#  Stochastic Recursive Optimal Control of McKean-Vlasov Type: A Viscosity
Solution Approach

Liangquan Zhang
School of Mathematics,
Renmin University of China, Beijing 100872, China L. Zhang acknowledges the
financial support partly by the National Nature Science Foundation of China
(Grant No. 12171053, 11701040, 11871010 &61871058) and the Fundamental
Research Funds for the Central Universities, and the Research Funds of Renmin
University of China (No. 23XNKJ05).

###### Abstract

In this paper, we study a kind of optimal control problem for forward-backward
stochastic differential equations (FBSDEs for short) of McKean–Vlasov type via
the dynamic programming principle (DPP for short) motivated by studying the
infinite dimensional Hamilton–Jacobi–Bellman (HJB for short) equation derived
from the decoupling field of the FBSDEs posed by Carmona and Delarue (_Ann
Probab_ , 2015, [30]). At the beginning, by considering the cost functional
defined by the backward component of the solution of the controlled FBSDEs
alluded to earlier, on one hand, we can prove the value function is
deterministic function with respect to the initial random variable; On the
other hand, we can show that the value function is _law-invariant_ , i.e.,
depend on only via its distribution by virtue of BSDE property. Afterward,
utilizing the notion of differentiability with respect to probability measures
introduced by P.L. Lions [50], we are able to establish a DPP for the value
function in the Wasserstein space of probability measures based on the
application of BSDE approach, particularly, employing the notion of stochastic
_backward semigroups_ associated with stochastic optimal control problems and
Itô’s formula along a flow property of the conditional law of the controlled
forward state process. We prove that the value function is the unique
viscosity solutions of the associated generalized HJB equations in some
sparable Hilbert space. Finally, as an application, we formulate an optimal
control problem for linear stochastic differential equations with quadratic
cost functionals of McKean-Vlasov type under nonlinear expectation,
$g$-expectation introduced by Peng [43] and derive the optimal feedback
control explicitly by means of several groups of Riccati equations.

AMS subject classifications: 93E20, 60H15, 60H30.

Key words: Dynamic programming principle, Forward-backward McKean-Vlasov
stochastic differential equations, Hamilton-Jacobi-Bellman equation,
Wasserstein space, Value function, Viscosity solution.

###### Contents

  1. 1 Introduction
  2. 2 Stochastic Recursive Control Systems of McKean-Vlasov Type
  3. 3 Hamilton-Jacobi-Bellman Equation
  4. 4 LQ stochastic McKean-Vlasov control problem via $g$-expectation
  5. 5 Concluding remark
  6. 6 Appendix: Technique Lemmas
    1. 6.1 Proof of Lemma 2.3
    2. 6.2 Proof of Lemma 2.6
    3. 6.3 Proof of Lemma 2.6
    4. 6.4 Proof of Lemma 2.8
    5. 6.5 Proof of Lemma 2.14
    6. 6.6 Proof of Lemma 3.1
  7. 7 References

##  1 Introduction

Since the fundamental works on McKean–Vlasov equations were considered by
McKean Jr. [40] and Kac [47, 48], there are huge literature focusing on
uncontrolled SDEs and obtaining the general propagation of chaos results.
Sustained academic attention in this field in the past decade was embodied in
the connection with the so-called mean-field game (MFG for short) theory,
introduced independently and simultaneously by Lasry and Lions in [54] and on
Huang, Caines and Malhamé [37]. As a matter of fact, the McKean–Vlasov
equation naturally happens whenever one tries to interpret the mechanism of
the behavior of many symmetric agents, all of which interact via the empirical
distribution of their states, to find a Nash equilibrium (competitive
equilibrium) or a Pareto equilibrium (cooperative equilibrium) (see [9, 31]).

As for the optimal control problem in the framework of the McKean–Vlasov type
(e.g. the classical mean–variance portfolio selection problem in finance, see
[63]) also called mean field type, it is hard to obtain the DPP for short due
to the appearance of the law of the process in the coefficients and nonlinear
dependency with respect to it. Therefore, problems like this actually belongs
to a time inconsistent issue (see Björk, Khapko and Murgoci, [12] and T.
Björk, M. Khapko and A. Murgoci, [13], Hernández and Possamaï [39] references
therein). It is necessary to point out that, though the problem itself is time
inconsistent, it is possible to capture some form of the DPP by extending the
state space, for instance, Laurière and Pironneau [53] (see also Bensoussan,
Frehse and Yam [9, 10, 11]) adopted the hypothesis that the existence at all
times of a density for the marginal distribution of the state process and
transformed the prime problem into a density control problem with a family of
deterministic controls. Afterward, they were able to prove a DPP and derive a
kind of Bellman’s equation in the space of density functions. Subsequently,
Pham and Wei [44] without the density assumption, obtained the DPP for closed-
loop controls. For open-loop controls, the related topics can be found in
Cosso and Pham [22] for McKean–Vlasov differential games and in Bayraktar,
Cosso and Pham [6] for the so-called randomised DPP. If containing the common
noise, Pham and Wei [45] proved a DPP where the control process is adapted to
the common noise filtration. Besides, Bouchard, Djehiche and Kharroubi [7]
investigated a stochastic McKean– Vlasov target problem, in which the
controlled process satisfies some target marginal constraints and established
a general geometric dynamic programming (see also [57]). Djete, Possamaï and
Tan [33], however, employed the measurable selection techniques to prove the
DPP. This approach can be used under very general conditions, avoiding for
instance the Markovian property or regularity assumptions mentioned in the
above literature. Alternatively, stochastic maximum principle (SMP for short)
for the control of McKean-Vlasov dynamics has been considered in [30, 17, 3]
for state dynamics depending upon marginal distribution and in [19, 26] for
conditional McKean-Vlasov dynamics. Carmona and Delarue, [30] established the
SMP and provided the sufficient conditions for existence of an optimal
control. Both necessary and sufficient parts of the stochastic Pontryagin
principle require the adjoint backward equation, and seeking the optimal
control problem turns de facto to the analysis of FBSDEs in which the marginal
distributions of the solutions appear in the coefficients of the equations. In
the same way, Li [51] studied a SMP for the mean-field controls. For mean
field linear-quadratic (LQ for short) problem, see Yong [64]. A stochastic
optimal control problem with delay and of mean-field type was considered by
Shen, Meng and Shi [58].

In history, nonlinear BSDEs of McKean–Vlasov type were investigated first by
Buckdahn, Djehiche Li and Peng [15]. Since then, the related results on the
theory of FBSDEs of McKean–Vlasov type, as well as that of the associated
partial differential equations (PDEs for short) have been widely studied.
Buckdahn, Li and Peng [16] proved not only an existence and uniqueness
theorem, but also a comparison theorem for expectation type. Employing a BSDE
approach, introduced by Peng [42], the authors also explored a probabilistic
interpretation to related nonlocal PDEs in finite dimensional (comparison of
our paper see Remark 3.2). Subsequently, with the rapid development of the
theory of mean field FBSDEs, many researchers focused on stochastic control
problems in the McKean–Vlasov type. For instance, whenever FBSDEs involving
the value function, with frozen partial initial values, Hao and Li [38]
considered an optimal control problem with systems of fully coupled controlled
mean field FBSDEs. Many other works in the mean-field area, see, e.g.,
Kotelenez and Kurtz [49] for stochastic PDES and Lasry and Lions [54] for
Economics, Finance and game theory etc.

The objective of this paper is to study the optimal control of FBSDEs of
McKean–Vlasov type, establish the DPP by virtue of BSDE approach and then
derive the corresponding HJB equation. However, as mentioned before, from a
traditional point of view, the DPP for FBSDEs of McKean–Vlasov type does not
hold true anymore due to the appearance of expectation terms in coefficients.
To overcome this difficulty, one possible direction is pointing to the
generalized state space. In a nutshell, the idea behind this is that we seize
the flow property of the controlled conditional distribution measurable
process and transform the prime BSDE into one depending on the conditional
distribution of forward variable, under which we are able to announce the DPP
via backward semigroup. Nonetheless, there are serval key points should be
stated as follows:

  * •

At first, unlike the cost functional in [45, 30, 22, 33] defined by taking the
expectation both on running function and terminal function, which means the
value function is deterministic no matter of the selection of initial state is
random or not (see Proposition 3.1 in [22]). Whereas, the cost functional in
our paper is defined by the solution to BSDE, so it can be random whenever the
initial state is measurable to the filtration $\mathcal{F}_{t}$ generated by
Brown motion at time $t\geq 0$. However, if it is measurable to a “rich
enough” (see the interpretation below) $\sigma$-algebra $\mathcal{G}$
independent of $\mathcal{F}_{t},$ we are able to prove the value function is
deterministic.

  * •

The value function will be depend on random variable $\xi$, but we hope it
depends on its distribution instead. Therefore, we should prove a crucial law
invariance property of the value function (defined by solution to BSDE) (see
Theorem 2.1), which implies that the value function can be considered as a
function on the Wasserstein space of probability measures on
$\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$.

  * •

In an earlier work Barles, Buckdahn and Pardoux [4], introduced certain space
of continuous functions and proved for in this space the uniqueness of the
viscosity solution of an integro-partial differential equation111This wide
class of PDEs has some apparent features in contrast to those derived from
forward stochastic controlled systems (see [45, 33]), for instance, the
inhomogeneous term depends not only on the value function but also its
gradient. These new changes make the investigation more comlicated. associated
with decoupled FBSDEs with jumps. Whereafter, Buckdahn, Li and Peng [16]
proved the uniqueness for PDEs associated with a decoupled mean field FBSDEs
(see Remark 2.7) without jumps in the space of continuous functions with at
most polynomial growth. However, in our framework, we have to consider the
PDEs in infinite dimensional set and corresponding result may be borrowed from
[35].

  * •

All the works in [45, 30, 22, 33] consider the optimal control problem under
the classical expectation. In reality, the objective expectation doesn’t
reflect people’s preferences (for instance, [1, 34]). There is a general
consensus to define a certain non-linear expectation, such as $g$-expectation
(cf. Peng [43]). In this paper, we will revisit McKean-Vlasov LQ problem under
$g$-expectation. We will see, on one hand, the admissible control set needs
more stringent integrability requirements due to the well-posedness for BSDE
regarding the terminal condition. On the other hand, the control problem under
generalized expectation is not necessarily equivalent to the classical
expectation. Hence, this topic is interesting in its own way.

The outline of the paper is organized as follows. After recalling briefly some
notations and stochastic framework to present an adequate and precise
definition of the tools that are used throughout the paper, we introduce in
Section 2 the controlled FBSDEs of the McKean–Vlasov type of stochastic
control problem and study the properties of corresponding value function.
Next, in Section 3, we derive an infinite dimensional HJB equation, and prove
the viscosity property together with a uniqueness result for the value
function. In Section 4, we revisit a stochastic McKean-Vlasov control problem
under a non-linear expectation, $g$-expectation. Section 5 states some
concluding remarks. At last, Some proofs are displayed in Appendix 6.

Let us introduce the following notations which are needed in what follows:

  * •

For $r\geq 1,$ $\mathcal{P}_{r}\left(\mathbb{R}^{n}\right)$ denotes the set
probability measure $\mu$ on $\mathbb{R}^{n}$, which satisfies square
integrable, i.e.
$\left\|\mu\right\|_{r}=\left[\int_{\mathbb{R}^{n}}\left|x\right|^{r}\mu\left(\mathrm{d}x\right)\right]^{\frac{1}{r}}<\infty.$

  * •

For any $\mu\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right),$ we denote by
$L_{\mu}^{2}\left(\mathbb{R}^{d}\right)$ the set of measure functions
$\phi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{d},$ which are square integrable
w.r.t. $\mu$ and define
$\mu\left(\phi\right):=\int_{\mathbb{R}^{n}}\phi\left(x\right)\mu\left(\mathrm{d}x\right).$
Similarly, $L_{\mu\otimes\mu}^{2}\left(\mathbb{R}^{d}\right)$ denotes the set
of measure functions
$\varphi:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{d},$ which
are square integrable w.r.t. $\mu\otimes\mu.$ We define
$\mu\otimes\mu\left(\varphi\right):=\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\varphi\left(x,\bar{x}\right)\mu\left(\mathrm{d}x\right)\mu\left(\mathrm{d}\bar{x}\right).$

  * •

$C_{q}\left(\mathbb{R}^{n}\right)$ denotes the set of continuous functions on
$\mathbb{R}^{n}$ with quadratic growth, and for any $\psi\in
C_{q}\left(\mathbb{R}^{n}\right),$ we can extend $\psi$ to
$\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$ by
$\Pi_{\psi}\left(\mu\right)=\int_{\mathbb{R}^{n}}\psi\left(x\right)\mu\left(\mathrm{d}x\right).$

  * •

$L_{\mu}^{\infty}\left(\mathbb{R}^{d}\right)$ denotes the subset of elements
$\phi\in L_{\mu}^{2}\left(\mathbb{R}^{d}\right)$ which are bounded w.r.t.
$\mu$ with $\left\|\phi\right\|_{\infty}$ being their essential supremum.
Similarly, $L_{\mu\otimes\mu}^{\infty}\left(\mathbb{R}^{d}\right)$ denotes the
subset of elements $\varphi\in
L_{\mu\otimes\mu}^{2}\left(\mathbb{R}^{d}\right)$ which are bounded w.r.t.
$\mu\otimes\mu$ with $\left\|\varphi\right\|_{\infty}$ being their essential
supremum.

  * •

Let $(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbb{P})$ be a
complete filtered probability space assume of the form
$(\Omega^{0}\times\Omega^{1},\mathcal{F}^{0}\otimes\mathcal{G},\mathbb{P}^{0}\otimes\mathbb{P}^{1}),$
where $\left(\Omega^{0},\mathcal{F}^{0},\mathbb{P}^{0}\right)$ supports a
$d$-dimensional Brownian motion $W.$ Any element $\omega\in\Omega$ can be
expressed as
$\omega=\left(\omega^{0},\omega^{1}\right)\in\Omega^{0}\times\Omega^{1}.$
Moreover, $W$ on $\Omega$ can be written as
$W\left(\omega^{0},\omega^{1}\right)=W\left(\omega^{0}\right),$ etc. Besides,
we suppose that $\Omega^{1}$ is a Polish space, $\mathcal{G}$ its Borel
$\sigma$-field, $\mathbb{P}^{1}$ an atomless probability measure on
$\left(\Omega^{1},\mathcal{G}\right).$ $\mathcal{G}\supset\mathcal{N},$ where
$\mathcal{N}$ is the set of all $\mathbb{P}$-null sets. We denote by
$\mathbb{E}^{0}$ (resp. $\mathbb{E}^{1}$) the expectation under
$\mathbb{P}^{0}$ (resp. $\mathbb{P}^{1}$). Let
$\mathcal{F}^{0}=\left(\mathcal{F}_{t}^{0}\right)_{0\leq t\leq T}$ is the
natural filtration generated by $W,$ while $\mathbb{F}$
$=\left(\mathcal{F}_{t}\right)_{0\leq t\leq T}$ is be the filtration generated
by $W$, completed and augmented by a $\sigma$-field $\mathcal{G}$, i.e.,

$\mathcal{F}_{t}^{0}=\sigma\left\\{W_{s}|s\leq t\right\\},\text{
}\mathcal{F}_{t}=\mathcal{F}_{t+}^{0}\vee\mathcal{G},$

where

$\mathcal{F}_{t+}^{0}=\bigcap_{s\geq t}\mathcal{F}_{s}^{0}.$

  * •

We assume that the above sub-$\sigma$-field $\mathcal{G}$ of $\mathcal{F}$
which is independent of $\mathcal{F}_{\infty}^{0}$ and “rich enough” in the
following sense:

$\mathcal{P}_{r}\left(\mathbb{R}^{n}\right)=\left\\{\mathbb{P}_{\xi}\left|\xi\in
L^{r}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1}\right)\right.\right\\},$ (1)

where $\mathbb{P}_{\xi}$ or $\mathcal{L}\left(\xi\right)$ denotes the law of
$\xi.$ From Lemma 2.1 in [25], $\mathcal{G}$ is “rich enough” if and only if
there exists a $\mathcal{G}$-measurable random variable
$U^{\mathcal{G}}:\Omega^{1}\rightarrow\mathbb{R}$ having uniform distribution
on $\left[0,1\right]$. Particularly, if the probability space
$\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1}\right)$ is _atomless_ (namely,
for any $A\in\mathcal{G}$ such that $\mathbb{P}^{1}\left(A\right)>0$ there
exists $B\in\mathcal{G}$, $B\subset A$, such that
$0<\mathbb{P}^{1}\left(B\right)<\mathbb{P}^{1}\left(A\right)$), then these two
mentioned properties holds. From Corollary 7.16.1 in [8], we know that, for
each probability space
$\left(\tilde{\Omega},\mathcal{\tilde{F}},\mathbb{\tilde{P}}\right),$ together
with $\tilde{\Omega}$ uncountable, separable, complete metric space,
$\mathcal{\tilde{F}}$ its Borel $\sigma$-field, $\mathbb{\tilde{P}}$ an
atomless probability, satisfies (1). For instance, (see Theorem 3.19 in [56]),
the probability space
$\left(\tilde{\Omega},\mathcal{\tilde{F}},\mathbb{\tilde{P}}\right)$ can be
chosen as
$\left(\left[0,1\right],B\left(\left[0,1\right]\right),\lambda\right)$ where
$\lambda$ denotes the Lebesgue measure.

  * •

We define

$\left(\mathcal{F}^{0}\right)_{s}^{t}\triangleq\sigma\left(W_{r}-W_{t}|t\leq
r\leq s\right)\vee\mathcal{N}\text{ },\text{ }s\in\left[t,T\right].$

  * •

$L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$ (resp.
$L^{2}\left(\mathcal{F}_{t}^{0}\vee\mathcal{G};\mathbb{R}^{n}\right)$) denotes
the set of $\mathbb{R}^{n}$-valued square integrable random variables on
$\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1}\right)$ (resp.
$\left(\Omega,\mathcal{F}_{t}^{0}\vee\mathcal{G},\mathbb{P}^{0}\right)$).

  * •

As we have known, $\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$ is a metric
space equipped with the $2$-Wasserstein distance

$\begin{array}[]{lll}\mathcal{W}_{2}\left(\mu^{1},\mu^{2}\right)&\triangleq&\inf\Bigg{\\{}\left(\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\left|x-y\right|^{2}\pi\left(\mathrm{d}x,\mathrm{d}y\right)\right)^{\frac{1}{2}}:\pi\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\times\mathbb{R}^{n}\right)\text{
}\\\ &&\text{with marginals }\mu^{1}\text{ and }\mu^{2}\Bigg{\\}}\\\
&&=\inf\Bigg{\\{}\left(\mathbb{E}\left|X-X^{\prime}\right|^{2}\right)^{\frac{1}{2}}:X,\text{
}X^{\prime}\in L^{2}\left(\mathcal{O};\mathbb{R}^{n}\right)\\\ &&\text{with
}\mathbb{P}_{X}=\mu^{1},\text{
}\mathbb{P}_{X^{\prime}}=\mu^{2}\Bigg{\\}}\end{array}$

endowed with Borel $\sigma$-field
$\mathcal{B}\left(\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\right).$ It
follows from Proposition 7.1.5 in [2] that
$\left(\mathcal{P}_{2}\left(\mathbb{R}^{n}\right),\mathcal{W}_{2}\right)$ is a
complete separable metric space222By Theorem 7.12 in [59], for
$\left(\mu_{n}\right)_{n}$,
$\mu_{n}\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$, we have that
$\mathcal{W}_{2}(\mu_{n},\mu)\rightarrow 0$ if and only if, for every
$\phi\in$ $C_{q}\left(\mathbb{R}^{n}\right),$
$\phi\left(\mu_{n}\right)\rightarrow\phi\left(\mu\right).$.

