# Singular Limit of Two Scale Stochastic Optimal Control Problems in Infinite
Dimensions by Vanishing Noise Regularization

Giuseppina Guatteri,
Dipartimento di Matematica,
Politecnico di Milano,
Piazza Leonardo da Vinci 32,
20133 Milano, Italia.
e-mail<EMAIL_ADDRESS>

Gianmario Tessitore,
Dipartimento di Matematica e Applicazioni
Università Milano-Bicocca,
via R. Cozzi 53 - Edificio U5,
20125 Milano, Italia.
e-mail<EMAIL_ADDRESS>
###### Abstract

In this paper we study the limit of the value function for a two-scale,
infinite-dimensional, stochastic controlled system with cylindrical noise and
possibly degenerate diffusion. The limit is represented as the value function
of a new reduced control problem (on a reduced state space). The presence of a
cylindrical noise prevents representation of the limit by viscosity solutions
of HJB equations as in [16] while degeneracy of diffusion coefficients
prevents representation as a classical BSDE as in [10]. We use a ˝vanishing
noise” regularization technique.

Keywords: Stochastic equations in infinite dimensions, optimal control, two
scale systems, vanishing noise.

##  1 Introduction

This paper studies the limit of the value functions for a sequence of optimal
control problems with state equation represented by the following system of
stochastic differential equations

$\begin{cases}dX_{t}=AX_{t}\,dt+b(X_{t},Q_{t},u_{t})dt\,+R(X_{t})d{W}^{1}_{t},&X_{0}=x_{0},\\\
\varepsilon
dQ_{t}=(BQ_{t}+F(X_{t},Q_{t})+G\rho(u_{t}))\;dt+\varepsilon^{1/2}G\,d{W}^{2}_{t},&Q_{0}=q_{0}.\end{cases}$
(1.1)

and cost represented by the following functional:

$J^{\epsilon}(x_{0},q_{0},u)=\mathbb{E}\left[\int_{0}^{1}l(X_{t},Q_{t},u_{t})dt+h(X_{1})\right].$

We notice that the small constant $\varepsilon$ in the second equation
modelizes the fact that $Q$ evolves quicker than $X$, with a ratio
$\frac{1}{\varepsilon}$ between the two velocities. Our goal is to represent
the limit of the value functions of these problems as the speed factor
diverges.

In (1.1) both $X$ and $Q$ take values in an Hilbert space. Moreover $A$ and
$B$ are unbounded linear operators, $u$ represents the control,
$({W}^{1}_{t})_{t\geq 0}$, $({W}^{2}_{t})_{t\geq 0}$ are infinite dimensional
independent cylindrical Wiener processes, $b$, $F$, $\rho$ are $R$ are
functions and $G$ is a bounded linear operator satisfying suitable assumptions
including dissipativity of $B+F(x,\cdot)$. The main feature of this paper is
that we allow both $W^{1}$ and $W^{2}$ to be cylindrical while we do not
require any regularizing or nondegeneracy assumptions to hold on $G$ or
$R(\cdot)$. We mention here that throughout the paper the control problems are
formulated in a weak form particularly suitable to be studied by Backward
Stochastic Differential Equations (BSDEs for short).

Several papers have been devoted to the characterization of limits of singular
stochastic control problems in finite dimensional spaces. Beside the
pioneering results based on direct computations in specific situations (see,
for instance, [11] and [12]) the general approach in finite dimensional cases
(see [1], [2], [3], [4], [5]) relies on the representation of the value
function as viscosity solution of a suitable HJB equation and on a convergence
result, as $\epsilon\rightarrow 0$, for the solution of such HJB equations to
a reduced nonlinear parabolic equation. The value of the (viscosity) solution
of the limit PDE is then the desired limit. The well known technical
difficulties nested in the proof of comparison principle for solution of
infinite dimensional HJB equations prevents a direct extension of the previous
results to the case of Hilbert-valued, two scale, controlled stochastic
systems.

At our best knowledge the first paper to address the problem in an infinite
dimensional framework is [10] where the value function is represented through
a BSDE and the result is obtained by convergence of a class of singularly
perturbed BSDEs. The main limitation of the results in [10] is that we need to
assume non degeneracy of the noise in the slowly evolving equation (in [10]
$R$ is indeed independent of $x$ and, more important, is invertible). Then in
[16] the viscosity solution approach is adapted to the Hilbert space case by a
deep analysis of the necessary technical assumptions. A rather general class
of two scales systems can be considered in this last paper with the only
remaining obstruction on the covariance of the noises that must be of finite
trace.

As we have already mentioned here both $(W^{1})$ and $(W^{2})$ are cilindrical
and we do not assume non degeneracy nor on $R$ nor on $G$ (nevertheless in the
equation for $Q$ the structure condition, allowing to apply Girsanov
transform, has to hold). Since the two previously mentioned possible
representations of our singular limit (the one through a BSDE and the one
through a viscosity solution of a HJB equation) seem not to be available in
the present case we try to represent it as the value function of a reduced
control problem. By the way we notice that such a representation somehow lies
underneath both the above mentioned ones. It is also worth mentioning that any
notion of solution of HJB equation stronger than viscosity, and consequently
any direct representation of the limit value function by a standard BSDE,
appears here to be excluded by the lack of regularity that one has to expect
for the value functions of degenerate stochastic control problem.

To compare the class of state equations that fall into the framework of this
paper with the ones treatable in [10] and [16] let us consider the following
two scale system of controlled reaction diffusion SPDEs in one space dimension
driven by space time noises. We refer, for instance to [14] Section 11.2 for
the abstract formulation and precise assumptions on the coefficients. We just
have to mention that $m$ is a positive constant and the Lipschitz constant of
$f$ with respect to $\mathcal{Q}$ is smaller then $m$.

$\left\\{\begin{array}[]{l}\displaystyle\frac{\partial}{\partial
t}\mathcal{X}^{\varepsilon}(t,x)=\frac{\partial^{2}}{\partial
x^{2}}\mathcal{X}^{\varepsilon}(t,x)+b(\mathcal{X}^{\varepsilon}(t,x),\mathcal{Q}^{\varepsilon}(t,x),u(t,x))+\sigma(x,\mathcal{X}^{\varepsilon}(t,x))\frac{\partial}{\partial
t}{\mathcal{W}}^{1}(t,x),\\\ \\\
\displaystyle\varepsilon\frac{\partial}{\partial
t}\mathcal{Q}^{\varepsilon}(t,x)\\!=\\!(\frac{\partial^{2}}{\partial
x^{2}}-m)\mathcal{Q}^{\varepsilon}(t,x)\\!+\\!f(\mathcal{X}^{\varepsilon}(t,x),\mathcal{Q}^{\varepsilon}(t,x))+\rho(x)r(u(t,x))+\varepsilon^{1/2}\rho(x)\frac{\partial}{\partial
t}{\mathcal{W}}^{2}(t,x),\\\ \\\
\mathcal{X}^{\varepsilon}(t,0)=\mathcal{X}^{\varepsilon}(t,1)=\mathcal{Q}^{\varepsilon}(t,0)=\mathcal{Q}^{\varepsilon}(t,1)=0,\\\
\\\ \mathcal{X}^{\varepsilon}(0,x)=\mathcal{X}^{0}(x),\
\mathcal{Q}^{\varepsilon}(0,x)=\mathcal{Q}^{0}(x),\qquad\qquad\qquad\qquad
t\in[0,1],\ x\in[0,1],\end{array}\right.$ (1.2)

To fit the assumptions in [10] one should assume $\sigma$ to be independent of
$\mathcal{X}$ and bounded away from 0 while to fit the assumptions in [16] one
should assume $({\mathcal{W}}^{i})$, $i=1,2$ to be colored in space. Here we
can take a general $\sigma$ (possibly vanishing ) and space-time white noises
$({\mathcal{W}}^{1})$, $i=1,2$.

The approach we present here is to regularize equation (1.1) adding an extra
noise with small parameter in the equation for $X$. We obtain a non degenerate
singular control problem (see equation (4.1)) that can be treated using the
results in [10]. The point is that, in this way, we have two handle a system
depending on two parameters (the original speed ratio $\varepsilon$ and the
new small noise parameter). We show that we can interchange the limits and let
first $\varepsilon\rightarrow 0$ for a fixed small noise parameter. Proceeding
like that (adapting the arguments in [10]) we end up with a forward backward
system depending on the small parameter $\eta$.

$\begin{cases}dX_{t}=AX_{t}\,dt+R(X_{t})\,d{W}^{1}_{t}+\eta\,d{B}_{t},\qquad
t\in[s,1],\\\
-dY_{t}=\lambda({X}_{t},\eta^{-1}{Z}_{t})\,dt-\,{Z}_{t}\,d{W}^{1}_{t}+\,{Z}_{t}\,d{B}^{1}_{t},\\\
X_{s}=x,\quad{Y}_{1}=h(X_{1}).\end{cases}$ (1.3)

We notice that in the above equation the nonlinearity $\lambda$ is itself the
value function of a suitable ergodic optimal control problem, roughly
speaking, for the second equation in (1.1) with frozen $X$.

The idea is then to see, by Fenchel duality, the $Y$ in equation (1.3) as the
value function of an auxiliary control problem. The key issue, at this level,
is to construct this new control problem in such a way that its running cost
has enough regularity to allow the final passage to the limit as the small
noise regularization vanishes and, eventually, to get the main result of this
paper (see Theorem 6.3). In addition the value function of our reduced control
problem can be shown to coincide with the minimal solution of a Backward
Stochastic Differential Equation with constraints on the martingale term (see
Remark 6.5 here and [13], [6]).

The paper is organized as follows: in Section 2 we introduce general notation,
in Section 3 we formulate the control problems introducing the weak
formulation that will be used throughout the paper, in Section 4 we introduce
the small noise regularization of the system, in Section 5 we prove that we
can change order between the limit with respect to the speed ratio parameter
$\varepsilon$ and the small noise parameter $\eta$, finally in Section 6 we
prove our mail result.

##  2 Notation

Given a Banach space $E$, the norm of its elements $x$ will be denoted by
$|x|_{E}$, or even by $|x|$ when no confusion is possible. If $F$ is another
Banach space, $L(E,F)$ denotes the space of bounded linear operators from $E$
to $F$, endowed with the usual operator norm. When $F=\mathbb{R}$ the dual
space $L(E,\mathbb{R})$ will be denoted by $E^{*}$. The letters $\Xi$, $H$ and
$K$ will always be used to denote Hilbert spaces. The scalar product is
denoted $\langle\cdot,\cdot\rangle$, equipped with a subscript to specify the
space, if necessary. All Hilbert spaces are assumed to be real and separable
and the dual of a Hilbert space will never be identified with the space
itself. By $L_{2}(\Xi,H)$ and $L_{2}(\Xi,K)$ we denote the spaces of Hilbert-
Schmidt operators from $\Xi$ to $H$ and to $K$, respectively. Finally
$\mathcal{G}(K,H)$ is the space of all Gateaux differentiable mappings $\phi$
from $K$ to $H$ such that the map $(k,v)\rightarrow\nabla\phi(k)v$ is
continuous from $K\times K$ to $H$; see [8] for details.

Next we define the following classes of stochastic processes with values in a
Hilbert space $V$. Given an arbitrary time horizon $T$, constant $p\geq 1$ and
a generic filtered space
$(\Omega^{0},\mathcal{E}^{0},(\mathcal{F}^{0}_{t})_{t\in[0,T]},\mathbb{P}^{0})$:

  * •

$L^{p}_{\mathcal{\mathcal{F}}^{0}}(\Omega^{0}\times[0,T];V)$ denotes the space
of equivalence classes of processes $Y\in L^{p}(\Omega\times[0,T];V)$
admitting a predictable version. It is endowed with the norm

$|Y|_{p}=\Big{(}\mathbb{E}^{0}\int_{0}^{T}|Y_{s}|^{p}\,ds\Big{)}^{1/p}.$

  * •

$L^{p}_{\mathcal{\mathcal{F}}^{0}}(\Omega^{0};C([0,T];V))$ denotes the space
of adapted processes $Y$ with continuous paths in $V$, such that the norm

$\|Y\|_{p}=(\mathbb{E}^{0}\sup_{s\in[0,T]}|Y_{s}|^{p})^{1/p}$

is finite. The elements of
$L^{p}_{\mathcal{\mathcal{F}}^{0}}(\Omega^{0};C([0,T];V))$ are identified up
to indistinguishability.

