# Representation Formula for Viscosity Solutions to a class of Nonlinear
Parabolic PDEs

Marco Pozza Dipartimento di Matematica “G. Castelnuovo”, Sapienza Università
di Roma. Piazzale Aldo Moro 5, 00185 Roma, Italy<EMAIL_ADDRESS>
###### Abstract

We provide a representation formula for viscosity solutions to a class of
nonlinear second order parabolic PDEs given as a sup–envelope function. This
is done through a dynamic programming principle derived from [7]. The formula
can be seen as a nonlinear extension of the Feynman–Kac formula and is based
on the backward stochastic differential equations theory.

_2010 Mathematics Subject Classification:_ 35K55, 60H30.

_Keywords:_ Nonlinear Feynman–Kac formula; Viscosity solutions; Backward
stochastic differential equations (BSDE).

##  1 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
_parabolic partial differential equations_ , PDEs for short, and getting
representation formulas, 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,
[15, 14, 1, 13, 10, 18] 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 _backward
stochastic differential equations_ , BSDEs for short, which was introduced by
Pardoux and Peng in [20] (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 _stochastic differential equations_ , SDEs for short, to
semilinear and quasilinear equations, see for example [6, 5, 16, 19]. See also
[22] 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,

$\begin{cases}\begin{aligned}
\frac{1}{2}\left\langle\sigma\sigma^{\dagger},D^{2}_{x}u\right\rangle(t,x)+(\nabla_{x}ub)(t,x)&\\\
+\partial_{t}u(t,x)+f(t,x,u,\nabla_{x}u\sigma)&=0\end{aligned}&t\in[0,T],x\in\mathds{R}^{N},\\\
u(T,x)=g(x),&x\in\mathds{R}^{N},\end{cases}$ (1)

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 _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 _backward stochastic differential equation_ , BSDE
for short, depending on the first one:

$\left\\{\begin{aligned}
\\!X^{t,x}_{s}\\!=&x+\int_{t}^{s}\sigma\left(r,X^{t,x}_{r}\right)dW_{r}+\int_{t}^{s}b\left(r,X^{t,x}_{r}\right)dr,\\\
\\!Y^{t,x}_{s}\\!=&g\left(X^{t,x}_{T}\right)\\!+\\!\\!\int_{s}^{T}\\!\\!f\\!\left(r,X_{r}^{t,x},Y_{r}^{t,x},Z_{r}^{t,x}\right)\\!dr\\!-\\!\\!\int_{s}^{T}\\!\\!Z_{r}^{t,x}dW_{r},\end{aligned}\right.s\\!\in\\![t,T],x\\!\in\\!\mathds{R}^{N}.$

As can be intuitively seen, the SDE takes care of the linear operator defined
by $\sigma$ and $b$, also called the _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 [23] (2006) the notion of _$G$ –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 [24], where he highlights the role of the so–called
_sublinear expectations_ , namely $G$–expectations generated by sublinear
operators given as a sup–envelope function. Finally in [7], Peng provides
representation formulas for viscosity solutions using these expectations. More
precisely, given a sublinear operators $G$ and the $G$–heat equation

$\begin{cases}\partial_{t}u(t,x)+G\left(D_{x}^{2}u(t,x)\right)=0,&t\in[0,T],x\in\mathds{R}^{N},\\\
u(T,x)=g(x),&x\in\mathds{R}^{N}.\end{cases}$ (2)

he represents the viscosity solution as

$u(t,x)=\sup_{\sigma\in\mathcal{A}}\mathds{E}\left(g\left(x+\int_{t}^{T}\sigma_{s}dW_{s}\right)\right),$

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 _dynamic programming
principle_ that we will illustrate in the paper, see section 3. We point out
that this formula has been established in [15] using a different method and
that 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 [7] in order to give representation formulas of
solutions to PDE problems of the type

$\begin{cases}\partial_{t}u(t,x)+F\left(t,x,u,\nabla_{x}u,D^{2}_{x}u\right)=0,&t\in[0,T],x\in\mathds{R}^{N},\\\
u(T,x)=g(x),&x\in\mathds{R}^{N},\end{cases}$ (3)

where $F$ is a sup–envelope function of semilinear operators as in (1). This
problem is clearly a blend between (1) and (2) which generalize both.

This is hopefully just a first step to further extend the Feynman–Kac formula
to fully nonlinear problems. In this regard the method here employed has been
successfully used in [28] to provide a representation formula for a particular
Dirichlet type problem. 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 [3] (2007). Then, in
2011, Soner, Touzi and Zhang [29] 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 [27, 29, 8, 9, 17, 12] and in
particular [26], which performs its analysis replacing the Lipschitz condition
on $y$ with monotonicity as we do here.

This paper is organized as follows: in section 2 we introduce the problem
along with some preliminary facts and in section 3 we develop a dynamic
programming principle which is the core of our theory. Then, in section 4, we
perform the essential part of our analysis, and obtain in this way our main
results. In appendix A we summarily analyze the connection between 2BSDEs and
our representation formula, while appendix B briefly gives some probability
results about stochastic differential equations needed to our analysis.
Finally in appendix C is proved a comparison result for parabolic problems.

We proceed setting the notation used in the paper.

#### Notation

We will work on the filtered probability space
$\left(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\in[0,\infty)},\mathds{P}\right)$,

  * •

$\mathcal{F}$ is a complete $\sigma$–algebra on $\Omega$;

  * •

the stochastic process $\\{W_{t}\\}_{t\in[0,\infty)}$ will denote the $N$
dimensional Brownian motion under $\mathds{P}$;

  * •

$\\{\mathcal{F}_{t}\\}_{t\in[0,\infty)}$ is the filtration defined by
$\\{W_{t}\\}_{t\in[0,\infty)}$ which respects the usual condition of
completeness and right continuity;

  * •

$\\{W_{s}^{t}\\}_{s\in[t,\infty)}:=\\{W_{s}-W_{t}\\}_{s\in[t,\infty)}$ is a
Brownian motion independent from $\\{W_{s}\\}_{s\in[0,t]}$ by the _strong
Markov property_ ;

  * •

$\\{\mathcal{F}^{t}_{s}\\}_{s\in[t,\infty)}$ is the filtration generated by
$\\{W_{s}^{t}\\}_{s\in[t,\infty)}$ which we assume respect the usual condition
and is independent from $\mathcal{F}_{t}$;

  * •

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 $t\in[0,\infty)$;

  * •

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
$\mathcal{B}([0,T])\times\mathcal{F}_{T}$;

  * •

a function on $\mathds{R}$ is called _cadlag_ if is right continuous and has
left limit everywhere;

  * •

a cadlag (in time) process is progressive if and only if is adapted;

  * •

$B_{\delta}(x)$ will denote an open ball centered in $x$ with radius $\delta$;

  * •

if $A\in\mathds{R}^{N\times M}$ then $A^{\dagger}$ will denote its transpose;

  * •

(Frobenius product) if $A,B\in\mathds{R}^{N\times M}$ then $\langle
A,B\rangle:=\operatorname{tr}\left(AB^{\dagger}\right)$ and $|A|$ is the norm
$\sqrt{\langle
A,A\rangle}=\sqrt{\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{M}A_{i,j}^{2}}$;

  * •

$\mathds{S}^{N}$ 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.

Acknowledgement: We would like to thank an anonymous referee to a previous
version for suggesting us the comparison theorem C.2 as well as other comments
which helped us improve this paper.

##  2 Setting of the Problem

Here we introduce the problem, which is object of investigation, as well as
basic preliminary facts.

We start giving the assumption of the PDE operator which we will analyze in
this paper.

######  Assumptions 2.1.

We fix a $T>0$, hereafter called _terminal time_ , the constants $\ell\geq 0$,
$\lambda\geq 0$, $\mu\in\mathds{R}$ and let $K_{F}$ be a set made up by
functions

$(b,\sigma):[0,T]\times\mathds{R}^{N}\longrightarrow\mathds{R}^{N}\times\mathds{R}^{N\times
M}$

and which satisfies, for any $(t,x)\in[0,T]\times\mathds{R}^{N}$, the
following conditions:

  1. i)

(local equiboundedness) $\sup\limits_{(b,\sigma)\in
K_{F}}|b(t,x)|+|\sigma(t,x)|<\infty$;

  2. ii)

(equicontinuity) for each $\varepsilon>0$ there is a neighborhood $U$ of
$(t,x)$ such that
$|b(t,x)-b(t^{\prime},x^{\prime})|+|\sigma(t,x)-\sigma(t^{\prime},x^{\prime})|<\varepsilon$
for any $(t^{\prime},x^{\prime})\in U$ and $(b,\sigma)\in K_{F}$;

  3. iii)

(equiLipschitz continuity) for any $(b,\sigma)\in K_{F}$ and $t\in[0,T]$

$|b(t,x)-b(t,x^{\prime})|\leq\ell|x-x^{\prime}|,\qquad|\sigma(t,x)-\sigma(t,x^{\prime})|\leq\ell|x-x^{\prime}|.$

Now let

$f:[0,T]\times\mathds{R}^{N}\times\mathds{R}^{N}\times\mathds{R}^{N\times
M}\times\mathds{R}\times\mathds{R}^{N}\longrightarrow\mathds{R}$

be a continuous function such that, for any $t\in[0,T]$,
$x,x^{\prime},q,q^{\prime},z,z^{\prime}\in\mathds{R}^{N}$,
$A,A^{\prime}\in\mathds{R}^{N\times M}$ and $y,y^{\prime}\in\mathds{R}$,

  1. iv)

$|f(t,x,q,A,y,z)-f(t,x^{\prime},q^{\prime},A^{\prime},y,z^{\prime})|\leq\ell(|x-x^{\prime}|+|q-q^{\prime}|+|A-A^{\prime}|+|z-z^{\prime}|)$;

  2. v)

$|f(t,x,q,A,y,z)|\leq\ell(1+|x|+|q|+|A|+|y|+|z|)$;

  3. vi)

$(y-y^{\prime})(f(t,x,q,A,y,z)-f(t,x,q,A,y^{\prime},z))\leq\mu|y-y^{\prime}|^{2}$;

  4. vii)

fixed a $(t,y)\in[0,T]\times\mathds{R}$, a compact set
$K\subset\mathds{R}^{N}\times\mathds{R}^{N\times M}\times\mathds{R}^{N}$ and a
positive $\varepsilon$ there is a $\delta>0$ such that

$|f(t,x,q,A,y,z)-f(t^{\prime},x,q,A,y^{\prime},z)|<\varepsilon$

for any $(t^{\prime},y^{\prime})\in
B_{\delta}((t,y))\cap[0,T]\times\mathds{R}$ and $(q,A,z)\in K$.

In what follows we will focus on nonlinear operator of this type

$F(t,x,y,p,S):=\sup_{(b,\sigma)\in
K_{F}}\frac{1}{2}\left\langle\sigma\sigma^{\dagger}(t,x),S\right\rangle+p^{\dagger}b(t,x)+f_{(b,\sigma)}\left(t,x,y,p^{\dagger}\sigma\right),$

where
$f_{(b,\sigma)}\left(t,x,y,p^{\dagger}\sigma\right):=f\left(t,x,b(t,x),\sigma(t,x),y,p^{\dagger}\sigma(t,x)\right)$.