  * •

We need the following spaces:

$\begin{array}[]{lll}\mathcal{S}_{\mathcal{F}}^{2}\left(t,T;\mathbb{R}^{n}\right)&\triangleq&\Big{\\{}\left.\varphi\right|\varphi:\Omega\times\left[t,T\right]\rightarrow\mathbb{R}^{n}\text{
is an }\mathcal{F}_{t}^{0}-\text{adapted process}\\\ &&\text{ satisfying
}\mathbb{E}\left[\sup_{0<s<T}\left|\varphi_{s}\right|^{2}\right]<\infty\Big{\\}};\\\
\mathcal{M}_{\mathcal{F}}^{2}\left(t,T;\mathbb{R}^{n\times
d}\right)&&\triangleq\Big{\\{}\left.\phi\right|\phi:\Omega\times\left[t,T\right]\rightarrow\mathbb{R}^{n\times
d}\text{ is }\mathcal{F}_{t}^{0}\text{-predictable process }\\\
&&\text{satisfying
}\mathbb{E}\left[\int_{t}^{T}\left|\phi_{s}\right|^{2}\mathrm{d}s\right]<\infty\Big{\\}}.\end{array}$

###### Remark 1.1

It is straightforward to check that the map $\Pi_{\psi}$ is
$\mathcal{B}\left(\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\right)$-measurable,
for any $\psi\in C_{q}\left(\mathbb{R}^{n}\right)$, by virtue of a monotone
class argument since it holds true whenever $\psi\in
C_{q}\left(\mathbb{R}^{n}\right)$. Besides,
$\mathcal{B}\left(\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\right)$ coincides
with the cylindrical $\sigma$-field $\sigma\left(\Pi_{\psi},\psi\in
C_{q}\left(\mathbb{R}^{n}\right)\right).$ Therefore, given a measurable space
$\left(G,\mathcal{G}\right)$ and a map
$\rho:G\rightarrow\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$, $\rho$ is
measurable if and only if the map $\Pi_{\psi}\circ\rho:G\rightarrow\mathbb{R}$
is measurable, for any $\psi\in C_{q}\left(\mathbb{R}^{n}\right)$.

##  2 Stochastic Recursive Control Systems of McKean-Vlasov Type

In this section, we consider the following recursive optimal control problem.
For any $\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right),$ the dynamics are governed by
McKean-Vlasov controlled FBSDEs in the form:

$\left\\{\begin{array}[]{rcl}\mathrm{d}X_{s}^{t,\xi;u}&=&b\left(X_{s}^{t,\xi;u},\mathbb{P}_{X_{s}^{t,\xi;u}}^{W},u_{s}\right)\mathrm{d}s+\sigma\left(X_{s}^{t,\xi;u},\mathbb{P}_{X_{s}^{t,\xi;u}}^{W},u_{s}\right)\mathrm{d}W_{s},\\\
\mathrm{d}Y_{s}^{t,\xi;u}&=&-\mathbb{E}^{\mathcal{F}_{s}^{0}}\left[f\left(X_{s}^{t,\xi,u},\mathbb{P}_{X_{s}^{t,\xi;u}}^{W},\Theta_{s}^{t,\xi,u},\mathbb{P}_{\Theta_{s}^{t,\xi;u}},u_{s}\right)\right]\mathrm{d}s+Z_{s}^{t,\xi;u}\mathrm{d}W_{s},\\\
X_{t}^{t,\xi;u}&=&\xi,\text{
}Y_{T}^{t,\xi;u}=\mathbb{E}^{\mathcal{F}_{T}^{0}}\left[\Phi\left(X_{T}^{t,\xi;u},\mathbb{P}_{X_{s}^{t,\xi;u}}^{W}\right)\right],\end{array}\right.$
(2)

where $\Theta_{s}^{t,\xi,u}=\left(Y_{s}^{t,\xi,u},Z_{s}^{t,\xi,u}\right)$ and
$\mathbb{E}^{\mathcal{F}_{s}^{0}}$ denotes the conditional expectation w.r.t.
$\mathcal{F}_{s}^{0}.$ Hereafter, $\mathbb{P}_{X_{s}^{t,\xi,u}}^{W}$ denotes
the regular conditional distribution of $X_{s}^{t,\xi,u}$ given
$\mathcal{F}^{0}$, and its realization at some $\omega^{0}\in\Omega^{0}$ reads
as the law under $\mathbb{P}^{1}$ of the random variable
$X_{s}^{t,\xi,u}\left(\omega^{0},\cdot\right)$ on
$\left(\Omega^{1},\mathcal{F}^{1}\right)$, i.e.
$\mathbb{P}_{X_{s}^{t,\xi,u}}^{W}\left(\omega^{0}\right)=\mathbb{P}_{X_{s}^{t,\xi,u}\left(\omega^{0},\cdot\right)}^{1}$.
As for $\mathbb{P}_{\Theta_{s}^{t,\xi;u}},$ it represents as the law under
$\mathbb{P}^{0}$ of the random variable $\left(Y^{t,\xi,u},Z^{t,\xi,u}\right)$
on $\left(\Omega^{0},\mathcal{F}^{0}\right)$, i.e.
$\mathbb{P}_{\Theta_{s}^{t,\xi;u}}=\mathbb{P}_{\Theta_{s}^{t,\xi;u}}^{0}.$

###### Remark 2.1

Note that $X^{t,\xi;u}$ in (2) is $\mathcal{F}$-adapted process, while
$Y^{t,\xi;u}$ is $\mathcal{F}^{0}$-predictable process since the conditional
expectation $\mathbb{E}^{\mathcal{F}_{s}^{0}}$ appearing in generator $f$ and
$\mathbb{E}^{\mathcal{F}_{T}^{0}}$ in the terminal function $\Phi$. We will
prove that $Y_{t}^{t,\xi;u}$ is deterministic for any
$\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$ rather than random (see Lemma
2.7 below).

We explain the following notations:

  * •

The control process $u,$ satisfying $u_{t}\in U,$ for $0\leq t\leq T$. Here
$U$, the subset of $\mathbb{R}^{k}$, called the control set333As a matter of
fact, the control set can be any Polish set $U$ equipped with the distance
$d_{U}$, satisfying w.l.o.g. $d_{U}<1.$ Recall Theorem 6.1 in [63] for
classical LQ problem, the optimal control process can be represented as a
linear function in state variable. Inspired by this result, in Section 4, we
will postulate the control set to be a function space satisfying Lipschitz
condition., is a _compact_ metric space. Define the regular admissible control
set as follows ($p>0$):

$\begin{array}[]{rcl}\mathcal{U}_{ad}^{p}\left[0,T\right]&\triangleq&\Bigg{\\{}u\left(\cdot\right):u_{s}\in
U\text{ is an }\mathcal{F}_{s}\text{-progressively measurable process }\\\
&&\text{satisfying
}\mathbb{E}^{0}\left[\int_{0}^{T}\left|u_{s}\right|^{2}\mathrm{d}s\right]^{\frac{p}{2}}<\infty\Bigg{\\}}.\end{array}\text{
}$ (3)

Notice that $\mathcal{U}_{ad}^{2}\left[0,T\right]$ is separable metric space
under the distance

$\left|u-v\right|_{ad}^{2}=\mathbb{E}^{0}\left[\int_{0}^{T}\left|u_{s}\right|^{2}\mathrm{d}s\right]^{\frac{1}{2}}.$

  * •

Denote by $\Lambda_{t,\xi},$ the set of admissible control starting from
$\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$ being an
$\mathcal{F}_{s}$-progressively measurable process valued in $U$ such that
$u\in\mathcal{U}_{ad}^{2}\left[0,T\right].$

Now suppose that $\Omega^{0}=C\left(\mathbb{R}_{+},\mathbb{R}^{n}\right),$ the
set of continuous functions from $\mathbb{R}_{+}$ into $\mathbb{R}^{n}$,
$\omega^{0}$ is the canonical process, and $\mathbb{P}^{0}$ is the Wiener
measure. We introduce the shifted control processes (cf. [24]) constructed by
concatenation of paths:

###### Definition 2.1

Given any $u\in\mathcal{U}_{ad}^{2}\left[0,T\right],$
$\left(t,\tilde{\omega}^{0}\right)\in\left[0,T\right]\times\Omega^{0},$ we
define

$u_{s}^{t,\tilde{\omega}^{0}}\left(\omega^{0}\right)=u_{s}\left(\tilde{\omega}^{0}\otimes_{t}\omega^{0}\right),\text{
for }\left(s,\omega^{0}\right)\in\left[0,T\right]\times\Omega^{0}$

where

$\left(\tilde{\omega}^{0}\otimes_{t}\omega^{0}\right)_{s}=\tilde{\omega}_{s}^{0}\mathbb{I}_{s<t}+\left(\tilde{\omega}_{s}^{0}+\omega_{s}^{0}-\omega_{t}^{0}\right)\mathbb{I}_{s\geq
t}.$

Notice that $\tilde{\omega}^{0}\otimes_{t}\omega^{0}\in\Omega^{0}.$ Moreover,
$u^{t,\tilde{\omega}^{0}}\in\mathcal{U}_{t}^{2},$ where $\mathcal{U}_{t}^{2}$
denotes the subset of $\mathcal{U}_{ad}^{2}\left[0,T\right]$ which are
independent of $\mathcal{F}^{0}$ under $\mathbb{P}^{0}.$

On the coefficients $b$ and $\sigma$, and also on the generator function $f$
and terminal function $\Phi$ in (2), we impose the following assumptions:

  1. (A1)

The coefficients
$b:\mathbb{R}^{n}\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\times
U\rightarrow\mathbb{R}^{n}$ and
$\sigma:\mathbb{R}^{n}\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\times
U\rightarrow\mathbb{R}^{n\times d}$ are measurable functions.

  2. (A2)

The functions $b,$ $\sigma$ are Lipschitz continuous in
$\left(x,\mu,u\right)$, i.e., there exists a positive constant $C$ such that,
for any $x,$ $x^{\prime}\in\mathbb{R}^{n},$ $\mu,$
$\mu^{\prime}\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right),$ and $u,$
$u^{\prime}\in U$

$\displaystyle\left|b\left(x,\mu,u\right)-b\left(x^{\prime},\mu^{\prime},u^{\prime}\right)\right|+\left|\sigma\left(x,\mu,u\right)-\sigma\left(x^{\prime},\mu^{\prime},u^{\prime}\right)\right|$
$\displaystyle\leq$ $\displaystyle
C\left(\left|x-x^{\prime}\right|+\mathcal{W}_{2}\left(\mu,\mu^{\prime}\right)+\left|u-u^{\prime}\right|\right)$

and

$\left|b\left(0,\delta_{0},u\right)\right|+\left|\sigma\left(0,\delta_{0},u\right)\right|\leq
C.$

###### Remark 2.2

It is necessary to point out that in [62], Peng (1997) first studied the
optimal control for FBSDEs under Markov framework assuming that the
coefficients $b$ and $\sigma$ have the linear growth with respect to
$\left(x,u\right)$, are Lipschitz continuous in $x$ but Hölder continuous with
respect to $u$. Unlike the maximum principle needing some smoothness of
coefficients, we postulate that the coefficients are Lipschitz continuous in
$u.$ In the literature, $b$ and $\sigma$ fulfill the the Lipchitz condition,
uniformly w.r.t. $U$ and $\mathcal{P}\left(U\right)$ (see [22]) here
$\delta_{0}$ stands for the Dirac measure with mass at $0\in\mathbb{R}^{n}.$

In order to ensure the well-defined of solution to (2), we add the following
conditions:

  1. (A3)

The generator
$f:\mathbb{R}^{n}\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\times\mathbb{R\times
R}^{d}\times\mathcal{P}_{2}\left(\mathbb{R}\times\mathbb{R}^{d}\right)\times
U\rightarrow\mathbb{R}$ and the terminal cost function
$\Phi:\mathbb{R}^{n}\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\rightarrow\mathbb{R}$
are continuous and measurable functions.

  2. (A4)

The functions $f,$ $\Phi$ satisfy the Lipschitz condition: there exists a
positive constant $C$ such that, for any $x,x^{\prime}\in\mathbb{R}^{n},$
$y,y^{\prime}\in\mathbb{R},z,z^{\prime}\in\mathbb{R}^{d},$
$\mu,\mu^{\prime}\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$
$\nu,\nu^{\prime}\in\mathcal{P}_{2}\left(\mathbb{R}\times\mathbb{R}^{d}\right)$and
$u,u^{\prime}\in U$

$\displaystyle\left|f\left(x,\mu,y,z,\nu,u\right)-f\left(x^{\prime},\mu^{\prime},y^{\prime},z^{\prime},\nu^{\prime},u^{\prime}\right)\right|+\left|\Phi\left(x,\mu\right)-\Phi\left(x^{\prime},\mu^{\prime}\right)\right|$
$\displaystyle\leq$ $\displaystyle
C\big{(}\left|x-x^{\prime}\right|+\left|y-y^{\prime}\right|+\left|z-z^{\prime}\right|+\mathcal{W}_{2}\left(\mu,\mu^{\prime}\right)+\mathcal{W}_{2}\left(\nu,\nu^{\prime}\right)+\left|u-u^{\prime}\right|\big{)}$

and

$\left|f\left(x,\delta_{x},0,0,\delta_{\left(0,0\right)},u\right)\right|+\left|\Phi\left(x,\delta_{0}\right)\right|\leq
C\left(1+\left|x\right|+\mathcal{W}_{2}\left(\delta_{x},\delta_{0}\right)\right).$

  3. (A5)

Assume that the $\sigma$-algebra $\mathcal{G}$ is countably generated upto
null sets.

###### Remark 2.3

In order to prove the continuity of value function, Cosso et al. [22], imposed
a continuity assumption on the payoff functions $f$ and $\Phi.$ As a matter of
fact, The continuity of the value functions there is only used to investigate
the viscosity properties of the value functions. The proof of the dynamic
programming principle nevertheless is not employed. However, in our paper,
notice in particular the value function is defined via the solution of BSDE,
the first issue is to ensure the existence and uniqueness of solution. Thus,
the Lipschitz continuity of $f$ and $\Phi$ is generally required. Besides, In
order to prove the law-invariant of value function, authors in [33] present
the following important condition: The functions $f$ and $\Phi$ are locally
uniformly continuous in $\left(x,\mu,y,z,\nu\right)$ uniformly with respect to
$u$. That is, for each $\varepsilon>0$ and $n\in N,$ there exists
$\delta=\delta_{\left(\varepsilon,n\right)}$ such that, for every $u\in U,$
$\left(x,\mu,y,z,\nu\right),\left(x^{\prime},\mu^{\prime},y^{\prime},z^{\prime},\nu^{\prime}\right)\in\mathbb{R}^{n}\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\times\mathbb{R\times
R}^{d}\times\mathcal{P}_{2}\left(\mathbb{R}\times\mathbb{R}^{d}\right),$
whenever
$\left|x\right|+\mathcal{W}_{2}\left(\left(\mu,\nu\right),\delta_{\left(0,0\right)}\right)\leq
n$ and
$\left|x^{\prime}\right|+\mathcal{W}_{2}\left(\left(\mu^{\prime},\nu^{\prime}\right),\delta_{\left(0,0\right)}\right)\leq
n$,

$\left|x-x^{\prime}\right|+\left|y-y^{\prime}\right|+\left|z-z^{\prime}\right|+\mathcal{W}_{2}\left(\left(\mu,\nu\right),\left(\mu^{\prime},\nu^{\prime}\right)\right)\leq\delta,$

we have

$\displaystyle\left|f\left(x,y,z,\mu,u\right)-f\left(x^{\prime},\mu^{\prime},y^{\prime},z^{\prime},\nu^{\prime}\right)\right|$
$\displaystyle\leq$ $\displaystyle\varepsilon,$
$\displaystyle\left|\Phi\left(x,\mu\right)-\Phi\left(x^{\prime},\mu^{\prime}\right)\right|$
$\displaystyle\leq$ $\displaystyle\varepsilon.$

Whilst, in [33] they put the following assumption: There exists a constant
$C>0$ such that

$\displaystyle\left|f\left(x,y,z,\mu,u\right)\right|+\left|\Phi\left(x,\mu\right)\right|$
$\displaystyle\leq$ $\displaystyle
C\bigg{(}1+\left|x\right|^{2}+\left|y\right|^{2}+\left|z\right|^{2}$
$\displaystyle+\int_{\mathbb{R}^{n}}\left|\Theta_{1}\right|^{2}\mu\left(\mathrm{d}\Theta_{1}\right)+\int_{\mathbb{R}\times\mathbb{R}^{d}}\left|\Theta_{2}\right|^{2}\nu\left(\mathrm{d}\Theta_{2}\right)+\left|u-u^{\prime}\right|^{2}\bigg{)}.$

This condition is weaker then (A4).

###### Remark 2.4

The reason why we assume that the $\sigma$-algebra $\mathcal{G}$ is countably
generated upto null set, is to ensures that the Hilbert space
$L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$ is
separable (cf. [32], Page. 92).