##  3 Setting of the problem and statement of the main result

Let $H$, $K$ and $\Xi$ separable Hilbert spaces and $U$ a separable metric
space. We denote by $\mathbb{S}^{1,2}$ ($\mathbb{S}$ stands for Setting) the
class of all $6$-uples
$\mathbb{U}=(\Omega,(\mathcal{F}_{t}),\mathbb{P},(W^{1}_{t}),(W^{2}_{t}),(u_{t}))$,
where $(\Omega,\mathcal{F},(\mathcal{F}_{t}))$ is a filtered complete
probability space, $(W^{1}_{t}),(W^{2}_{t})$ are two independent,
$\Xi$-valued, $(\mathcal{F}_{t})$-Wiener processes and $u$ is an
$(\mathcal{F}_{t})$ predictable process taking values in $U$. When needed, we
will add the mark $\mathbb{U}$ to each term to avoid confusion.

Given $x_{0}\in H,q_{0}\in K,\varepsilon>0$, and
$\mathbb{U}\in\mathbb{S}^{1,2}$, we consider the following two scale state
equation in $H\times K$:

$\begin{cases}dX_{t}=AX_{t}\,dt+b(X_{t},Q_{t},u_{t})dt\,+R(X_{t})d{W}^{1}_{t},&X_{0}=x_{0},\\\
\\\ \varepsilon
dQ_{t}=(BQ_{t}+F(X_{t},Q_{t})+G\rho(u_{t}))\;dt+\varepsilon^{1/2}G\,d{W}^{2}_{t},&Q_{0}=q_{0}.\end{cases}$
(3.1)

that has, under Hypothesis 3.1-3.6 listed below, a unique mild solution
belonging to
$L^{p}_{\mathcal{\mathcal{F}}^{\mathbb{U}}}(\Omega^{\mathbb{U}};C([0,T];H))$,
$p\geq 1$ that we denote by $X^{\varepsilon,\mathbb{U}}$, see [10, Lemma 3.9
and Lemma 3.10]. We omit reference to initial state $(x_{0},q_{0})$ trying to
ease the notation (when we will need to show such dependence we will
explicitly mention it). We introduce the following cost functional to minimize

$J^{\varepsilon}(x_{0},q_{0},\mathbb{U})=\mathbb{E}^{\mathbb{U}}\left[\int_{0}^{1}l(X^{\varepsilon,\mathbb{U}}_{t},Q^{\varepsilon,\mathbb{U}}_{t},u^{\mathbb{U}}_{t})dt+h(X^{\varepsilon,\mathbb{U}}_{1})\right],$
(3.2)

where $\mathbb{E}^{\mathbb{U}}$ denotes the expectation with respect to the
probability $\mathbb{P}$ in $\mathbb{U}$.

We make the following general assumptions fixing, in the mean time, three
constants $M>0$, $L>0$ and $\gamma\in[0,1/2)$ that will not be changed
throughout the paper.

###### Hypothesis 3.1

$A:D(A)\subset H\to H$ is a linear, unbounded operator that generates a
$C_{0}$\- semigroup $\\{e^{tA}\\}_{t\geq 0}$, such that $|e^{tA}|_{L(H,H)}\leq
M_{A}e^{\omega_{A}t},t\geq 0$ for some positive constants $M_{A}$ and
$\omega_{A}$. $B:D(B)\subset K\to K$ is a linear, unbounded operator that
generates a $C_{0}$\- semigroup $\\{e^{tB}\\}_{t\geq 0}$ such that
$|e^{tB}|_{L(K,K)}\leq M_{B}e^{\omega_{B}t},t\geq 0$ for some
$M_{B},\omega_{B}>0$.
Moreover there exist $C>0$ s.t.:

$\displaystyle|e^{sA}|_{L_{2}(H,H)}+|e^{sB}|_{L_{2}(K,K)}\leq
Cs^{-\gamma},\quad\forall s\in[0,1].$

###### Hypothesis 3.2

The functions $b:H\times K\times U\to H$ and $F:H\times K\to K$ are measurable
and:

$|b(x,q,u)|\leq M,\quad|b(x,q,u)-b(x^{\prime},q^{\prime},u)|\leq
L(|x-x^{\prime}|+|q-q^{\prime}|),\qquad\qquad\forall\,q,q^{\prime}\in
K,x,x^{\prime}\in H,\,u\in U,$ $|F(x,q)-F(x^{\prime},q^{\prime})|_{K}\leq
L(|x-x^{\prime}|_{H}+|q-q^{\prime}|_{K})\qquad\qquad\forall\,q,q^{\prime}\in
K,x,x^{\prime}\in H.$

Moreover we assume that, $F(x,\cdot)$ is Gateaux differentiable, more
precisely, $F(x,\cdot)\in\mathcal{G}^{1}(K,K)$, $\forall x\in H$.

###### Hypothesis 3.3

$B+F$ is dissipative i.e. there exists some $\mu>0$ such that:

$\langle
Bq+F(x,q)-(Bq^{\prime}+F(x,q^{\prime})),q-q^{\prime}\rangle\leq-\mu|q-q^{\prime}|^{2},$

for all $x\in H,q,q^{\prime}\in D(B)$.

###### Hypothesis 3.4

$R\colon H\rightarrow L(\Xi,H)$ is a bounded Lipschitz map. Moreover for all
$x,x^{\prime}\in H$:

$\big{|}e^{sA}R(x)-e^{sA}R(x^{\prime})\big{|}_{L_{2}(\Xi,H)}\ \leq\
\frac{L}{s^{\gamma}}|x-x^{\prime}|_{H},\qquad\hbox{ for all $s\in(0,1)$.}$

###### Hypothesis 3.5

$G\in{L}(\Xi;K)$.

###### Hypothesis 3.6

The functions $l:H\times K\times U\rightarrow\mathbb{R}$ and
$h:H\rightarrow\mathbb{R}$ are measurable and satisfy the assumptions below,
moreover

$|l(x,q,u)-l(x^{\prime},q^{\prime},u)|\leq
L(|x-x^{\prime}|+|q-q^{\prime}|),\qquad\qquad\forall\,q,q^{\prime}\in
K,x,x^{\prime}\in H,\,u\in U,$ $|h(x)-h(x^{\prime})|\leq
L|x-x^{\prime}|,\qquad\qquad\forall x,x^{\prime}\in H,$
$|l(x,q,u)|,|\rho(u)|,|h(x)|\leq M,\qquad\qquad\forall q\in K,x\in H,\,u\in
U.$

We are interested in studying the limit of the value function
$V^{\varepsilon}(x_{0},q_{0})$,

${V}^{\varepsilon}(x_{0},q_{0})=:\inf_{\mathbb{U}\in\mathbb{S}^{1,2}}J^{\varepsilon}(x_{0},q_{0},\mathbb{U})$
(3.3)

as the ratio $\varepsilon$ between the speed of the slow component and the
speed of the fast one tends to $0$. Namely we shall provide a representation
of this limit by a  reduced  stochastic control problem.

##  4 Small noise approximations of the two scale problem

In order to regularize our initial problem we introduce a vanishing noisy term
in (3.1). To do that we have to modify our class of settings.

Namely we denote by $\mathbb{S}^{1,2,B}$ the class of $7$-uples
$\mathbb{U}^{B}=(\Omega,(\mathcal{F}_{t}),\mathbb{P},(W^{1}_{t}),(W^{2}_{t}),(B_{t}),(u_{t}))$,
where, beside the forementioned elements, there is a third
$(\mathcal{F}_{t})$-Wiener process $(B_{t})$, independent of
$(W^{1}_{t},W^{2}_{t})$.

Then given $x_{0},q_{0}$ and a setting $\mathbb{U}^{B}\in\mathbb{S}^{1,2,B}$,
for every $\eta\geq 0$, let us consider the following regularized two scale
state equation:

$\begin{cases}dX_{t}=AX_{t}\,dt+b(X_{t},Q_{t},u_{t})dt\,+R(X_{t})dW^{1}_{t}+\eta\,dB_{t},&X_{0}=x_{0},\\\
\\\ \varepsilon
dQ_{t}=(BQ_{t}+F(X_{t},Q_{t})+G\rho(u_{t}))\;dt+\varepsilon^{1/2}G\,dW^{2}_{t},&Q_{0}=q_{0}.\end{cases}$
(4.1)

Such system has a unique mild solution, indeed following [10, Lemma 3.9 and
Lemma 3.10] we have that for every $\varepsilon>0,\eta>0$ there exists a
unique couple of processes
$(X^{\varepsilon,\eta,\mathbb{U}^{B}},Q^{\varepsilon,\eta,\mathbb{U}^{B}})$,
with $X^{\varepsilon,\eta,\mathbb{U}^{B}}$ belonging to
$L^{p}_{\mathcal{F}^{\mathbb{U}^{B}}}(\Omega^{\mathbb{U}^{B}};C([0,1];H))$ and
$Q^{\varepsilon,\eta,\mathbb{U}^{B}}$ in
$L^{p}_{\mathcal{F}^{\mathbb{U}^{B}}}(\Omega^{\mathbb{U}^{B}};C([0,1];K))$ .

Moreover, see again [10, Lemma 3.9 and Lemma 3.10], also see the proof of
Theorem 4.13 below, the following estimates hold:

$\mathbb{E}^{\mathbb{U}^{B}}(\sup_{t\in[0,1]}|X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}|^{p})\leq
c_{p}(1+|x_{0}|^{p}),\qquad x_{0}\in H,\qquad\forall p\geq 1$ (4.2)
$\sup_{t\in[0,1]}\mathbb{E}^{\mathbb{U}^{B}}|{Q}^{\varepsilon,\eta,\mathbb{U}^{B}}_{s}|^{p}\leq
k_{p}(1+|q_{0}|^{p}),\qquad q_{0}\in K,\qquad\forall p\geq 1.$ (4.3)

with constant $c_{p}$ and $k_{p}$ independent from $\varepsilon$ and $\eta$.

We also consider the analogue of our control problem in this enriched and
regularized situation and the corresponding value function
$V^{\varepsilon,\eta}$. Namely:

$\begin{split}&J^{\varepsilon,\eta}(x_{0},q_{0},\mathbb{U}^{B})=\mathbb{E}^{\mathbb{U}^{B}}\left[\int_{0}^{1}l(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},u^{\mathbb{U}^{B}}_{t})dt+h(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{1})\right].\\\
&V^{\varepsilon,\eta}(x_{0},q_{0})=\inf_{\mathbb{U}^{B}\in\mathbb{S}^{1,2,B}}J^{\varepsilon,\eta}(x_{0},q_{0},\mathbb{U}^{B})\end{split}$
(4.4)

It is straightforward to verify that, fixed $(x_{0},p_{0})$, the set
$\\{V^{\varepsilon,\eta}(x_{0},q_{0}):\varepsilon>0,\eta>0\\}$ is bounded.

###### Remark 4.1

Given a setting $\mathbb{U}^{B}$ in $\mathbb{S}^{1,2,B}$ we define the setting
$P\mathbb{U}^{B}$ in $\mathbb{S}^{1,2}$ as the setting obtained by omitting
the process $B$. Namely

$P(\Omega,(\mathcal{F}_{t}),\mathbb{P},(W^{1}_{t}),(W^{2}_{t}),(B_{t}),(u_{t}))=(\Omega,(\mathcal{F}_{t}),\mathbb{P},(W^{1}_{t}),(W^{2}_{t}),(u_{t}))$

In particular the control $u$ is the same in $\mathbb{U}^{B}$ and
$P\mathbb{U}^{B}$, moreover
$(X^{\varepsilon,0,\mathbb{U}^{B}},Q^{\varepsilon,0,\mathbb{U}^{B}})\equiv(X^{\varepsilon,P\mathbb{U}^{B}},Q^{\varepsilon,P\mathbb{U}^{B}}).$
On the other hand given $\mathbb{U}\in\mathbb{S}^{1,2}$ we define by
$R\mathbb{U}\subset\mathbb{S}^{1,2,B}$ the set, always non empty, of all
settings $\mathbb{U}^{B}$ obtained from $\mathbb{U}$, by choosing any
$(\Omega^{0},\mathcal{F}^{0},\mathbb{P}^{0},(\mathcal{F}^{0}_{t}),B)$, where
$(\Omega^{0},\mathcal{F}^{0},\mathbb{P}^{0},(\mathcal{F}^{0}_{t}))$, is a
filtered complete probability space and $(B)$ is an
$(\mathcal{F}^{0}_{t})$-Wiener process, and setting

$U^{B}=(\Omega\times\Omega^{0},(\mathcal{F}\otimes\mathcal{F}^{0}),(\mathcal{F}_{t}\otimes\mathcal{F}^{0}_{t}),\mathbb{P}\otimes\mathbb{P}^{0},(W^{\prime
1}_{t}),(W^{\prime 2}_{t}),(B^{\prime}_{t}),(u^{\prime}_{t})).$

In the above formula ${W}^{\prime i}(\omega,\omega^{0}):=W^{i}(\omega)$,
$i=1,2$; $u^{\prime}(\omega,\omega^{0})=u(\omega)$,
$B^{\prime}(\omega,\omega^{0}):=B(\omega^{0})$, for every
$(\omega,\omega^{0})\in\Omega\times\Omega^{0}$. We notice that if
$\mathbb{U}^{B}\in R\mathbb{U}$ then the law of
$(X^{\varepsilon,\mathbb{U}},Q^{\varepsilon,\mathbb{U}},u^{\mathbb{U}})$ under
$\mathbb{P}^{\mathbb{U}}$ coincides with the law of
$(X^{\varepsilon,0,\mathbb{U}^{B}},Q^{\varepsilon,0,\mathbb{U}^{B}},u^{\mathbb{U}^{B}})$
under $\mathbb{P}^{\mathbb{U}^{B}}$.