It is a simple verification that $F$ is continuous. Furthermore we have that
$F$ is an elliptic and convex (for the second order term) operator in the
sense that, for any $t\in[0,T]$, $x\in\mathds{R}^{N}$, $y\in\mathds{R}$,
$p\in\mathds{R}^{N}$ and $S\in\mathds{S}^{N}$,

  1. i)

_(Convexity)_ for any $\delta\in[0,1]$

$F(t,x,y,p,\delta S+(1-\delta)S^{\prime})\leq\delta
F(t,x,y,p,S)+(1-\delta)F(t,x,y,p,S^{\prime});$

  2. ii)

_(Ellipticity)_ there is a $\lambda\geq 0$ such that, for any $S^{\prime}\geq
0$,

$F(t,x,y,p,S+S^{\prime})-F(t,x,y,p,S)\geq\lambda|S^{\prime}|.$

The $\lambda$ in item ii) is actually the lowest eigenvalue of the matrix
$\sigma\sigma^{\dagger}$.

######  Remark 2.2.

We point out that items ii) and i) in assumptions 2.1 and the Arzelà–Ascoli
theorem imply that $K_{F}$ is relatively compact in the compact–open topology
and it is easy to see that its closure $\overline{K_{F}}$ satisfies items ii),
iii) and i). Moreover the map

$(b,\sigma)\mapsto\frac{1}{2}\left\langle\sigma\sigma^{\dagger}(t,x),S\right\rangle+p^{\dagger}b(t,x)+f_{(b,\sigma)}\left(t,x,y,p^{\dagger}\sigma\right)$

is continuous, thus we have that

$F(t,x,y,p,S)=\max_{(b,\sigma)\in\overline{K_{F}}}\frac{1}{2}\left\langle\sigma\sigma^{\dagger}(t,x),S\right\rangle+p^{\dagger}b(t,x)+f_{(b,\sigma)}\left(t,x,y,p^{\dagger}\sigma\right).$

For this reason we will always assume that $K_{F}$ is closed.

######  Problem 2.3.

Let $T$ be a terminal time, $F$ a sublinear operator as in assumptions 2.1 and
$g:\mathds{R}^{N}\to\mathds{R}$ a continuous function such that, for any
$x,x^{\prime}\in\mathds{R}^{N}$,

  1. i)

$|g(x)-g(x^{\prime})|\leq\ell|x-x^{\prime}|$;

  2. ii)

$|g(x)|\leq\ell(1+|x|)$,

where $\ell$ is the same constant in assumptions 2.1.
Find the solution $u$ to the parabolic PDE

$\begin{cases}\partial_{t}u(t,x)+F\left(t,x,u,\nabla_{x}u,D^{2}_{x}u\right)=0,&t\in(0,T),x\in\mathds{R}^{N},\\\
u(T,x)=g(x),&x\in\mathds{R}^{N}.\end{cases}$

The solutions to 2.3 are to be intended in the viscosity sense, hence let us
define what we mean with viscosity solution. For a detailed overview of the
viscosity solution theory we refer to [4].

######  Definition 2.4.

Given an upper semicontinuous function $u$ we say that a function $\varphi$ is
a _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 _subtangent_ to a lower
semicontinuous function $v$ at $(t,x)$ if $(t,x)$ is a local minimizer of
$v-\psi$.

######  Definition 2.5.

An upper semicontinuous function $u$ is called a _viscosity subsolution_ to
2.3 if, for any suitable $(t,x)$ and $C^{1,2}$ supertangent $\varphi$ to $u$
at $(t,x)$,

$\partial_{t}\varphi(t,x)+F\left(t,x,u,\nabla_{x}\varphi,D^{2}\varphi\right)\geq
0.$

Similarly a lower semicontinuous function $v$ is called a _viscosity
supersolution_ to 2.3 if, for any suitable $(t,x)$ and $C^{1,2}$ subtangent
$\psi$ to $v$ at $(t,x)$,

$\partial_{t}\psi(t,x)+F\left(t,x,v,\nabla_{x}\psi,D^{2}\psi\right)\leq 0.$

Finally a continuous function $u$ is called a _viscosity solution_ to 2.3 if
it is both a super and a subsolution to 2.3.

For viscosity sub and supersolutions to 2.3 holds a comparison result which we
will prove in appendix C.

######  Theorem 2.6.

Let $u$ and $v$ be respectively a sub and a supersolution to 2.3. If
$u(T,x)\leq v(T,x)$ for any $x\in\mathds{R}^{N}$ and both $u$ and $v$ have
polynomial growth, then $u\leq v$ on $(0,T]\times\mathds{R}^{N}$.

Our method to obtain representation formulas relies on a _dynamic programming
principle_ , which is based on a construction on a broader set than $K_{F}$.
This set, which we call $\mathcal{A}_{F}$, is made up of the controls

$(b,\sigma):[0,T]\times\Omega\times\mathds{R}^{N}\longrightarrow\mathds{R}^{N}\times\mathds{R}^{N\times
M},$

where $\Omega$ is a filtered probability space as in the notation section.
These controls are cadlag, i.e. right continuous with finite left limits
everywhere, on $[0,T]$ and such that, for any $t\in[0,T]$ and
$\omega\in\Omega$, there is a neighborhood $(t_{1},t_{2})$ of $t$ and a pair
$(b^{\prime},\sigma^{\prime})\in K_{F}$ such that
$(b,\sigma)|_{(t_{1},t_{2})}(\omega)\equiv(b^{\prime},\sigma)^{\prime}|_{(t_{1},t_{2})}$.
We also require that, for any $x\in\mathds{R}^{N}$ and
$(b,\sigma)\in\mathcal{A}_{F}$, the process
$\\{(b,\sigma)(t,x)\\}_{t\in[0,T]}$ is progressively measurable.
$\mathcal{A}_{F}$ is trivially non empty, since it contains $K_{F}$. From the
definition we have that the elements of $\mathcal{A}_{F}$ are uniformly
Lipschitz continuous in $\mathds{R}^{N}$ with Lipschitz constant $\ell$ and
the eigenvalues of $(\sigma\sigma^{\dagger})(t,\omega,x)$ belong to
$[\lambda,\infty)^{N}$ for any $t\in[0,T]$, $\omega\in\Omega$,
$x\in\mathds{R}^{N}$. In what follows we will usually omit the dependence on
$\Omega$.

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 $\left\\{\mathcal{F}_{t}^{\tau}\right\\}_{t\in[0,\infty)}$.
Clearly $\mathcal{A}^{0}_{F}=\mathcal{A}_{F}$ and

$F(t,x,y,p,S)=\max_{(b,\sigma)\in\mathcal{A}_{F}^{t}}\frac{1}{2}\left\langle\sigma\sigma^{\dagger}(t,x),S\right\rangle+p^{\dagger}b(t,x)+f_{(b,\sigma)}\left(t,x,y,p^{\dagger}\sigma\right).$

######  Remark 2.7.

By remark 2.2, for any bounded subset $A$ of $\mathds{R}^{N}$ and $p>0$ we
have that

$\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(\int_{0}^{T}\int_{A}\left(|b(t,x)|^{p}+|\sigma(t,x)|^{p}\right)dxds\right)<\infty.$

Thus $\mathcal{A}_{F}$ is contained in $L^{p}\left([0,T]\times\Omega\times
A;\mathds{R}^{N}\times\mathds{R}^{N\times M}\right)$ for any bounded subset
$A$ of $\mathds{R}^{N}$ and $p>0$.

##  3 Dynamic Programming Principle

The scope of this section is to provide the necessary instruments to prove the
theorem 4.4, which will be used to derive representation formulas for the
viscosity solution to 2.3. 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 (1) it means that

$\mathds{E}\left(u\left(s,X^{t,x}_{s}\right)+\int_{s}^{T}f\left(r,X_{r}^{s,X^{t,x}_{s}},Y_{r}^{s,X^{t,x}_{s}},Z_{r}^{s,X^{t,x}_{s}}\right)dr\right)\\\
\begin{aligned}
=&\mathds{E}\left(Y^{s,X^{t,x}_{s}}_{s}+\int_{s}^{T}f\left(r,X_{r}^{s,X^{t,x}_{s}},Y_{r}^{s,X^{t,x}_{s}},Z_{r}^{s,X^{t,x}_{s}}\right)dr\right)\\\
=&\mathds{E}\left(Y^{t,x}_{s}+\int_{s}^{T}f\left(r,X_{r}^{t,x},Y_{r}^{t,x},Z_{r}^{t,x}\right)dr\right)\\\
=&\mathds{E}\left(Y^{t,x}_{t}\right)=u(t,x),\end{aligned}$

which is just a simple consequence of the uniqueness of the solutions to the
FBSDE, while for the $G$–heat equation (2) this means that

$\sup_{\sigma\in\mathcal{A}_{G}}\mathds{E}\left(u\left(s,x+\int_{t}^{s}\sigma(r)dW_{r}\right)\right)\\\
\begin{aligned}
=&\sup_{\sigma\in\mathcal{A}_{G}}\sup_{\sigma^{\prime}\in\mathcal{A}_{G}}\mathds{E}\left(g\left(x+\int_{t}^{s}\sigma(r)dW_{r}+\int_{s}^{T}\sigma^{\prime}(r)dW_{r}\right)\right)\\\
=&\sup_{\sigma\in\mathcal{A}_{G}}\mathds{E}\left(g\left(x+\int_{t}^{T}\sigma(r)dW_{r}\right)\right)\\\
=&u(t,x).\end{aligned}$ (4)

The proof of (4) is contained in [7, Subsection 3.1]. This also intuitively
explain why we ask to the elements of $\mathcal{A}_{F}$ to be cadlag in time.
We point out that in [7] the authors ask to the elements of $\mathcal{A}_{F}$
to only be measurable in time, but for the analysis of the more general 2.3 we
will use right continuity.

The dynamic programming principle exposed in theorem 4.4 is a generalization
of the one presented by Denis, Hu and Peng in [7] and will be obtained in a
similar way.

We start endowing $\mathcal{A}_{F}$ with the topology of the
$L^{2}$–convergence on compact sets, which is to say that a sequence converges
in $\mathcal{A}_{F}$ if and only if it converges in
$L^{2}([0,T]\times\Omega\times K)$ for any compact $K\subset\mathds{R}^{N}$,
see remark 2.7.

Let $\varphi$ be a continuous map from
$[0,T]\times\mathds{R}^{N}\times\mathcal{A}_{F}$ into
$L^{2}(\Omega,\mathcal{F}_{T};\mathds{R})$ such that, for any stopping time
$\tau$ with values in $[0,T]$, $x\in\mathds{R}^{N}$ and
$(b,\sigma)\in\mathcal{A}_{F}$, $\varphi_{\tau}(x,b,\sigma)$ is affected only
by the values that $(b,\sigma)$ assumes on $[\tau,T]$, i.e.