Under Assumptions (A1)-(A5), for any
$\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$, one can prove the uniqueness
and existences of solution to (2) by classical approach (see Theorem A.1 in
[52]), denoted by
$\left(X_{s}^{t,\xi;u},Y_{s}^{t,\xi;u},Z_{s}^{t,\xi;u}\right)\in\mathcal{S}_{\mathcal{F}}^{2}\left(t,T;\mathbb{R}^{n}\right)\times\mathcal{S}_{\mathcal{F}^{0}}^{2}\left(t,T;\mathbb{R}\right)\times\mathcal{M}_{\mathcal{F}^{0}}^{2}\left(t,T;\mathbb{R}^{d}\right)$.
Moreover, applying standard estimates, Burkholder–Davis–Gundy inequality, we
have

$\mathbb{E}\left[\sup_{t\leq s\leq
T}\left|X_{s}^{t,\xi;u}\right|^{2}\right]\leq
C\left(1+\mathbb{E}\left|\xi\right|^{2}\right)$ (4)

where the constant $C$ is independent on $u,$ $\xi$ and $t$ (see Lemma 2.1 in
[22]). Under the Lipschitz condition in (A1)-(A2), employing the standard
arguments, namely, Burkholder–Davis–Gundy and Gronwall lemma, we get the
following estimate similar to the ones for controlled diffusion processes (cf.
Theorem 5.9 and Corollary 5.10, Chapter 2, in [46]):

$\mathbb{E}\left[\sup_{t\leq\theta\leq
T}\left|X_{\theta}^{t,\xi;u}-X_{\theta}^{s,\xi^{\prime};u}\right|^{2}\right]\leq
C\left[\mathbb{E}\left|\xi-\xi^{\prime}\right|^{2}+\left(1+\mathbb{E}\left|\xi\right|^{2}+\mathbb{E}\left|\xi^{\prime}\right|^{2}\right)\left|s-t\right|\right].$
(5)

In fact, we have a more delicate estimate as follows:

###### Lemma 2.1

Under the Assumptions _(A1)-(A2)_ , for $p\geq 2$, there exists a positive
constant $C_{p}$ depending on the Lipschitz constants of $b$, $\sigma$ and
$T$, such that for all $t\in[0,T]$, $\xi,\xi^{\prime}\in
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right),$
$u,u^{\prime}\in\mathcal{U}_{ad}^{2}\left[t,T\right],$ we have

$\mathbb{E}\left[\sup_{t\leq\theta\leq
T}\left|X_{\theta}^{t,\xi;u}-X_{\theta}^{t,\xi^{\prime};u^{\prime}}\right|^{p}\right]\leq
C_{p}\left[\mathcal{W}_{2}^{p}\left(\mathbb{P}_{\xi},\mathbb{P}_{\xi^{\prime}}\right)+\Upsilon_{0,T}^{u,u^{\prime};p}\right],$
(6)

where

$\Upsilon_{0,T}^{u,u^{\prime};p}=\mathbb{E}\left[\int_{0}^{T}\left|u_{s}-u_{s}^{\prime}\right|^{p}\mathrm{d}\theta\right].$
(7)

Proof For read’s convenience, we display the main step as follows. To this
end, denote
$\hat{X}_{\theta}=X_{\theta}^{t,\xi;u}-X_{\theta}^{t,\xi^{\prime};u^{\prime}},$
$\hat{\xi}=\xi-\xi^{\prime}$ and Applying Burkholder-Davis-Gundy inequality,
Lipschitz condition (A2) and Gronwall lemma, we derive that

$\mathbb{E}\left[\sup_{t\leq\theta\leq
s}\left|\hat{X}_{\theta}\right|^{p}\right]\leq
C_{p}\left|\hat{\xi}\right|^{p}+C_{p}\int_{t}^{s}\Big{[}\left|u_{s}-u_{s}^{\prime}\right|^{p}+\mathcal{W}_{2}^{p}\left(\mathbb{P}_{X_{\theta}^{t,\xi;u}}^{W},\mathbb{P}_{X_{\theta}^{t,\xi^{\prime};u^{\prime}}}^{W}\right)\Big{]}\mathrm{d}\theta.$
(8)

From the definition of $\mathcal{W}_{2},$ we have

$\mathcal{W}_{2}^{2}\left(\mathbb{P}_{X_{\theta}^{t,\xi;u}}^{W},\mathbb{P}_{X_{\theta}^{t,\xi^{\prime};u^{\prime}}}^{W}\right)\leq\mathbb{E}\left[\left|\hat{X}_{\theta}\right|^{2}\right].$

Immediately, for $\theta\in\left[t,T\right]$

$\mathcal{W}_{2}^{2}\left(\mathbb{P}_{X_{\theta}^{t,\xi;u}}^{W},\mathbb{P}_{X_{\theta}^{t,\xi^{\prime};u^{\prime}}}^{W}\right)\leq
C\mathbb{E}\left[\left|\hat{\xi}\right|^{2}\right]+C\int_{t}^{\theta}\mathcal{W}_{2}^{2}\left(\mathbb{P}_{X_{\theta}^{t,\xi;u}}^{W},\mathbb{P}_{X_{\theta}^{t,\xi^{\prime};u^{\prime}}}^{W}\right)\mathrm{d}\theta+\Upsilon_{0,T}^{u,u^{\prime};p}.$

Gronwall lemma yields

$\sup_{\theta\in\left[t,T\right]}\mathcal{W}_{2}^{2}\left(\mathbb{P}_{X_{\theta}^{t,\xi;u}}^{W},\mathbb{P}_{X_{\theta}^{t,\xi^{\prime};u^{\prime}}}^{W}\right)\leq
C\mathbb{E}\left[\left|\hat{\xi}\right|^{2}+\Upsilon_{0,T}^{u,u^{\prime};p}\right]$

and taking into account the arbitrariness of $\xi,\xi^{\prime}\in
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right),i=1,2,$ we get

$\sup_{\theta\in\left[t,T\right]}\mathcal{W}_{2}^{2}\left(\mathbb{P}_{X_{\theta}^{t,\xi;u}}^{W},\mathbb{P}_{X_{\theta}^{t,\xi^{\prime};u^{\prime}}}^{W}\right)\leq
C\left[\mathcal{W}_{2}^{2}\left(\mathbb{P}_{\xi},\mathbb{P}_{\xi^{\prime}}\right)+\Upsilon_{0,T}^{u,u^{\prime};p}\right].$

This together with (8) completes the proof.

###### Remark 2.5

Immediately, we have

$\sup_{\theta\in\left[t,T\right]}\mathcal{W}_{2}^{2}\left(\mathbb{P}_{X_{\theta}^{t,\xi;u}}^{W},\mathbb{P}_{X_{\theta}^{t,\xi^{\prime};u}}^{W}\right)\leq
C\mathcal{W}_{2}^{2}\left(\mathbb{P}_{\xi},\mathbb{P}_{\xi^{\prime}}\right),$

which means for uncontrolled forward system, its solution depends on the law
of $\xi.$

For any $u\in\Lambda_{t,\xi},$ we know that
$\left\\{X_{s}^{t,\xi;u}\right\\}_{t\leq s\leq T},$ which is
$\mathcal{F}$-adapted and $W$ is a $\left(\Omega,\mathcal{F}\right)$ Wiener
process. Therefore, it derives that

$\mathbb{P}_{X_{s}^{t,\xi;u}}^{W}\left(\mathrm{d}x\right)=\mathbb{P}\left(\left.X_{s}^{t,\xi;u}\in\mathrm{d}x\right|\mathcal{F}^{0}\right)=\mathbb{P}\left(\left.X_{s}^{t,\xi;u}\in\mathrm{d}x\right|\mathcal{F}_{s}^{0}\right)$

thus for any $\psi\in C_{q}\left(\mathbb{R}^{n}\right)$

$\mathbb{P}_{X_{s}^{t,\xi;u}}^{W}\left(\psi\right)=\mathbb{E}\left[\left.\psi\left(X_{s}^{t,\xi;u}\right)\right|\mathcal{F}^{0}\right]=\mathbb{E}\left[\left.\psi\left(X_{s}^{t,\xi;u}\right)\right|\mathcal{F}_{s}^{0}\right],\text{
}t\leq s\leq T,$

which indicates that
$\left\\{\mathbb{P}_{X_{s}^{t,\xi;u}}^{W}\left(\psi\right)\right\\}_{t\leq
s\leq T}$ is $\left(\mathcal{F}^{0}\right)_{t\leq s\leq T}$-measurable. We
define the map
$\mathbb{P}_{X_{s}^{t,\xi;u}\left(\omega\right)}^{W}:\omega\rightarrow\mathcal{P}_{2}\left(\mathbb{R}^{n}\right).$
Then, $\mathbb{P}_{X_{s}^{t,\xi;u}}^{W}$ is $\mathcal{F}_{s}^{0}$-measurable
if and only if $\mathbb{P}_{X_{s}^{t,\xi;u}}^{W}\left(\psi\right)$ is
$\mathcal{F}_{s}^{0}$-measurable. Therefore, we report that
$\left\\{\mathbb{P}_{X_{s}^{t,\xi;u}}^{W}\right\\}_{t\leq s\leq T}$ is
$\left(\mathcal{F}_{s}^{0}\right)_{t\leq s\leq T}$-adapted. Moreover, we have
the following flow property for forward SDE in (2)

$X_{r}^{t,\xi;u}=X_{r}^{s,X_{s}^{t,\xi;u};u},\text{ for any
}r\in\left[s,T\right],\text{ }\mathbb{P}\text{-a.s.,}$

which implies that

$\mathbb{P}_{X_{r}^{t,\xi;u}}^{W}=\mathbb{P}_{X_{r}^{s,X_{s}^{t,\xi;u};u}}^{W},\text{
for any }r\in\left[s,T\right],\text{ }\mathbb{P}\text{-a.s.}$

###### Definition 2.2

Given any $s\in\left[t,T\right],$ $\xi\in
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right),$ with $\mathbb{P}_{\xi}^{W}=\mu$
and $u\in\mathcal{U}_{ad}^{2}\left[0,T\right],$ we define

$\rho_{s}^{t,\mu;u}=\mathbb{P}_{X_{s}^{t,\xi;u}}^{W}.$ (9)

From Lemma 3.1 in [45], we know that $\rho_{s}^{t,\mu;u}$ defines a square
integrable $\mathcal{F}^{0}$-progressive continuous process in
$\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$. The map
$\left(s,t,\omega^{0},\mu,u\right)\in\left[0,T\right]\times\left[0,T\right]\times\Omega^{0}\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\times\mathcal{U}_{ad}^{2}\left[0,T\right]\rightarrow\rho_{s}^{t,\mu;u}\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$
is measurable and admits the following flow property:

$\rho_{s}^{t,\mu;u}=\rho_{s}^{\tau,\rho_{\tau}^{t,\mu;u};u^{\tau}},$ (10)

where $u^{\tau}$ is defined in Definition 2.1 for all
$\mathcal{F}^{0}$-stopping time $\tau$ valued in $\left[t,T\right],$ such a
set is denoted by $\mathbb{T}_{t,T}^{0}.$

We now deal with

$\displaystyle\mathbb{E}^{\mathcal{F}_{s}^{0}}\left[f\left(X_{s}^{t,\xi,u},\mathbb{P}_{X_{s}^{t,\xi;u}}^{W},\Theta_{s}^{t,\xi,u},\mathbb{P}_{\Theta_{s}^{t,\xi;u}},u_{s}\right)\right]$
$\displaystyle=$
$\displaystyle\rho_{s}^{t,\mu;u}\left(f\left(\cdot,\rho_{s}^{t,\mu;u},\Theta_{s}^{t,\xi,u},\mathbb{P}_{\Theta_{s}^{t,\xi;u}},u_{s}\right)\right)$
$\displaystyle=$ $\displaystyle
f^{0}\left(\rho_{s}^{t,\mu;u},\Theta_{s}^{t,\xi,u},\mathbb{P}_{\Theta_{s}^{t,\xi;u}},u_{s}\right),$

and

$\displaystyle\mathbb{E}^{\mathcal{F}_{T}^{0}}\left[\Phi\left(X_{T}^{t,\xi;u},\mathbb{P}_{X_{s}^{t,\xi;u}}^{W}\right)\right]$
$\displaystyle=$
$\displaystyle\rho_{T}^{t,\mu;u}\left(\Phi\left(\cdot,\rho_{T}^{t,\mu;u}\right)\right)$
$\displaystyle=$ $\displaystyle\Phi^{0}\left(\rho_{T}^{t,\mu;u}\right)$

with the functions
$f^{0}:\left(\mu,y,z,\mathbb{\pi},u\right)\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\times\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{P}_{2}\left(\mathbb{R}\times\mathbb{R}^{d}\right)\times
U\rightarrow\mathbb{R}$ and
$\Phi^{0}:\mu\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\rightarrow\mathbb{R},$
defined by

$\displaystyle f^{0}\left(\mu,y,z,\mathbb{\pi},u\right)$ $\displaystyle=$
$\displaystyle\mu\left(f\left(\cdot,\mu,y,z,\mathbb{\pi},u\right)\right)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}f\left(x,\mu,y,z,\mathbb{\pi},u\right)\mu\left(\mathrm{d}x\right)$

and

$\Phi^{0}\left(\mu\right)=\mu\left(\Phi\left(\cdot,\mu\right)\right)=\int_{\mathbb{R}^{n}}\Phi\left(x,\mu\right)\mu\left(\mathrm{d}x\right).$

Then FBSDEs (2) can be equivalently expressed as follows:

$\left\\{\begin{array}[]{lll}\mathrm{d}X_{s}^{t,\xi;u}&=&b\left(X_{s}^{t,\xi;u},\mathbb{P}_{X_{s}^{t,\xi;u}}^{W},u_{s}\right)\mathrm{d}s+\sigma\left(X_{s}^{t,\xi;u},\mathbb{P}_{X_{s}^{t,\xi;u}}^{W},u_{s}\right)\mathrm{d}W_{s},\\\
X_{t}^{t,\xi;u}&=&\xi,\\\
\mathrm{d}Y_{s}^{t,\xi;u}&=&-f^{0}\left(\rho_{s}^{t,\mu;u},\Theta_{s}^{t,\xi,u},\mathbb{P}_{\Theta_{s}^{t,\xi;u}},u_{s}\right)\mathrm{d}s+Z_{s}^{t,\xi;u}\mathrm{d}W_{s},\\\
Y_{T}^{t,\xi;u}&=&\Phi^{0}\left(\rho_{T}^{t,\mu;u}\right).\end{array}\right.$
(11)

However, the new expression of BSDE in (11) means that the solution
$Y^{t,\xi;u}$ depends on $\xi$ only through its distribution
$\mu=\rho_{t}^{t,\mu;u}=\mathcal{L}\left(\xi\right).$ So (11) can be written
as:

$\left\\{\begin{array}[]{lll}\mathrm{d}X_{s}^{t,\xi;u}&=&b\left(X_{s}^{t,\xi;u},\mathbb{P}_{X_{s}^{t,\xi;u}}^{W},u_{s}\right)\mathrm{d}s+\sigma\left(X_{s}^{t,\xi;u},\mathbb{P}_{X_{s}^{t,\xi;u}}^{W},u_{s}\right)\mathrm{d}W_{s},\\\
X_{t}^{t,\xi;u}&=&\xi,\\\
\mathrm{d}Y_{s}^{t,\mu;u}&=&-f^{0}\left(\rho_{s}^{t,\mu;u},\Theta_{s}^{t,\mu;u},\mathbb{P}_{\Theta_{s}^{t,\mu;u}},u_{s}\right)\mathrm{d}s+Z_{s}^{t,\mu;u}\mathrm{d}W_{s},\\\
Y_{T}^{t,\xi;u}&=&\Phi^{0}\left(\rho_{T}^{t,\mu;u}\right).\end{array}\right.$
(12)

###### Remark 2.6

The merit to write the dependence on $\mu$ in $Y_{s}^{t,\mu;u}$ is embodied in
the convenience to derive the DPP by means of the flow property of
$\rho^{t,\mu;u}$ (see 2.13). As claimed in Introduction, the DPP for mean-
field FBSDEs (13) does not hold true anymore because of the presence of
expectation terms in coefficients. Nevertheless, it is possible to capture
some form of the DPP by extending the state space, namely, the flow of the
conditional law of the controlled forward state process. So this formulation
is different from [15, 16]. In addition, if employing the conditional law, one
must handle the dependence of value function on conditional law, i.e. law-
invariant (see Theorem 2.1 below).

###### Remark 2.7

Buckdahn, Li and Peng [16] (see also [38] for fully coupled case) considered
mean-field FBSDEs in a more general framework, with general coefficient like

$\left\\{\begin{array}[]{lll}\mathrm{d}X_{s}^{t,\zeta}&=&\mathbb{E}^{\prime}\left[b\left(s,\left(X_{s}^{0,x_{0}}\right)^{\prime},X_{s}^{t,\zeta}\right)\right]\mathrm{d}s+\mathbb{E}^{\prime}\left[\sigma\left(s,\left(X_{s}^{0,x_{0}}\right)^{\prime},X_{s}^{t,\zeta}\right)\right]\mathrm{d}W_{s},\\\
X_{t}^{t,\zeta}&=&\zeta\in
L^{2}\left(\Omega,\mathcal{F}_{t}^{0},\mathbb{P};\mathbb{R}^{n}\right),\\\
\mathrm{d}Y_{s}^{t,\zeta}&=&-\mathbb{E}^{\prime}\left[f\left(s,\left(X_{s}^{0,x_{0}}\right)^{\prime},X_{s}^{t,\zeta},\left(Y_{s}^{0,x_{0}}\right)^{\prime},Y_{s}^{t,\zeta},Z_{s}^{t,\zeta}\right)\right]\mathrm{d}s+Z_{s}^{t,\zeta}\mathrm{d}W_{s},\\\
Y_{T}^{t,\zeta}&=&\mathbb{E}^{\prime}\left[\Phi\left(\left(X_{s}^{0,x_{0}}\right)^{\prime},X_{T}^{t,\zeta}\right)\right],\end{array}\right.$
(13)

where the driving coefficient of (13) has to be interpreted as follows (the
other coefficients can be done in the same way):

$\displaystyle\mathbb{E}^{\prime}\left[f\left(s,\left(X_{s}\right)^{\prime},X_{s},\left(Y_{s}\right)^{\prime},Y_{s},Z_{s}\right)\right]$
$\displaystyle=$
$\displaystyle\int_{\Omega}f\left(\omega^{\prime},\omega,s,X_{s}\left(\omega^{\prime}\right),X_{s}\left(\omega\right),Y_{s}\left(\omega^{\prime}\right),Y_{s}\left(\omega\right),Z_{s}\left(\omega\right)\right)P\left(\mathrm{d}\omega^{\prime}\right).$

Clearly, $f$ does not dependent on the law of solutions. For the convenience
of readers, let us depict briefly the procedure to solve (13). Under certain
assumptions (for instance, Lipschitz and linear growth conditions), SDE in
(13) with $\left(t,\zeta\right)=\left(0,x_{0}\right)$ admits a unique strong
solution. Once we know $X_{s}^{0,x_{0}}$, SDE in (13) with
$\left(t,\zeta\right)$ becomes a classical equation. Analogously, we first
deal with BSDE in (13) for $\left(t,\zeta\right)=\left(0,x_{0}\right),$ from
Theorem 3.1 in [16] that there exists a unique solution
$\left(Y_{s}^{0,x_{0}},Z_{s}^{0,x_{0}}\right)_{t\leq s\leq T}.$ Once we have
$\left(Y_{s}^{0,x_{0}},Z_{s}^{0,x_{0}}\right)_{t\leq s\leq T},$ BSDE in (13)
reduces to a classical BSDE of the following type.

Pardoux and Peng ([41], 1990) introduced the following nonlinear BSDE:

$Y_{t}=\xi+\int_{t}^{T}f\left(s,Y_{s},Z_{s}\right)\mathrm{d}s-\int_{t}^{T}Z_{s}\mathrm{d}W_{s}$
(14)

where $\xi\in L_{\mathcal{F}_{T}}^{2}\left(0,T;\mathbb{R}\right),$ under
certain conditions for the generator $f$, BSDE (14) admits a unique strong
adapted solution
$\left(Y\left(\cdot\right),Z\left(\cdot\right)\right)\in\mathcal{S}_{\mathcal{F}^{0}}^{2}\left(t,T;\mathbb{R}\right)\times\mathcal{M}_{\mathcal{F}^{0}}^{2}\left(t,T;\mathbb{R}^{d}\right)$.
Besides, we need the following estimations for BSDE, whose proof can be seen
in Proposition 3.2 of Briand et al. [5].