Thus
$J^{\varepsilon,0}(x_{0},q_{0},\mathbb{U}^{B})=J^{\varepsilon}(x_{0},q_{0},\mathbb{U})$,
for every $\mathbb{U}\in\mathbb{S}^{1,2}$ and $\mathbb{U}^{B}\in
R\mathbb{U}\subset\mathbb{S}^{1,2,B}$.

The above remark entitles us to replace our original control problem with the
enriched one in the trivial case $\eta=0$. Namely we have the not very
surprising equality:

###### Lemma 4.2

$V^{\varepsilon,0}(x_{0},q_{0})={V}^{\varepsilon}(x_{0},q_{0}),\qquad\forall
x_{0}\in H,\;q_{0}\in K.$

Proof. Thanks to the previous remark, for every
$\mathbb{U}^{B}\in\mathbb{S}^{1,2,B}$ we have that
$J^{\varepsilon}(x_{0},q_{0},P\mathbb{U}^{B})=J^{\varepsilon,0}(x_{0},q_{0},\mathbb{U}^{B})$.
Conversely if $\mathbb{U}\in\mathbb{S}^{1,2}$ and $\mathbb{U}^{B}\in
R\mathbb{U}$ we again have that
$J^{\varepsilon}(x_{0},q_{0},\mathbb{U})=J^{\varepsilon}(x_{0},q_{0},P\mathbb{U}^{B})=J^{\varepsilon,0}(x_{0},q_{0},\mathbb{U}^{B})$.
Thus the sets
$\\{J^{\varepsilon}(x_{0},q_{0},\mathbb{U}):\mathbb{U}\in\mathbb{S}^{1,2}\\}$
and
$\\{J^{\varepsilon,0}(x_{0},q_{0},\mathbb{U}^{B}):\mathbb{U}^{B}\in\mathbb{S}^{1,2,B}\\}$
coincide and obviously the same is true for their infimum.

Now we fix a reference setting
$(\bar{\Omega},\bar{\mathcal{F}},\bar{\mathbb{P}},(\bar{\mathcal{F}_{t}}),\bar{W}^{1},\bar{W}^{2},\bar{B})$,
we denote with
$\bar{\mathcal{F}}^{1,2,B}_{t},\bar{\mathcal{F}}^{1,B}_{t},\bar{\mathcal{F}}^{2}_{t}$
the natural filtration generated, respectively, by the processes
$(\bar{W}^{1},\bar{W}^{2},\bar{B}),(\bar{W}^{1},\bar{B})$, and $\bar{W}^{2}$
(always completed with the $\bar{\mathbb{P}}$ -negligible sets in
$\bar{\mathcal{F}}$), eventually by $\bar{\mathbb{E}}$ we will denote the
expectation with respect to $\bar{\mathbb{P}}$.

We then consider the following system:

$\begin{cases}dX_{t}=AX_{t}\,dt+R(X_{t})\,d\bar{W}^{1}_{t}+\eta\,d\bar{B}_{t},&X_{0}=x_{0},\\\
\\\ \varepsilon
dQ_{t}=(BQ_{t}+F(X_{t},Q_{t})dt+\varepsilon^{1/2}G\,d\bar{W}^{2}_{t},&Q^{\varepsilon}_{0}=q_{0}.\end{cases}$
(4.5)

That has a unique mild solution $(X^{\eta},Q^{\varepsilon,\eta})$ with
$X^{\eta}\\!\in L^{p}_{\bar{\mathcal{F}}^{1,B}}(\Omega;C([0,1];H))$,
$Q^{\varepsilon,\eta}\\!\in
L^{p}_{\bar{\mathcal{F}}^{1,2,B}}(\Omega;C([0,1];K))$, $p\geq 1$.

We introduce here the BSDE:

$\begin{cases}-dY_{t}=\psi^{\epsilon,\eta}(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t},{Z}^{1}_{t},{Z}^{2}_{t},\Xi_{t})dt-{Z}^{1}_{t}d\bar{W}^{1}_{t}-{Z}^{2}_{t}d\bar{B}_{t}-{\Xi}_{t}d\bar{W}^{2}_{t},\\\
\quad Y_{1}\ =h(X_{1}).\end{cases}$ (4.6)

with

$\displaystyle\psi^{\varepsilon,\eta}(x,q,z_{2},v):=\inf_{\\{u\in{U}\\}}\\{l(x,q,u)+\frac{z_{2}b(x,q,u)}{\eta}+\frac{v}{\sqrt{\varepsilon}}\rho(u)\\}=\psi(x,q,\frac{z_{2}}{\eta},\frac{v}{\sqrt{\varepsilon}}),$

where
$\displaystyle\psi(x,q,z_{2},v)=\inf_{u\in{U}}\\{l(x,q,u)+z_{2}b(x,q,u)+v\rho(u)\\}$
for every $x,z_{2}\in H,q\in K,v\in\Xi$.

We notice that $\psi$ is Lipschitz in $z_{2}$ and $v$ uniformly with respect
to $x$ and $q$, moreover it is Lipschitz with respect to $x$ and $q$ with
Lipschitz constant linearly growing in $|z_{2}|$.

By standard BSDE theory (see for instance [8]) equation (4.6) has a unique
solution
$(Y^{\varepsilon,\eta},Z^{1,\varepsilon,\eta},Z^{2,\varepsilon,\eta},\Xi^{\varepsilon,\eta})$
with $Y^{\varepsilon,\eta}\\!\in
L^{2}_{\bar{\mathcal{F}}^{1,2,B}}(\Omega;C([0,1];\mathbb{R}))$,
$Z^{1,\varepsilon,\eta}\\!\in
L^{2}_{\bar{\mathcal{F}}^{1,2,B}}(\Omega\times[0,1];\Xi^{*})$,
$Z^{2,\varepsilon,\eta}\\!\in
L^{2}_{\bar{\mathcal{F}}^{1,2,B}}(\Omega\times[0,1];H^{*})$ and
$\Xi^{\varepsilon,\eta}\\!\in
L^{2}_{\bar{\mathcal{F}}^{1,2,B}}(\Omega\times[0,1];\Xi^{*})$.

We introduce the space of $U$-valued processes $\mathcal{U}^{1,2,B}$ being
progressively measurable w.r.t. $\bar{\mathcal{F}}_{t}^{1,2,B}$.

The following identification is a standard result in BSDE theory we report the
proof in order to take into account the two different (weak) formulations of
the control problem that will be needed below.

###### Lemma 4.3

It holds that:

$V^{\varepsilon,\eta}(x_{0},q_{0})=\inf_{u\in\mathcal{U}^{1,2,B}}\mathbb{E}^{u}\Big{(}\int_{0}^{1}l(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t}))\,dt+h(X^{\eta}_{1})\Big{)}=Y^{\varepsilon,\eta}_{0}.$
(4.7)

where $\mathbb{E}^{u}$ denotes expectation with respect to the probability
$\mathbb{P}^{u}$ on $(\bar{\Omega},\bar{\mathcal{F}})$ under which

$(\bar{W}^{1}_{t},-\int_{0}^{1}\varepsilon^{-1/2}\rho(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t})\,dt+\bar{W}^{2}_{t},-\int_{0}^{1}\eta^{-1}b(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t},u_{t})\,dt+\bar{B}_{t})$

is a Wiener process on $\Xi\times\Xi\times H$.

Proof. We begin noticing that, given $u\in\mathcal{U}^{1,2,B}$, we can,
starting from the reference setting, build the following setting
$\bar{\mathbb{U}}^{B}\in\mathbb{S}^{1,2,B}$ by:

$\bar{\mathbb{U}}^{B}:=(\bar{\Omega},\bar{\mathcal{F}},{\mathbb{P}^{u}},({\bar{\mathcal{F}}^{1,2,B}_{t}}),\bar{W}^{1}_{t},-\int_{0}^{1}\varepsilon^{-1/2}\rho(u_{t})\,dt+\bar{W}^{2}_{t},-\int_{0}^{1}\eta^{-1}b(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t},u_{t})\,dt+\bar{B}_{t},u_{t}).$

$\bar{\mathbb{U}}^{B}$ has been constucted in such a way that the
corresponding solution to (4.1) $X^{\varepsilon,\eta,{\mathbb{U}}^{B}}$
coincides with the solution $X^{\eta}$ of (4.5). We then have that

$J^{\varepsilon,\eta}(x_{0},q_{0},\bar{\mathbb{U}}^{B})=\mathbb{E}^{u}\Big{(}\int_{0}^{1}l(X^{\eta}_{s},Q^{\varepsilon,\eta}_{s})\,ds+h(X^{\eta}_{1})\Big{)}.$

So, by definition, we have that:

$V^{\varepsilon,\eta}(x_{0},q_{0})\leq\inf_{u\in\mathcal{U}^{1,2,B}}\mathbb{E}^{u}\Big{(}\int_{0}^{1}l(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t})\,dt+h(X^{\eta}_{1})\Big{)}.$
(4.8)

To prove the converse we add and subtract the terms
$\displaystyle\int_{0}^{1}l(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t},u_{t})\,dt$
to system (4.5) and compute the mean value with respect to $\mathbb{P}^{u}$ to
get the usual fundamental relation:

$\displaystyle Y^{\varepsilon,\eta}_{0}$
$\displaystyle=\mathbb{E}^{u}\Big{[}\int_{0}^{1}(\psi^{\varepsilon,\eta}(X^{\eta}_{t},Q^{\varepsilon,\eta}_{s},Z^{1,\varepsilon,\eta}_{t},Z^{2,\varepsilon,\eta}_{t},\Xi^{\varepsilon,\eta}_{t})-l(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t},u_{t})-\frac{1}{\eta}Z^{2,\varepsilon,\eta}_{t}b(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t},u_{t})-\frac{1}{\sqrt{\varepsilon}}\Xi^{\varepsilon,\eta}_{t}\rho(u_{t}))\,dt\Big{]}$
$\displaystyle+\mathbb{E}^{u}\Big{[}\int_{0}^{1}l(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t},u_{t})\,dt+h(X^{\eta}_{1})\Big{]}$

Choosing, in a measurable way, a minimizing sequence $u^{n}$ such that

$|\psi^{\varepsilon,\eta}(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t},Z^{1,\varepsilon,\eta}_{t},Z^{2,\varepsilon,\eta}_{t},\Xi^{\varepsilon,\eta}_{t})-l(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t},u^{n}_{t})-\frac{1}{\eta}Z^{2,\varepsilon,\eta}_{t}b(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t},u^{n}_{t})-\frac{1}{\sqrt{\varepsilon}}\Xi^{\varepsilon,\eta}_{t}\rho(u^{n}_{t}))|\displaystyle\leq\frac{1}{n}$

(notice that $(X^{\eta},Q^{\varepsilon,\eta})$ do not depend on $u$) we end
up, see for instance [8], with:

$Y^{\varepsilon,\eta}_{0}=\inf_{u\in\mathcal{U}^{1,2,B}}\mathbb{E}^{u}\Big{(}\int_{0}^{1}l(X^{\eta}_{t},Q^{\varepsilon,\eta}_{t}))\,dt+h(X^{\eta}_{1})\Big{)}$
(4.9)

To complete the proof it is enough to show that:

$J^{\varepsilon,\eta}(x_{0},q_{0},\mathbb{U}^{B})\geq
Y^{\varepsilon,\eta}_{0},\quad\text{for all}\
\mathbb{U}^{B}\in\mathbb{S}^{1,2,B}.$