$\varphi_{\tau}:\mathds{R}^{N}\times\mathcal{A}_{F}|_{[\tau,T]}\longrightarrow
L^{2}(\Omega,\mathcal{F}_{T};\mathds{R}),$

and if $(b,\sigma)\in\mathcal{A}_{F}^{\tau}$, then
$\varphi_{\tau}(x,b,\sigma)$ is $\mathcal{F}^{\tau}_{T}$–measurable. We
further assume that, for any stopping time $\tau$ with value in $[0,T]$ and
$\zeta$ in $L^{2}\left(\Omega,\mathcal{F}_{\tau};\mathds{R}^{N}\right)$,

$\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}(|\varphi_{\tau}(\zeta,b,\sigma)|)<\infty,$
(5)

thus the function

$\Phi_{\tau}(\zeta):=\operatorname*{ess\,sup}_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}(\varphi_{\tau}(\zeta,b,\sigma)|\mathcal{F}_{\tau})$

is well defined, as we will prove in 3.1.

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.

######  Lemma 3.1.

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 a
$(b,\sigma)\in\mathcal{A}_{F}$ such that

$\mathds{E}(\varphi_{\tau}(\zeta,b,\sigma)|\mathcal{F}_{\tau})=\mathds{E}(\varphi_{\tau}(\zeta,b_{1},\sigma_{1})|\mathcal{F}_{\tau})\vee\mathds{E}(\varphi_{\tau}(\zeta,b_{2},\sigma_{2})|\mathcal{F}_{\tau}).$
(6)

Therefore there exists a sequence $\\{(b_{i},\sigma_{i})\\}_{i\in\mathds{N}}$
in $\mathcal{A}_{F}$ such that a.e.

$\mathds{E}(\varphi_{\tau}(\zeta,b_{i},\sigma_{i})|\mathcal{F}_{\tau})\uparrow\Phi_{\tau}(\zeta).$
(7)

We also have

$\mathds{E}(|\Phi_{\tau}(\zeta)|)\leq\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}(|\varphi_{\tau}(\zeta,b,\sigma)|)<\infty,$
(8)

and, for any stopping time $\tau^{\prime}\leq\tau$,

$\mathds{E}\left(\operatorname*{ess\,sup}_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}(\varphi_{\tau}(\zeta,b,\sigma)|\mathcal{F}_{\tau})\middle|\mathcal{F}_{\tau^{\prime}}\right)=\operatorname*{ess\,sup}_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}(\varphi_{\tau}(\zeta,b,\sigma)|\mathcal{F}_{\tau^{\prime}}).$
(9)

###### Proof.

Given $(b_{1},\sigma_{1}),(b_{2},\sigma_{2})\in\mathcal{A}_{F}$, we define

$A:=\\{\omega\in\Omega:\mathds{E}(\varphi_{\tau}(\zeta,b_{1},\sigma_{1})|\mathcal{F}_{\tau})(\omega)\geq\mathds{E}(\varphi_{\tau}(\zeta,b_{2},\sigma_{2})|\mathcal{F}_{\tau})(\omega)\\},$

which belong to $\mathcal{F}_{\tau}$, and
$(b,\sigma):=(\chi_{A}(b_{1},\sigma_{1})+\chi_{A^{c}}(b_{2},\sigma_{2}))\chi_{\\{t\geq\tau\\}}$.
We thus have $(b,\sigma)\in\mathcal{A}_{F}$ and (6). From this, by the
properties of the essential supremum, we obtain the existence of a sequence
$\\{(b_{i},\sigma_{i})\\}_{i\in\mathds{N}}$ in $\mathcal{A}_{F}$ such that (7)
is true. Furthermore, from (6), (5) and [31, Theorem 1], (8) and (9) follow. ∎

######  Remark 3.2.

To prove 3.1 the randomness of the controls of $\mathcal{A}_{F}$ is crucial,
this is the reason why we consider a set of stochastic process instead of a
deterministic one.

The next one is a density result on the control set $\mathcal{A}_{F}$.

######  Lemma 3.3.

The set

$\mathcal{J}^{\tau}:=\left\\{\begin{aligned}
&(b,\sigma)\in\mathcal{A}_{F}:(b,\sigma)|_{[\tau,T]}=\sum_{i=1}^{n}\chi_{A_{i}}(b_{i},\sigma_{i})|_{[\tau,T]},\\\
&\text{ where
}\\{(b_{i},\sigma_{i})\\}_{i=1}^{n}\subset\mathcal{A}_{F}^{\tau}\text{and
}\\{A_{i}\\}_{i=0}^{n}\text{ is a }\mathcal{F}_{\tau}\text{--partition of
}\Omega\end{aligned}\right\\}$

is dense in $\mathcal{A}_{F}$ for any stopping time $\tau$ with value in
$[0,T]$.

###### Proof.

Preliminarily we will prove the section when the stopping time $\tau$ is
identically equal to a $t\in[0,T]$.
Notice that, fixed a $k\in\mathds{N}$, 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
$\mathcal{B}_{k}:=\mathcal{B}([0,T]\times B_{k}(0))$ 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 $B\in\mathcal{B}_{k}$ is a $\pi$–system which
contains the complementary of its sets and generate
$\sigma(\mathcal{F}_{T}\times\mathcal{B}_{k})$, by [30, Dynkin’s lemma A1.3]
each set in $\sigma(\mathcal{F}_{T}\times\mathcal{B}_{k})$, 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
$s_{\varepsilon}(t,\omega,x)=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}s^{j}_{i}(t,x)\chi_{A_{i}}(\omega)\chi_{A^{\prime}_{j}}(\omega)$
where $\\{A_{i}\\}_{i=1}^{n}$ and $\\{A^{\prime}_{j}\\}_{j=1}^{m}$ are
respectively a $\mathcal{F}_{t}$–partition and a
$\mathcal{F}^{t}_{T}$–partition of $\Omega$ and

$\mathds{E}\left(\int_{0}^{T}\int_{B_{k}(0)}|(b,\sigma)(t,x)-s_{\varepsilon}(t,x)|^{2}dxdt\right)<\varepsilon.$

From this follows that for each $A_{i}$ and $A^{\prime}_{j}$ there is a
$\omega_{i}^{j}$, which belongs to $A_{i}\cap A^{\prime}_{j}$ if $A_{i}\cap
A^{\prime}_{j}\neq\emptyset$, such that

$\sum_{i=1}^{n}\sum_{j=1}^{m}\mathds{E}\left(\int_{0}^{T}\int_{B_{k}(0)}\left|(b,\sigma)\left(t,\omega_{i}^{j},x\right)-s_{i}^{j}(t,x)\right|^{2}\chi_{A_{i}}\chi_{A_{j}^{\prime}}dxdt\right)<\varepsilon.$

If we let
$\left(b_{i}^{k},\sigma_{i}^{k}\right):=\sum\limits_{j=1}^{m}(b,\sigma)\left(\omega_{i}^{j}\right)\chi_{A^{\prime}_{j}}$,
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

$\mathds{E}\left(\int_{0}^{T}\int_{B_{k}(0)}\left|(b,\sigma)(t,x)-\left(b^{k}_{\varepsilon},\sigma^{k}_{\varepsilon}\right)(t,x)\right|^{2}dxdt\right)<4\varepsilon.$

This proves that the sequence
$\left\\{\left(b_{k^{-1}}^{k},\sigma_{k^{-1}}^{k}\right)\right\\}_{k\in\mathds{N}}$
converges to $(b,\sigma)$ and thus $\mathcal{J}^{t}$ is dense in
$\mathcal{A}_{F}$.
Now let $\tau$ be a stopping time with values in $[0,T]$ and define

$\tau_{n}:=\left\\{\begin{aligned} &\frac{j}{n}T,&&\text{if
}\frac{j-1}{n}T\leq\tau<\frac{j}{n}T,j\in\\{1,\dotsc,n\\},\\\ &T,&&\text{if
}\tau=T.\end{aligned}\right.$

Clearly $\tau$ is the decreasing limit of $\\{\tau_{n}\\}_{n\in\mathds{N}}$
and for any fixed a $(b,\sigma)\in\mathcal{A}_{F}$ the previous step shows
that there is a sequence
$\left\\{\left(b_{i}^{j,n},\sigma_{i}^{j,n}\right)\right\\}_{i\in\mathds{N}}\subset\mathcal{J}^{\frac{j}{n}T}$
converging to $(b,\sigma)$ for any $n\in\mathds{N}$ and
$j\in\\{1,\cdots,n\\}$. Finally we define, for any $n\in\mathds{N}$ and
$(t,x)\in[0,T]\times\mathds{R}^{N}$,

$\displaystyle(b_{n},\sigma_{n})(t,x):=$
$\displaystyle(b,\sigma)(t,x)\chi_{\\{t<\tau\\}}+(b_{n}^{n,n},\sigma^{n,n}_{n})(T,x)\chi_{\\{\tau=T=t\\}}$
$\displaystyle+\sum_{j=1}^{n}\left(b_{n}^{j,n},\sigma^{j,n}_{n}\right)(\tau_{n}-\tau+t,x)\chi_{\left\\{\frac{j-1}{n}\leq\tau<\frac{j}{n},t\geq\tau\right\\}}.$

It is easy to see that $(b_{n},\sigma_{n})$ belongs to $\mathcal{J}^{\tau}$
for any $n\in\mathds{N}$ and it is a simple verification that
$(b_{n},\sigma_{n})$ converges to $(b,\sigma)$. The arbitrariness of
$(b,\sigma)$ hence proves that $\mathcal{J}^{\tau}$ is dense in
$\mathcal{A}_{F}$. ∎

######  Lemma 3.4.

For any stopping time $\tau$ with values in $[0,T]$ and $x\in\mathds{R}^{N}$,
$\Phi_{\tau}(x)$ is deterministic. Furthermore

$\Phi_{\tau}(x)=\operatorname*{ess\,sup}_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}(\varphi_{\tau}(x,b,\sigma)|\mathcal{F}_{\tau})=\operatorname*{ess\,sup}_{(b,\sigma)\in\mathcal{A}_{F}^{\tau}}\mathds{E}(\varphi_{\tau}(x,b,\sigma)|\mathcal{F}_{\tau}).$
(10)

###### Proof.

By 3.3 $\mathcal{J}^{\tau}$ is dense in $\mathcal{A}_{F}$, hence, by 3.1,
there is a sequence

$\\{(b_{i},\sigma_{i})\\}_{i\in\mathds{N}}=\left\\{\sum_{j=1}^{n_{i}}\chi_{A_{j}}(b_{i,j},\sigma_{i,j})\right\\}_{i\in\mathds{N}}$

which belongs to $\mathcal{J}^{\tau}$ such that
$\mathds{E}(\varphi_{\tau}(x,b_{i},\sigma_{i})|\mathcal{F}_{\tau})\uparrow\Phi_{\tau}(x)$
a.e.. But