###### Lemma 2.2

Let $\left(y^{i},z^{i}\right),$ $i=1,2,$ be the solution to the following

$y^{i}\left(t\right)=\xi^{i}+\int_{t}^{T}f^{i}\left(s,y_{s}^{i},z_{s}^{i}\right)\mathrm{d}s-\int_{t}^{T}z_{s}^{i}\mathrm{d}W_{s},$
(15)

where $\xi^{i}\in L^{2}\left(\mathcal{F}_{T};\mathbb{R}\right)$ satisfying
$\mathbb{E}\left[\left|\xi^{i}\right|^{\beta}\right]<\infty,$
$f^{i}=f^{i}\left(s,y^{i},z^{i}\right):\left[0,T\right]\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
satisfies Lipschitz in $\left(y,z\right)$ and

$\mathbb{E}\left[\left(\int_{t}^{T}\left|f^{i}\left(s,y_{s}^{i},z_{s}^{i}\right)\right|\mathrm{d}s\right)^{\beta}\right]<\infty.$

Then, for some $\beta\geq 2,$ there exists a positive constant $C_{\beta}$
(depending on $\beta,$ $T$ and Lipschitz constant) such that

$\displaystyle\mathbb{E}\left[\sup_{0\leq t\leq
T}\left|y_{t}^{1}-y_{t}^{2}\right|^{\beta}+\left(\int_{0}^{T}\left|z_{s}^{1}-z_{s}^{2}\right|^{2}\mathrm{d}s\right)^{\frac{\beta}{2}}\right]$
(16) $\displaystyle\leq$ $\displaystyle
C_{\beta}\mathbb{E}\Bigg{[}\left|\xi^{1}-\xi^{2}\right|^{\beta}+\left(\int_{t}^{T}\left|f^{1}\left(s,y_{s}^{1},z_{s}^{1}\right)-f^{2}\left(s,y_{s}^{1},z_{s}^{1}\right)\right|\mathrm{d}s\right)^{\beta}\Bigg{]}.$

Particularly, whenever putting $\xi^{2}=0,$ $f^{2}=0,$ one has

$\mathbb{E}\left[\sup_{0\leq t\leq
T}\left|y_{t}^{1}\right|^{\beta}+\left(\int_{0}^{T}\left|z_{s}^{1}\right|^{2}\mathrm{d}s\right)^{\frac{\beta}{2}}\right]\leq
C_{\beta}\mathbb{E}\Bigg{[}\left|\xi^{1}\right|^{\beta}+\left(\int_{t}^{T}\left|f^{1}\left(s,0,0\right)\right|\mathrm{d}s\right)^{\beta}\Bigg{]}.$
(17)

###### Remark 2.8

Notice that Briand et al. [5] considered the BSDE (14) under rather weak
assumptions on the data. They proved the existence and uniqueness of solutions
in $L^{p},$ for $p>1$, extending the condition of Lipschitz to the case where
the monotonicity conditions are satisfied. Moreover, under an additional
assumption, they reported an existence and uniqueness result for BSDE on a
fixed time interval, when the data $\xi$ are only in $L^{1}$. However, in our
paper, we only study the case $L^{p},$ for $p\geq 2.$

###### Lemma 2.3

Suppose that the Assumptions _(A1)-(A4)_ are in force. Then, for $p\geq 2,$
there exists a constant $C_{p}>0$ depending on the Lipschitz constant of
$b,\sigma,\Phi,f,$ such that, for $\xi,\xi^{\prime}\in
L^{2}\left(\mathcal{F}_{t};\mathbb{R}^{n}\right)$

$\mathbb{E}\left[\sup_{t\leq s\leq
T}\left|Y_{s}^{t,\xi;u}\right|^{p}+\left(\int_{t}^{T}\left|Z_{s}^{t,\xi;u}\right|^{2}\mathrm{d}s\right)^{\frac{p}{2}}\right]\leq
C_{p},$ (18) $\displaystyle\mathbb{E}\left[\sup_{t\leq s\leq
T}\left|Y_{s}^{t,\xi;u}-Y_{s}^{t,\xi^{\prime};u^{\prime}}\right|^{p}+\left(\int_{t}^{T}\left|Z_{s}^{t,\xi;u}-Z_{s}^{t,\xi^{\prime};u^{\prime}}\right|^{2}\mathrm{d}s\right)^{\frac{p}{2}}\right]$
(19) $\displaystyle\leq$ $\displaystyle
C_{p}\left[\Upsilon_{0,T}^{u,u^{\prime};p}+\mathcal{W}_{2}^{p}\left(\mathbb{P}_{\xi},\mathbb{P}_{\xi^{\prime}}\right)\right],$

where $\Upsilon_{0,T}^{u,u^{\prime};p}$ is defined in (8), and

$\int_{t}^{T}\mathcal{W}_{2}^{2}\left(\mathbb{P}_{\Theta_{s}^{t,\xi,u}},\mathbb{P}_{\Theta_{s}^{t,\xi^{\prime},u}}\right)\leq
C_{p}\left[\Upsilon_{0,T}^{u,u^{\prime};p}+\mathcal{W}_{2}^{p}\left(\mathbb{P}_{\xi},\mathbb{P}_{\xi^{\prime}}\right)\right].$
(20)

The proof is displayed in Appendix.

We now state a comparison theorem for BSDE which is vital to derive the DPP.

######  Lemma 2.4 (Comparison theorem)

Let $\left(Y^{i},Z^{i}\right),$ $i=1,2,$ be the solution to the following

$Y_{t}^{i}=\zeta^{i}+\int_{t}^{T}g^{i}\left(Y_{s}^{i},Z_{s}^{i},\mathbb{P}_{\left(Y_{s}^{i},Z_{s}^{i}\right)}\right)\mathrm{d}s-\int_{t}^{T}Z_{s}^{i}W_{s}\mathrm{d}s,\text{
}i=1,2,$ (21)

where $\zeta^{i}\in L^{2}\left(\mathcal{F}_{T};\mathbb{R}^{n}\right)$ and
$g^{i}:\mathbb{R\times
R}^{d}\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\times\mathbb{R}^{n\times
d}\right)\rightarrow\mathbb{R}$ satisfies the conditions (A3)-(A4). Then,
under Assumptions _(A3)-(A4)_ , BSDE (21) admits a unique adapted solution
$\left(Y^{i},Z^{i}\right)\in\mathcal{S}_{\mathcal{F}}^{2}\left(t,T;\mathbb{R}\right)\times\mathcal{M}_{\mathcal{F}}^{2}\left(t,T;\mathbb{R}^{n\times
d}\right)$, respectively, for $i=1,2$. Furthermore, if the following
conditions hold, (i) $\zeta^{1}\geq\zeta^{2},$ $\mathbb{P}$-a.s.; (ii)
$g^{1}\left(Y,Z,\gamma\right)\geq g^{2}\left(Y,Z,\gamma\right),$ for any
$\left(Y,Z,\gamma\right)\in\mathbb{R\times R}^{n\times
d}\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\times\mathbb{R}^{n\times
d}\right).$ Then, we have $Y_{t}^{1}\geq Y_{t}^{2},$ $\mathbb{P}$-a.s., for
all $t\in\left[0,T\right].$

The proof can be found in Appendix.

In [45], the authors applied the measurable selection arguments and shifted
control technique to derive DPP, while Cosso et al. [22] proved the DPP
inspired by the classical approach for deterministic differential games. Under
general conditions (Markovian or non-Markovian), Djete et al. (cf. [33])
recently developed the classical measurable selection, conditioning and
concatenation arguments for common noise framework, and established the DPP.
In our paper, we will deal with the control problem driven by FBSDEs of
McKean-Vlasov type. The application of BSDE approach, in particular, the use
of the notion of stochastic backward semigroups introduced by Peng [42] allows
a straightforward proof of a DPP for value functions associated with
stochastic optimal control problems. We define the family of backward
semigroups associated with the FBSDEs (12).

Given the initial data $\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$, a positive number $\delta\leq
T-t$, an admissible control process $u\in\Lambda_{t,\xi}$ and a real-valued
random variable $\eta\in
L^{2}\left(\mathcal{F}_{t+\delta}^{0};\mathbb{R}^{n}\right)$, we set

$\mathbb{G}_{s,t+\delta}^{t,\xi;u}\left[\eta\right]:=\mathcal{Y}_{s}^{t,\xi;u},\text{
}t\leq s\leq t+\delta,$

where $\left(\mathcal{Y}_{s}^{t,\xi;u},\mathcal{Z}_{s}^{t,\xi;u}\right)_{t\leq
s\leq t+\delta}$ is the solution to the following BSDE of McKean-Vlasov type
with the time horizon $t+\delta:$

$\displaystyle\mathcal{Y}_{s}^{t,\xi;u}$ $\displaystyle=$
$\displaystyle\eta+\int_{s}^{t+\delta}f^{0}\left(\rho_{r}^{t,\xi;u},\mathcal{Y}_{r}^{t,\xi;u},\mathcal{Z}_{r}^{t,\xi;u},\mathbb{P}_{\left(\mathcal{Y}_{r}^{t,\xi;u},\mathcal{Z}_{r}^{t,\xi;u}\right)},u_{r}\right)\mathrm{d}r$
(22)
$\displaystyle-\int_{s}^{t+\delta}\mathcal{Z}_{r}^{t,\xi;u}\mathrm{d}W_{r},$

where $\rho_{r}^{t,\xi;u}$ is defined in (9).

Apparently, the solution of
$\left(Y_{s}^{t,\mu,u},Z_{s}^{t,\mu,u}\right)_{t\leq s\leq T}$ of BSDE in (12)
admits

$\mathbb{G}_{t,T}^{t,\mu;u}\left[\Phi^{0}\left(\rho_{T}^{t,\mu;u}\right)\right]=\mathbb{G}_{t,t+\delta}^{t,\mu;u}\left[Y_{t+\delta}^{t,\mu;u}\right].$
(23)

On the one hand, we are in the Markovian case; On the other hand, Lemma 2.1
and Lemma 2.2 once again indicates that, the processes
$\left(X_{s}^{t,\xi,u},Y_{s}^{t,\xi,u},Z_{s}^{t,\xi,u}\right)$ depend on $\xi$
only through its distribution, which means
$\left(X_{s}^{t,\xi,u},Y_{s}^{t,\xi,u},Z_{s}^{t,\xi,u}\right)_{t\leq s\leq T}$
and
$\left(X_{s}^{t,\xi^{\prime},u},Y_{s}^{t,\xi^{\prime},u},Z_{s}^{t,\xi^{\prime},u}\right)_{t\leq
s\leq T}$ are indistinguishable as long as $\xi$ and $\xi^{\prime}$ have the
same distribution. Besides, from (23), we observe that
$Y_{t+\delta}^{t,\mu;u}$ is
$\mathcal{F}_{t+\delta}^{0}\vee\mathcal{G}$-adapted.

Next we will prove that $Y_{t}^{t,\xi,u}$ is not random whenever
$\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right).$ To this end, we first state the
following result.

###### Lemma 2.5

Assume that $\tilde{f}$ satisfies _(A3)-(A4)_. For any fixed
$t_{0}\in\left[0,T\right]$, if for every
$\left(y,z,\mu\right)\in\mathbb{R\times
R}^{d}\times\mathcal{P}_{2}\left(\mathbb{R}\times\mathbb{R}^{d}\right)$,
$\tilde{f}\left(\cdot,y,z,\mu\right)$-adapted on the interval
$\left[t_{0},T\right]$ and $\zeta\in
L^{2}\left(\left(\mathcal{F}^{0}\right)_{T}^{t_{0}};\mathbb{R}\right),$ then
the solution $\left(y_{s},z_{s}\right)_{t_{0}\leq s\leq T}$ to the following
BSDE is also $\mathcal{F}_{s}^{t_{0}}$-adapted on the interval
$\left[t_{0},T\right]$:

$y_{s}=\zeta+\int_{s}^{T}\tilde{f}\left(s,y_{s},z_{s},\mathbb{P}_{\left(y_{s},z_{s}\right)}\right)\mathrm{d}s-\int_{s}^{T}z_{s}\mathrm{d}W_{s}.$
(24)

Particularly, $y_{t_{0}}$ is a deterministic constant.

Proof We set $W_{s}^{t_{0}}=W_{s}-W_{t_{0}}.$ Consider the following BSDE on
the interval $\left[t_{0},T\right]$:

$\tilde{y}_{s}=\zeta+\int_{s}^{T}\tilde{f}\left(s,\tilde{y}_{s},\tilde{z}_{s},\mathbb{P}_{\left(\tilde{y}_{s},\tilde{z}_{s}\right)}\right)\mathrm{d}s-\int_{s}^{T}\tilde{z}_{s}\mathrm{d}W_{s}^{t_{0}}.$
(25)

It is easy to check that
$\int_{t}^{T}z_{s}\mathrm{d}W_{s}=\int_{s}^{T}\tilde{z}_{s}\mathrm{d}W_{s}^{t_{0}}.$
Then, according to existence and uniqueness of (24), the solution of
$\left(y_{s},z_{s}\right)_{t_{0}\leq s\leq T}$ coincides with
$\left(\tilde{y}_{s},\tilde{z}_{s}\right)_{t_{0}\leq s\leq T}$ which indicates
that $\left(y_{s},z_{s}\right)_{t_{0}\leq s\leq T}$ is
$\mathcal{F}_{s}^{t_{0}}$-adapted.

Note that if the initial state $\tilde{\xi}\in
L^{2}\left(\mathcal{F}_{t}^{0}\vee\mathcal{G};\mathbb{R}^{n}\right)$ with
$\mathcal{L}\left(\tilde{\xi}\right)=\mu,$ and $u\in\Lambda_{t,\xi},$ then we
claim that $Y_{t}^{t,\tilde{\xi};u}$ is random.

###### Lemma 2.6

For any $\left(t,\tilde{\xi}\right)\in\left[0,T\right]\times
L^{2}\left(\mathcal{F}_{t}^{0}\vee\mathcal{G};\mathbb{R}^{n}\right)$ with
$\mathcal{L}\left(\tilde{\xi}\right)=\mu,$ and $u\in\Lambda_{t,\tilde{\xi}},$
$Y_{t}^{t,\tilde{\xi};u}$ is $\mathcal{F}_{t}^{0}\vee\mathcal{G}$ measurable.

The proof can be found in Appendix.

However, whenever we are in the Markovian case (deterministic coefficients).
The following lemma claims that $Y_{t}^{t,\xi;u}$ is _deterministic_ if __
$\xi\in L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$.

###### Lemma 2.7

Under the Assumptions _(A1)-(A5)_ , for any
$\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right),$
$\left.Y_{s}^{t,\xi;u}\right|_{s=t}$ is deterministic.

Proof For any $\left(t,x\right)\in\left[0,T\right]\times\mathbb{R}^{n}$ and
$u\in\Lambda_{t,\xi},$ $Y_{t}^{t,x;u}$ is deterministic. Indeed, the solution
$X_{s}^{t,x;u}$ $\left(\rho_{s}^{t,\mu;u}\right)$ of SDE in (11) is
$\left(\mathcal{F}^{0}\right)_{s}^{t}$ -adapted which also means
$X_{T}^{t,x;u}$ $\left(\rho_{T}^{t,\mu;u}\right)$ is
$\left(\mathcal{F}^{0}\right)_{T}^{t}$-measurable and consequently, the term
$f^{0}\left(\rho_{s}^{t,\mu;u},\Theta_{s}^{t,x,u},\mathbb{P}_{\Theta_{s}^{t,x;u}},u_{s}\right)$
in (11) is also $\left(\mathcal{F}^{0}\right)_{s}^{t}$-adapted according to
property of BSDE. Then from Lemma 2.5, the solution
$\left\\{\Theta_{s}^{t,x,u}\right\\}_{t\leq s\leq T}$ is also
$\left(\mathcal{F}^{0}\right)_{s}^{t}$-adapted. In particular, $Y_{t}^{t,x;u}$
is deterministic. Repeating the method in Lemma 2.6, consider a simple random
variable $\xi$ of the form: $\xi=\sum_{i=1}^{N}I_{A_{i}}x^{i}$, and a control
process $u$ of the form
$u_{\left(\cdot\right)}=\sum_{i=1}^{N}I_{A_{i}}u_{\cdot}^{i}$, where
$\left\\{A_{i}\right\\}_{i=1,\ldots,N}$ is a finite partition of
$\left(\Omega,\mathcal{G}\right),$ $x^{i}\in\mathbb{R}^{n}$ and
$u_{\cdot}^{i}$ is $\left(\mathcal{F}^{0}\right)_{\cdot}^{t}$-adapted, for
$1\leq i\leq N.$ For each $i,$ set
$\left(X_{s}^{i},Y_{s}^{i},Z_{s}^{i}\right)=\left(X_{s}^{i},\Theta_{s}^{i}\right)=\left(X_{s}^{t,x^{i},u^{i}},Y_{s}^{t,x^{i},u^{i}},Z_{s}^{t,x^{i},u^{i}}\right),$
for $s\in\left[t,T\right]$. Then $\left(X^{i},Y^{i},Z^{i}\right)$ is the
solution to the FBSDEs

$\left\\{\begin{array}[]{lll}\mathrm{d}X_{s}^{i}&=&b\left(X_{s}^{i},\mathbb{P}_{X_{s}^{i}}^{W},u_{s}^{i}\right)\mathrm{d}s+\sigma\left(X_{s}^{i},\mathbb{P}_{X_{s}^{i}}^{W},u_{s}^{i}\right)\mathrm{d}W_{s},\\\
\mathrm{d}Y_{s}^{i}&=&-f^{0}\left(\mathbb{P}_{X_{s}^{i}}^{W},\Theta_{s}^{i},\mathbb{P}_{\Theta_{s}^{i}},u_{s}^{i}\right)\mathrm{d}s+Z_{s}^{i}\mathrm{d}W_{s},\\\
X_{t}^{i}&=&x^{i},\text{
}Y_{T}^{i}=\Phi^{0}\left(X_{T}^{i},\mathbb{P}_{X_{T}^{i}}^{W}\right).\end{array}\right.$
(26)

Therefore,

$Y_{t}^{t,\xi;u}=\sum_{i=1}^{N}I_{A_{i}}Y_{t}^{i}.$ (27)

Clearly, from Lemma 2.5, each $Y_{t}^{i}$ is deterministic. Now taking the
conditional expectation $\mathbb{E}^{\mathcal{F}_{t}^{0}}$ on both sides of
(27) yields

$\displaystyle\mathbb{E}^{\mathcal{F}_{t}^{0}}\left[Y_{t}^{t,\xi;u}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}^{\mathcal{F}_{t}^{0}}\left[\sum_{i=1}^{N}I_{A_{i}}Y_{t}^{i}\right]$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N}Y_{t}^{i}\mathbb{E}^{\mathcal{F}_{t}^{0}}\left[I_{A_{i}}\right]$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N}Y_{t}^{i}P\left(A_{i}\right)\text{ since
}\mathcal{G}\text{ is independent of }\mathcal{F}_{t}^{0}\text{.}$

Therefore, for simple random variables, we have the desired result. For a
general $\xi\in L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right),$ we can take a
sequence of simple random variables $\left\\{\xi_{i}\right\\}_{i\in N}$ which
converge to $\xi$ in $L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$.
Consequently, from the estimates (19), we have

$\mathbb{E}\left[\left|Y_{t}^{t,\xi_{i};u}-Y_{t}^{t,\xi;u}\right|^{2}\right]\leq
C\mathbb{E}\left[\left|\xi_{i}-\xi\right|^{2}\right],\text{ as
}i\rightarrow\infty.$

Thus we end the proof.