The key point is the well known observation that the law of the solution to
the forward backward system (4.5) does not depend on the specific setting (the
solution is obtained by a Picard iteration argument that conserves the law).
Given
${\mathbb{U}^{B}}=(\Omega^{\mathbb{U}^{B}},(\mathcal{F}^{\mathbb{U}^{B}}_{t}),\mathbb{P}^{\mathbb{U}^{B}},(W^{1,\mathbb{U}^{B}}_{t}),(W^{2,\mathbb{U}^{B}}_{t}),(B^{\mathbb{U}^{B}}_{t}),(u^{\mathbb{U}^{B}}_{t}))\in\mathbb{S}^{1,2,B}$,
we set

$\widetilde{B}_{t}=\eta^{-1}\int_{0}^{t}b(X^{\eta}_{s},Q^{\varepsilon,\eta}_{s},u^{\mathbb{U}^{B}}_{s})\,ds+{B}^{\mathbb{U}^{B}}_{t}\hbox{
and
}\widetilde{W}^{2}=\varepsilon^{-1/2}\int_{0}^{t}\rho(u^{\mathbb{U}^{B}}_{s})\,ds+{W}_{t}^{2,\mathbb{U}^{B}}.$

Hence
$(X^{\varepsilon,\eta,\mathbb{U}^{B}},Q^{\varepsilon,\eta,\mathbb{U}^{B}})$
solves:

$\begin{cases}dX_{t}=AX_{t}\,dt+R(X_{t})d{W}^{1,\mathbb{U}^{B}}_{t}+\eta\,d\widetilde{B}_{t},&X_{0}=x_{0},\\\
\\\ \varepsilon
dQ_{t}=(BQ_{t}+F(X_{t},Q_{t})dt+\varepsilon^{1/2}G\,d\widetilde{W}^{2}_{t},&Q^{\varepsilon}_{0}=q_{0}.\end{cases}$
(4.10)

Now we associate to such forward system the backward equation

$\begin{cases}-dY_{t}=\psi^{\epsilon,\eta}(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},{Z}^{1}_{t},{Z}^{2}_{t},\Xi_{t})dt-{Z}^{1}_{t}d{W}^{1,\mathbb{U}^{B}}_{t}-{Z}^{2}_{t}d\widetilde{B}_{t}-{\Xi}_{t}d\widetilde{W}^{2}_{t},\\\
\quad Y_{1}\ =h(X_{1}).\end{cases}$ (4.11)

Since the law of the solution does not depend on the particular setting, we
get that $Y_{0}=Y^{\varepsilon,\eta}_{0}$, then rewriting (4.11) with respect
to ${B}^{\mathbb{U}^{B}}$ and ${W}^{2,\mathbb{U}^{B}}$, computing the
expectation with respect to $\mathbb{P}^{\mathbb{U}^{B}}$ and recalling the
definition of $\psi^{\varepsilon,\eta}$, we get:

$Y_{0}=Y^{\varepsilon,\eta}_{0}\leq\mathbb{E}^{{\mathbb{U}^{B}}}\Big{[}\int_{0}^{1}l(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},u^{\mathbb{U}^{B}}_{t})\,dt+h(X^{\eta,\mathbb{U}^{B}}_{1})\Big{]}=J^{\varepsilon,\eta}(x_{0},q_{0},\mathbb{U}^{B}).$
(4.12)

Thus the proof is completed.

By simple considerations on the control problems we can prove the following
uniform convergence.

###### Theorem 4.4

Under 3.1—3.6 we have that:

$\lim_{\eta\to
0}\sup_{\varepsilon>0}|V^{\varepsilon,\eta}(x_{0},q_{0})-V^{\varepsilon,\eta}(x_{0},q_{0})|=0$
(4.13)

Proof. First of all we recall, see Lemma 4.2, that
$V^{\varepsilon}(x_{0},q_{0})=V^{\varepsilon,0}(x_{0},q_{0})$.

Moreover, by definition,
$V^{\varepsilon}(x_{0},q_{0})=\inf_{\mathbb{U}^{B}\in\mathbb{S}^{1,2,B}}J^{\varepsilon,\eta}(x_{0},q_{0},\mathbb{U}^{B})$,
for every $\eta\geq 0$. Therefore, our claim follows if we prove that:

$\lim_{\eta\to
0}\sup_{\varepsilon>0}\sup_{\mathbb{U}^{B}\in\mathbb{S}^{1,2,B}}|J^{\varepsilon,\eta}(x_{0},q_{0},\mathbb{U}^{B})-J^{\varepsilon,0}(x_{0},q_{0},\mathbb{U}^{B})|=0.$
(4.14)

We fix then $\varepsilon>0$ and
$\mathbb{U}^{B}=(\Omega,(\mathcal{F}_{t}),\mathbb{P},(W^{1}_{t}),(W^{2}_{t}),(B_{t}),(u_{t}))\in\mathbb{S}^{1,2,B}$
and consider:

$\begin{cases}d(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}-X^{\varepsilon,0,\mathbb{U}^{B}}_{t})=A(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}-X^{\varepsilon,0,\mathbb{U}^{B}}_{t})\,dt+[b(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},u_{t})-b(X^{\varepsilon,0,\mathbb{U}^{B}}_{t},Q^{\varepsilon,0,\mathbb{U}^{B}}_{t},u_{t})]dt&\\\
\qquad\qquad\qquad\qquad+[R(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t})-R(X^{\varepsilon,0,\mathbb{U}^{B}}_{t})]dW^{1}_{t}+\eta\,dB_{t},&\\\
X^{\varepsilon,\eta,\mathbb{U}^{B}}_{0}-X^{\varepsilon,0,\mathbb{U}^{B}}_{0}=0,\\\
d(Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}-Q^{\varepsilon,0,\mathbb{U}^{B}}_{t})=\varepsilon^{-1}B(Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}-Q^{\varepsilon,0,\mathbb{U}^{B}}_{t})dt+\varepsilon^{-1}[F(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},u_{t})-F(X^{\varepsilon,0,\mathbb{U}^{B}}_{t},Q^{\varepsilon,0,\mathbb{U}^{B}}_{t},u_{t})]dt,&\\\
Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{0}-Q^{\varepsilon,0,\mathbb{U}^{B}}_{0}=0.&\end{cases}$

Taking into account the second equation we have by Hypothesis 3.3 and standard
estimates (see for instance [15])

$|Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}-Q^{\varepsilon,0,\mathbb{U}^{B}}_{t}|\leq\frac{C}{\varepsilon}\int_{0}^{t}e^{-\displaystyle\frac{\mu}{\varepsilon}(t-s)}|X^{\varepsilon,\eta,\mathbb{U}^{B}}_{s}-X^{\varepsilon,0,\mathbb{U}^{B}}_{s}|\,ds,$
(4.15)

and consequently

$\sup_{s\leq
t}|Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{s}-Q^{\varepsilon,0,\mathbb{U}^{B}}_{s}|\leq\frac{C}{\mu}\sup_{s\leq
t}|X^{\varepsilon,\eta,\mathbb{U}^{B}}_{s}-X^{\varepsilon,0,\mathbb{U}^{B}}_{s}|$
(4.16)

where $C$ is a constant independent of $\epsilon$ and $\eta$ with value that
can change from line to line.

As far as the first equation is concerned we have:

$\displaystyle
X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}-X^{\varepsilon,0,\mathbb{U}^{B}}_{t}=$
$\displaystyle\int_{0}^{t}e^{(t-s)A}[b(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{s},Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{s},u_{s})-b(X^{\varepsilon,0,\mathbb{U}^{B}}_{s},Q^{\varepsilon,0,\mathbb{U}^{B}}_{s},u_{s})]\,ds+\int_{0}^{t}e^{(t-s)A}\eta\,dB_{s}$
$\displaystyle+\int_{0}^{t}e^{(t-s)A}[R(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{s})-R(X^{\varepsilon,0,\mathbb{U}^{B}}_{s})]\,dW^{1}_{s}.$

Thanks to standard estimates and the factorization method, see [9], we get for
$p>\frac{2}{1-2\gamma}$, $\alpha\in(\frac{1}{p},\frac{1}{2}-\gamma)$,
($\gamma\in(0,1/2)$ is the constant appearing in Assumption 3.1 and Assumption
3.4) and any $\rho\in[0,1]$:

$\displaystyle\mathbb{E}^{\mathbb{U}^{B}}\sup_{t\in[0,\rho]}|X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}-X^{\varepsilon,0,\mathbb{U}^{B}}_{t}|^{p}\leq
C\int_{0}^{\rho}[\mathbb{E}^{\mathbb{U}^{B}}\\!\\!\\!\sup_{s\in[0,r]}|X^{\varepsilon,\eta,\mathbb{U}^{B}}_{s}-X^{\varepsilon,0,\mathbb{U}^{B}}_{s}|^{p}+\mathbb{E}^{\mathbb{U}^{B}}|Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{r}-Q^{\varepsilon,0,\mathbb{U}^{B}}_{r}|^{p}]\,dr$
$\displaystyle+\int_{0}^{\rho}\mathbb{E}^{\mathbb{U}^{B}}\Big{[}\int_{0}^{r}(r-l)^{-2(\alpha+\gamma)}|X^{\varepsilon,\eta,\mathbb{U}^{B}}_{l}-X^{\varepsilon,0,\mathbb{U}^{B}}_{l}|^{2}\,dl\Big{]}^{p/2}\,dr+|\eta|^{p}$
$\displaystyle\leq
C\left(1+\left(\int_{0}^{1}\sigma^{-2(\alpha+\gamma)}\,d\sigma\right)^{p/2}\right)\int_{0}^{\rho}\mathbb{E}^{\mathbb{U}^{B}}\\!\\!\\!\sup_{s\in[0,r]}|X^{\varepsilon,\eta,\mathbb{U}^{B}}_{s}-X^{\varepsilon,0,\mathbb{U}^{B}}_{s}|^{p}\,dr+\int_{0}^{\rho}\mathbb{E}^{\mathbb{U}^{B}}|Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{r}-Q^{\varepsilon,0,\mathbb{U}^{B}}_{r}|^{p}\,dr+|\eta|^{p}.$

Recalling (4.16) we also get:

$\mathbb{E}^{\mathbb{U}^{B}}\sup_{t\in[0,\rho]}|X^{\varepsilon,\eta,{\mathbb{U}^{B}}}_{t}-X^{\varepsilon,0,{\mathbb{U}^{B}}}_{t}|^{p}\leq
C\Big{[}\int_{0}^{\rho}\mathbb{E}^{\mathbb{U}^{B}}\sup_{s\in[0,r]}|X^{\varepsilon,\eta,{\mathbb{U}^{B}}}_{s}-X^{\varepsilon,0,{\mathbb{U}^{B}}}_{s}|^{p}\,dr+|\eta|^{p}\Big{]};$

and applying the Gromwall Lemma to
$v(r)=:\mathbb{E}^{\mathbb{U}^{B}}\sup_{s\in[0,r]}|X^{\varepsilon,\eta,\mathbb{U}^{B}}_{s}-X^{\varepsilon,0,\mathbb{U}^{B}}_{s}|^{p}$,
we conclude

$\displaystyle\mathbb{E}^{\mathbb{U}^{B}}\sup_{t\in[0,\rho]}|X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}-X^{\varepsilon,0,\mathbb{U}^{B}}_{t}|^{p}\leq
C|\eta|^{p},\qquad\qquad\forall\varepsilon>0$

and applying once again (4.16)

$\displaystyle\mathbb{E}^{\mathbb{U}^{B}}\sup_{t\in[0,\rho]}|Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}-Q^{\varepsilon,0,\mathbb{U}^{B}}_{t}|^{p}\leq
C|\eta|^{p},\qquad\qquad\forall\varepsilon>0$

Finally if we consider the difference between the value functions:

$\displaystyle|V^{\varepsilon,\eta}(x_{0},q_{0})-V^{\varepsilon,0}(x_{0},q_{0})|$
$\displaystyle\leq\sup_{\mathbb{U}^{B}\in\mathbb{S}^{1,2,B}}\Big{[}\mathbb{E}^{\mathbb{U}^{B}}\int_{0}^{1}|l(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{t},u_{t})-l(X^{\varepsilon,0,\mathbb{U}^{B}}_{t},Q^{\varepsilon,0,\mathbb{U}^{B}}_{t},u_{t})|\,dt$
$\displaystyle\qquad\qquad+\mathbb{E}^{\mathbb{U}^{B}}|h(X^{\varepsilon,\eta,\mathbb{U}^{B}}_{1})-h(X^{\varepsilon,0,\mathbb{U}^{B}}_{1})|\Big{]}$
(4.17) $\displaystyle\leq
C\sup_{\mathbb{U}^{B}\in\mathbb{S}^{1,2,B}}\mathbb{E}^{\mathbb{U}^{B}}\Big{[}\sup_{t\in[0,1]}|X^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}-X^{\varepsilon,0,\mathbb{U}^{B}}_{t}|+\int_{0}^{1}|Q^{\varepsilon,\eta,\mathbb{U}^{B}}_{t}-Q^{\varepsilon,0,\mathbb{U}^{B}}_{t}|\,dt\Big{]}$
$\displaystyle\leq C|\eta|,\qquad\forall\varepsilon>0.$

Thus our claim holds.