$\displaystyle\mathds{E}(\varphi_{\tau}(x,b_{i},\sigma_{i})|\mathcal{F}_{\tau})\\!=$
$\displaystyle\\!\sum_{j=1}^{n_{i}}\\!\chi_{A_{j}}\mathds{E}(\varphi_{\tau}(x,b_{i,j},\sigma_{i,j})|\mathcal{F}_{\tau})\\!=\\!\sum_{j=1}^{n_{i}}\\!\chi_{A_{j}}\mathds{E}(\varphi_{\tau}(x,b_{i,j},\sigma_{i,j}))$
$\displaystyle\leq$
$\displaystyle\max_{j=1}^{n_{i}}\mathds{E}(\varphi_{\tau}(x,b_{i,j},\sigma_{i,j}))=\mathds{E}(\varphi_{\tau}(x,b_{i,j_{i}},\sigma_{i,j_{i}})),$

where $j_{i}$ is the index that realizes the maximum in the previous formula.
This implies that
$\lim\limits_{i\to\infty}\mathds{E}(\varphi_{\tau}(x,b_{i,j_{i}},\sigma_{i,j_{i}}))=\Phi_{\tau}(x)$,
therefore $\Phi_{\tau}(x)$ is deterministic, and the inequality

$\lim_{i\to\infty}\mathds{E}(\varphi_{\tau}(x,b_{i,j_{i}},\sigma_{i,j_{i}}))\leq\operatorname*{ess\,sup}_{(b,\sigma)\in\mathcal{A}_{F}^{\tau}}\mathds{E}(\varphi_{\tau}(x,b,\sigma)|\mathcal{F}_{\tau})\leq\Phi_{\tau}(x)$

confirms the equation (10). ∎

######  Remark 3.5.

Since $\Phi_{\tau}(x)$ is deterministic for any stopping time $\tau$ with
values in $[0,T]$ and $x\in\mathds{R}^{N}$ we have, thanks to (9),

$\Phi_{\tau}(x)=\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}(\varphi_{\tau}(x,b,\sigma)).$

######  Lemma 3.6.

We define the function

$\displaystyle u:[0,T]\times\mathds{R}^{N}$
$\displaystyle\xrightarrow{\qquad}\mathds{R}$ $\displaystyle(t,x)$
$\displaystyle\xmapsto{\qquad}\Phi_{t}(x)$

and assume that it is continuous. Then, for each stopping time $\tau$ with
value in $[0,T]$ and $\zeta\in
L^{2}\left(\Omega,\mathcal{F}_{\tau};\mathds{R}^{N}\right)$, we have that
$u_{\tau}(\zeta)=\Phi_{\tau}(\zeta)$ a.e..

###### Proof.

By the continuity of $u$, and consequently of $\Phi$, we only need to prove
the section when $\tau=\sum\limits_{j=1}^{n}\chi_{A_{j}}t_{j}$ and
$\zeta=\sum\limits_{j=1}^{n}\chi_{A_{j}}x_{j}$, where $t_{j}\in[0,T]$,
$x_{j}\in\mathds{E}^{N}$ and $\\{A_{j}\\}_{j=1}^{n}$ is a
$\mathcal{F}_{\tau}$–partition of $\Omega$. As seen in the proof of 3.4, for
each $(t_{j},x_{j})$ there is a sequence
$\\{(b_{i,j},\sigma_{i,j})\\}_{i\in\mathds{N}}$ in $\mathcal{A}^{t_{j}}_{F}$
such that

$\lim_{i\to\infty}\mathds{E}\left(\varphi_{t_{j}}(x_{j},b_{i,j},\sigma_{i,j})\right)=\Phi_{t_{j}}(x_{j})=u_{t_{j}}(x_{j}).$

Setting
$(b_{i},\sigma_{i}):=\sum\limits_{j=1}^{n}\chi_{A_{j}}(b_{i,j},\sigma_{i,j})$,
we have

$\displaystyle\Phi_{\tau}(\zeta)\geq$
$\displaystyle\mathds{E}(\varphi_{\tau}(\zeta,b_{i},\sigma_{i})|\mathcal{F}_{\tau})$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\chi_{A_{j}}\mathds{E}\left(\varphi_{t_{j}}(x_{j},b_{i,j},\sigma_{i,j})|\mathcal{F}_{t_{j}}\right)\xrightarrow[i\to\infty]{}\sum_{j=1}^{n}\chi_{A_{j}}u_{t_{j}}(x_{j})=u_{\tau}(\zeta).$

On the other hand, for any $(b,\sigma)\in\mathcal{A}_{F}$,

$\mathds{E}(\varphi_{\tau}(\zeta,b,\sigma)|\mathcal{F}_{\tau})=\sum_{j=1}^{n}\chi_{A_{j}}\mathds{E}\left(\varphi_{t_{j}}(x_{j},b,\sigma)|\mathcal{F}_{t_{j}}\right)\leq\sum_{j=1}^{n}\chi_{A_{j}}u_{t_{j}}(x_{j})=u_{\tau}(\zeta),$

thus
$\operatorname*{ess\,sup}\limits_{b,\sigma\in\mathcal{A}_{F}}\mathds{E}(\varphi_{\tau}(\zeta,b,\sigma)|\mathcal{F}_{\tau})\leq
u_{\tau}(\zeta)$. This completes the proof. ∎

######  Remark 3.7.

3.6 says, as a consequence of (10), that

$\operatorname*{ess\,sup}_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}(\varphi_{\tau}(\zeta,b,\sigma)|\mathcal{F}_{\tau})=\operatorname*{ess\,sup}_{(b,\sigma)\in\mathcal{A}_{F}^{\tau}}\mathds{E}(\varphi_{\tau}(\zeta,b,\sigma)|\mathcal{F}_{\tau})$

for any stopping time $\tau$ with value in $[0,T]$ and $\zeta\in
L^{2}\left(\Omega,\mathcal{F}_{\tau};\mathds{R}^{N}\right)$.

##  4 Representation Formula

When $F$ is a linear operator it is known that the representation formula of
its viscosity solution is built from a stochastic system, see (1). To adapt
this method to our case we will build a candidate viscosity solution using a
stochastic system, defined below, and we will prove, thanks to the theorem
4.4, that it is actually a solution to 2.3.

######  Definition 4.1.

Consider the _forward backward stochastic differential equation_ , FBSDE for
short,

$\left\\{\begin{aligned}
X^{t,\zeta}_{s}=&\zeta+\int_{t}^{s}\sigma\left(r,X^{t,\zeta}_{r}\right)dW_{r}+\int_{t}^{s}b\left(r,X^{t,\zeta}_{r}\right)dr,\\\
Y^{t,\zeta}_{s}=&g\left(X^{t,\zeta}_{T}\right)+\int_{s}^{T}f_{(b,\sigma)}\left(r,X_{r}^{t,\zeta},Y_{r}^{t,\zeta},Z_{r}^{t,\zeta}\right)dr\\\
&-\int_{s}^{T}Z_{r}^{t,\zeta}dW_{r},\end{aligned}\right.\,s\in[t,T],$ (11)

where $\zeta\in L^{2}\left(\Omega,\mathcal{F}_{t};\mathds{R}^{N}\right)$,
$(b,\sigma)\in\mathcal{A}_{F}$ and the functions $f$ and $g$ are as in the
assumptions of 2.3. Notice that the first and the second equation of (11) are,
respectively, an SDE and a BSDE, see appendix B for a brief review about these
topics.
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 $(t,\zeta)\in[0,T]\times
L^{2}\left(\Omega,\mathcal{F}_{t};\mathds{R}^{N}\right)$.

Theorems B.2 and B.6 yield that, under our assumptions, there is a unique
solution to (11). Due to remark B.3, this is true even if $t$ is a stopping
time. In particular from this follows the next crucial result.

######  Proposition 4.2.

For a fixed $(b,\sigma)\in\mathcal{A}_{F}$, let $(X,Y,Z)$ be the solution to
(11). Then, for any $0\leq t\leq r\leq s\leq T$,

$\left(X^{r,X^{t,\zeta}_{r}}_{s},Y^{r,X^{t,\zeta}_{r}}_{s},Z^{r,X^{t,\zeta}_{r}}_{s}\right)=\left(X^{t,\zeta}_{s},Y^{t,\zeta}_{s},Z^{t,\zeta}_{s}\right),\qquad\text{a.e.}.$

This holds true even if $t$, $r$ and $s$ are stopping times.

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 in 4.1, for
$(b,\sigma)$ that varies in $\mathcal{A}_{F}$. To ease notation we will omit
the dependence of $X$, $Y$ and $Z$ on $(b,\sigma)$.

We will prove that
$u(t,x):=\operatorname*{ess\,sup}\limits_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(Y^{t,x}_{t}\middle|\mathcal{F}_{t}\right)$
is a viscosity solution to 2.3, breaking the proof in several steps. Before
that however we stress out that since $Y^{t,x}_{t}$ is $\mathcal{F}_{t}$
measurable and thanks to remark 3.5 we have that for $u$ the following
identities holds true

$u(t,x):=\operatorname*{ess\,sup}_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(Y^{t,x}_{t}\middle|\mathcal{F}_{t}\right)=\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(Y^{t,x}_{t}\right)=\operatorname*{ess\,sup}_{(b,\sigma)\in\mathcal{A}_{F}}Y^{t,x}_{t}.$

######  Proposition 4.3.

The function
$u(t,x):=\sup\limits_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(Y^{t,x}_{t}\right)$
is $\frac{1}{2}$–Hölder continuous in the first variable and Lipschitz
continuous in the second one. Furthermore we have that there is a constant
$c$, which depends only on $\ell$, $\mu$ and $T$, such that

$\mathds{E}\left(|u(\tau,\zeta)|^{2}\right)\leq\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(\left|Y^{\tau,\zeta}_{\tau}\right|^{2}\right)\leq
c\left(1+\mathds{E}\left(|\zeta|^{2}\right)\right),$ (12)

for any stopping time $\tau$ with values in $[0,T]$ and $\zeta\in
L^{2}\left(\Omega,\mathcal{F}_{t};\mathds{R}^{N}\right)$.

We point out that this proposition permits us to use the results of section 3
on $u$. In particular the function $(t,x,b,\sigma)\mapsto Y^{t,x}_{t}$
satisfies the same conditions of $\varphi$ in section 3, thanks to theorems
B.2, B.6, B.4 and B.7. Furthermore we prove here that it satisfies (5) and the
continuity of $u$, which is needed for 3.6. See also remark 3.5.

