cool gift that your child may want but your son or daughter probably has many things that the Birthday Girl doesn't have. Life doesn't always seem fair but generally speaking, things have a way of balancing themselves out in the end.
If it's not abundantly clear by this point, I strongly believe that kids should open their birthday presents at their birthday parties. Can we change this growing trend against opening gifts at parties and go back to "The Good Old Days?"


text:
We arrange regular scheduled trips to various destinations as listed below, those listed below, are either fully focussed upon or include an element of capturing images of the black-legged kittiwake.
Puffins of the Arctic Circle 7 nights - 13th - 20th June 2022Iceland Full
Join Dan Trim, a veteran of the NaturesLens guiding team, & winner of the BWPA competition in 2017, in during June 2022 for 7 nights; take the opportunity to capture images of the puffins of the arctic circle; the NaturesLens Puffins of the Arctic Circle photography trip is offered on a full board, non-shared room basis, & costs £2249 per participant.
Species: Auk, Black Guillemot, Black Murre, Brunnich's Guillemot, Common Murre, Northern Wheatear, Razorbill, Red-Necked Phalarope, Thick-billed Murre, White Wagtail, atlantic puffin, black-legged kittiwake, common guillemot, northern fulmar, snow bunting
Puffins of the Arctic Circle 7 nights - 19th - 26th June 2023Iceland £2199 per participant
Join Dan Trim, a veteran of the NaturesLens guiding team, & winner of the BWPA competition in 2017, in during June 2023 for 7 nights; take the opportunity to capture images of the puffins of the arctic circle; the NaturesLens Puffins of the Arctic Circle photography holiday is offered on a full board, non-shared room basis, & costs £2199 per participant.
The Polar Wildlife of Svalbard 12 nights - 18th - 30th July 2023Norway £12,500 £10500 * per participant
Join David & Pui Hang Miles, the co-founders of NaturesLens, in Longyearbyen during July 2023 for 12 nights; take the opportunity to capture images of the the polar wildlife of svalbard; the NaturesLens The Polar Wildlife of Svalbard photography trip is offered on a full board, shared room basis, & costs £12,500 £10500 per participant.
Species: arctic fox, arctic skua, arctic tern, barnacle goose, bearded seal, black-legged kittiwake, brent goose, common eider, common guillemot, glaucous gull, grey phalarope, harbour seal, ivory gull, little auk, northern fulmar, pink-footed goose, polar bear, purple sandpiper, red-throated diver, ringed seal, sibling vole, snow bunting, svalbard reindeer, svalbard rock ptarmigan, walrus
Whatever your query regarding our photography holiday, Puffins of the Arctic Circle, please fill out your details below & we will get back to you.


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text:
Q: Why does a C program show more memory with .so of libmath than with .a of libmath? Based on the comments in this C tutorial page which I'm using to review C after some time not working with it, I expect that a simple program compiled with a shared version of library will appear to use less memory than a version of that program using the static version of the library.
Here is a simple example program, which requests user input just so the program will be idle while I go use top and ps to inspect it. The goal is to compile the program with just -lm (linking libmath), then later compile it with -lm --static. When I run each program, I should see the static option taking up less memory associated with its running process.
/* lib_a_vs_so.c */
#include <math.h>
#include <stdio.h>

void main()
{
    double x = sin(3.14);
    int user_input;

    printf("Enter a number: ");
    scanf("%d", &user_input);
    printf("You entered %d and sin(3.14) is %.2f\n", user_input, x);
}

The steps to compile two different versions:
c99 -o so_version lib_a_vs_so.c -lm
c99 -o a_version lib_a_vs_so.c -lm --static

and the output from top on my system (gcc version 4.8.4 (Ubuntu 4.8.4-2ubuntu1~14.04.1)) when I run both programs.
top - 19:06:51 up 10:20,  5 users,  load average: 0.06, 0.24, 0.25
Tasks:   2 total,   0 running,   2 sleeping,   0 stopped,   0 zombie
%Cpu(s):  2.4 us,  0.8 sy,  0.0 ni, 96.4 id,  0.3 wa,  0.0 hi,  0.0 si,  0.0 st
KiB Mem:  16327932 total,  7522656 used,  8805276 free,   435692 buffers
KiB Swap:        0 total,        0 used,        0 free.  2836848 cached Mem

  PID USER      PR  NI    VIRT    RES    SHR S  %CPU %MEM     TIME+ COMMAND
 7010 ely       20   0    4192    356    276 S   0.0  0.0   0:00.00 so_version
 7025 ely       20   0    1080    264    212 S   0.0  0.0   0:00.00 a_version

Why does so_version appear to use more memory?