###### Theorem 4.5

For every fixed $x\in H$ and $z\in H^{*}$, let us consider the following
ergodic control problem with state equation in $K$, driven by an arbitrary
$\Xi$-valued cylindrical Wiener process $(\hat{{W}})$, with control
$\beta:[0,\infty[\times\Omega\rightarrow U$ varying in the set
$\mathcal{U}_{\infty}$ of progressively measurable $U$-valued processes with
respect to the natural filtration of $(\hat{{W}})$.

$d\hat{Q}^{\beta}_{s}=B\hat{Q}^{\beta}_{s}ds+F(x,\hat{Q}^{\beta}_{s})\,ds+G\rho(\beta_{s})ds+Gd\hat{W}^{2}_{s},\qquad\hat{Q}^{\beta}_{0}=0$
(4.18)

and ergodic cost functional:

$\check{J}(x,z,\beta)=\liminf_{\delta\to
0}\hat{\mathbb{E}}\,\delta\int_{0}^{\frac{1}{\delta}}[z\,b(x,\hat{Q}^{\beta}_{s},\beta_{s})+l(x,\hat{Q}^{\beta}_{s},\beta_{s})]ds.$
(4.19)

Let $\lambda(x,z)$ be is the value function of the above ergodic control
problem, that is:

$\lambda(x,z)=\inf_{\beta\in\mathcal{U}^{2}}\check{J}(x,z,\beta).$ (4.20)

Under 3.1—3.6 we have that $\lambda(x,\,\cdot\,)$ is concave moreover:

$\begin{array}[]{l}|\lambda(x,z)-\lambda(x,z^{\prime})|\leq M|z-z^{\prime}|\\\
|\lambda(x,z)-\lambda(x^{\prime},z)|\leq L(1+|z|)|x-x^{\prime}|\\\
|\lambda(x,z)|\leq M(1+|z|)\end{array}$ (4.21)

Moreover for every $\eta>0$:

$\lim_{\varepsilon\to 0}Y^{\varepsilon,\eta}_{0}=\lim_{\varepsilon\to
0}V^{\varepsilon,\eta}(x_{0},q_{0})=Y^{\eta}_{0}$ (4.22)

where $Y^{\eta}_{0}$ is defined as part of the solution to the reduced BSDE
(for the definition of $X^{\eta}$ see (4.5)).

$\begin{cases}-dY_{t}=\lambda({X}^{\eta}_{t},\eta^{-1}{Z}^{2}_{t})\,dt-\,{Z}^{1}_{t}\,d\bar{W}^{1}_{t}-\,{Z}^{2}_{t}\,d\bar{B}_{t},\\\
\quad{Y}_{1}=h(X^{\eta}_{1}).\end{cases}$ (4.23)

where $Y^{\eta}\in
L^{2}_{\bar{\mathcal{F}}^{1,B}}(\Omega;C([0,1];\mathbb{R}))$, $Z^{1,\eta}\in
L^{2}_{\bar{\mathcal{F}}^{1,B}}(\Omega\times[0,1];\Xi^{*})$ and $Z^{2,\eta}\in
L^{2}_{\bar{\mathcal{F}}^{1,B}}(\Omega\times[0,1];H^{*})$.

Proof: The proof follows the one in [10, Theorem 5.4]. The only point to check
is the discretization procedure of the forward component $X^{\eta}$ since now
in the limit equation (4.23) a multiplicative noise appears. As in [10,
Theorem 5.4] we introduce, for all $N\in\mathbb{N}$ a partition $\pi$ of
$[0,1]$, $\pi=\\{\frac{k}{2^{N}}:k:0,\dots,2^{N}-1\\}$ and we define a
sequence $X^{\eta,N}$

$X^{\eta,N}_{t}=\sum_{k=0}^{2^{N}-1}X^{\eta}_{\frac{k}{2^{N}}}I_{[\frac{k}{2^{N}},\frac{k+1}{2^{N}}[}(t)+X^{\eta}_{1}\delta_{1}(t),\qquad
t\in[0,1]$ (4.24)

We need to arrive to the following

$\lim_{N\to\infty}\bar{\mathbb{E}}\sup_{t\in[0,1]}|X^{\eta,N}_{t}-X^{\eta}_{t}|^{4}=0,$
(4.25)

in order to exploit the procedure of [10, Theorem 5.4].

First of all, using the factorization method, see [8] and [9], we can find,
for any $p>\frac{2}{1-2\gamma}$ (and consequently for any $p\geq 1$), a
constant $C_{p}$ independent of $\eta$ such that:

$\bar{\mathbb{E}}\sup_{t\in[0,1]}|X^{\eta}_{t}|^{p}\leq
C_{p}(|x_{0}|^{p}+|\eta|^{p}).$ (4.26)

For $t\in[\frac{k}{2^{N}},\frac{k+1}{2^{N}}[$ we evaluate the difference:

$\displaystyle
X^{\eta,N}_{t}-X^{\eta}_{t}=\int_{\frac{k}{2^{N}}}^{t}e^{(t-s)A}R(X^{\eta}_{s})\,d\bar{W}^{1}_{s}+\int_{\frac{k}{2^{N}}}^{t}e^{(t-s)A}\eta\,d\bar{B}_{s}.$
(4.27)

again using the factorization method [9] for any $p>\frac{2}{1-2\gamma}$ any
$\alpha\in(\frac{1}{p},\frac{1}{2}-\gamma)$ we can find a constant $M$, that
depends on $\alpha$ but not on $N$ such that, for $q=\frac{p}{p-1}$, we get:

$\displaystyle\bar{\mathbb{E}}\sup_{t\in[\frac{k}{2^{N}},\frac{k+1}{2^{N}}]}|X^{N,\eta}_{t}-X^{\eta}_{t}|^{p}\leq$
$\displaystyle
M\Big{(}\int_{0}^{\frac{1}{2^{N}}}s^{{q}(\alpha-1)}\,dr\Big{)}^{p/q}\Big{(}\int_{0}^{1}(s-r)^{-2(\alpha+\gamma)}\,dr\Big{)}^{p/2}\int_{0}^{1}(1+\bar{\mathbb{E}}\sup_{t\in[0,1]}|X^{\eta}_{t}|^{p}+|\eta|^{p})\,dt,\qquad\forall
k\in\\{0,\cdots,2^{N}-1\\}$ (4.28)

For the reader convenience we write some details regarding the first term at
the R.H.S. in (4.27).

$\displaystyle\int_{\frac{k}{2^{N}}}^{t}e^{(t-s)A}R(X^{\eta}_{s})\,d\bar{W}^{1}_{s}$
$\displaystyle=\frac{1}{B(\alpha,1-\alpha)}\int_{\frac{k}{2^{N}}}^{t}e^{(t-r)A}(t-r)^{1-\alpha}\int_{\frac{k}{2^{N}}}^{r}e^{(r-s)A}(r-s)^{-\alpha}R(X^{\eta}_{s})\,d\bar{W}^{1}_{s}\,dr$
$\displaystyle=\frac{1}{B(\alpha,1-\alpha)}\int_{\frac{k}{2^{N}}}^{t}e^{(t-r)A}(t-r)^{1-\alpha}Y(r)\,dr$

where by $B(\alpha,1-\alpha)$ we denote the normalization constant of the beta
distribution and by $Y(r)$ the random variable
$Y(r):=\int_{\frac{k}{2^{N}}}^{r}e^{(r-s)A}(r-s)^{-\alpha}R(X^{\eta}_{s})\,dW^{1}_{s}$.

Thus for any $p>\frac{2}{1-2\gamma}$ and
$\frac{1}{2}-\gamma>\alpha>\frac{1}{p}$

$\displaystyle\bar{\mathbb{E}}\sup_{t\in[\frac{k}{2^{N}},\frac{k+1}{2^{N}}]}\Big{|}\int_{\frac{k}{2^{N}}}^{t}e^{(t-s)A}R(X^{\eta}_{s})\,dW^{1}_{s}\Big{|}^{p}\leq{\Big{(}\frac{M_{A}e^{\omega_{A}}}{B(\alpha,1-\alpha)}\Big{)}^{p}}\Big{(}\int_{\frac{k}{2^{N}}}^{\frac{k+1}{2^{N}}}(t-r)^{(1-\alpha)q}\,dr\Big{)}^{p/q}\bar{\mathbb{E}}\int_{\frac{k}{2^{N}}}^{\frac{k+1}{2^{N}}}|Y(r)|^{p}\,dr$
$\displaystyle\leq{C}\Big{(}\int_{0}^{\frac{1}{2^{N}}}s^{{q}(\alpha-1)}\,dr\Big{)}^{p/q}\int_{\frac{k}{2^{N}}}^{\frac{k+1}{2^{N}}}\bar{\mathbb{E}}\left|\int_{\frac{k}{2^{N}}}^{r}e^{(r-s)A}(r-s)^{-\alpha}R(X^{\eta}_{s})\,dW^{1}_{s}\right|^{p}\,dr$
$\displaystyle\leq{C}\Big{(}\int_{0}^{\frac{1}{2^{N}}}s^{{q}(\alpha-1)}\,dr\Big{)}^{p/q}\int_{\frac{k}{2^{N}}}^{\frac{k+1}{2^{N}}}\Big{(}\int_{\frac{k}{2^{N}}}^{r}(r-s)^{-2\alpha-2\gamma}\bar{\mathbb{E}}|R(X^{\eta}_{s})|^{2}_{L(H)}\,ds\Big{)}^{p/2}\,dr$
$\displaystyle\leq
C\Big{(}\int_{0}^{\frac{1}{2^{N}}}s^{{q}(\alpha-1)}\,dr\Big{)}^{p/q}\Big{(}\int_{0}^{1}\sigma^{-2\alpha-2\gamma}\,d\sigma\Big{)}^{p/2}\Big{(}\int_{0}^{1}\bar{\mathbb{E}}\sup_{r\in[0,1]}(1+|X^{\eta}_{r}|^{p})\,dr\Big{)}$

Where ${C}$ may change from line to line but is always independent of $N$.
Thus thanks to (4.26) and (4) we get the thesis.  .

##  5 Interchanging limits

We now prove general result allowing us to interchange the limit with respect
to $\varepsilon$ and the one with respect to $\eta$ as well.

###### Theorem 5.1

Let $v^{\eta}(x_{0})=Y^{\eta}_{0}$ (see (4.23) for the definition of
$Y^{\eta}$) then, for all $x_{0}\in H$ and $q_{0}\in K$, it holds:

$\lim_{\varepsilon\rightarrow
0}V^{\varepsilon}(x_{0},q_{0})=\lim_{\eta\rightarrow
0}v^{\eta}(x_{0}):=V(x_{0}).$

Moreover $V^{\varepsilon}$ is Lipschitz uniformly with respect to
$\varepsilon$, $v^{\eta}$ is Lipschitz uniformly with respect to $\eta$ and
$V$ is Lipschitz.