###### Proof.

Notice that Jensen’s inequality yields

$\displaystyle|u(t,x)-u(s,y)|=$
$\displaystyle\left|\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(Y^{t,x}_{t}\right)-\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(Y^{s,y}_{s}\right)\right|$
$\displaystyle\leq$
$\displaystyle\sup_{(b,\sigma)\in\mathcal{A}_{F}}\left(\mathds{E}\left(\left|Y^{t,x}_{t}-Y^{s,y}_{s}\right|^{2}\right)\right)^{\frac{1}{2}}$

and

$|u(t,\zeta)|=\left|\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(Y^{t,\zeta}_{t}\right)\right|\leq\left(\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(\left|Y^{t,\zeta}_{t}\right|^{2}\right)\right)^{\frac{1}{2}},$

for any $t,s\in[0,T]$, $x,y\in\mathds{R}^{N}$ and $\zeta\in
L^{2}\left(\Omega,\mathcal{F}_{t};\mathds{R}^{N}\right)$.
In order to prove that $u$ is Hölder continuous in the first variable we
assume that $s\leq t$, hence we obtain from theorems B.6 and B.2 that there
exist two constants $c_{1}$ and $c_{2}$, which depend on $\ell$, $\mu$ and
$T$, such that

$\mathds{E}\left(\left|Y^{t,x}_{t}-Y^{s,x}_{s}\right|^{2}\right)\leq
2\mathds{E}\left(\left|Y^{s,x}_{t}-Y^{s,x}_{s}\right|^{2}\right)+2\mathds{E}\left(\left|Y^{t,x}_{t}-Y^{s,x}_{t}\right|^{2}\right)\\\
\begin{aligned}
=&2\mathds{E}\left(\left|Y^{s,x}_{t}-Y^{s,x}_{s}\right|^{2}\right)+2\mathds{E}\left(\left|Y^{t,x}_{t}-Y^{t,X^{s,x}_{t}}_{t}\right|^{2}\right)\\\
\leq&c_{1}\mathds{E}\biggl{(}\begin{aligned}
&\int_{s}^{t}(|f_{(b,\sigma)}\left(r,X^{s,x}_{r},Y^{s,x}_{r},Z^{s,x}_{r})|^{2}+|Z_{r}^{s,x}|^{2}\right)dr\\\
&+\left|g\left(X^{t,x}_{T}\right)-g\left(X^{t,X^{s,x}_{t}}_{T}\right)\right|^{2}\\\
&+\left.\int_{t}^{T}\left|f_{(b,\sigma)}\left(r,X^{t,x}_{r},Y^{t,x}_{r},Z^{t,x}_{r}\right)-f_{(b,\sigma)}\left(r,X^{t,X^{s,x}_{t}}_{r},Y^{t,x}_{r},Z^{t,x}_{r}\right)\right|^{2}dr\right)\end{aligned}\\\
\leq&c_{2}\mathds{E}\biggl{(}\begin{aligned}
&\int_{s}^{t}\left(1+\left|X^{s,x}_{r}\right|^{2}+\left|Y^{s,x}_{r}\right|^{2}+\left|Z^{s,x}_{r}\right|^{2}\right)dr\\\
&+\left.\left|X^{t,x}_{T}-X^{t,X^{s,x}_{t}}_{T}\right|^{2}+\int_{t}^{T}\left|X^{t,x}_{r}-X^{t,X^{s,x}_{t}}_{r}\right|^{2}dr\right).\end{aligned}\end{aligned}$

Using once again theorems B.6 and B.2, we have that there exist five constants
$c_{3}$, $c_{4}$, $c_{5}$, $c_{6}$ and $c_{7}$ depending upon $\mu$, $\ell$
and $T$ such that

$\displaystyle\mathds{E}\left(\int_{s}^{t}\left(|Y^{s,x}_{r}|^{2}+|Z^{s,x}_{r}|^{2}\right)dr\right)\leq$
$\displaystyle
c_{3}\mathds{E}\left(\int_{s}^{t}\left|f_{(b,\sigma)}(r,X^{s,x}_{r},0,0)\right|^{2}dr\right)$
(13) $\displaystyle\leq$ $\displaystyle
c_{4}\mathds{E}\left(\int_{s}^{t}\left(1+|X^{s,x}_{r}|^{2}\right)dr\right)$
$\displaystyle\leq$ $\displaystyle c_{5}(t-s)\left(1+|x|^{2}\right),$

and

$\mathds{E}\left(\sup_{r\in[t,T]}\left|X^{t,x}_{r}-X^{t,X^{s,x}_{t}}_{r}\right|^{2}\right)\leq
c_{6}\mathds{E}\left(|x-X^{s,x}_{t}|^{2}\right)\leq c_{7}(t-s).$

These inequalities together imply that

$\mathds{E}\left(\left|Y^{t,x}_{t}-Y^{s,x}_{s}\right|^{2}\right)\leq
c_{8}(t-s)\left(1+|x|^{2}\right),$

where $c_{8}$ is a constants depending only on $\mu$, $\ell$ and $T$, showing
that $u$ is $\frac{1}{2}$–Hölder continuous in the first variable. The rest
can be proved similarly. ∎

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

######  Theorem 4.4 (Dynamic programming principle).

For any $(b,\sigma)\in\mathcal{A}_{F}$ we let
$\left(\overline{Y},\overline{Z}\right)$ be the solution to the BSDE

$\overline{Y}_{s}=u\left(\tau,X_{\tau}^{t,x}\right)+\int_{s\wedge\tau}^{\tau}f_{(b,\sigma)}\left(r,X^{t,x}_{r},\overline{Y}_{r},\overline{Z}_{r}\right)dr-\int_{s\wedge\tau}^{\tau}\overline{Z}_{r}dW_{r},$
(14)

where $s\in[t,T]$ and $\tau$ is a stopping time with values in $[t,T]$. Then
we have
$\sup\limits_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(\overline{Y}_{t}\right)=u(t,x)$.

###### Proof.

Fixed $(\overline{b},\overline{\sigma})\in\mathcal{A}_{F}$ in (14) we 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 3.6 we know that

$\operatorname*{ess\,sup}_{(b,\sigma)\in\overline{\mathcal{A}}_{F}}\mathds{E}\left(Y_{\tau}^{t,x}\middle|\mathcal{F}_{\tau}\right)=\operatorname*{ess\,sup}_{(b,\sigma)\in\overline{\mathcal{A}}_{F}}\mathds{E}\left(Y_{\tau}^{\tau,\overline{X}^{t,x}_{\tau}}\middle|\mathcal{F}_{\tau}\right)=u\left(\tau,\overline{X}_{\tau}^{t,x}\right)$

and 3.1 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

$\lim_{n\to\infty}\mathds{E}\left(Y_{n,\tau}^{t,x}\middle|\mathcal{F}_{\tau}\right)=\operatorname*{ess\,sup}_{(b,\sigma)\in\overline{\mathcal{A}}_{F}}\mathds{E}\left(Y_{\tau}^{t,x}\middle|\mathcal{F}_{\tau}\right)=u\left(\tau,\overline{X}_{\tau}^{t,x}\right).$

Then, by theorem B.6 and the dominated convergence theorem, there exists a
constant $c$ such that

$\lim_{n\to\infty}\mathds{E}\left(\left|\overline{Y}_{t}-Y^{t,x}_{n,t}\right|^{2}\right)\leq\lim_{n\to\infty}c\mathds{E}\left(\left|u\left(\tau,\overline{X}_{\tau}^{t,x}\right)-Y^{t,x}_{n,\tau}\right|^{2}\right)=0,$

hence, up to subsequences,

$\lim\limits_{n\to\infty}\mathds{E}\left(Y_{n,t}^{t,x}\right)=\mathds{E}\left(\overline{Y}_{t}\right).$
(15)

Furthermore, thanks to theorem B.8, $Y^{t,x}_{t}\leq\overline{Y}_{t}$ for any
$(b,\sigma)\in\overline{\mathcal{A}}_{F}$, which together with (15) 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:

$\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(\overline{Y}_{t}\right)=\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(Y_{t}^{t,x}\right)=u(t,x).$

∎

We conclude this section with our main statement.

######  Theorem 4.5.

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 2.3 satisfying polynomial growth condition
such that $u(T,x)=g(x)$ for any $x$ in $\mathds{R}^{N}$.

###### Proof.

By (12) $u$ has polynomial growth, thus the uniqueness of $u$ follows from
theorem 2.6. Moreover, from 4.3, we know that $u$ is continuous 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 that it is a viscosity solution.
We start showing that $u$ is a subsolution. In order to do so we preliminarily
fix $(t,x)\in(0,T)\times\mathds{R}^{N}$ and
$(b,\sigma)\in\mathcal{A}_{F}^{t}$, and observe that $(b,\sigma)(t)$ is
$\mathcal{F}_{t}^{t}$–measurable. By Blumenthal 0–1 law then it is a.e.
deterministic, which is to say that there is a
$(b^{\prime},\sigma^{\prime})\in K_{F}$ such that
$(b,\sigma)(t,y)=(b^{\prime},\sigma^{\prime})(t,y)$ a.e. for any
$y\in\mathds{R}^{N}$. Similarly for the corresponding $Y$ we have
$Y^{t,y}_{t}=\mathds{E}\left(Y^{t,y}_{t}\right)$ a.e. for any
$y\in\mathds{R}^{N}$. We will prove that
$v(s,y):=\mathds{E}\left(Y^{s,y}_{s}\right)$ is a subsolution at $(t,x)$ and,
since by remarks 3.7 and 3.5
$u(t,x)=\sup\limits_{(b,\sigma)\in\mathcal{A}_{F}^{t}}\mathds{E}\left(Y^{t,x}_{t}\right)$,
well known properties of viscosity solution yield that $u$ is a subsolution at
$(t,x)$. From the arbitrariness of $(t,x)$ will then follows that $u$ is a
subsolution to 2.3.
Let $\varphi$ be a supertangent to $v$ at $(t,x)$ which we assume, without
loss of generality, equal to $v$ at $(t,x)$ and, by contradiction, such that

$\partial_{t}\varphi(t,x)+\frac{1}{2}\left\langle\sigma\sigma^{\dagger},D^{2}_{x}\varphi\right\rangle(t,x)+(\nabla_{x}\varphi
b)(t,x)+f_{(b,\sigma)}(t,x,v,\nabla_{x}\varphi\sigma)\\\
\leq\partial_{t}\varphi(t,x)+F\left(t,x,v,\nabla_{x}\varphi,D^{2}_{x}\varphi\right)<0.$

We define the stopping time

$\tau:=\inf\left\\{s\in[t,T]:Y^{t,x}_{s}=Y^{s,X^{t,x}_{s}}_{s}>\varphi\left(s,X_{s}^{t,x}\right)\right\\}$
(16)

and observe that $\tau>t$ on a set of positive measure, since otherwise we
would have that
$\varphi\left(s,X^{t,x}_{s}\right)<v\left(s,X^{t,x}_{s}\right)$ for any $s>t$
in some neighborhood of $t$. Since the controls in $\mathcal{A}_{F}$ are right
continuous we also assume, eventually taking a smaller stopping time $\tau$,
that, for any $s\in[t,T]$,

$\partial_{t}\varphi\left(s\wedge\tau,X^{t,x}_{s\wedge\tau}\right)+\frac{1}{2}\left\langle\sigma\sigma^{\dagger},D^{2}_{x}\varphi\right\rangle\left(s\wedge\tau,X^{t,x}_{s\wedge\tau}\right)\\\
+(\nabla_{x}\varphi
b)\left(s\wedge\tau,X^{t,x}_{s\wedge\tau}\right)+f_{(b,\sigma)}\left(s\wedge\tau,X^{t,x}_{s\wedge\tau},v,\nabla_{x}\varphi\sigma\right)<0.$
(17)

Now define
$\left(\overline{Y}_{s},\overline{Z}_{s}\right):=\left(Y^{t,x}_{s\wedge\tau},Z^{t,x}_{s\wedge\tau}\right)$,
which solves the BSDE