Now the cost functional corresponding to the control $u\in\Lambda_{t,\xi}$ is
defined to be

$J\left(t,\xi;u\right)\triangleq\left.Y_{s}^{t,\xi;u}\right|_{s=t}\text{ }$
(28)

for $\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$ with $\rho_{t}^{t,\mu;u}=\mu.$

It follows from the uniqueness of the solution of (12) and flow property of
$\rho^{t,\mu;u}$ (see (10)),

$Y_{t+\delta}^{t,\mu;u}=Y_{t+\delta}^{t+\delta,\rho_{t+\delta}^{t,\mu;u};u}=J\left(t+\delta,\rho_{t+\delta}^{t,\mu;u};u\right).$
(29)

###### Remark 2.9

When $\xi=x\in\mathbb{R}^{n},$ let $\delta_{x}$ be the Dirac measure with mass
at $x\in\mathbb{R}^{n},$ we instantly have
$J\left(t,\delta_{x};u\right)=Y_{t}^{t,\delta_{x};u}$ since
$Y_{t}^{t,\delta_{x};u}$ is deterministic by virtue of Lemma 2.5, which
reduces to the classical recursive stochastic optimal control problem when
$f^{0}$ does not contain the distributions of $X$ and $\left(Y,Z\right)$ (see
[62, 68]).

###### Remark 2.10

A special case is when $f$ doesn’t depend on
$\left(y,z,\pi\right)\in\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{P}_{2}\left(\mathbb{R}\times\mathbb{R}^{d}\right)$,
i.e., $f=f\left(x,\mu,u\right)$. Now if we still suppose (A1)-(A2) hold then
the cost functional (28) turns into

$\displaystyle J\left(t,\xi;u\right)$ $\displaystyle=$ $\displaystyle
Y_{t}^{t,\xi;u}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\Phi^{0}\left(\rho_{T}^{t,\mu;u}\right)+\int_{t}^{T}f^{0}\left(\rho_{s}^{t,\mu;u},u_{s}\right)\mathrm{d}s\right],$

which has been studied by Djete et al. [33] with $W\equiv 0,$ Pham and Wei
[45] when $B\left(\cdot\right)\equiv 0,$ Cosso and Pham [22] with single
control process. If

$\displaystyle J\left(t,\xi;u\right)$ $\displaystyle=$ $\displaystyle
Y_{t}^{t,\xi;u}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\Phi^{0}\left(\rho_{T}^{t,\mu;u}\right)+\int_{t}^{T}c\left(\rho_{s}^{t,\mu;u}\right)Y_{s}^{t,\xi;u}+f^{0}\left(\rho_{s}^{t,\mu;u},u_{s}\right)\mathrm{d}s\right],$

which is a kind of extended Feynman–Kac formula.

We now consider the optimal control problem as follows:

Problem MVS. The McKean-Vlasov recursive control problem is to derive, over
admissible control set $\Lambda_{t,\xi}$, the following value function

$V\left(t,\xi\right)=\inf_{u\in\Lambda_{t,\xi}}J\left(t,\xi;u\right),\text{ }$
(30)

for $\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$ with
$\rho_{t}^{t,\mu;u}=\mathbb{P}_{X_{t}^{t,\xi;u}}^{W}=\mu.$

Before introducing the PDE by virtue of DPP for the value function, we at
present study some properties of the value function, which is useful to
characterize the viscosity solution.

###### Lemma 2.8

Assume that _(A1)-(A5)_ hold. There exists a constant $C>0$ such that, for any
$t,$ $t^{\prime}\in\left[0,T\right],$ $\xi,$ $\xi^{\prime}\in
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$

$\displaystyle\left|V\left(t,\xi\right)-V\left(t,\xi^{\prime}\right)\right|$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}\left[\left|\xi-\xi^{\prime}\right|\right],$ (31)
$\displaystyle\left|V\left(t,\xi\right)-V\left(t^{\prime},\xi\right)\right|$
$\displaystyle\leq$ $\displaystyle
C\left(1+\mathbb{E}\left[\left|\xi\right|\right]\right)\left|t-t^{\prime}\right|^{\frac{1}{2}},$
(32) $\displaystyle\left|V\left(t,\xi\right)\right|$ $\displaystyle\leq$
$\displaystyle C\left(1+\mathbb{E}\left[\left|\xi\right|\right]\right).$ (33)

The proof can be seen in the Appendix.

Apparently, for any $u\in\Lambda_{t,\xi},$ the control processes $u$ may
depend on $\xi$ since it is measurable with respect to $\mathcal{G}$.
Therefore, we can only say that the law $\mathbb{P}_{X_{s}^{t,\xi;u}}^{W}$ of
$X_{s}^{t,\xi;u}$ depends on the joint law of
$\left(\xi,u,(W_{s}-W_{t})_{s\in\left[t,T\right]}\right)$. The next lemma will
show that the value function $V$ is _law-invariant_ , i.e., depend on $\xi$
only via its distribution.

###### Remark 2.11

For the classical stochastic McKean–Vlasov control problem, for instance,
Cosso and Pham [22], Cosso et al. [25] they defined the value function as a
function of $\left(t,\xi\right)$ and then proved the so called “law invariance
property”, stating that the value function does not depend on $\xi$ but rather
on its law. In [25], the authors shown that one can’t expect this invariance
property to hold without any regularity conditions on the reward functionals
and terminal function (see [25], Remark 3.7). Djete et al. [33] studied the
same topic via measurable selection techniques and presented weaker conditions
than the ones in [25] (see Remark 6 in [25]).

###### Theorem 2.1

We assume that the Assumptions _(A1)-(A5)_ are in force. For any $\xi,$
$\xi^{\prime}\in
L^{2}\left(\mathcal{F}_{t}^{0}\vee\mathcal{G};\mathbb{R}^{n}\right)$ with
$\mathbb{P}_{\xi}=\mathbb{P}_{\xi^{\prime}},$ we have

$V\left(t,\xi\right)=V\left(t,\xi^{\prime}\right),\text{
}\mathbb{P}\text{-a.s.}$

###### Remark 2.12

Note that in (28) and (30), the initial condition $\xi$ belongs to
$L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right).$ On one hand, we can prove that
$Y_{t}^{t,\xi;u}$ is a constant (see Lemma 2.7) which is corresponding to a
version of deterministic HJB equation (see Section 3); As a matter of fact, in
[45, 33, 22, 25, 63] etc, the reward functional is defined in the following
way: No matter which spaces the initial condition $\xi$ belongs to, the cost
functional is all performed after taking the absolute expectation
$\mathbb{E},$ which immediately reduces that the value function is always non-
random; Notwithstanding, the value function in our paper is defined via the
solution to BSDE. Hence very different from the aforementioned works, the
current value function is random whenever $\xi\in
L^{2}\left(\mathcal{F}_{t}^{0}\vee\mathcal{G};\mathbb{R}^{n}\right)$ (see
Lemma 2.6). On the other hand, as pointed out in [25], Remark 3.7, the main
reason why we take $\xi\in
L^{2}\left(\mathcal{F}_{t}^{0}\vee\mathcal{G};\mathbb{R}^{n}\right)$ is to
state and prove the DPP (more details see Theorem 3.4 in [25]). In our paper,
for instance, the initial condition at time $t$ is $\xi\in
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$, nevertheless, $X_{s}^{t,\xi;u}$
in general, is not $\mathcal{G}$ measurable!

For reader’s convenience, we exhibit two kinds of lemmata (Lemma 2.9 and Lemma
2.10) from [25] to proceed our proof, which are crucial to prove the law-
invariant. In [25], the authors deal with the state space in a Hilbert space
$H$ given by the set of $C\left(\left[0,T\right];H\right)$ of continuous
$H$-valued functions on $\left[0,T\right].$ Hence, reduced to the finite-
dimensional case, i.e., $H=\mathbb{R}^{n},$ and in the Markovian case, we have

###### Lemma 2.9

For any $u\in\mathcal{U}_{ad}^{2}\left[0,T\right]$ and $\xi\in
L^{2}\left(\mathcal{F}_{t}^{0}\vee\mathcal{G};\mathbb{R}^{n}\right)$, assume
that there exists an $\mathcal{F}_{t}^{0}\vee\mathcal{G}$-measurable random
variable $U_{\xi}$, having uniform distribution on $[0,1]$ and being
independent of $\xi$. Then, there exists a measurable function

$\mathbf{a:}\left(\left[0,T\right]\times\Omega\times\mathbb{R}^{n}\times\left[0,1\right],\mathcal{B}\left(\left[0,T\right]\right)\otimes\mathcal{F}_{t}^{0}\otimes\mathcal{B}\left(\mathbb{R}^{n}\right)\otimes\mathcal{B}\left(\left[0,1\right]\right)\right)\rightarrow\left(U,\mathscr{U}\right),$

where $\mathscr{U}$ denotes the Borel $\sigma$-field of $U$ such that
$\left\\{\mathbf{a}_{s}\left(\xi,U_{\xi}\right)\right\\}_{0\leq s\leq
T}\in\mathcal{U}_{ad}^{2}\left[0,T\right],$
$\mathbf{a}_{s}\left(\cdot,\cdot\right)$ is constant for every $s<t,$
moreover,

$\left(\xi,\left(\mathbf{a}_{s}\left(\xi,U_{\xi}\right)\right)_{0\leq s\leq
T},\left(W_{s}-W_{t}\right)_{0\leq s\leq
T}\right)\overset{\mathscr{Law}}{=}\left(\xi,\left(u_{s}\right)_{0\leq s\leq
T},\left(W_{s}-W_{t}\right)_{0\leq s\leq T}\right),$

where $\mathscr{Law}$ denotes the equality in law.

###### Lemma 2.10

For any $\xi\in
L^{2}\left(\mathcal{F}_{t}^{0}\vee\mathcal{G};\mathbb{R}^{n}\right)$, assume
that there exists $\left\\{x_{1},\ldots,x_{m}\right\\}\subset\mathbb{R}^{n},$
with $x_{i}\neq x_{j},$ if $i\neq j,$ such that

$\mathbb{P}_{\xi}=\sum_{i=1}^{m}p_{i}\delta_{x_{i}},$

where $\delta_{x_{i}}$ is the Dirac measure at $x_{i}$ and $p_{i}>0$, with
$\sum_{i=1}^{m}p_{i}=1$. Then, there exists an
$\mathcal{F}_{t}^{0}\vee\mathcal{G}$-measurable random variable $U_{\xi}$
having uniform distribution on $\left[0,1\right]$ and being independent of the
$\mathcal{F}_{t}^{0}\vee\mathcal{G}$-measurable map

$\tilde{\xi}:\Omega\rightarrow\mathbb{R}^{n},\omega\rightarrow\xi\left(\omega\right).$

Proof We borrow the idea from [25] and treat the proof in the following steps.

_Step 1._ We begin by assuming that there exists two random variables
$U_{\xi}$ and $U_{\xi^{\prime}}$ which are
$\mathcal{F}_{t}^{0}\vee\mathcal{G}$-measurable with uniform distribution on
$\left[0,1\right]$, such that $\xi$ and $U_{\xi}$ (resp. $\xi^{\prime}$ and
$U_{\xi^{\prime}}$) are independent. Given any
$u\in\mathcal{U}_{ad}^{2}\left[0,T\right],$ from Lemma 2.9, we are able to
find a function
$\mathbf{a:}\left[0,T\right]\times\Omega\times\mathbb{R}^{n}\times\left[0,1\right]\rightarrow
U$ such that

$\left(\xi,\left(\mathbf{a}_{s}\left(\xi,U_{\xi}\right)\right)_{0\leq s\leq
T},\left(W_{s}-W_{t}\right)_{0\leq s\leq
T}\right)\overset{\mathscr{Law}}{=}\left(\xi,\left(u_{s}\right)_{0\leq s\leq
T},\left(W_{s}-W_{t}\right)_{0\leq s\leq T}\right).$

We notice that $\xi,$ $\xi^{\prime}$ have the same law. Therefore, whenever
taking $u^{\prime}=\left(\mathbf{a}_{s}\left(\xi,U_{\xi}\right)\right)_{0\leq
s\leq T}$ which admits $u^{\prime}\in\mathcal{U}_{ad}^{2}\left[0,T\right],$ we
immediately, have

$\left(\xi,\left(\mathbf{a}_{s}\left(\xi,U_{\xi}\right)\right)_{0\leq s\leq
T},\left(W_{s}-W_{t}\right)_{0\leq s\leq
T}\right)\overset{\mathscr{Law}}{=}\left(\xi^{\prime},\left(u_{s}^{\prime}\right)_{0\leq
s\leq T},\left(W_{s}-W_{t}\right)_{0\leq s\leq T}\right)$

As we know, under the Assumptions (A1)-(A4), FBSDEs (12) admits a unique
solution. So from inequalities (6) and (19), we imply that

$\left(\left(\Upsilon_{s}^{t,\xi;u}\right)_{0\leq s\leq
T},\left(\mathbf{a}_{s}\left(\xi,U_{\xi}\right)\right)_{0\leq s\leq
T},\right)\overset{\mathscr{Law}}{=}\left(\left(\Upsilon_{s}^{t,\xi^{\prime};u^{\prime}}\right)_{0\leq
s\leq T},\left(u_{s}^{\prime}\right)_{0\leq s\leq T},\right),$

where
$\Upsilon_{s}^{t,\xi;u}\triangleq\left(X_{s}^{t,\xi;u},Y_{s}^{t,\xi;u},Z_{s}^{t,\xi;u}\right)$
and
$\Upsilon_{s}^{t,\xi^{\prime};u^{\prime}}\triangleq\left(X_{s}^{t,\xi^{\prime};u^{\prime}},Y_{s}^{t,\xi^{\prime};u^{\prime}},Z_{s}^{t,\xi^{\prime};u^{\prime}}\right).$
As a result, from the definition of value function, we get

$V\left(t,\xi\right)=\inf_{u\in\Lambda_{t,\xi}}J\left(t,\xi;u\right)\leq\inf_{u^{\prime}\in\Lambda_{t,\xi}}J\left(t,\xi^{\prime};u^{\prime}\right)=V\left(t,\xi^{\prime}\right).$
(34)

Now interchanging the roles of $\xi$ and $\xi^{\prime},$ we derive the other
part, i,e, $V\left(t,\xi\right)\geq V\left(t,\xi^{\prime}\right)$.

_Step 2._ In order to remove the condition ($U_{\xi}$ and $U_{\xi^{\prime}}$
are $\mathcal{F}_{t}^{0}\vee\mathcal{G}$-measurable with uniform distribution
on $\left[0,1\right]$, such that $\xi$ and $U_{\xi}$ (resp. $\xi^{\prime}$ and
$U_{\xi^{\prime}}$) are independent). We first consider the discrete case of
$\xi$. Suppose that there are some
$\left\\{x_{1},\ldots,x_{m}\right\\}\subset\mathbb{R}^{n},$ with $x_{i}\neq
x_{j},$ if $i\neq j,$ such that

$\mathbb{P}_{\xi}=\sum_{i=1}^{m}p_{i}\delta_{x_{i}}.$

Instantly, from Lemma 2.10, there exist two random variables $U_{\xi}$ and
$U_{\xi^{\prime}}$ which are $\mathcal{F}_{t}^{0}\vee\mathcal{G}$-measurable
with uniform distribution on $\left[0,1\right]$, such that $\xi$ and $U_{\xi}$
(resp. $\xi^{\prime}$ and $U_{\xi^{\prime}}$) are independent. The desired
conclusion is established from Step 1.

_Step 3._ For the general case, the continuity of value function will play an
important rule as we adopt the technique of approximation of $\xi$ and $\eta$.
Now consider a decomposition of $\mathbb{R}^{n}$, namely, $\sum_{i\geq
1}\mathcal{O}_{i}^{k}=\mathbb{R}^{n}$ such that
diam$\left(\mathcal{O}_{i}^{k}\right)\leq 2^{-k},$ for each $k\geq 1.$ For
each $k\in\mathbb{N},$ pick up $\chi_{i}^{k}\in\mathcal{O}_{i}^{k}$ and then
define

$\check{\xi}_{k}\triangleq\sum_{i\geq
1}\chi_{i}^{k}\mathbb{I}_{\mathcal{O}_{i}^{k}}\left(\xi\right),\text{
}\check{\eta}_{k}\triangleq\sum_{i\geq
1}\chi_{i}^{k}\mathbb{I}_{\mathcal{O}_{i}^{k}}\left(\eta\right).$

It is easy to check that $\check{\xi}_{k}$, $\check{\eta}_{k}\in
L^{2}\left(\mathcal{F}_{t}^{0}\vee\mathcal{G};\mathbb{R}^{n}\right)$ and
$\check{\xi}_{k}\rightarrow\xi,$ $\check{\eta}_{k}\rightarrow\eta$ as $\
k\rightarrow\infty,$ uniformly w.r.t. $\omega\in\Omega.$ Besides,
$\check{\xi}_{k}$ and $\check{\eta}_{k}$ have the same law. By virtue of the
diagonal approach, for each $k\in\mathbb{N},$ there exists $K_{k}$ such that
we are able to seek sequences $\xi_{k}$, $\eta_{k},$ denoted by

$\xi_{k}\triangleq\sum_{i=1}^{K_{k}}\chi_{i}^{k}\mathbb{I}_{\mathcal{O}_{i}^{k}}\left(\xi\right),\text{
}\eta_{k}\triangleq\sum_{i=1}^{K_{k}}\chi_{i}^{k}\mathbb{I}_{\mathcal{O}_{i}^{k}}\left(\eta\right),$

which converge respectively to $\xi$ and $\eta.$ Next, from Step 2, we derive
that

$V\left(t,\xi_{k}\right)=V\left(t,\eta_{k}\right),\text{
}\mathbb{P}\text{-a.s. for each }k\in\mathbb{N}.$

At last, we get desired result by passing the limit as $k\rightarrow\infty$ by
means of the continuity of value function (see Lemma 2.8).

###### Remark 2.13

In [25], the authors impose an assumption, so-called
$\left(A_{f,g}\right)_{\text{cont}}$444The running function $f$ is locally
uniformly continuous in $\left(x,\mu\right)$ uniformly with respect to
$\left(t,u,\nu\right)$. Similarly, the terminal function $g$ is locally
uniformly continuous. is not assumed in our paper since the value function is
defined by solution of BSDE. The assumption (A3)-(A4) can ensure the existence
and uniqueness of the solution.

###### Remark 2.14

From now on, the solution of FBSDEs (12) can be written as
$\Theta_{s}^{t,\mu;u}$ to underline the dependence on the law
$\mu=\mathbb{P}_{\xi},$ which is important to establish DPP.

Lemma 2.1 shows that the value function $V$ is _law-invariant_. Therefore, we
can define the inverse-lifted function (see Section 3 for the definition) of
value function $V.$

###### Definition 2.3

Given any
$\left(t,\mu\right)\in\left[0,T\right]\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right),$
we define

$\mathcal{V}\left(t,\mu\right)=V\left(t,\xi\right),\text{
}\mathbb{P}\text{-a.s.}$ (35)

where $\xi\in
L^{2}\left(\mathcal{F}_{t}^{0}\vee\mathcal{G};\mathbb{R}^{n}\right)$ with
$\mathbb{P}_{\xi}=\mu.$

Clearly, under the assumptions (A1)-(A5), the value function
$\mathcal{V}\left(t,\mu\right)$ is well-defined and a priori it is a bounded
$\mathcal{F}_{t}^{0}\vee\mathcal{G}$-measurable random variable. But it turns
out that $\mathcal{V}\left(t,\mu\right)$ is even deterministic. We have the
following result.