Proof.  Since, fixed $x_{0}\in H$ the sequence $v^{\eta}(x_{0})$ is bounded
(see (4.22) and (4.4)) then there exists a sequence $\eta_{n}\searrow 0$
(depending on $x_{0}$ but we omit this information in the notation since it is
not relevant here) such that the sequence $v^{\eta_{n}}(x_{0})$ converges to a
limit that we denote by $V(x_{0})$. By standard adding and subtracting

$|V^{\varepsilon,0}(x_{0},q_{0})-V(x_{0})|\leq|V^{\varepsilon,0}(x_{0},q_{0})-V^{\varepsilon,\eta_{n}}(x_{0},q_{0})|+|V^{\varepsilon,\eta_{n}}(x_{0},q_{0})-v^{\eta_{n}}(x_{0})|+|v^{\eta_{n}}(x_{0})-V(x_{0})|$
(5.1)

Fix $\delta>0$. By (4.13) there exists $n_{\delta}$ such that

$|V^{\varepsilon,\eta_{n}}(x_{0},q_{0})-V^{\varepsilon,0}(x_{0},q_{0})|+|v^{\eta_{n}}(x_{0})-V(x_{0})|\leq\delta\quad\forall\varepsilon>0,\;\forall
n\geq n_{\delta}$

We fix an arbitrary $\bar{n}\geq n_{\delta}$ and notice that by Theorem 4.5
there exists $\varepsilon_{\delta}>0$ such that

$|V^{\varepsilon,\eta_{\bar{n}}}(x_{0}.q_{0})-v^{\eta_{\bar{n}}}(x_{0})|<\delta,\qquad\forall\varepsilon\in(0,\varepsilon_{\delta}).$

It is then straightforward to conclude:

$\lim_{\varepsilon\rightarrow 0}V^{\varepsilon}(x_{0}.q_{0})=V(x_{0})$

Moreover, by the same argument, from any sequence $\hat{\eta}_{n}\searrow 0$
we can extract a subsequence $\hat{\eta}_{n_{k}}\searrow 0$ such that

$\lim_{k\rightarrow\infty}v^{\hat{\eta}_{n_{k}}}(x_{0})=\lim_{\varepsilon\rightarrow
0}V^{\varepsilon}(x_{0},q_{0})=V(x_{0})$

and this implies that

$\lim_{\eta\rightarrow 0}v^{\eta}(x_{0})=\lim_{\varepsilon\rightarrow
0}V^{\varepsilon}(x_{0},q_{0})=V(x_{0}).$

It lasts to show that $V$ is Lipschitz continuous. Clearly it is enough to
show that $V^{\varepsilon,\eta}$ is Lipschitz (with respect to $x_{0}$)
uniformly in $\varepsilon>0$ and $\eta>0$.

First we notice that, by the definition of $V^{\varepsilon,\eta}(x,q)$ (we
here indicate dependence of the solution of equation 3.1 on initial data
$(x,q)$ and $(x^{\prime},q^{\prime})$) and hypothesis (3.6) :

$\displaystyle|V^{\varepsilon,\eta}(x,q)-V^{\varepsilon,\eta}(x^{\prime},q^{\prime})|\leq\sup_{\mathbb{U}^{B}\in\mathbb{S}^{1,2,B}}\Big{|}\mathbb{E}^{\mathbb{U}^{B}}\int_{0}^{1}l(X^{\varepsilon,\eta,\mathbb{U}^{B},x,q}_{s},Q^{\varepsilon,\eta,\mathbb{U}^{B},x,q}_{s},u_{s})-l(X^{\varepsilon,\eta,\mathbb{U}^{B},x^{\prime},q^{\prime}}_{s},Q^{\varepsilon,\eta,\mathbb{U}^{B},x^{\prime},q^{\prime}}_{s},u_{s})\,ds\Big{|}$
$\displaystyle+|\mathbb{E}^{\mathbb{U}^{B}}[h(X^{\varepsilon,\eta,\mathbb{U}^{B},x,q}_{1})-h(X^{\varepsilon,\eta,\mathbb{U}^{B},x^{\prime},q^{\prime}}_{1})]|,$
$\displaystyle\leq
L[\sup_{\mathbb{U}^{B}\in\mathbb{S}^{1,2,B}}\mathbb{E}^{\mathbb{U}^{B}}\sup_{s\in[0,1]}|X^{\varepsilon,\eta,\mathbb{U}^{B},x,q}_{s}-X^{\varepsilon,\eta,\mathbb{U}^{B},x^{\prime},q^{\prime}}_{s}|+\mathbb{E}^{\mathbb{U}^{B}}\int_{0}^{1}|Q^{\varepsilon,\eta,\mathbb{U}^{B},x,q}_{s}-Q^{\varepsilon,\eta,\mathbb{U}^{B},x^{\prime},q^{\prime}}_{s}|\,ds].$

Then, arguing as in Theorem 4.13, we can use the factorization method and the
dissipativity condition to get that $\forall r\in[0,1]$

$\mathbb{E}^{\mathbb{U}^{B}}\sup_{s\in[0,r]}|Q^{\varepsilon,\eta,{\mathbb{U}^{B}},x,q}_{s}-Q^{\varepsilon,\eta,{\mathbb{U}^{B}},x^{\prime},q^{\prime}}_{s}|\leq
C(\mathbb{E}^{\mathbb{U}^{B}}\sup_{s\in[0,r]}|X^{\varepsilon,\eta,{\mathbb{U}^{B}},x,q}_{s}-X^{\varepsilon,\eta,{\mathbb{U}^{B}},x^{\prime},q^{\prime}}_{s}|+|q-q^{\prime}|),$

and consequently that

$\mathbb{E}^{\mathbb{U}^{B}}\sup_{s\in[0,1]}|X^{\varepsilon,\eta,{\mathbb{U}^{B}},x,q}_{s}-X^{\varepsilon,\eta,{\mathbb{U}^{B}},x^{\prime},q^{\prime}}_{s}|\leq
C(|x-x^{\prime}|+|q-q^{\prime}|),$

for some positive constants $C$ independent from $u$ $\varepsilon$ and $\eta$.

##  6 Main characterizations

For $\eta$ fixed we consider the limit system (starting now at an arbitrary
time $s\in[0,1]$ from state $x\in H$ and written with respect to the reference
setting).

$\begin{cases}dX_{t}=AX_{t}\,dt+R(X_{t})\,d\bar{W}^{1}_{t}+\eta\,d\bar{B}_{t},\qquad
t\in[s,1],\\\
-dY_{t}=\lambda({X}_{t},\eta^{-1}{Z}^{2}_{t})\,dt-\,{Z}^{1}_{t}\,d\bar{W}^{1}_{t}+\,{Z}^{2}_{t}\,d\bar{B}_{t},\\\
X_{s}=x,\quad{Y}_{1}=h(X_{1}).\end{cases}$ (6.1)

Again by standard BSDE theory equation (6.1) has a unique solution
$(Y^{\eta,s,x},Z^{1,\eta,s,x},Z^{2,\eta,s,x},\Xi^{\eta,s,x})$ with
$Y^{\eta,s,x}\\!\in
L^{2}_{\bar{\mathcal{F}}^{1,B}}(\Omega;C([s,1];\mathbb{R}))$,
$Z^{1,\eta,s,x}\\!\in
L^{2}_{\bar{\mathcal{F}}^{1,B}}(\Omega\times[s,1];\Xi^{*})$,
$Z^{2,\eta,s,x}\\!\in
L^{2}_{\bar{\mathcal{F}}^{1,B}}(\Omega\times[s,1];H^{*})$ and
$\Xi^{\eta,s,x}\\!\in
L^{2}_{\bar{\mathcal{F}}^{1,B}}(\Omega\times[s,1];\Xi^{*})$, where
$\lambda(x,z)$ is defined in (4.20) (notice that in this section we need to
indicate the dependence on the initial time $s$ and state $x$).

The above system follows within the framework of [7, Theorem 4.1] (indeed
assumptions 2.1, 3.1-3.7 and 4.1 are verified and hypothesis 3.4 can be
relaxed as pointed out in [7, pag. 443]). Thus if we denote with
$v^{\eta}(s,x)=Y^{\eta,s,x}_{s}$, we have that

$(Z^{1,\eta,s,x}_{t},Z^{2,\eta,s,x}_{t})=\left(\nabla_{x}v^{\eta}(t,X^{\eta,s,x}_{t})R(X^{\eta,s,x}_{t}),\eta\nabla_{x}v^{\eta}(t,X^{\eta,s,x}_{t})\right)$

In particular $Z^{2,\eta,s,x}_{t}=\eta\nabla_{x}v^{\eta}(t,X^{\eta,s,x}_{t})$,
$\mathbb{P}\times ds$ -almost surely.

Taking into account the representation in Lemma 4.3 (with initial time $s$
instead of 0) and proceeding as in the proof of Theorem 5.1, we get that
$v^{\eta}(s,\cdot)$ is Lipschitz uniformly with respect to $s\in[0,1]$ and
$\eta>0$. Thus (see also [7, Theorem 4.1] ) we have that:

$\displaystyle\Big{|}{Z^{2,\eta,s,x}_{t}}\Big{|}_{H^{*}}\leq a\eta,\qquad
d\mathbb{P}\times dt-a.e..$ (6.2)

where $a\in\mathbb{R}^{+}$ is independent on $\eta$, $s$ and $x$.

We now introduce the following function, $\tilde{\lambda}$ defined, for a
constant $k>M$ large enough, by:

$\tilde{\lambda}(x,z):=\lambda(x,z)\wedge[-(M+1)|z|+\kappa]$ (6.3)

By (4.21) we get

$\tilde{\lambda}(x,z):=\begin{cases}\lambda(x,z),&\hbox{ if
}|z|\leq\displaystyle\frac{\kappa-M}{2M+1},\\\ \kappa-(M+1)|z|,&\hbox{ if
}|z|\geq\displaystyle k+M.\end{cases}$

Choosing $k$ large enough we can assume that $(k-M)/(2M+1)>a$ so that
$\tilde{\lambda}(x,z)=\lambda(x,z)$ when $|z|\leq a$. Moreover
$\tilde{\lambda}$ remains concave being the minimum of concave functions and,
by (4.21), is Lipschitz with respect to $z$, uniformly with respect to $x$
with Lipschitz constant $M+1$.

Finally we have:

$|\tilde{\lambda}(x,z)-\tilde{\lambda}(x^{\prime},z)|\;\begin{cases}=|\lambda(x,z)-\lambda(x^{\prime},z)|,&|z|\leq
a,\\\ \leq|\lambda(x,z)-\lambda(x^{\prime},z)|,&a<|z|<\kappa+M\\\
=0,&|z|\geq\kappa+M.\end{cases}$

therefore, thanks again to (4.21), $\tilde{\lambda}$ is Lipschitz continuous
with respect to $x$, uniformly w.r.t. $z$, with Lipschitz constant equal to
$\tilde{L}=L(1+\kappa+M)$.

Taking into account (6.2), system (6.1) can be written, replacing $\lambda$ by
$\tilde{\lambda}$ and choosing $s=0$, as follows:

$\begin{cases}dX_{t}=AX_{t}\,dt+R(X_{t})\,d\bar{W}^{1}_{t}+\eta\,d\bar{B}_{t},\qquad
t\in[0,1],\\\
-dY_{t}=\tilde{\lambda}({X}_{t},\eta^{-1}{Z}^{2}_{t})\,dt-\,{Z}^{1}_{t}\,d\bar{W}^{1}_{t}-\,{Z}^{2}_{t}\,d\bar{B}_{t},\\\
X_{0}=x,\quad{Y}_{1}=h(X_{1}).\end{cases}$ (6.4)

We denote by $\tilde{\lambda}_{*}$ the Legendre transform of
$\tilde{\lambda}$, that is, for $x$ and $\alpha$ in $H$ (recall that
$\tilde{\lambda}$ is concave, this justifies the negative signs):

$\tilde{\lambda}_{*}(x,\alpha):=\inf_{z\in
H^{*}}\\{-z\alpha-\tilde{\lambda}{(x,z)}\\}$ (6.5)

It turns out that $\tilde{\lambda}_{*}$ is Lipschitz continuous with respect
to $x$, uniformly w.r.t. $\alpha$, as well. Indeed:

$|\tilde{\lambda}_{*}(x,\alpha)-\tilde{\lambda}_{*}(x^{\prime},\alpha)|\leq\sup_{z\in
H^{*}}|\tilde{\lambda}(x,z)-\tilde{\lambda}(x^{\prime},z)|\leq\tilde{L}\,|x-x^{\prime}|,\qquad\forall
x,x^{\prime},\alpha\in H.$ (6.6)

Moreover taking into account Lipschitzianity with respect to $z$ of
$\tilde{\lambda}$ we get:

$\tilde{\lambda}_{*}(x,\alpha)=-\infty\;\hbox{ if }|\alpha|>M+1$

That yields the following simplification in the Fenchel duality:

$\tilde{\lambda}(x,z):=\inf_{\alpha\in
H}\\{-z\alpha-\tilde{\lambda}_{*}{(x,\alpha)}\\}=\inf_{\alpha\in
H:|\alpha|\leq M+1}\\{-z\alpha-\tilde{\lambda}_{*}{(x,\alpha)}\\}$ (6.7)

The solution $(Y^{\eta})$ can then be represented by a reduced control problem
that has the needed regularity to eventually allow the passage to the limit as
$\eta\rightarrow 0$ giving the final representation of
$\lim_{\varepsilon\rightarrow 0}V^{\varepsilon}$ .