$\overline{Y}_{s}=Y_{\tau}^{t,x}+\int_{s\wedge\tau}^{\tau}f_{(b,\sigma)}\left(r,X^{t,x}_{r},\overline{Y}_{r},\overline{Z}_{r}\right)dr-\int_{s\wedge\tau}^{\tau}\overline{Z}_{r}dW_{r},\quad
s\in[t,T],$

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, for any $s\in[t,T]$, to

$\displaystyle\hat{Y}_{s}=$
$\displaystyle\varphi\left(\tau,X^{t,x}_{\tau}\right)-\int_{s\wedge\tau}^{\tau}\hat{Z}_{r}dW_{r}$
$\displaystyle-\int_{s\wedge\tau}^{\tau}\\!\left(\partial_{t}\varphi\left(r,X^{t,x}_{r}\right)+\frac{1}{2}\left\langle\sigma\sigma^{\dagger},D^{2}_{x}\varphi\right\rangle\left(r,X^{t,x}_{r}\right)+(\nabla_{x}\varphi
b)\left(r,X^{t,x}_{r}\right)\right)dr.$

Thanks to (16) and (17), we obtain from theorems B.8 and B.9 that
$\varphi(t,x)>v(t,x)$, in contradiction to our assumptions. Therefore $u$ is a
subsolution to 2.3.
We will now prove that $u$ is also a supersolution. Fixed a $(t,x)$ in
$(0,T)\times\mathds{R}^{N}$, let $\psi$ be a subtangent to $u$ in $(t,x)$ such
that $\psi(t,x)=u(t,x)$. We know, thanks to remark 2.2, that there is a
continuous and deterministic $(b,\sigma)\in\mathcal{A}_{F}$ for which

$\displaystyle F\left(t,x,u,\nabla_{x}\psi,D^{2}_{x}\psi\right)=$
$\displaystyle\frac{1}{2}\left\langle\sigma\sigma^{\dagger},D^{2}_{x}\psi\right\rangle(t,x)+(\nabla_{x}\psi
b)(t,x)$ $\displaystyle+f_{(b,\sigma)}(t,x,u,\nabla_{x}\psi\sigma),$

thus we assume by contradiction

$\partial_{t}\psi(t,x)+\frac{1}{2}\left\langle\sigma\sigma^{\dagger},D^{2}_{x}\psi\right\rangle(t,x)+(\nabla_{x}\psi
b)(t,x)+f_{(b,\sigma)}(t,x,u,\nabla_{x}\psi\sigma)\\\
=\partial_{t}\psi(t,x)+F\left(t,x,u,\nabla_{x}\psi,D^{2}_{x}\psi\right)>0.$

We then have by our assumptions that there is a $\delta>0$ such that

$\partial_{t}\psi(s,y)+\frac{1}{2}\left\langle\sigma\sigma^{\dagger},D^{2}_{x}\psi\right\rangle(s,y)+(\nabla_{x}\psi
b)(s,y)+f_{(b,\sigma)}(s,y,u,\nabla_{x}\psi\sigma)\\!<\\!0$ (18)

and

$\psi(s,y)\leq u(s,y)$ (19)

for any $(s,y)\in[t,t+\delta)\times B_{\delta}(x)$, therefore the stopping
time

$\tau:=(t+\delta)\wedge\inf\left\\{s\in[t,T]:\left|X_{s}^{t,x}-x\right|\geq\delta\right\\}$

is a.e. bigger than $t$. Let $X$ be the solution to the SDE $(b,\sigma)$,
$\left(\overline{Y}_{s},\overline{Z}_{s}\right)$ be the solution to the BSDE

$\overline{Y}_{s}=u\left(\tau,X_{\tau}^{t,x}\right)+\int_{s\wedge\tau}^{\tau}f_{(b,\sigma)}\left(r,X^{t,x}_{r},\overline{Y}_{r},\overline{Z}_{r}\right)dr-\int_{s\wedge\tau}^{\tau}\overline{Z}_{r}dW_{r},\quad
s\in[t,T]$

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, for any $[s\in[t,T]$, to

$\displaystyle\hat{Y}_{s}=$
$\displaystyle\psi\left(\tau,X^{t,x}_{\tau}\right)-\int_{s\wedge\tau}^{\tau}\hat{Z}_{r}dW_{r}$
$\displaystyle-\int_{s\wedge\tau}^{\tau}\\!\left(\partial_{t}\psi\left(r,X^{t,x}_{r}\right)+\frac{1}{2}\left\langle\sigma\sigma^{\dagger},D^{2}_{x}\psi\right\rangle\left(r,X^{t,x}_{r}\right)+(\nabla_{x}\psi
b)\left(r,X^{t,x}_{r}\right)\right)\\!dr.$

The theorem 4.4 finally yields

$\sup_{(b,\sigma)\in\mathcal{A}_{F}}\mathds{E}\left(\overline{Y}_{t}\right)=u(t,x)=\psi(t,x),$
(20)

which is in contradiction with (18), since together with (19) it implies,
thanks to theorems B.8 and B.9, that $\overline{Y}_{t}>\psi(t,x)$ a.e.. Then
the arbitrariness of $(t,x)$ proves that $u$ is a supersolution to 2.3 and
concludes the proof. ∎

##  Appendix A Connection with 2BSDEs

In [3] it is established a connection between second order BSDEs, 2BSDEs for
short, and parabolic PDEs similar to the ones studied here, which can be used
to obtain representation formulas. The link between this approach and this
method can be seen, in particular, in the dynamic programming principle we
have developed. We emphasize, on the other side, starting from the dynamic
programming principle, we can provide a new angle for looking at 2BSDE. We
present this point of view in the appendix. This is clearly just a short note
and not a complete analysis.

We start giving the formulation of 2BSDE as in [29, 26]. 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

$\hat{a}_{t}:=\lim_{\varepsilon\downarrow
0}\frac{[W]_{t}-[W]_{t-\varepsilon}}{\varepsilon}$

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

$X^{\alpha}_{t}:=\int_{0}^{t}\alpha^{1/2}_{s}dW_{s},\qquad\mathds{P}_{0}\text{--a.e.},$

and $\alpha$ is a progressive process in $\mathds{S}^{N}_{+}$ such that, for
two fixed $\underline{a},\overline{a}$ in $\mathds{S}^{N}_{+}$,
$\underline{a}\leq\alpha\leq\overline{a}$, $\mathds{P}_{0}$–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 $a\in\mathds{S}^{N}_{+}$

$f(t,\omega,y,z,a):=\sup_{\gamma\in
D_{h}}\left(\frac{1}{2}\langle\gamma,a\rangle-h(t,\omega,y,z,\gamma)\right).$

Furthermore let $\hat{f}(t,y,z):=f(t,y,z,\hat{a}_{t})$,

$\mathcal{P}_{h}^{2}:=\left\\{\mathds{P}\in\mathcal{P}_{S}:\mathds{E}^{\mathds{P}}\left(\int_{0}^{T}\left|\hat{f}(t,0,0)\right|^{2}dt\right)<\infty\right\\}$

and
$\mathcal{P}_{h}^{2}(t,\mathds{P}):=\left\\{\mathds{P}^{\prime}\in\mathcal{P}_{h}^{2}:\mathds{P}^{\prime}=\mathds{P}\text{
on }\mathcal{F}_{t}\right\\}$.

A pair of progressive processes $(Y,Z)$ is solution to the 2BSDE

$Y_{t}=\xi+\int_{t}^{T}\hat{f}(s,Y_{s},Z_{s})ds-\int^{T}_{t}Z_{s}dW_{s}+K_{T}-K_{t}$
(21)

if, for any $\mathds{P}\in\mathcal{P}_{h}^{2}$,

  1. i)

$Y_{T}=\xi$, $\mathds{P}$–a.e.;

  2. ii)

the process $K^{\mathds{P}}$ defined below has non decreasing path
$\mathds{P}$–a.e.,

$K_{t}^{\mathds{P}}=Y_{0}-Y_{t}-\int_{0}^{t}\hat{f}(s,Y_{s},Z_{s})ds+\int^{t}_{0}Z_{s}dW_{s},\quad
t\in[0,T],\,\mathds{P}\text{--a.e.};$

  3. iii)

the family $\left\\{K^{\mathds{P}},\mathds{P}\in\mathcal{P}^{2}_{h}\right\\}$
satisfies the minimum condition

$K_{t}^{\mathds{P}}=\operatorname*{ess\,inf}_{\mathds{P}^{\prime}\in\mathcal{P}_{h}^{2}(t,\mathds{P})}\mathds{E}^{\mathds{P}}\left(K_{T}^{\mathds{P}^{\prime}}\middle|\mathcal{F}_{t}\right),\qquad\mathds{P}\text{--a.e.}.$
(22)

Under suitable conditions the 2BSDE (21) 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

$Y_{t}=\xi+\int_{t}^{r}\hat{f}(s,Y_{s},Z_{s})ds-\int^{r}_{t}Z_{s}dW_{s},\qquad
t\in[0,r],\,\mathds{P}\text{--a.e.},$

it can be proved that
$Y_{t}=\operatorname*{ess\,sup}\limits_{\mathds{P}^{\prime}\in\mathcal{P}_{h}^{2}(t,\mathds{P})}Y_{t}^{\mathds{P}^{\prime}}(r,Y_{r})$
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 a control set, made up by the
progressive processes in $L^{2}([0,T]\times\Omega,\mathds{P}_{0};B)$, where
$B$ is a Banach space, and, for any $t\in[0,T]$,

$\mathcal{A}(t,\alpha):=\left\\{\alpha^{\prime}\in\mathcal{A}:\alpha^{\prime}_{s}=\alpha_{s}\text{
for any }s\in[0,t]\right\\}.$

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

$|f(t,y,z,\alpha)-f(t,y,z,\alpha^{\prime})|\leq C|\alpha-\alpha^{\prime}|.$

We will also assume that, for each $\alpha\in\mathcal{A}$,
$f_{\alpha}(t,y,z):=f(t,y,z,\alpha_{t})$ satisfies assumptions B.5 uniformly
with respect to $\alpha$. We point out that these conditions are not intended
to be minimal. We then have that the BSDEs

$Y^{\alpha}_{t}=\xi_{\alpha}+\int_{t}^{T}f_{\alpha}(s,Y_{s}^{\alpha},Z_{s}^{\alpha})ds-\int_{t}^{T}Z^{\alpha}_{s}dW_{s},\qquad
t\in[0,T],$

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 reasoning as in
the proof of theorem 4.4,

$\overline{Y}_{t}^{\alpha}:=\operatorname*{ess\,sup}_{\alpha^{\prime}\in\mathcal{A}(t,\alpha)}Y^{\alpha^{\prime}}_{t}=\operatorname*{ess\,sup}_{\alpha^{\prime}\in\mathcal{A}(t,\alpha)}Y^{\alpha^{\prime}}_{t}\left(r,\overline{Y}_{r}^{\alpha^{\prime}}\right),\quad\text{for
any }0\leq t\leq r\leq T,$ (23)

where $Y^{\alpha^{\prime}}\left(r,\overline{Y}_{r}^{\alpha^{\prime}}\right)$
is solution to the BSDE
$\left(\overline{Y}^{\alpha^{\prime}}_{r},f_{\alpha},r\right)$. 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 [26] we have 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