###### Proposition 2.1

Suppose that the assumptions _(A1)-(A5)_ are in force. The value function
$\mathcal{V}\left(t,\mu\right)$ is a deterministic function.

The main idea for this proof is to construct a sequence of admissible controls
used by Peng in [62] with help of Lemma 2.7. So we omit it.

It is necessary to point out that the value function defined in (35) can be
taken over the subset $\mathcal{U}_{t}^{2}$ of elements in
$\mathcal{U}_{ad}^{2}\left[0,T\right],$ which are independent of
$\mathcal{F}^{0}$ under $\mathbb{P}^{0}.$

###### Lemma 2.11

Under the Assumptions _(A1)-(A5)_ , for any
$\left(t,\mu\right)\in\left[0,T\right]\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right),$
we have

$\mathcal{V}\left(t,\mu\right)=\inf_{u\in\mathcal{U}_{t}^{2}}J\left(t,\mu;u\right).\text{
}$ (36)

Proof _Step 1._ Given any $\mathcal{F}^{0}$-stopping time
$\tau\in\mathbb{T}_{t,T}^{0},$ according to the flow property (10), backward
semigroup (23) and noting $\rho_{s}^{t,\mu;u}$ is $\mathcal{F}_{\tau}^{0}$
-measurable, we get for $\mathbb{P}^{0}$-a.s. $\omega^{0}\in\Omega^{0},$

$\displaystyle
J\left(\tau,\rho_{\tau}^{t,\mu;u};u^{\tau}\right)\left(\omega^{0}\right)$
$\displaystyle=$ $\displaystyle Y_{\tau}^{t,\mu;u}\left(\omega^{0}\right)$
(37) $\displaystyle=$
$\displaystyle\mathbb{G}_{\tau,T}^{t,\mu;u}\left[\Phi^{0}\left(\rho_{T}^{t,\mu;u}\right)\right]\left(\omega^{0}\right)$
$\displaystyle=$
$\displaystyle\left.\mathbb{G}_{r,T}^{r,\pi;\upsilon}\left[\Phi^{0}\left(\rho_{T}^{r,\pi;\upsilon}\right)\right]\left(\omega^{0}\right)\right|_{r=\tau\left(\omega^{0}\right),\pi=\rho_{\tau\left(\omega^{0}\right)}^{t,\mu;u}\left(\omega^{0}\right),\upsilon=u^{r,\omega^{0}}}$
$\displaystyle=$
$\displaystyle\left.J\left(r,\pi;\upsilon\right)\right|_{r=\tau\left(\omega^{0}\right),\pi=\rho_{\tau\left(\omega^{0}\right)}^{t,\mu;u}\left(\omega^{0}\right),\upsilon=u^{r,\omega^{0}}},\text{
}\mathbb{P}^{0}\text{-a.s.}$

in which the process $u^{r,\omega^{0}}\in\mathcal{U}_{r}^{2}$ is independent
of $\mathcal{F}_{r}^{0}$, which implies that $X_{s}^{t,\vartheta;u}$ (due to
$\mathcal{G}$ “rich enough”, there exists a random variable $\vartheta\in
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$ such that
$\mathcal{L}\left(\vartheta\right)=\pi$) is independent of
$\mathcal{F}_{r}^{0}$, and consequently $\rho_{s}^{r,\pi;\upsilon}$ is also
independent of $\mathcal{F}_{r}^{0}$ for $r\leq s$. Therefore, we get the
relation (37) by virtue of Lemma 2.5.

_Step 2._ For the sake of simplicity, we put

$\mathcal{\tilde{V}}\left(t,\mu\right)=\inf_{u\in\mathcal{U}_{t}^{2}}J\left(t,\mu;u\right).$

From the fact
$\mathcal{U}_{r}^{2}\subset\mathcal{U}_{ad}^{2}\left[0,T\right]$, it yields
that $\mathcal{V}\left(t,\mu\right)\leq\mathcal{\tilde{V}}\left(t,\mu\right).$
To obtain the opposite relationship, we let $\tau=t$ in (37), then get

$J\left(t,\mu;u^{t,\omega^{0}}\right)=\left.J\left(t,\mu;u\right)\right|_{t,\mu,u^{t,\omega^{0}}},\text{
}\mathbb{P}^{0}\text{-a.s.}$ (38)

Immediately,

$\int_{\Omega^{0}}J\left(t,\mu;u^{t,\omega^{0}}\right)\mathbb{P}^{0}\left(\mathrm{d}\omega^{0}\right)=J\left(t,\mu;u\right).$

Besides, noting however $u^{t}\in\mathcal{U}_{t}^{2},$ we have

$J\left(t,\mu;u^{t,\omega^{0}}\right)\geq\mathcal{\tilde{V}}\left(t,\mu\right),$

from which we get the desired result since the arbitrary of
$u\in\mathcal{U}_{ad}^{2}\left[0,T\right].$

We can get some properties of the value function
$\mathcal{V}\left(t,\mu\right)$ which is an immediate consequence of its
definitions (35), (30) and the Lipschitz property (19) of the cost functional.

###### Lemma 2.12

Under the Assumptions _(A1)-(A5)_ , there exists a constant $C>0$ such that,
for all $t\in\left[0,T\right]$,
$\mu,\mu^{\prime}\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$

$\displaystyle\left|\mathcal{V}\left(t,\mu\right)-\mathcal{V}\left(t,\mu^{\prime}\right)\right|$
$\displaystyle\leq$ $\displaystyle
C\mathcal{W}_{2}\left(\mu,\mu^{\prime}\right),$ (39)
$\displaystyle\left|\mathcal{V}\left(t,\mu\right)\right|$ $\displaystyle\leq$
$\displaystyle C\left[1+\mathcal{W}_{2}\left(\mu,\delta_{0}\right)\right],$
(40)

where $\delta_{0}$ is the Dirac measure with mass at $0\in\mathbb{R}^{n}.$

###### Lemma 2.13

Under the Assumptions _(A1)-(A5)_ , for any
$\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\mathcal{G};\mathbb{R}^{n}\right)$ with $\mathbb{P}_{\xi}=\mu$ and
all $u\in\mathcal{U}_{ad}^{2}\left[0,T\right]$ and $\mathcal{F}^{0}$-stopping
time $\tau\in\mathbb{T}_{t,T}^{0},$ we have

$\mathcal{V}\left(t,\mu\right)\leq\mathbb{G}_{t,\tau}^{t,\mu;u}\left[\mathcal{V}\left(\tau,\rho_{\tau}^{t,\mu;u}\right)\right],$
(41)

Meanwhile, for any $\varepsilon>0,$ there exists an admissible control
$u^{\varepsilon}\in\mathcal{U}_{ad}^{2}\left[0,T\right]$ for all
$\tau\in\mathbb{T}_{t,T}^{0},$ such that

$\mathcal{V}\left(t,\mu\right)\geq\mathbb{G}_{t,\tau}^{t,\mu;u^{\varepsilon}}\left[\mathcal{V}\left(\tau,\rho_{\tau}^{t,\mu;u^{\varepsilon}}\right)\right]-\varepsilon.$
(42)

###### Remark 2.15

The inequality in (42) indicates that, for any given $\varepsilon>0,$ there
exists a near-optimal control $u^{\varepsilon}.$ It has nice structure and
large availability as well as flexibility. So it is possible to choose among
them which are easier for analysis and implementation (see [69]). Beyond that,
this result will be employed to prove the viscosity super-solution property of
the value function.

Proof From (37) and (36), we have for all $\tau\in\mathbb{T}_{t,T}^{0}$

$\displaystyle J\left(t,\mu;u\right)$ $\displaystyle=$
$\displaystyle\mathbb{G}_{t,\tau}^{t,\mu;u}\left[J\left(\tau,\rho_{\tau}^{t,\mu;u};u^{\tau}\right)\right]$
(43) $\displaystyle\geq$
$\displaystyle\mathbb{G}_{t,\tau}^{t,\mu;u}\left[\mathcal{V}\left(\tau,\rho_{\tau}^{t,\mu;u}\right)\right].$

The arbitrary of $u$ implies

$\mathcal{V}\left(t,\mu\right)\geq\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\sup_{\tau\in\mathbb{T}_{t,T}^{0}}\mathbb{G}_{t,\tau}^{t,\mu;u}\left[\mathcal{V}\left(\tau,\rho_{\tau}^{t,\mu;u}\right)\right].$
(44)

Now for any $\varepsilon>0$ and $\omega^{0}\in\Omega^{0},$ by Lemma 37, there
exists certain $u^{\varepsilon}\in\mathcal{U}_{\tau}^{2}$ such that

$\mathcal{V}\left(\tau\left(\omega^{0}\right),\rho_{\tau\left(\omega^{0}\right)}^{t,\mu;u}\right)+\varepsilon\geq
J\left(\tau\left(\omega^{0}\right),\rho_{\tau\left(\omega^{0}\right)}^{t,\mu;u}\left(\omega^{0}\right);u^{\varepsilon}\left(\omega^{0}\right)\right),\text{
}\mathbb{P}^{0}\text{-a.s.}$ (45)

The measurable selection arguments ([60]) combining to _Lemma continuous_
yields that the map $\omega^{0}\in\Omega^{0}\rightarrow
u^{\varepsilon}\left(\omega^{0}\right)$ is measurable. We now define a shifted
control process based on $u^{\varepsilon}$

$\tilde{u}_{s}^{\varepsilon}\left(\omega^{0}\right)=u_{s}\left(\omega^{0}\right)\mathbb{I}_{s<\tau\left(\omega^{0}\right)}+u^{\varepsilon}\left(\omega^{0}\right)\mathbb{I}_{s\geq\tau\left(\omega^{0}\right)},\text{
}s\in\left[0,T\right].$

Clearly, $\rho_{s}^{t,\mu;\tilde{u}^{\varepsilon}}=\rho_{s}^{t,\mu;u}$
whenever $s\leq\tau$ and $\rho^{t,\mu;u}$ has continuous trajectories. Notice
that $\mathcal{U}_{ad}^{2}\left[0,T\right]$ is a separable metric space, by
Lemma 2.1 in [57], the control process $\tilde{u}_{s}^{\varepsilon}$ is
$\mathcal{F}_{s}^{0}$-adapted, and thus
$\tilde{u}^{\varepsilon}\in\mathcal{U}_{ad}^{2}\left[0,T\right].$ Therefore,
(45) can be expressed as

$\mathcal{V}\left(\tau,\rho_{\tau}^{t,\mu;u}\right)+\varepsilon\geq
J\left(\tau,\rho_{\tau}^{t,\mu;u};\tilde{u}^{\varepsilon}\right),\text{
}\mathbb{P}^{0}\text{-a.s.}$ (46)

Then

$\displaystyle\mathcal{V}\left(t,\mu\right)$ $\displaystyle\leq$
$\displaystyle J\left(t,\mu;\tilde{u}^{\varepsilon}\right)$ $\displaystyle=$
$\displaystyle\mathbb{G}_{t,\tau}^{t,\mu;u}\left[J\left(\tau,\rho_{\tau}^{t,\mu;u};\left(\tilde{u}^{\varepsilon}\right)^{\tau}\right)\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{G}_{t,\tau}^{t,\mu;u}\left[\mathcal{V}\left(\tau,\rho_{\tau}^{t,\mu;u}\right)+\varepsilon\right],$

where the third inequality holds by using the comparison theorem (see Lemma
2.4). More preciously, set

$\zeta^{1}=J\left(\tau,\rho_{\tau}^{t,\mu;u};\left(\tilde{u}^{\varepsilon}\right)^{\tau}\right),\text{
}\zeta^{2}=\mathcal{V}\left(\tau,\rho_{\tau}^{t,\mu;u}\right)+\varepsilon,$

in Lemma 2.4, from the fact $\zeta^{1}\leq\zeta^{2},$ $\mathbb{P}^{0}$-a.s.,
we get the desired result.

Now since $\tau,u$ and $\varepsilon$ are arbitrary, it follows that

$\mathcal{V}\left(t,\mu\right)\leq\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\inf_{\tau\in\mathbb{T}_{t,T}^{0}}\mathbb{G}_{t,\tau}^{t,\mu;u}\left[\mathcal{V}\left(\tau,\rho_{\tau}^{t,\mu;u}\right)+\varepsilon\right].$
(47)

At last, the inequalities (44) and (47) yield

$\displaystyle\mathcal{V}\left(t,\mu\right)$ $\displaystyle\leq$
$\displaystyle\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\inf_{\tau\in\mathbb{T}_{t,T}^{0}}\mathbb{G}_{t,\tau}^{t,\mu;u}\left[\mathcal{V}\left(\tau,\rho_{\tau}^{t,\mu;u}\right)\right]$
$\displaystyle=$
$\displaystyle\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\sup_{\tau\in\mathbb{T}_{t,T}^{0}}\mathbb{G}_{t,\tau}^{t,\mu;u}\left[\mathcal{V}\left(\tau,\rho_{\tau}^{t,\mu;u}\right)\right],$

which support the DPP property. We complete the proof.

###### Remark 2.16

Note that in [22], the authors considered the initial condition $\xi\in
L^{2}\left(\mathcal{F}_{t}\vee\mathcal{G};\mathbb{R}^{n}\right).$ Whenever the
coefficients in [22] are only dependent on the state and control rather than
the law of state, moreover $\mathcal{G=}\left\\{\varnothing,\Omega\right\\}$,
nevertheless the initial state is still a random variable, then $\xi\in
L^{2}\left(\mathcal{F}_{t};\mathbb{R}^{n}\right)$. Observe that the cost
functional defined in [22] and [45] by means of taking the absolute
expectation which instantly implies that the value function is deterministic.
Thereafter, one can retrieve the standard dynamic principle in the non-
McKean–Vlasov situation (see Proposition 4.1 in [22]). However, under the
similar assumptions, Problem MVS defined in our paper actually becomes a kind
of standard stochastic recursive optimal control problem (cf. [62] for more
details). From Lemma 2.6, one can easily derive that
$\left.Y_{s}^{t,\xi;u}\right|_{s=t}$ is random, which is completely different
from the framework studied in [22] and [45].

###### Lemma 2.14

Under the Assumptions _(A1)-(A5)_ , there exists a positive constant $C$ such
that

$\left|\mathcal{V}\left(t,\mu\right)-\mathcal{V}\left(t+\delta,\mu\right)\right|\leq
C\left(1+\mathcal{W}_{2}\left(\mu,\delta_{0}\right)\right)\delta^{\frac{1}{2}},$

where $\mu\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right),$
$t\in\left[0,T\right)$ and $\delta>0$ with $0<\delta+t\leq T.$

The proof can be found in Appendix.

##  3 Hamilton-Jacobi-Bellman Equation

After obtaining the DPP, as usual, we will derive the corresponding Bellman
equation with help of Itô’s formula in Wasserstein space and the viscosity PDE
characterization. The value function will be studied on the Hilbert space or
the Wasserstein space of probability measures via the associated inverse-
lifted identification. To this end, let us briefly recall the following
notions of derivatives w.r.t. a probability measure, as introduced by Lions
(see [50]).

The following notations are mainly taken from [45, 23]. Let $\vartheta$ be a
real-valued function defined on $\mathcal{P}_{2}\left(\mathbb{R}^{n}\right).$
We say $\upsilon$ is the lifted version of $\vartheta$, namely, the function
defined on $L^{2}\left(\Omega,\mathcal{G},\mathbb{P};\mathbb{R}^{n}\right)$ by
$\upsilon\left(\xi\right)=\vartheta\left(\mathbb{P}_{\xi}\right).$ Conversely,
for any function $\upsilon$ defined on
$L^{2}\left(\Omega,\mathcal{G},\mathbb{P};\mathbb{R}^{n}\right)$, we call
inverse-lifted function of $\upsilon$ the function $\vartheta$ defined on
$\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$ by
$\upsilon\left(\xi\right)=\vartheta\left(\mathbb{\mu}\right)$ for
$\mu=\mathcal{L}\left(\xi\right).$ Clearly, $\vartheta$ exists if and only if
$\upsilon$ depends only on the distribution of $\xi$ for any $\xi\in
L^{2}\left(\Omega,\mathcal{G},\mathbb{P};\mathbb{R}^{n}\right).$ In the
sequel, we set $\upsilon=\vartheta$ often for simplicity.

In the literatures, $\vartheta$ is said to be $C^{1}$ on
$\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$ if the lift $\upsilon$ is Fréchet
differentiable with continuous derivatives on
$L^{2}\left(\Omega,\mathcal{G},\mathbb{P};\mathbb{R}^{n}\right).$ By the Riesz
representation theorem, the Fréchet derivative $\left[D\upsilon\right](\xi)$,
regarded as an element $D\upsilon(\xi)$ of
$L^{2}\left(\Omega,\mathcal{G},\mathbb{P};\mathbb{R}^{n}\right),$ can be
expressed as:

$\left[D\upsilon\right](\xi)\left(\zeta\right)=\mathbb{E}^{1}\left[\left\langle
D\upsilon(\xi),\zeta\right\rangle\right]$

with
$D\upsilon(\xi)=\partial_{\mu}\vartheta\left(\mathcal{L}\left(\xi\right)\right)\left(\xi\right)$
for $\zeta\in L^{2}\left(\Omega,\mathcal{G},\mathbb{P};\mathbb{R}^{n}\right),$
in which
$\partial_{\mu}\vartheta\left(\mathcal{L}\left(\xi\right)\right):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$
is called the derivative of $\vartheta$ at $\mu=\mathcal{L}\left(\xi\right).$
Note that $\mathcal{G}$ is assumed to be “rich enough”. Therefore,
$\mu\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)=\left\\{\left.\mathcal{L}\left(\xi\right)\right|\xi\in
L^{2}\left(\Omega,\mathcal{G},\mathbb{P};\mathbb{R}^{n}\right)\right\\}.$ We
say that $\vartheta$ is fully $C^{2}$ if it is $C^{1}$, and one can find, for
any $\mu\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$, a continuous version
of the mapping
$x\in\mathbb{R}^{n}\rightarrow\partial_{\mu}\vartheta\left(\mathcal{\mu}\right)\left(x\right)$,
such that the mapping
$\left(x,\mu\right)\in\mathbb{R}^{n}\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\rightarrow\partial_{\mu}\vartheta\left(\mathcal{\mu}\right)\left(x\right)$
is continuous at any point $\left(x,\mu\right)$ such that
$x\in$Supp$\left(\mu\right),$ and (a) for any
$\mu\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$, the mapping
$x\in\mathbb{R}^{n}\rightarrow\partial_{\mu}\vartheta\left(\mathcal{\mu}\right)\left(x\right)$
is differentiable in the standard sense, with a gradient denoted by
$\partial_{x}\partial_{\mu}\vartheta\left(\mathcal{\mu}\right)\left(x\right)\in\mathbb{R}^{n\times
n}$, and s.t. the mapping
$\left(x,\mu\right)\in\mathbb{R}^{n}\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\rightarrow\partial_{x}\partial_{\mu}\vartheta\left(\mathcal{\mu}\right)\left(x\right)$
is continuous; (b) for any fixed$\ x\in\mathbb{R}^{n}$, the mapping
$\mu\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\rightarrow\partial_{\mu}\vartheta\left(\mathcal{\mu}\right)\left(x\right)$
is differentiable in the above lifted sense. Its derivative, interpreted thus
as a mapping
$\tilde{x}\in\mathbb{R}^{n}\rightarrow\partial_{\mu}\left[\partial_{\mu}\vartheta\left(\mathcal{\mu}\right)\left(x\right)\right]\left(\tilde{x}\right)\in\mathbb{R}^{n\times
n}$ in $L_{\mu}^{2}\left(\mathbb{R}^{n\times n}\right)$, is denoted by
$\tilde{x}\in\mathbb{R}^{n}\rightarrow\partial_{\mu}^{2}\vartheta\left(\mathcal{\mu}\right)\left(x,\tilde{x}\right)$,
and s.t. the mapping
$(\mu,x,\tilde{x})\in\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\times\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\partial_{\mu}^{2}\vartheta\left(\mathcal{\mu}\right)\left(x,\tilde{x}\right)$
is continuous.