We denote by $\mathcal{U}^{1,B}_{H}$ the set of all processes
$(\alpha_{t})_{t\in[0,1]}$ taking values in the ball $\\{\alpha\in
H:|\alpha|\leq M+1\\}$ and being progressively measurable with respect to the
filtration $(\bar{\mathcal{F}}^{1,B})$.

###### Lemma 6.1

We have:

$Y^{\eta}_{t}=\inf_{\alpha\in\,\mathcal{U}^{1,B}_{H}}\mathbb{E}^{\alpha}\biggl{(}h(X^{\eta}_{1})-\int_{t}^{1}\tilde{\lambda}_{*}(X^{\eta}_{\ell},{\alpha}_{\ell})d\ell\bigg{|}\mathcal{F}^{1,B}_{t}\biggr{)},$
(6.8)

where ${\mathbb{E}}^{{\alpha}}$ denotes the mean value with respect the
probability $\mathbb{P}^{{\alpha}}$ under which

$\left(\bar{W}^{1}_{t},\displaystyle\int_{0}^{t}\frac{{\alpha}_{\ell}}{\eta}d\ell+\bar{B}_{t}\right):=(\bar{W}^{1}_{t},\bar{B}^{\alpha}_{t})$

is a Wiener process.

Notice that, with respect to $(\bar{W}^{1},\bar{B}^{{\alpha}})$ process
$(X^{\eta})$ solves the controlled stochastic differential equation:

$dX_{t}=AX_{t}dt-\,{\alpha}_{t}dt+R(X_{t})d\bar{W}^{1}_{t}+\eta\bar{B}^{\alpha}_{t},\quad
X_{0}=x.$ (6.9)

Proof: To start with we point out that in the (6.7) the infimum can be
restricted to a bounded subset of $H$, as a consequence the choice of controls
$\alpha$ in (6.9) can be restricted to bounded (by $M+1$) controls and we are
allowed to apply Girsanov transform to see perturbation by $\alpha$ as a
change of probability.

Taking into account (6.7), equation (6.4) evaluated at its solution
$(X^{\eta},Y^{\eta},Z^{1,\eta},Z^{2,\eta})$ yields:

$\begin{array}[]{rcl}{Y}^{\eta}_{t}&=&\displaystyle
h(X^{\eta}_{1})+\int_{t}^{1}\tilde{\lambda}(X^{\eta}_{\ell},\eta^{-1}{Z}_{\ell}^{2,\eta})ds-\int_{t}^{1}{Z}^{1,\eta}_{\ell}d\bar{W}^{1}_{\ell}-\int_{t}^{1}{Z}^{2,\eta}_{\ell}d\bar{B}_{\ell}\\\
&\leq&h(X^{\eta}_{1})\displaystyle-\int_{t}^{1}\left(\eta^{-1}{Z}^{2,\eta}_{\ell}{\alpha}_{\ell}+\tilde{\lambda}_{*}(X^{\eta}_{\ell},{\alpha}_{\ell})\right)d\ell-\int_{t}^{1}Z^{1,\eta}_{\ell}d\bar{W}^{1}_{\ell}-\int_{t}^{1}{Z}^{2,\eta}_{\ell}d\bar{B}_{\ell}.\end{array}$
(6.10)

and by the definition of $(\bar{B}^{\alpha})$:

${Y}^{\eta}_{t}\leq
h(X^{\eta}_{1})\displaystyle-\int_{t}^{1}\tilde{\lambda}_{*}(X^{\eta}_{\ell},{\alpha}_{\ell})d\ell-\int_{t}^{1}Z^{1,\eta}_{\ell}d\bar{W}^{1}_{\ell}-\int_{t}^{1}{Z}^{2,\eta}_{\ell}d\bar{B}^{\alpha}_{\ell},$

which shows that for all $\alpha\in\mathcal{U}_{H}^{1,B}$,

${Y}^{\eta}_{t}\leq\bar{\mathbb{E}}^{\alpha}\biggl{(}h(X^{\eta}_{1})-\int_{t}^{1}\tilde{\lambda}_{*}(X^{\eta}_{\ell},{\alpha}_{\ell})d\ell\bigg{|}\bar{\mathcal{F}}^{1,B}_{t}\biggr{)}.$

Conversely, by measurable selection, we may choose a minimizing sequence of
controls, $(\bar{\alpha}^{n})_{n\in\mathbb{N}}\subset\mathcal{U}^{1,B}$, such
that, for all $\ell\in[0,1]$, $\mathbb{P}$-a.s.:

$-\frac{Z^{2,\eta}_{\ell}}{\eta}\bar{\alpha}_{\ell}^{n}-\tilde{\lambda}_{*}(X^{\eta}_{\ell},\bar{\alpha}_{\ell}^{n})-1/n\leq\tilde{\lambda}\left(X^{\eta}_{\ell},\frac{Z^{2,\eta}_{\ell}}{\eta}\right).$
(6.11)

Proceeding as in (6.10) and taking into account (6.11) to obtain the reverse
inequality we get:

${Y}^{\eta}_{t}\geq
h(X_{1}^{\eta})-\int_{t}^{1}\left(\frac{{Z}^{2,\eta}_{\ell}}{\eta}\bar{\alpha}^{n}_{\ell}+\tilde{\lambda}_{*}(X^{\eta}_{\ell},\bar{\alpha}^{n}_{\ell})+\frac{1}{n}\right)d\ell-\int_{t}^{1}{Z}^{1,\eta}_{\ell}d\bar{W}^{1}_{s}-\int_{t}^{1}{Z}^{2,\eta}_{\ell}d\bar{B}_{\ell},$

and rewriting the above in terms of $\bar{B}^{{\bar{\alpha}}^{n}}$:

$\begin{split}{Y}^{\eta}_{t}+\frac{1-t}{n}&\geq
h(X^{\eta}_{1})-\int_{t}^{1}\tilde{\lambda}_{*}(X^{\eta}_{\ell},\bar{\alpha}^{n}_{\ell})d\ell-\int_{t}^{1}{Z}^{1,\eta}_{\ell}d\bar{W}^{1}_{\ell}-\int_{t}^{1}{Z}^{2,\eta}_{\ell}d\bar{B}^{\bar{\alpha}^{n}}_{\ell}.\end{split}$

Therefore we can conclude that:

${Y}^{\eta}_{t}+1/n\geq\bar{\mathbb{E}}^{{\bar{\alpha}^{n}}}\biggl{(}h(X^{\eta}_{1})-\int_{t}^{1}\tilde{\lambda}_{*}(X^{\eta}_{\ell},\bar{\alpha}^{n}_{\ell})d\ell\bigg{|}\bar{\mathcal{F}}^{1,B}_{t}\biggr{)}$

and the claim is proved.

To complete the circle and give sense to the limit as $\eta\rightarrow 0$ we
just have to come back to a control representation of $Y^{\eta}_{0}$ that does
not relay on Girsanov transform (meaningless if $\eta=0$). This was already
done for a different control problem but by general techniques in Lemma 4.3.

Namely if we denote by $\mathbb{S}^{1,B}_{H}$ the class of $7$-uples
$\mathbb{U}^{B}_{H}=(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P},(W^{1}_{t}),(B_{t}),(\alpha_{t}))$,
similar to the settings $\mathbb{S}^{1,2,B}$ defined in Section 4 with the
only differences are that here the noise $(W^{2})$ is omitted and $(\alpha)$
is now any $(\mathcal{F}_{t})$-progressive process with values in the closed
ball $\\{x\in H:|x|\leq M+1\\}$ (the subscript $H$ in the notation
$\mathbb{S}_{H}$ indicates that we are now considering $H$-valued controls).
It holds:

###### Lemma 6.2

$Y^{\eta}_{0}=\inf_{\mathbb{U}^{B}_{H}\in\mathbb{S}^{1,B}_{H}}\bar{\mathbb{E}}^{\mathbb{U}^{B}_{H}}\biggl{(}h(X^{\eta,\mathbb{U}^{B}_{H}}_{1})-\int_{t}^{1}\tilde{\lambda}_{*}(X^{\eta,\mathbb{U}^{B}_{H}}_{\ell},{\alpha}_{\ell})d\ell\biggr{)}$
(6.12)

where given $\mathbb{U}^{B}_{H}\in\mathbb{S}^{1,B}_{H}$ as above
$X^{\eta,\mathbb{U}^{B}_{H}}$ solves:

$d{X}_{s}=A{X}_{s}ds-\,{\alpha}_{s}ds+R({X}_{s})dW^{1}_{s}+\eta\,dB_{t},\quad{X}_{0}=x_{0}.$
(6.13)

Proof.

The proof follows exactly as in the cited Lemma 4.3.

We are now able to prove the main result of the paper namely the
characterization of the limit, as $\varepsilon\rightarrow 0$, of the value
function $V^{\varepsilon}(x_{0},q_{0})$ of the original control problem in
terms the value function of a reduced control problem on a reduced state
space.

Let $\mathbb{S}^{1}_{H}$ the class of $6$-uples
$\mathbb{U}_{H}=(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P},(W^{1}_{t}),(\alpha_{t}))$
identical to the ones in $\mathbb{S}^{1,B}_{H}$ with the only difference that
$(B)$ is not present.

###### Theorem 6.3

It holds:

$\displaystyle\lim_{\varepsilon\rightarrow
0}V^{\varepsilon}(x_{0},q_{0})={V}(x_{0})$
$\displaystyle=\inf_{\mathbb{U}_{H}\in\,\mathbb{S}^{1}_{H}}\mathbb{E}^{\mathbb{U}_{H}}\left(h({X}^{\mathbb{U}_{H}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{\mathbb{U}_{H}}_{s},{\alpha}_{s}^{\mathbb{U}_{H}})ds\right).$

where, given $\mathbb{U}_{H}\in\mathbb{S}^{1}_{H}$ as above,
$X^{\eta,\mathbb{U}_{H}}$ solves the state equation:

$d{X}_{s}=A{X}_{s}ds-\,{\alpha}_{s}ds+R({X}_{s})dW^{1}_{s},\quad{X}_{0}=x_{0}.$

Proof: First we notice that as in Lemma 4.2 it holds:

$\inf_{\mathbb{U}_{H}\in\,\mathbb{S}^{1}_{H}}\mathbb{E}^{\mathbb{U}_{H}}\left(h({X}^{\mathbb{U}_{H}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{\mathbb{U}_{H}}_{s},{\alpha}_{s}^{\mathbb{U}_{H}})ds\right)=\inf_{\mathbb{U}^{B}_{H}\in\,\mathbb{S}^{1,B}_{H}}\mathbb{E}^{\mathbb{U}^{B}_{H}}\left(h({X}^{0,\mathbb{U}^{B}_{H}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{0,\mathbb{U}^{B}_{H}}_{s},{\alpha}_{s}^{\mathbb{U}^{B}_{H}})ds\right).$

where, we recall ${X}^{0,\mathbb{U}^{B}_{H}}$ solves equation (6.13) with
$\eta=0$.