$\overline{Y}_{t}^{\alpha}=\xi_{\alpha}+\int_{t}^{T}f_{\alpha}\left(s,\overline{Y}^{\alpha}_{s},\overline{Z}_{s}^{\alpha}\right)ds-\int^{T}_{t}\overline{Z}_{s}^{\alpha}dW_{s}+K^{\alpha}_{T}-K^{\alpha}_{t},\qquad
t\in[0,T].$

We also have that, as in (22),

$K_{t}^{\alpha}=\operatorname*{ess\,inf}_{\alpha^{\prime}\in\mathcal{A}(t,\alpha)}\mathds{E}\left(K_{T}^{\alpha^{\prime}}\middle|\mathcal{F}_{t}\right),\qquad\text{a.e.
for any }t\in[0,T],\,\alpha\in\mathcal{A}.$

If we let $X$ be as in (11) and $\left(\overline{Y},\overline{Z},K\right)$ be
the solution to the 2BSDE (we omit the dependence on the control set
$\mathcal{A}_{F}$)

$\overline{Y}_{s}^{t,x}=g\left(X^{t,x}_{T}\right)+\int_{s}^{T}f_{(b,\sigma)}\left(r,X^{t,x}_{T},\overline{Y}_{r}^{t,x},\overline{Z}_{r}^{t,x}\right)ds-\int^{T}_{s}\overline{Z}_{r}^{t,x}dW_{r}+K^{t,x}_{T}-K^{t,x}_{s}$

for any $x\in\mathds{R}^{N}$ and $0\leq t\leq s\leq T$, we then have that the
function $u(t,x):=\overline{Y}^{t,x}_{t}$ is the viscosity solution to 2.3.

##  Appendix B Stochastic Differential Equations

In this appendix are given some results on _stochastic differential equations_
, SDEs for short, we use in this paper.
Consider the SDE $(b,\sigma)$

$X^{t,\zeta}_{s}=\zeta+\int_{t}^{s}\sigma\left(r,X^{t,\zeta}_{r}\right)dW_{r}+\int_{t}^{s}b\left(r,X^{t,\zeta}_{r}\right)dr,\qquad
s\in[t,\infty),$ (24)

under the following assumptions:

######  Assumptions B.1.

Let $t\in[0,\infty)$, $\zeta\in
L^{2}\left(\Omega,\mathcal{F}_{t};\mathds{R}^{N}\right)$ and

$b:[0,\infty)\times\Omega\times\mathds{R}^{N}\to\mathds{R}^{N}\text{ and
}\sigma:[0,\infty)\times\Omega\times\mathds{R}^{N}\to\mathds{R}^{N\times M}.$

Assume there exists a positive constant $\ell$ such that

  1. i)

$\\{(b,\sigma)(t,x)\\}_{t\in[0,\infty)}$ is a progressive process belonging to
$L^{2}([0,T]\times\Omega)$ for any $x\in\mathds{R}^{N}$ and $T\in[0,\infty)$

and a.e., for any $r\in[0,\infty)$, $x,x^{\prime}\in\mathds{R}^{N}$,

  1. ii)

$|b(r,x)-b(r,x^{\prime})|+|\sigma(r,x)-\sigma(r,x^{\prime})|\leq\ell|x-x^{\prime}|$.

A solution to this SDE is a continuous progressive process $X$ as in (24) 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 [13].

######  Theorem B.2.

Under assumptions B.1 there exists a unique solution to the SDE (24).
Moreover, if $\overline{X}$ is solution to the SDE
$\left(\overline{b},\overline{\sigma}\right)$, there is for each
$T\in[t,\infty)$ a constant $c$, depending only on $\ell$ and $T$ such that

$\mathds{E}\left(\sup_{s\in[t,T]}\left|X^{t,\zeta}_{s}-\overline{X}^{t,\zeta^{\prime}}_{s}\right|^{2}\right)\\!\leq\\!c\mathds{E}\\!\left(|\zeta-\zeta^{\prime}|^{2}\\!+\\!\int_{t}^{T}\\!\left|b\\!\left(s,X_{s}^{t,\zeta}\right)\\!-\overline{b}\left(s,X_{s}^{t,\zeta}\right)\right|^{2}\\!ds\\!\right)\\\
+c\mathds{E}\left(\int_{t}^{T}\left|\sigma\left(s,X_{s}^{t,\zeta}\right)-\overline{\sigma}\left(s,X_{s}^{t,\zeta}\right)\right|^{2}ds\right).$

######  Remark B.3.

The results obtained in this section hold even for SDEs with an a.e. finite
stopping time $\tau$ as starting time. Indeed if for any $\zeta$ in
$L^{2}\left(\Omega,\mathcal{F}_{\tau};\mathds{R}^{N}\right)$ we define

$\overline{b}(t,x):=b(t,x+\zeta)\chi_{\\{\tau\leq t\\}}\text{ and
}\,\overline{\sigma}(t,x):=\sigma(t,x+\zeta)\chi_{\\{\tau\leq t\\}},$

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.

######  Remark B.4.

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 $W_{\tau+t}$.

###  B.1 Backward Stochastic Differential Equations

Here we give some results on _backward stochastic differential equations_ ,
BSDEs for short, used in our investigation. Most of them are well known and
actually hold under more general assumptions. We refer to [21, 25, 22, 2] for
a detailed overview on this topic.

We will work under the followings assumptions:

######  Assumptions B.5.

Let $T\in[0,\infty)$, $\xi\in
L^{2}\left(\Omega,\mathcal{F}_{T},\mathds{P};\mathds{R}\right)$ and

$f:[0,T]\times\Omega\times\mathds{R}\times\mathds{R}^{N}\to\mathds{R}.$

Assume there exists a positive constant $\ell$ and a real number $\mu$ such
that

  1. i)

$\\{f(s,y,z)\\}_{s\in[0,T]}$ is a progressive process belonging to
$L^{2}([0,T]\times\Omega)$ for any $y\in\mathds{R}$ and $z\in\mathds{R}^{N}$

and a.e., for any $t\in[0,T]$, $y,y^{\prime}\in\mathds{R}$ and
$z,z^{\prime}\in\mathds{R}^{N}$,

  1. ii)

$|f(t,y,z)|\leq|f(t,0,0)|+\ell(1+|y|+|z|)$;

  2. iii)

$|f(t,y,z)-f(t,y,z^{\prime})|\leq\ell|z-z^{\prime}|$;

  3. iv)

$(y-y^{\prime})(f(t,y,z)-f(t,y^{\prime},z))\leq\mu|y-y^{\prime}|^{2}$;

  4. v)

$v\mapsto f(t,v,z)$ is continuous.

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 continuous
progressive processes belonging to $L^{2}([0,T]\times\Omega)$ such that

$Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds-\int_{t}^{T}Z_{s}dW_{s},\quad\text{for
any }t\in[0,T].$ (25)

The followings are classical results of BSDE theory, see for example theorems
1.2 and 1.3, and proposition 1.3 in [21].

######  Theorem B.6.

Under the assumptions B.5 the BSDE (25) has a unique solution $(Y,Z)$.
Furthermore, if $(Y^{\prime},Z^{\prime})$ is the solution to the BSDE
$(\xi^{\prime},f^{\prime},T)$, there exists a constant $c$, which depends on
$T$, $\mu$ and $\ell$, such that

$\mathds{E}\left(\sup_{t\in[0,T]}|Y_{t}-Y^{\prime}_{t}|^{2}+\int_{0}^{T}|Z_{t}-Z^{\prime}_{t}|^{2}dt\right)\\\
\leq
c\mathds{E}\left(|\xi-\xi^{\prime}|^{2}+\int_{0}^{T}|f(t,Y^{\prime}_{t},Z^{\prime}_{t})-f^{\prime}(t,Y^{\prime}_{t},Z^{\prime}_{t})|^{2}dt\right).$

######  Remark B.7.

Also for BSDEs holds a measurability result similar to the one seen in remark
B.4. Indeed, if $\tau$ is a stopping time with values in $[0,T]$, $\xi\in
L^{2}\left(\Omega,\mathcal{F}_{T}^{\tau},\mathds{P};\mathds{R}^{M}\right)$ and
$f$ is progressive with respect to the filtration
$\\{\mathcal{F}^{\tau}_{t}\\}_{t\in[0,\infty)}$, the solution $(Y,Z)$ to the
BSDE $(\xi,f,T)$ restricted to $[\tau,T]$ is progressive with respect to the
filtration $\\{\mathcal{F}^{\tau}_{t}\\}_{t\in[0,\infty)}$.

######  Theorem B.8.

Let $(Y,Z)$ be the solution to the BSDE $(\xi,f,T)$ under the assumptions B.5
and

$Y^{\prime}_{t}=\xi^{\prime}+\int_{t}^{T}V_{s}ds-\int_{t}^{T}Z^{\prime}_{s}dW_{s},\qquad
t\in[0,T],$

where $\xi^{\prime}\in
L^{2}\left(\Omega,\mathcal{F}_{T},\mathds{P};\mathds{R}\right)$,
$Y^{\prime},V\in L^{2}([0,T]\times\Omega)$ and $Z^{\prime}\in
L^{2}([0,T]\times\Omega)$. Suppose that $\xi\leq\xi^{\prime}$ a.e. and
$f(t,Y^{\prime}_{t},Z^{\prime}_{t})\leq V_{t}$ a.e. for the $dt\times
d\mathds{P}$ measure. Then, for any $t\in[0,T]$, $Y_{t}\leq Y^{\prime}_{t}$
a.e..
Furthermore, if $f$ is right continuous on $[0,T]$ and $Y_{0}=Y^{\prime}_{0}$
a.e., then $Y_{t}=Y^{\prime}_{t}$ a.e. for any $t\in[0,T]$. In particular,
whenever either $\mathds{P}(\\{\xi<\xi^{\prime}\\})>0$ or
$f(s,Y^{\prime}_{s},Z^{\prime}_{s})<V_{s}$, for any
$(y,z)\in\mathds{R}\times\mathds{R}^{N}$ on a set of positive $dt\times
d\mathds{P}$ measure, then $Y_{0}<Y^{\prime}_{0}$.

######  Proposition B.9.

Let $(Y,Z)$ be the solution to the BSDE (25) under the assumptions B.5 and
assume that there exists a stopping time $\tau$ such that $\tau\leq T$, $\xi$
is $\mathcal{F}_{\tau}$–measurable and $f(t,y,z)=0$ on the set $\\{\tau\leq
t\\}$. Then $Y_{t}=Y_{\tau\wedge t}$ a.e. and $Z_{t}=0$ a.e. on the set
$\\{\tau\leq t\\}$. In particular

$Y_{\tau\wedge t}=\xi+\int_{\tau\wedge
t}^{\tau}f(s,Y_{s},Z_{s})ds-\int_{\tau\wedge
t}^{\tau}Z_{s}dW_{s},\qquad\text{for any }t\in[0,T].$

##  Appendix C Comparison Theorem

Consider the parabolic problem

$\partial_{t}u(t,x)+G\left(t,x,u,\nabla_{x}u,D_{x}^{2}u\right)=0,\quad
t\in(0,T),x\in\mathds{R}^{N},$ (26)

under the following assumptions.