Let $C_{b}^{2}\left(\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\right)$ denotes
the set of functions if it is partially
$C^{2},\partial_{x}\partial_{\mu}\vartheta\left(\mathcal{\mu}\right)\in
L_{\mu}^{\infty}\left(\mathbb{R}^{n\times n}\right),$ and for any compact set
$G$ of $\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$, it admits

$\sup_{\mu\in
G}\left[\int_{G}\left|\partial_{\mu}\vartheta\left(\mathcal{\mu}\right)\left(x\right)\right|^{2}\mu\left(\mathrm{d}x\right)+\left\|\partial_{x}\partial_{\mu}\vartheta\left(\mathcal{\mu}\right)\right\|_{\infty}+\left\|\partial_{\mu}^{2}\vartheta\left(\mathcal{\mu}\right)\right\|_{\infty}\right]<\infty.$

We now ready to state the Itô’s formula (see [45, 23]). Consider the following
type of Itô process

$x_{t}=x_{0}+\int_{0}^{t}b_{r}\mathrm{d}r+\int_{0}^{t}\sigma_{r}\mathrm{d}W_{r},\text{
}0\leq t\leq T,$ (48)

where $x_{0}$ is independent of $W,$ while $b,\sigma$ are are progressively
measurable processes w.r.t. the natural filtration $\mathcal{F}$ generated by
$\left(x_{0},W\right)$, and satisfying the following condition:

$\mathbb{E}^{1}\left[\int_{0}^{T}\left(\left|b_{t}\right|^{2}+\left|\sigma_{t}\right|^{2}\right)\mathrm{d}t\right]<\infty.$

As mentioned before, $\mathbb{P}_{x_{t}}^{W}$ denotes the conditional law of
$x_{t}$, $0\leq t\leq T$, given the $\sigma$-field $\mathcal{F}$ generated by
the whole filtration of $W$, namely,
$\mathbb{P}_{x_{t}}^{W}\left(\omega^{0}\right)=\mathbb{P}_{x_{t}\left(\omega^{0},\cdot\right)}^{1}$.
Then, for any $\vartheta\in
C_{b}^{1}\left(\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\right),$ we have

$\displaystyle\vartheta\left(\mathbb{P}_{x_{t}}^{W}\right)$ $\displaystyle=$
$\displaystyle\vartheta\left(\mathbb{P}_{x_{0}}\right)+\int_{0}^{t}\Big{\\{}\mathbb{E}^{1}\big{[}\left\langle\partial_{\mu}\vartheta\left(\mathbb{P}_{x_{s}}^{W}\right)\left(x_{s}\right),b_{s}\right\rangle$
(49)
$\displaystyle+\frac{1}{2}\text{tr}\left(\partial_{x}\partial_{\mu}\vartheta\left(\mathbb{P}_{x_{s}}^{W}\right)\left(x_{s}\right)\sigma_{s}\sigma_{s}^{\top}\right)\big{]}$
$\displaystyle+\mathbb{E}^{1}\Big{[}\mathbb{\tilde{E}}^{1}\big{[}\frac{1}{2}\text{tr}\left(\partial_{\mu}^{2}\vartheta\left(\mathbb{P}_{x_{s}}^{W}\right)\left(x_{s},\tilde{x}_{s}\right)\sigma_{s}\tilde{\sigma}_{s}^{\top}\right)\big{]}\Big{\\}}\mathrm{d}s$
$\displaystyle+\int_{0}^{t}\mathbb{E}^{1}\big{[}\partial_{\mu}\vartheta\left(\mathbb{P}_{x_{s}}^{W}\right)\left(x_{s}\right)^{\top}\sigma_{s}\big{]}\mathrm{d}W_{s},$

where $\tilde{x},\tilde{\sigma}$ are copies of $x$ and $\sigma$ on anther
probability space
$\tilde{\Omega}=\big{(}\Omega^{0}\times\tilde{\Omega}^{1},\mathcal{F}^{0}\otimes\mathcal{\tilde{F}}^{1},\mathbb{P}^{0}\otimes\mathbb{\tilde{P}}^{1}\big{)}.$
It will be useful to apply Itô’s formula for the lifted function on
$L^{2}\left(\tilde{\Omega}^{1},\mathcal{\tilde{F}}^{1},\mathbb{\tilde{P}}^{1};\mathbb{R}^{n}\right).$
To this end, we postulate that $\upsilon\in
C^{2}\left(L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\right),$
then we have the following connection between derivatives in the Wasserstein
space $\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$ and in the Hilbert space
$L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$:

$\displaystyle D^{2}\upsilon\left(\xi\right)\left[\eta,\eta\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}^{1}\big{[}\mathbb{\tilde{E}}^{1}\left[\text{tr}\left(\partial_{\mu}^{2}\vartheta\left(\mathcal{L}\left(\xi\right)\right)\left(\xi,\tilde{\xi}\right)\eta\tilde{\eta}^{\top}\right)\right]$
(50)
$\displaystyle+\mathbb{\tilde{E}}^{1}\left[\text{tr}\left(\partial_{x}\partial_{\mu}\vartheta\left(\mathcal{L}\left(\xi\right)\right)\left(\xi\right)\eta\eta^{\top}\right)\right],$

where $\xi,\eta\in
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$ and
$\left(\tilde{\xi},\tilde{\eta}\right)$ is a copy of $\left(\xi,\eta\right)$
on another Polish and atomless space
$\left(\tilde{\Omega}^{1},\mathcal{\tilde{F}}^{1},\mathbb{\tilde{P}}^{1}\right).$
Let $\tilde{x}_{0},\tilde{b},\tilde{\sigma}$ be copies of $x_{0},b,\sigma$ on
$\left(\Omega^{0}\times\tilde{\Omega}^{1},\mathcal{F}^{0}\otimes\mathcal{G},\mathbb{P}^{0}\otimes\mathbb{\tilde{P}}^{1}\right)$
and now consider the following SDE, a copy of (48)

$\tilde{x}_{t}=\tilde{x}_{0}+\int_{0}^{t}\tilde{b}_{r}\mathrm{d}r+\int_{0}^{t}\tilde{\sigma}_{r}\mathrm{d}W_{r},\text{
}0\leq t\leq T.$ (51)

Based on (51), we define the process as
$\breve{\phi}_{t}\left(\omega_{0}\right)=\tilde{\phi}_{t}\left(\omega_{0},\cdot\right),$
for $0\leq t\leq T,\omega_{0}\in\Omega^{0}.$ Clearly, if
$\breve{\phi}=\tilde{b},$ $\tilde{\sigma},$ and $\tilde{x},$
$\breve{\phi}_{t}$ is $\mathcal{F}^{0}$-progressive, valued in
$L^{2}\left(\tilde{\Omega}^{1},\mathcal{\tilde{F}}^{1},\mathbb{\tilde{P}}^{1};\mathbb{R}^{n}\right),L^{2}\left(\tilde{\Omega}^{1},\mathcal{\tilde{F}}^{1},\mathbb{\tilde{P}}^{1};\mathbb{R}^{n\times
d}\right)$ and
$L^{2}\left(\tilde{\Omega}^{1},\mathcal{\tilde{F}}^{1},\mathbb{\tilde{P}}^{1};\mathbb{R}^{n}\right),$
respectively. Based on (50) and the assumption $\upsilon\in
C^{2}\left(\tilde{\Omega}^{1},\mathcal{\tilde{F}}^{1},\mathbb{\tilde{P}}^{1};\mathbb{R}^{n}\right),$
Itô’s formula (49) can be rewritten as (see Proposition 6.3 in [30])

$\displaystyle\upsilon\left(\breve{x}_{t}\right)$ $\displaystyle=$
$\displaystyle\upsilon\left(\breve{x}_{0}\right)+\int_{0}^{t}\mathbb{E}^{1}\Big{[}\left\langle
D\upsilon\left(\breve{x}_{s}\right),\breve{b}_{s}\right\rangle+\frac{1}{2}\left\langle
D^{2}\upsilon\left(\breve{x}_{s}\right)\breve{\sigma}_{s},\breve{\sigma}_{s}\right\rangle\Big{]}\mathrm{d}s$
(52)
$\displaystyle+\int_{0}^{t}\mathbb{E}^{1}\left[D\upsilon\left(\breve{x}_{s}\right)^{\top}\breve{\sigma}_{s}\right]\mathrm{d}W_{s}.$

###### Remark 3.1

Note that Itô’s formula (49) holds even if the lift is not twice continuously
Fréchet differentiable as shown in [21].

Now suppose that $\mathcal{V}\left(t,\mu\right)$ is sufficiently smooth. For
simplicity of notations, we set

$\displaystyle b_{s}^{t,\mu,u}$ $\displaystyle=$ $\displaystyle
b\left(X_{s}^{t,\mu;u},\mathbb{P}_{X_{s}^{t,\mu;u}}^{W},u_{s}\right),\text{
}\sigma_{s}^{t,\mu,u}=\sigma\left(X_{s}^{t,\mu;u},\mathbb{P}_{X_{s}^{t,\mu;u}}^{W},u_{s}\right),$
$\displaystyle\tilde{b}_{s}^{t,\mu,u}$ $\displaystyle=$
$\displaystyle\tilde{b}\left(X_{s}^{t,\mu;u},\mathbb{P}_{X_{s}^{t,\mu;u}}^{W},u_{s}\right),\text{
}\tilde{\sigma}_{s}^{t,\mu,u}=\tilde{\sigma}\left(X_{s}^{t,\mu;u},\mathbb{P}_{X_{s}^{t,\mu;u}}^{W},u_{s}\right).$

By virtue of Lemma 2.13 (DPP), we have

$\mathcal{V}\left(t,\mu\right)=\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\mathbb{G}_{t,\tau}^{t,\mu;u}\left[\mathcal{V}\left(t+\delta,\rho_{t+\delta}^{t,\mu;u}\right)\right],$

namely

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\Bigg{[}\mathcal{V}\left(t+\delta,\rho_{t+\delta}^{t,\mu;u}\right)-\mathcal{V}\left(t,\mu\right)$
$\displaystyle+\int_{t}^{t+\delta}f^{0}\left(\rho_{s}^{t,\mu;u},\Theta_{s}^{t,\mu;u},\mathbb{P}_{\Theta_{s}^{t,\mu;u}},u_{s}\right)\mathrm{d}s-\int_{t}^{t+\delta}Z_{s}^{t,\mu;u}\mathrm{d}W_{s}\Bigg{]}.$

By applying Itô’s formula (49) to (41)-(42), we obtain the HJB equation: all
$0\leq t<s<T,$

$\displaystyle
0=\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\Bigg{\\{}\int_{t}^{t+\delta}f^{0}\left(\rho_{s}^{t,\mu;u},\Theta_{s}^{t,\mu;u},\mathbb{P}_{\Theta_{s}^{t,\mu;u}},u_{s}\right)\mathrm{d}s-\int_{t}^{t+\delta}Z_{s}^{t,\mu;u}\mathrm{d}W_{s}$
$\displaystyle+\int_{t}^{t+\delta}\Bigg{[}\mathbb{E}^{1}\Big{[}\left\langle\partial_{\mu}\mathcal{V}\left(s,\mathbb{P}_{X_{s}^{t,\mu;u}}^{W}\right)\left(X_{s}^{t,\mu;u}\right),b_{s}^{t,\mu,u}\right\rangle$
$\displaystyle+\frac{1}{2}\text{tr}\left(\partial_{x}\partial_{\mu}\mathcal{V}\left(s,\mathbb{P}_{X_{s}^{t,\mu;u}}^{W}\right)\left(X_{s}^{t,\mu;u}\right)\sigma_{s}^{t,\mu,u}\left(\sigma_{s}^{t,\mu,u}\right)^{\top}\right)\Big{]}$
$\displaystyle+\mathbb{E}^{1}\left[\mathbb{\tilde{E}}^{1}\left[\frac{1}{2}\text{tr}\left(\partial_{\mu}^{2}\mathcal{V}\left(s,\mathbb{P}_{X_{s}^{t,\mu;u}}^{W}\right)\left(X_{s}^{t,\mu;u},\tilde{X}_{s}^{t,\mu;u}\right)\sigma_{s}^{t,\mu,u}\left(\tilde{\sigma}_{s}^{t,\mu,u}\right)^{\top}\right)\right]\right]\Bigg{]}\mathrm{d}s$
$\displaystyle+\int_{t}^{t+\delta}\mathbb{E}^{1}\left[\partial_{\mu}\mathcal{V}\left(s,\mathbb{P}_{X_{s}^{t,\mu;u}}^{W}\right)\left(X_{s}^{t,\mu;u}\right)^{\top}\sigma_{s}^{t,\mu,u}\right]\mathrm{d}W_{s}\Bigg{\\}},$
(53)

from which, we derive that

$\left\\{\begin{array}[]{l}Y_{s}^{t,\mu;\bar{u}}=\mathcal{V}\left(s,\mathbb{P}_{X_{s}^{t,\mu;u}}^{W}\right),\text{
}\mathbb{P}\text{-a.s. }t\leq s\leq T,\\\
Z_{s}^{t,\mu;\bar{u}}=\mathbb{E}^{1}\left[\partial_{\mu}\mathcal{V}\left(t,\mathbb{P}_{X_{s}^{t,\mu;u}}^{W}\right)\left(X_{s}^{t,\mu;u}\right)^{\top}\sigma_{s}^{t,\mu,u}\right],\text{
}\mathbb{P}\text{-a.s. }t\leq s\leq T.\end{array}\right.$

Arranging the above derivations and combing the fact that $\mathcal{V}$
depends on $\xi$ only through distribution $\mathcal{L}\left(\xi\right)$, we
announce that HJB equation associated to the value function of stochastic
recursive McKean-Vlasov control problem appearing in the following type:

$\left\\{\begin{array}[]{rcl}0&=&-\partial_{t}\mathcal{V}\left(t,\mu\right)\\\
&&-\inf_{u\in
U}\Bigg{[}f^{0}\Big{(}\mu,\mathcal{V}\left(t,\mathbb{\mu}\right),\mathbb{\mu}\left(\partial_{\mu}\mathcal{V}\left(t,\mathbb{\mu}\right)x^{\top}\sigma\left(x,\mu,u\right)\right),\\\
&&\mathbb{P}_{\left(\mathcal{V}\left(t,\mathbb{\mu}\right),\partial_{\mu}\mathcal{V}\left(t,\mathbb{\mu}\right)x^{\top}\sigma\left(x,\mu,u\right)\right)},u\Big{)}\\\
&&+\mu\left(\mathscr{L}^{u}\mathcal{V}\left(t,\mathbb{\mu}\right)\right)+\mu\otimes\mu\left(\mathscr{M}^{u}\mathcal{V}\left(t,\mathbb{\mu}\right)\right)\Bigg{]},\\\
\mathcal{V}\left(T,\mathbb{\mu}\right)&=&\Phi^{0}\left(\mu\right),\text{
}\left(t,\mu\right)\in\left[0,T\right]\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right),\end{array}\right.$
(54)

where for $\varphi\in
C^{2}\left(\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\right),$ the functions
$\mathscr{L}^{u}\mathcal{\varphi}\left(\mathbb{\mu}\right):\mathbb{R}^{n}\rightarrow\mathbb{R}$
and
$\mathscr{M}^{u}\mathcal{\varphi}\left(\mathbb{\mu}\right):\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$
are defined by

$\displaystyle\mathscr{L}^{u}\mathcal{\varphi}\left(\mu\right)\left(x\right)=\left\langle\partial_{\mu}\mathcal{\varphi}\left(\mu\right)\left(x\right),b\left(x,\mu,u\right)\right\rangle+\frac{1}{2}\text{tr}\left(\partial_{x}\partial_{\mu}\mathcal{\varphi}\left(\mu\right)\left(x\right)\sigma\sigma^{\top}\left(x,\mu,u\right)\right),$
$\displaystyle\mathscr{M}^{u}\mathcal{\varphi}\left(\mu\right)\left(x,\tilde{x}\right)=\frac{1}{2}\text{tr}\left(\partial_{\mu}^{2}\mathcal{\varphi}\left(\mu\right)\left(x,\tilde{x}\right)\sigma\left(x,\mu,u\right)\sigma^{\top}\left(\tilde{x},\mu,u\right)\right).$

###### Remark 3.2

Comparing with the existing results, our paper makes the following
contributions: 1) There appears the term
$\mu\otimes\mu\left(\mathscr{M}^{u}\mathcal{V}\left(t,\mu\right)\right)$ in
(54) since we apply the Itô’s formula (49) with stochastic flow which is
slightly different from the PDE in [52, 16]. 2) Our methodology can deal with
the situation whenever the generator $f$ can depends on the value function and
its gradient, which actually extends the framework [22], [45] and [33] to a
more general case.

By means of the lifting identification, HJB equations (54) can be expressed in
$\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right).$ To
keep the compatibility of symbols, we still denote
$\mathcal{V}\left(t,\mathbb{\xi}\right)=\mathcal{V}\left(t,\mathcal{L}\left(\xi\right)\right)$
combing the fact that $\mathcal{V}$ depends on $\xi$ only through distribution
again. From the relation (50), alternatively, HJB equations (54) can be
reported as

$\left\\{\begin{array}[]{rcl}0&=&-\partial_{t}\mathcal{V}\left(t,\mathbb{\xi}\right)-\mathcal{H}\left(\xi,\mathcal{V}\left(t,\mathbb{\xi}\right),D\mathcal{V}\left(t,\mathbb{\xi}\right),D^{2}\mathcal{V}\left(t,\mathbb{\xi}\right)\right),\\\
&&\text{ }\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right),\\\
\mathcal{V}\left(T,\mathbb{\xi}\right)&=&\mathbb{E}^{1}\left[\Phi\left(\xi,\mathcal{L}\left(\xi\right)\right)\right],\text{
}\xi\in
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right),\end{array}\right.$
(55)

where
$\mathcal{H}:L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}\right)\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{S}^{n}\right)\rightarrow\mathbb{R}$
is defined as

$\displaystyle\mathcal{H}\left(\xi,p,q,Q\right)$ (56) $\displaystyle=$
$\displaystyle\inf_{u\in
U}\mathbb{E}^{1}\Bigg{[}f\Big{(}\xi,\mathcal{L}\left(\xi\right),p,\mathbb{E}^{1}\left[q^{\top}\sigma\left(\xi,\mathcal{L}\left(\xi\right),u\right)\right],\mathcal{L}_{\left(p,\mathbb{E}^{1}\left[q^{\top}\sigma\left(\xi,\mathcal{L}\left(\xi\right),u\right)\right]\right)},u\Big{)}$
$\displaystyle+\left\langle
q,b\left(\xi,\mathcal{L}\left(\xi\right),u\right)\right\rangle+\frac{1}{2}Q\left(\sigma\left(\xi,\mathcal{L}\left(\xi\right),u\right)\right)\sigma\left(\xi,\mathcal{L}\left(\xi\right),u\right)\Bigg{]}.$

As we have well-known, in the stochastic optimal control theory, the value
function is a solution to the corresponding H-J-B equation whenever it admits
classical solutions (smooth enough to the order of derivatives involved in the
equation)555We refer to the paper [36] for existence result of smooth solution
to the HJB equation (55) on small time horizon.. Unfortunately, this is not
necessarily the case even for some very simple situations.