Thus, by (6.12) and Th. 5.1 it is enough to prove that, if $\eta\rightarrow
0$:

$\inf_{\mathbb{U}^{B}_{H}\in\mathbb{S}^{1,B}_{H}}\bar{\mathbb{E}}^{\mathbb{U}^{B}_{H}}\biggl{(}h(X^{\eta,\mathbb{U}^{B}_{H}}_{1})-\int_{t}^{1}\tilde{\lambda}_{*}(X^{\eta,\mathbb{U}^{B}_{H}}_{\ell},{\alpha}_{\ell})d\ell\biggr{)}\rightarrow\inf_{\mathbb{U}^{B}_{H}\in\mathbb{S}^{1,B}_{H}}\bar{\mathbb{E}}^{\mathbb{U}^{B}_{H}}\biggl{(}h(X^{0,\mathbb{U}^{B}_{H}}_{1})-\int_{t}^{1}\tilde{\lambda}_{*}(X^{0,\mathbb{U}^{B}_{H}}_{\ell},{\alpha}_{\ell})d\ell\biggr{)}$

Fix any
$\mathbb{U}^{B}_{H}=(\Omega,(\mathcal{F}_{t}),\mathbb{P},(W^{1}_{t}),(B_{t}),(\alpha_{t}))\in\mathbb{S}^{1,B}_{H}$.
Thanks to the Lipschitzianity of $h$ (see hypotheses 3.6) and of
$\tilde{\lambda}_{*}$ we easily have that for a suitable constant $C$
independent on $\eta$:

$\displaystyle\Big{|}\mathbb{E}^{{\mathbb{U}^{B}_{H}}}\biggl{(}h(X^{\eta,{\mathbb{U}^{B}_{H}}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}(X^{\eta,{\mathbb{U}^{B}_{H}}}_{t},{\alpha}_{t})dt\biggr{)}-\mathbb{E}^{{\mathbb{U}^{B}_{H}}}\biggl{(}h({X}^{0,{\mathbb{U}^{B}_{H}}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{0,{\mathbb{U}^{B}_{H}}}_{t},{\alpha}_{t})dt\biggr{)}\Big{|}$
$\displaystyle\leq
C\,\mathbb{E}^{{\mathbb{U}^{B}_{H}}}\sup_{t\in[0,1]}|X^{\eta,{\mathbb{U}^{B}_{H}}}_{t}-{X}^{0,{\mathbb{U}^{B}_{H}}}_{t}|.$

We have :

$\begin{cases}d(X^{\eta,{\mathbb{U}^{B}_{H}}}_{t}-{X}^{0,{\mathbb{U}^{B}_{H}}}_{t})=A(X^{\eta,{\mathbb{U}^{B}_{H}}}_{t}-{X}^{0,{\mathbb{U}^{B}_{H}}}_{t})dt-[R(X^{\eta,{\mathbb{U}^{B}_{H}}}_{t})-R({X}^{0,\mathbb{U}^{B}}_{t})]dW^{1}_{t}+\eta
dB_{t},&t\in[0,1]\\\ \quad
X^{\eta,{\mathbb{U}^{B}_{H}}}_{0}-{X}^{0,{\mathbb{U}^{B}_{H}}}_{0}=0&\end{cases}$

By standard estimates based on the factorization method (see the proof of
Theorem 4.13), we end up with:

$\mathbb{E}^{{\mathbb{U}^{B}_{H}}}\sup_{t\in[0,1]}|X^{\eta,{\mathbb{U}^{B}_{H}}}_{t}-{X}^{0,{\mathbb{U}^{B}_{H}}}_{t}|^{p}\leq
C_{p}\eta^{p}.$

for all $p\geq 1$ and a suitable constant $C_{p}$ independent of $\alpha$.
Thus we can conclude that:

$\displaystyle\Big{|}\inf_{{\mathbb{U}^{B}_{H}}\in\,\mathbb{S}^{1,B}}\mathbb{E}^{\mathbb{U}^{B}_{H}}\biggl{(}h(X^{\eta,{\mathbb{U}^{B}_{H}}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}(X^{\eta,{\mathbb{U}^{B}_{H}}}_{t},\alpha_{t})dt\biggr{)}-\inf_{{\mathbb{U}^{B}_{H}}\in\,\mathbb{S}^{1,B}}\mathbb{E}^{\mathbb{U}^{B}_{H}}\biggl{(}h({X}^{0,\mathbb{U}^{B,\mathfrak{m}}}_{1}-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{0,{\mathbb{U}^{B}_{H}}}_{t},{\alpha}_{t})dt\biggr{)}\Big{|}$
$\displaystyle\leq\sup_{{\mathbb{U}^{B}_{H}}\in\,\mathbb{S}^{1,B}}\Big{|}\mathbb{E}^{{\mathbb{U}^{B}_{H}}}\biggl{(}h({X}^{\eta,{\mathbb{U}^{B}_{H}}})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{0,{\mathbb{U}^{B}_{H}}}_{t},{\alpha}_{t})dt\biggr{)}-\mathbb{E}^{\mathbb{U}^{B}_{H}}\biggl{(}h({X}^{0,\mathbb{U}^{B}_{H}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{0,{\mathbb{U}^{B}_{H}}}_{t},{\alpha}_{t})dt\biggr{)}\Big{|}$
$\displaystyle\leq C\eta,$

and the claim follows.

###### Remark 6.4

In the special case in which the slow evolution is not perturbed by the noise
$(W^{1})$ (equivalently $R\equiv 0$ in (3.1)), in Theorem 6.3 the following
characterization holds:

$V(x_{0})=\inf_{\mathbb{U}_{H}\in\,\mathbb{S}^{0}_{H}}\mathbb{E}^{\mathbb{U}^{0}_{H}}\left(h({X}^{\mathbb{U}^{0}_{H}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{\mathbb{U}^{0}_{H}}_{s},{\alpha}_{s}^{\mathbb{U}^{0}_{H}})ds\right)$

where $\mathbb{S}_{H}$ is the set of all
$\mathbb{U}^{0}_{H}=(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P},(\alpha_{t}))$
where $\alpha$ is any $(\mathcal{F}_{t})$ progressively measurable process
with values into the ball $B(0,M+1)\subset H$ of center 0 and radius $M+1$ and
$X^{\mathbb{U}^{0}_{H}}$ solves:

$d{X}_{s}=A{X}_{s}ds+\alpha_{s}\,ds,\quad{X}_{0}=x_{0}.$ (6.14)

It is natural to think that the stochastic framework is here pleonastic and
that the infimum above can be restricted to deterministic controls and trivial
settings, namely:

$V(x_{0})=\inf_{a}\left(h({X}^{a}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{a}_{s},a_{s})ds\right)$
(6.15)

where the above infimum is computed over all functions $a:[0,1]\rightarrow
B(0,M+1)$ ($B(0,M+1)$ being the ball of $H$ centered in the origin of radius
$M+1$) and $X^{a}$ is the mild solution of the deterministic evolution
equation:

$\frac{d}{dt}X_{t}=AX_{t}+a_{t},\qquad X_{0}=x_{0}$

It is straight forward to see that

$\inf_{\mathbb{U}^{0}_{H}\in\,\mathbb{S}_{H}}\mathbb{E}^{\mathbb{U}_{H}}\left(h({X}^{\mathbb{U}^{0}_{H}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{\mathbb{U}^{0}_{H}}_{s},{\alpha}_{s}^{\mathbb{U}^{0}_{H}})ds\right)\leq\inf_{a}\left(h({X}^{a}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{a}_{s},a_{s})ds\right)$

To prove the converse let, for any $\epsilon>0$, $\mathbb{U}_{H}\in\mathbb{S}$
such that:

$\mathbb{E}^{\mathbb{U}^{0}_{H}}\left(h({X}^{\mathbb{U}^{0}_{H}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{\mathbb{U}^{0}_{H}}_{s},{\alpha}_{s}^{\mathbb{U}^{0}_{H}})ds\right)\leq\inf_{\mathbb{U}^{0}_{H}\in\,\mathbb{S}_{H}}\mathbb{E}^{\mathbb{U}^{0}_{H}}\left(h({X}^{\mathbb{U}^{0}_{H}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{\mathbb{U}^{0}_{H}}_{s},{\alpha}_{s}^{\mathbb{U}^{0}_{H}})ds\right)+\epsilon.$
(6.16)

Then we have, recalling that equation (6.14) can be solved pathwise:

$\mathbb{P}^{\mathbb{U}^{0}_{H}}\left(\omega\in\Omega^{\mathbb{U}^{0}_{H}}:h({X}^{\mathbb{U}^{0}_{H}}_{1}(\omega))-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{\mathbb{U}^{0}_{H}}_{s}(\omega),{\alpha}_{s}^{\mathbb{U}^{0}_{H}}(\omega))ds\leq
V(x_{0})+\epsilon\right)>0$

To conclude is now enough to select $\bar{\omega}$ in the above set and choose
$\bar{a}=\alpha^{\mathbb{U}^{0}_{H}}(\bar{\omega})$.

###### Remark 6.5

Taking advantage of our main result (see Theorem 6.3) we can further represent
the singular limit $V(x_{0})$ as the value at time $0$ of the minimal solution
of a BSDE with constrints on the martingale term (see [13] for the definition
for and [6] for the infinite dimensional case). The bridge is given by the
results in [6] allowing to represent the value function of a control problem
by such a constrained BSDE without using viscosity solutions of the related
HJB equation.

First of all, to fit the framework in [6] we notice that the control problem
introduced in Theorem 6.3 can be rewritten considering an unbounded set of
controls. This is readily (and rather obviously) done by introduciong the
class $\widehat{\mathbb{S}}^{1}_{H}$ of $6$-uples
$\mathbb{U}_{H}=(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P},(W^{1}_{t}),(u_{t}))$
identical to the ones in $\mathbb{S}^{1}_{H}$ with the only difference that
here $u$ are not required to be bounded by $M+1$ and noticing that:

${V}(x_{0})=\inf_{\mathbb{U}_{H}\in\mathbb{S}^{1}_{H}}\mathbb{E}^{\mathbb{U}_{H}}\left(h({X}^{\mathbb{U}_{H}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{\mathbb{U}_{H}}_{s},{\alpha}_{s})ds\right)=\inf_{\widehat{\mathbb{U}}_{H}\in\widehat{\mathbb{S}}^{1}_{H}}\mathbb{E}^{\mathbb{U}_{H}}\left(h({X}^{\widehat{\mathbb{U}}_{H}}_{1})-\int_{0}^{1}\tilde{\lambda}_{*}({X}^{\widehat{\mathbb{U}}_{H}}_{s},\Gamma(u_{s}))ds\right)$
(6.17)

where $\Gamma(h):=\frac{h}{||h||}\min(||h||,M+1)$ for $h\in H$ and
$X^{\widehat{\mathbb{U}}_{H}}$ solves:

$d{X}_{s}=A{X}_{s}ds-\,\Gamma({u}_{s})ds+R({X}_{s})dW^{1}_{s},\quad{X}_{0}=x_{0}.$

Then we just have to remind the construction and the results in [6].

Given $x_{0}\in H$ we consider the following system of forward-backward
stochastic differential equations:

$\begin{cases}\mathcal{X}^{x_{0}}_{t}\ =\displaystyle\
e^{tA}x_{0}+\int_{0}^{t}e^{(t-s)A}\Gamma(S\widehat{\mathcal{W}}_{s})ds+\,\int_{0}^{t}e^{(t-s)A}R(\mathcal{X}^{x_{0}}_{s})d\mathcal{W}_{s},\\\
\mathcal{Y}^{x_{0}}_{t}=h(\mathcal{X}^{x_{0}}_{1})-\displaystyle\int_{t}^{1}\tilde{\lambda}_{*}(\mathcal{X}^{x_{0}}_{s},\Gamma(S\widehat{\mathcal{W}}_{s}))\,ds-K^{x_{0}}_{1}+K^{x_{0}}_{t}-\int_{t}^{1}\mathcal{Z}^{x_{0}}_{s}\,d\mathcal{W}_{s}\
,\end{cases}$ (6.18)

where $S:H\to H$ is an arbitrary trace class and injective linear operator
with dense image and $\mathcal{W}$ and $\widehat{\mathcal{W}}$ are two
independent cylindrical Wiener processes with values in $H$ defined on a
probability space satisfying the usual conditions. We denote by
$(\mathcal{F}^{0}_{t})$ the natural filtration of
$(\mathcal{W}_{t},\widehat{\mathcal{W}}_{t})$ augmented. Notice that, besides
the two typical terms in the backward component, the unknown $K$ appears. Such
process belongs to the set of real-valued $(\mathcal{F}^{0}_{t})$\- adapted
nondecreasing continuous processes $K$ on $[0,T]$ such that
$\mathbb{E}|K_{T}|^{2}<\infty$ and $K_{0}=0$.

Then, see [6] §4.2 the following holds:

  * •

the forward equation in (6.18) has a unique solution $(\mathcal{X}^{x_{0}})$
in $L^{2}_{\mathcal{F}^{0}}(\Omega;{C}([0,T];{H}))$.

  * •

the backward equation in system (6.18) has a maximal solution
$(\mathcal{Y}^{x_{0}},\mathcal{Z}^{x_{0}},K^{x_{0}})$ belonging to the space
$L^{2}_{\mathcal{F}^{0}}(\Omega;{C}([0,T];\mathbb{R}))\times
L^{2}_{\mathcal{F}^{0}}(\Omega\times[0,T];\Xi^{*})\times\mathcal{K}^{2}(0,T)$,
maximal in the sense that if there exists another solution
$(\mathcal{Y}^{\prime},\mathcal{Z}^{\prime},{K}^{\prime})$ belonging to the
same functional spaces then
$\mathcal{Y}_{t}^{x_{0}}\geq\mathcal{Y}^{\prime}_{t}$ for all $t\in[0,1]$,
$\mathbb{P}$-a.s.

  * •

the following characterization of the singular limit $V(x_{0})$ holds

$\mathcal{Y}^{x}_{0}={V}(x_{0}).$ (6.19)

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