######  Assumptions C.1.

$G$ is a continuous elliptic operator which admits, for any $t\in[0,T]$,
$x,p,p^{\prime}\in\mathds{R}^{N}$, $r,r^{\prime}\in\mathds{R}$ and
$S,S^{\prime}\in\mathds{S}^{N}$, a positive $\ell$ and a $\mu\in\mathds{R}$
such that

  1. i)

$|G(t,x,r,p,S)-G(t,x,r,p,S^{\prime})|\leq\ell\left(1+|x|^{2}\right)|S-S^{\prime}|$;

  2. ii)

$|G(t,x,r,p,S)-G(t,x,r,p^{\prime},S)|\leq\ell(1+|x|)|p-p^{\prime}|$;

  3. iii)

$(G(t,x,r,p,S)-G(t,x,r^{\prime},p,S))(r-r^{\prime})\leq\mu|r-r^{\prime}|^{2}$;

  4. iv)

for each $R>0$ there is a modulus of continuity $\omega_{R}$ for which

$G(t,x,r,\alpha(x-y),S)-G(t,y,r,\alpha(x-y),S^{\prime})\leq\omega_{R}\left(\alpha|x-y|^{2}+|x-y|\right)$

for any $\alpha>0$, $t\in[0,T]$, $x,y\in B_{R}(0)$ and
$S,S^{\prime}\in\mathds{S}^{N}$ such that

$-3\alpha\begin{pmatrix}I_{N}&0\\\
0&I_{N}\end{pmatrix}\leq\begin{pmatrix}S&0\\\ 0&-S^{\prime}\end{pmatrix}\leq
3\alpha\begin{pmatrix}I_{N}&-I_{N}\\\ -I_{N}&I_{N}\end{pmatrix},$ (27)

where $I_{N}$ is the $N\times N$ identity matrix.

For this problem holds a comparison result, which is a parabolic version of
[11, Theorem 7.4].

######  Theorem C.2.

Let $u$ and $v$ be respectively a sub and a supersolution to (26) under
assumptions C.1. If $u(T,x)\leq v(T,x)$ for any $x\in\mathds{R}^{N}$ and both
$u$ and $v$ have polynomial growth, then $u\leq v$ on
$(0,T]\times\mathds{R}^{N}$.

###### Proof.

First of all let $\gamma$ be a constant that we will fix later. If we define
$u_{\gamma}(t,x):=e^{\gamma t}u(t,x)$ and $v_{\gamma}(t,x):=e^{\gamma
t}v(t,x)$ we have that they are a sub and a supersolution, respectively, to

$\partial_{t}u(t,x)-\gamma
u(t,x)+G_{\gamma}\left(t,x,u,\nabla_{x}u,D^{2}_{x}u\right)=0,$ (28)

where $G_{\gamma}(t,x,r,p,S):=e^{\gamma t}G\left(t,x,e^{-\gamma t}r,e^{-\gamma
t}p,e^{-\gamma t}S\right)$. It is a simple verification that $G_{\gamma}$
still satisfies assumptions C.1 and in particular

$(r-r^{\prime})(G_{\gamma}(t,x,r,p,S)-\gamma
r-G_{\gamma}(t,x,r^{\prime},p,S)+\gamma
r^{\prime})\leq(\mu-\gamma)|r-r^{\prime}|^{2}.$ (29)

Since we assumed that $u$ and $v$ have polynomial growth, there is a constant
$k\geq 2$ such that, for any $t\in[0,T]$ and $\varepsilon>0$,

$\lim_{|x|\to\infty}u_{\gamma}(t,x)-v_{\gamma}(t,x)-2\varepsilon(1+|x|)^{k}=-\infty,$

thus we set $\phi(x):=(1+|x|)^{k}$,
$u_{\gamma,\varepsilon}:=u_{\gamma}-\varepsilon\phi$,
$v_{\gamma,\varepsilon}:=v_{\gamma}+\varepsilon\phi$ and we get

$\lim_{|x|\to\infty}u_{\gamma,\varepsilon}(t,x)-v_{\gamma,\varepsilon}(t,x)=-\infty.$
(30)

Notice that

$\displaystyle 0\leq$ $\displaystyle\partial_{t}u_{\gamma}(t,x)-\gamma
u_{\gamma}(t,x)+G_{\gamma}\left(t,x,u_{\gamma},\nabla_{x}u_{\gamma},D^{2}_{x}u_{\gamma}\right)$
$\displaystyle=$
$\displaystyle\partial_{t}u_{\gamma,\varepsilon}(t,x)-\gamma(u_{\gamma,\varepsilon}(t,x)+\varepsilon\phi(x))$
$\displaystyle+G_{\gamma}\left(t,x,u_{\gamma,\varepsilon}+\varepsilon\phi(x),\nabla_{x}u_{\gamma,\varepsilon}+\varepsilon\nabla\phi(x),D^{2}_{x}u_{\gamma,\varepsilon}+\varepsilon
D^{2}\phi(x)\right)$ $\displaystyle\leq$
$\displaystyle\partial_{t}u_{\gamma,\varepsilon}(t,x)-\gamma
u_{\gamma,\varepsilon}(t,x)-(\gamma-\mu)\varepsilon\phi(x)$
$\displaystyle+G_{\gamma}\left(t,x,u_{\gamma,\varepsilon},\nabla_{x}u_{\gamma,\varepsilon}+\varepsilon\nabla\phi(x),D^{2}_{x}u_{\gamma,\varepsilon}+\varepsilon
D^{2}\phi(x)\right)$

and since $|\nabla\phi(x)|$ and $\left|D^{2}\phi(x)\right|$ are polynomials of
degree $k-1$ and $k-2$, respectively, items i) and ii) in assumptions C.1
yield that there exists a constant $C>0$ such that

$\displaystyle 0\leq$
$\displaystyle\partial_{t}u_{\gamma,\varepsilon}(t,x)-\gamma
u_{\gamma,\varepsilon}(t,x)-(\gamma-\mu)\varepsilon\phi(x)$
$\displaystyle+G_{\gamma}\left(t,x,u_{\gamma,\varepsilon},\nabla_{x}u_{\gamma,\varepsilon},D^{2}_{x}u_{\gamma,\varepsilon}\right)+C\varepsilon\phi(x).$

Similarly we can prove that

$\displaystyle 0\geq$
$\displaystyle\partial_{t}v_{\gamma,\varepsilon}(t,x)-\gamma
v_{\gamma,\varepsilon}(t,x)+(\gamma-\mu)\varepsilon\phi(x)$
$\displaystyle+G_{\gamma}\left(t,x,v_{\gamma,\varepsilon},\nabla_{x}v_{\gamma,\varepsilon},D^{2}_{x}v_{\gamma,\varepsilon}\right)-C\varepsilon\phi(x).$

Choosing $\gamma=C+\mu>0$ we get that $u_{\gamma,\varepsilon}$ and
$v_{\gamma,\varepsilon}$ are respectively a sub and a supersolution to (28)
and, thanks to (29), $r\mapsto G_{\gamma}(t,x,r,p,S)-\gamma r$ is decreasing.
Finally observe that by (30) there is a $R>0$ such that
$u_{\gamma,\varepsilon}<v_{\gamma,\varepsilon}$ on
$(0,T]\times\mathds{R}^{N}\setminus B_{R}(0)$ and that the PDE (28) on
$(0,T]\times B_{R}(0)$ satisfies the assumptions of [4, Theorem 8.2],
therefore $u_{\gamma,\varepsilon}\leq v_{\gamma,\varepsilon}$ on
$(0,T]\times\mathds{R}^{N}$. The arbitrariness of $\varepsilon$ then implies
that $u\leq v$ on $(0,T]\times\mathds{R}^{N}$. ∎

It is easy to see that items i), iii) and ii) in assumptions C.1 hold true for
the operator $F$ in assumptions 2.1, therefore, if we can show that it also
satisfies item iv), then a comparison result holds for 2.3. In order to do so
let $S,S^{\prime}\in\mathds{S}^{N}$ be such that (27) holds, then, for any
$(b,\sigma)\in K_{F}$, $t\in[0,T]$ and $x\in\mathds{R}^{N}$, we have

$\left(\sigma^{\dagger}S\sigma\right)(t,x)-\left(\sigma^{\dagger}S^{\prime}\sigma\right)(t,y)\leq
3\alpha(\sigma(t,x)-\sigma(t,y))^{\dagger}(\sigma(t,x)-\sigma(t,y)),$

hence

$\displaystyle\operatorname{tr}\\!\left(\sigma^{\dagger}S\sigma\right)\\!(t,x)\\!-\\!\operatorname{tr}\\!\left(\sigma^{\dagger}S^{\prime}\sigma\right)\\!(t,y)\\!\leq$
$\displaystyle
3\alpha\\!\operatorname{tr}\\!\left((\sigma(t,x)\\!-\\!\sigma(t,y))^{\dagger}(\sigma(t,x)\\!-\\!\sigma(t,y))\right)$
$\displaystyle=$ $\displaystyle
3\alpha|\sigma(t,x)-\sigma(t,y)|^{2}=3\alpha\ell^{2}|x-y|^{2}$

and consequently the identity
$\operatorname{tr}\left(AB^{\dagger}\right)=\operatorname{tr}\left(B^{\dagger}A\right)$
yields

$\left\langle\sigma\sigma^{\dagger}(t,x),S\right\rangle-\left\langle\sigma\sigma^{\dagger}(t,y),S^{\prime}\right\rangle\leq
3\alpha\ell^{2}|x-y|^{2}.$

From this we get that, for any $\alpha>0$, $t\in[0,T]$, $x,y\in\mathds{R}^{N}$
and $S,S^{\prime}\in\mathds{S}^{N}$ as in (27), there exists a constant $C$
such that

$F(t,x,r,\alpha(x-y),S)-F(t,y,r,\alpha(x-y),S^{\prime})\\\ \begin{aligned}
\leq&\sup_{(b,\sigma)\in K_{F}}\begin{aligned}
&\left|\left\langle\sigma\sigma^{\dagger}(t,x),S\right\rangle-\left\langle\sigma\sigma^{\dagger}(t,y),S^{\prime}\right\rangle\right.\\\
&+\alpha(x-y)^{\dagger}b(t,x)-\alpha(x-y)^{\dagger}b(t,y)\\\
&\left.+f_{(b,\sigma)}\left(t,x,r,\alpha(x-y)^{\dagger}\sigma\right)-f_{(b,\sigma)}\left(t,y,r,\alpha(x-y)^{\dagger}\sigma\right)\right|\end{aligned}\\\
\leq&C\left(\alpha|x-y|^{2}+|x-y|\right),\end{aligned}$

thus $F$ satisfies assumptions C.1 and by theorem C.2 the next comparison
result follows.

See 2.6

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