A: Static Linking
When you statically link a library into a binary and run the binary, the operating system will only load the stuff it actually needs to use (when the execution of an instruction attempts to reference a location in virtual memory that is not loaded into physical memory, a page fault occurs and the OS loads the page and continues -- helpful).[1]
Dynamically Loading
When dynamically loading a library, the operating system uses mmap to load the shared object.[2] mmap naturally implements demand paging as well, so the entire object is not loaded into physical memory, however it is shared between multiple processes, so parts of it may be loaded into physical memory already.
Answer
I think there are two possibilities for the behavior your question addresses (which are not necessarily mutually exclusive):

*

*(more likely, IMO) When top calculates the memory usage, it counts the shared object space as being used by the process, which includes the entire shared object space that is physically loaded, even if this particular program didn't need to use it. This will make the memory used by this process appear larger than it's statically linked counterpart, which has less of it's statically linked library physically loaded.

*(less likely, IMO) The address space mapped for the shared object is guaranteed to contain the entire library, since at any time a new program could pop up that needs to use any part of the shared library. However, in static linking, the linker can potentially skip compilation units if they are never referenced by the rest of the program being linked (source). That could cause the size of the static library to be reduced during linking (and thus, smaller memory usage in runtime).

[1] This applies to all major operating systems (Linux-based, BSD-based (macOS, iOS), and Windows). I'm sure there is an OS out there that loads entire binaries into memory before running them, but that's outside the scope of this answer.
[2] This detail is something I'm aware that Linux and FreeBSD does. Windows probably does it with DLLs, but I don't know for sure.

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text:
Les Faviinae sont une sous-famille de coraux durs de la famille des Faviidae.

Taxonomie
Selon les sources ce taxon a été créé par Henri Milne-Edwards et Jules Haime en 1857 ou par John Walter Gregory en 1900.

Liste des genres
Selon  :
 genre Colpophyllia Milne Edwards & Haime, 1848
 genre Diplocoenia Duncan, 1868 †
 genre Diploria Milne Edwards & Haime, 1848
 genre Favia De Blainville, 1820
 genre Lamellastraea Duncan, 1868 †
 genre Manicina Ehrenberg, 1834
 genre Mussismilia Ortmann, 1890
 genre Pseudodiploria Fukami, Budd & Knowlton, 2012

Publication originale
 Milne Edwards & Haime, 1857 : Histoire naturelle des coralliaires ou polypes proprement dits. ,  (texte intégral) .

Notes et références

Liens externes

Mussidae

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Covers fundamental topics in fluid dynamics at the graduate level: Euler and Lagrange descriptions of continuum dynamics; conservation laws for inviscid and viscous flows; potential flows; exact solutions of the Navier-Stokes equation; boundary layer theory; gravity waves. Students cannot receive credit for this course and course 107. Enrollment restricted to graduate students; undergraduates may enroll by permission of the instructor.


See what's in-store at Myer when the four extraordinary performance artists of the Urban Dream Capsule take up residence inside the Bourke Street windows for the duration of the Games, living their day-to-day lives in full view! With their Richard Jeziorny-designed bedroom, living room, kitchen and bathroom visible to the world 24 hours a day, their simplest tasks - sleeping, washing, grooming, cooking – will become endurance events during this fourteen day performance marathon. Watch as our intrepid heroes turn everyday living into the most entertaining of artforms, and form relationships with their audience that break the pane-barrier.
Who can forget the debut of this now world-famous troupe with its sensational two-week installation in the Myer windows in Bourke Street for the 1996 Melbourne International Festival of the Arts? Since then the Urban Dream Capsule has insinuated itself - to widespread critical acclaim – all around the world. And now artists Neil Thomas, Andrew Morrish, Nick Papas and David Wells return to their hometown for a special Commonwealth Games tenth anniversary performance.