Normally, whenever the diffusion is possibly degenerate, the HJB equation may
in general have no classical solutions. To remove this obstacle, Crandall and
Lions introduced the so-called viscosity solutions in the early 1980s (cf.
[20]). Out of question, this idea makes the theory a powerful tool in studying
the optimal control problems.

Similarly, we will employ a notion of viscosity solutions via the lifting
identification in the Hilbert space
$L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$
instead of the Wasserstein space $\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)$.

For simplicity, we define, for $\left(\xi,r,p,P\right)\in
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}\right)\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{S}^{n}\right),$

$F\left(\xi,r,p,P\right)=-\mathcal{H}\left(\xi,r,p,P\right).$ (57)

Then the H-J-B equation (55) can be expressed as

$\left\\{\begin{array}[]{rcl}0&=&-\partial_{t}\mathcal{V}\left(t,\mathbb{\xi}\right)-F\left(\xi,\mathcal{V}\left(t,\mathbb{\xi}\right),D\mathcal{V}\left(t,\mathbb{\xi}\right),D^{2}\mathcal{V}\left(t,\mathbb{\xi}\right)\right),\\\
&&\text{ }\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right),\\\
\mathcal{V}\left(T,\mathbb{\xi}\right)&=&\mathbb{E}^{1}\left[\Phi\left(\xi,\mathcal{L}\left(\xi\right)\right)\right],\text{
}\xi\in
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right).\end{array}\right.$
(58)

The following definition of viscosity solution for infinite dimension is
similar as Definition 3.40 in [35] for time-dependent situation since HJB
equation (58) does not contain the dissipative operator.

###### Definition 3.1

Let $\mathscr{V}\left(t,\mu\right)\in
C\left(\left[0,T\right]\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\right)$
and $\tilde{\mathscr{V}}$ be lifted version of $\mathscr{V}.$ For every
$\varphi\in C^{1,2}\left(\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\right),$666Here
$C^{1,2}\left(\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\right)$
denotes the set of real-valued continuous functions on $\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$ which
are continuously differentiable in $t\in\left[0,T\right)$, and twice
continuously Fréchet differentiable on
$L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right).$

(1) for each local maximum point
$\left(t_{0},\xi_{0}\right)\in\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$ of
$\tilde{\mathscr{V}}-\varphi$ in the interior of $\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right),$ we
have

$-\frac{\partial}{\partial
t}\varphi+F\left(\xi_{0},\varphi,D\varphi,D^{2}\varphi\right)\left(t_{0},\xi_{0}\right)\leq
0$ (59)

and

$\tilde{\mathscr{V}}\left(T,\xi_{0}\right)-\Phi^{0}\left(\mathcal{L}\left(\xi_{0}\right)\right)\leq
0,$ (60)

i.e., $\mathscr{V}$ is a sub-solution to HJB equation _( 58)_;

(2) for each local minimum point
$\left(t_{0},\xi_{0}\right)\in\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$ of
$\tilde{\mathscr{V}}-\varphi$ in the interior of $\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right),$ we
have

$-\frac{\partial}{\partial
t}\varphi+F\left(\xi_{0},\varphi,D\varphi,D^{2}\varphi\right)\left(t_{0},\xi_{0}\right)\geq
0$ (61)

and

$\tilde{\mathscr{V}}\left(T,\xi_{0}\right)-\Phi^{0}\left(\mathcal{L}\left(\xi_{0}\right)\right)\geq
0,$ (62)

i.e., $\mathscr{V}$ is a super-solution to HJB equation _( 58)_.

(3) $\mathscr{V}\left(t,\mu\right)\in
C\left(\left[0,T\right]\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\right)$
is said to be a viscosity solution of _( 58)_ if it is both a viscosity sub-
and super-solution.

Note that the Hamiltonian function $\mathcal{H}$ define in (56) depends not
only on $\mathcal{V}$ but its derivative. In order to prove the uniqueness of
solution to HJB equation (55), we need the following hypothesis, under which
we say that $F$ is _proper_ whenever $\kappa>0.$

  1. (A6)

There exists a $\kappa\geq 0$ such that for every
$\left(x,\mu,z,\nu,u\right)\in\mathbb{R}^{n}\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right)\times\mathbb{R}^{d}\times\mathcal{P}_{2}\left(\mathbb{R}\times\mathbb{R}^{d}\right)\times
U$ and $r,$ $s\in\mathbb{R}$ with $r\geq s,$

$f\left(x,\mu,r,z,\nu,u\right)-f\left(x,\mu,s,z,\nu,u\right)\leq\kappa\left(r-s\right).$

We are at position to assert the main result:

###### Theorem 3.1

Assume that the Assumptions _(A1)-(A6)_ are in force. Then the value function
defined in (_35_) is a unique viscosity solution to HJB equation (_55_).

Proof We will employ the DPP of Lemma 2.13 for the value function now regarded
as a function on $\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right).$ To
this end, denoting
$\mathscr{X}_{s}^{t,\xi;u}\left(\omega^{0}\right)=X_{s}^{t,\xi;u}\left(\omega^{0},\cdot\right)$
for $t\leq s\leq T.$ Apparently,
$\left\\{\mathscr{X}_{s}^{t,\xi;u}\right\\}_{t\leq s\leq T}$ is
$\mathcal{F}^{0}$-adapted process, valued in
$L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$ and
$\rho_{s}^{t,\mu;u}=\mathbb{P}_{\mathscr{X}_{s}^{t,\xi;u}}^{1}$ for $t\leq
s\leq T$ with $\mu=\mathcal{L}\left(\xi\right)$. Immediately, we derive that
the value function satisfying
$\mathcal{V}\left(s,\mathscr{X}_{s}^{t,\xi;u}\right)=\mathcal{V}\left(s,\rho_{s}^{t,\mu;u}\right),$
namely, the lifted valued function on $\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$ equals
to the value function on
$\left[0,T\right]\times\mathcal{P}_{2}\left(\mathbb{R}^{n}\right).$ Therefore,
DPP in Lemma 2.13 can be written as, for any
$\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$ with
$\mu=\mathcal{L}\left(\xi\right),$

$\displaystyle\mathcal{V}\left(t,\xi\right)$ $\displaystyle=$
$\displaystyle\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\inf_{\tau\in\mathbb{T}_{t,T}^{0}}\mathbb{G}_{t,\tau}^{t,\xi;u}\left[\mathcal{V}\left(\tau,\mathscr{X}_{\tau}^{t,\xi;u}\right)\right]$
$\displaystyle=$
$\displaystyle\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\sup_{\tau\in\mathbb{T}_{t,T}^{0}}\mathbb{G}_{t,\tau}^{t,\xi;u}\left[\mathcal{V}\left(\tau,\mathscr{X}_{\tau}^{t,\xi;u}\right)\right],$

From Lemma 2.12 and Lemma 2.14, we know that the value function is continuous
on $\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right).$
Particularly, at the terminal time, it requires that
$\mathcal{V}\left(T,\xi\right)=\mathbb{E}^{1}\left[\Phi\left(\xi,\mathcal{L}\left(\xi\right)\right)\right].$
For the proof of this theorem, we take the following three steps.

_Step 1._ For arbitrarily chosen but fixed $\varphi\in
C_{l,b}^{2}\left(\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\right),$777Here
$C_{l,b}^{2}\left(\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\right)$
denotes the set of class $C^{2}$ (Lions’s sense) whose partial derivatives of
order less than or equal to $2$ are bounded (and hence the function itself
grows at most linearly at infinity). we define

$\displaystyle\mathscr{F}\left(t,\xi,y,z,u\right)$ $\displaystyle=$
$\displaystyle\partial_{t}\varphi\left(t,\xi\right)+\mathbb{E}^{1}\Bigg{\\{}\left\langle
D\varphi\left(t,\xi\right),b\left(\xi,\mathcal{L}\left(\xi\right),u\right)\right\rangle$
$\displaystyle+\frac{1}{2}D^{2}\varphi\left(t,\xi\right)\left(\sigma\left(\xi,\mathcal{L}\left(\xi\right),u\right)\right)\sigma\left(\xi,\mathcal{L}\left(\xi\right),u\right)$
$\displaystyle+f\Big{(}\xi,\mathcal{L}\left(\xi\right),y+\varphi\left(t,\xi\right),z+\mathbb{E}^{1}\left[D\varphi\left(t,\xi\right)^{\top}\sigma\left(\xi,\mathcal{L}\left(\xi\right),u\right)\right],$
$\displaystyle\mathcal{L}_{\left(y+\varphi\left(t,\xi\right),z+\mathbb{E}^{1}\left[q^{\top}\sigma\left(\xi,\mathcal{L}\left(\xi\right),u\right)\right]\right)},u\Big{)}\Bigg{\\}}$

$\left(t,\xi,y,z,u\right)\in\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\times\mathbb{R}\times\mathbb{R}^{d}\times
U,$ and we introduce the following BSDE defined on the interval
$\left[t,t+\delta\right]$ for $0<\delta\leq T-t,$

$\left\\{\begin{array}[]{rcl}-\mathrm{d}\mathscr{Y}_{s}^{1,u}&=&\mathscr{F}\left(s,\mathscr{X}_{s}^{t,\xi;u},\mathscr{Y}_{s}^{1,u},\mathscr{Z}_{s}^{1,u},u_{s}\right)\mathrm{d}s-\mathscr{Z}_{s}^{1,u}\mathrm{d}W_{s},\text{
}t\leq s\leq t+\delta,\\\ \mathscr{Y}_{t+\delta}^{1,u}&=&0,\end{array}\right.$
(63)

where $u\in\mathcal{U}_{ad}^{2}\left[0,T\right]$ and the process
$\mathscr{X}_{s}^{t,\xi;u}$ is defined by
$X_{s}^{t,\xi;u}\left(\omega^{0},\cdot\right).$ It is easy to check that
$\tilde{f}\left(s,y,z,u\right)$ fulfills ()-() in ?. Thus BSDE (63) admits a
unique solution. As a matter of fact, we are able to establish the
relationship between $\mathscr{Y}_{s}^{1,u}$ and $\mathscr{X}_{s}^{t,\xi;u}$
by virtue of Itô’s formula as follows:

$\mathscr{Y}_{s}^{1,u}=\mathbb{G}_{s,t+\delta}^{t,\xi;u}\left[\varphi\left(t+\delta,\mathscr{X}_{t+\delta}^{t,\xi;u}\right)\right]-\varphi\left(s,\mathscr{X}_{s}^{t,\xi;u}\right)$
(64)

Recall that
$\mathbb{G}_{s,t+\delta}^{t,\xi;u}\left[\varphi\left(t+\delta,\mathscr{X}_{t+\delta}^{t,\xi;u}\right)\right]$
is corresponding to the solution of the following BSDE:

$\left\\{\begin{array}[]{rcl}-\mathrm{d}\mathscr{Y}_{s}^{u}&=&f\left(s,\mathscr{X}_{s}^{t,\xi;u},\mathscr{Y}_{s}^{u},\mathscr{Z}_{s}^{u},u_{s}\right)\mathrm{d}s-\mathscr{Z}_{s}^{u}\mathrm{d}W_{s},\\\
\mathscr{Y}_{T}^{u}&=&\varphi\left(t+\delta,\mathscr{X}_{t+\delta}^{t,\xi;u}\right)\end{array}\right.$
(65)

by

$\mathbb{G}_{s,t+\delta}^{t,\xi;u}\left[\varphi\left(t+\delta,\mathscr{X}_{t+\delta}^{t,\xi;u}\right)\right]=\mathscr{Y}_{s}^{u},\text{
}t\leq s\leq t+\delta.$

The relation (64) can be verified by applying Itô’s formula to
$\varphi\left(s,\mathscr{X}_{s}^{t,\xi;u}\right).$ Afterward, we compare the
coefficients of the stochastic differentials of
$\mathscr{Y}_{s}^{u}-\varphi\left(s,\mathscr{X}_{s}^{t,\xi;u}\right)$ and
$\mathscr{Y}_{s}^{1,u},$ meanwhile we notice that at the terminal time
$t+\delta,$ the equality

$\mathscr{Y}_{t+\delta}^{u}-\varphi\left(s,\mathscr{X}_{t+\delta}^{t,\xi;u}\right)=0=\mathscr{Y}_{t+\delta}^{1,u}$

holds.

_Step 2._ Consider the following BSDE

$\left\\{\begin{array}[]{rcl}-\mathrm{d}\mathscr{Y}_{s}^{2,u}&=&\mathscr{F}\left(s,\xi,\mathscr{Y}_{s}^{2,u},\mathscr{Z}_{s}^{2,u},u_{s}\right)\mathrm{d}s-\mathscr{Z}_{s}^{2,u}\mathrm{d}W_{s},\text{
}t\leq s\leq t+\delta,\\\ \mathscr{Y}_{t+\delta}^{2,u}&=&0,\end{array}\right.$
(66)

where the driving process $\mathscr{X}_{s}^{t,\xi;u}$ in (63) are replaced by
the initial state $\xi.$

###### Lemma 3.1

For any $u\in\mathcal{U}_{ad}^{2}\left[0,T\right],$ there exists a constant
$C$ independent of $u\in\mathcal{U}_{ad}^{2}\left[0,T\right]$ such that

$\left|\mathscr{Y}_{t}^{1,u}-\mathscr{Y}_{t}^{2,u}\right|\leq
C\delta\rho_{1}\left(\delta\right),$

where $\rho_{1}\left(\delta\right):\mathbb{R}_{+}\rightarrow\mathbb{R}.$ When
$\delta\rightarrow 0,$ it has $\rho_{1}\left(\delta\right)\rightarrow 0$ and
$\rho_{1}\left(\delta\right)$ is independent of contorl
$u\in\mathcal{U}_{ad}^{2}\left[0,T\right].$

The proof can be seen in the Appendix.

###### Lemma 3.2

Consider the following ordinary differential equation with law (ODEL for
short) :

$\left\\{\begin{array}[]{rcl}-\dot{\mathscr{Y}}_{s}&=&\Upsilon\left(s,\xi,\mathscr{Y}_{s},0\right),\\\
\mathscr{Y}_{t+\delta}&=&0,\end{array}\right.$ (67)

where the function $\Upsilon$ is defined by

$\Upsilon\left(t,\xi,y,z\right)\triangleq\inf_{u\in
U}\mathscr{F}\left(t,\xi,y,z,u\right),$ (68)

in which $\left(t,\xi,y,z\right)\in\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\times\mathbb{R}\times\mathbb{R}^{d}.$
Then

$\underset{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}{\text{essinf}}\mathscr{Y}_{t}^{2,u}=\mathscr{Y}_{t},\text{
a.s.}$

The proof also can be seen in the Appendix.

Apparently, $\Upsilon\left(t,\xi,y,z\right)$ is Lipschitz in
$\left(y,z\right)$, uniformly with respect to $\left(t,\xi\right)$ This
guarantees the existence and uniqueness of ODEL (67). Consider the BSDE

$\left\\{\begin{array}[]{rcl}-\dot{\mathscr{Y}}_{s}&=&\Upsilon\left(s,\xi,\mathscr{Y}_{s},\mathscr{Z}_{s}\right)\mathrm{d}s-\mathscr{Z}_{s}\mathrm{d}W_{s}\\\
\mathscr{Y}_{t+\delta}&=&0,\text{ }t\leq s\leq t+\delta.\end{array}\right.$
(69)

From the definition of $\Upsilon$ (see (68)) we know

$\Upsilon\left(t,\xi,y,z\right)\leq\mathscr{F}\left(t,\xi,y,z,u_{s}\right),\text{
for all }u\in\mathcal{U}_{ad}^{2}\left[0,T\right].$

From Lemma 2.4 (comparison theorem), we have

$\mathscr{Y}_{t}\leq\underset{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}{\text{essinf}}\mathscr{Y}_{t}^{2,u}.$
(70)

Observe that $\Upsilon$ is a deterministic function of
$\left(t,\xi,y,z\right)$ (see the definition of $F$), therefore
$\left(\mathscr{Y}_{s},\mathscr{Z}_{s}\right)\equiv$
$\left(\mathscr{Y}_{s},0\right)$ where $\mathscr{Y}$ is the solution of the
ODEL (67). Now to prove the reverse inequality in (70), we introduce the
following ODEL

$\left\\{\begin{array}[]{rcl}-\dot{\mathscr{Y}_{s}^{2,\upsilon}}&=&\mathscr{F}\left(s,\xi,\mathscr{Y}_{s}^{2,\upsilon},0,\upsilon_{s}\right),\\\
\mathscr{Y}_{t+\delta}^{2,\upsilon}&=&0,\text{ }t\leq s\leq
t+\delta,\end{array}\right.$ (71)

where $\upsilon\in\mathscr{U}\left[0,T\right]$ (Here
$\mathscr{U}\left[0,T\right]$ denotes the set of all deterministic admissible
controls). Immediately, one can show

$\mathscr{Y}_{t}=\inf_{\upsilon\in\mathscr{U}\left[0,T\right]}\mathscr{Y}_{t}^{2,\upsilon}.$

_Step 3._ We will prove that $\mathcal{V}$ is a viscosity solution of (55). At
the beginning, we have
$\mathcal{V}\left(T,\xi\right)=\mathbb{E}^{1}\left[\Phi\left(\xi,\mathcal{L}\left(\xi\right)\right)\right].$
We only prove that W is a viscosity sub-solution since the proof is analogous
for the viscosity super-solution. For this we suppose that $\varphi\in
C^{1,2}\left(\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)\right),$
and $\left(t,\xi\right)\in\left[0,T\right]\times
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$ such
that $\mathcal{V}-\varphi$ attains a maximum at $\left(t,\xi\right).$ Without
loss of generality we may also suppose that
$\mathcal{V}\left(t,\xi\right)=\varphi\left(t,\xi\right)$. Then, by virtue of
the DPP (see Lemma 2.13 and Lemma 2.10),

$\varphi\left(t,\xi\right)=\mathcal{V}\left(t,\xi\right)=\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\mathbb{G}_{t,t+\delta}^{t,\mu;u}\left[\mathcal{V}\left(t+\delta,X_{t+\delta}^{t,\mu;u}\right)\right],\text{
}0\leq\delta\leq T-t,$

and from $\mathcal{V}\leq\varphi$ and the monotonicity property of
$\mathbb{G}_{t,t+\delta}^{t,\mu;u}\left[\mathcal{\cdot}\right]$ (see Lemma
2.4), we get

$\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\left\\{\mathbb{G}_{t,t+\delta}^{t,\mu;u}\left[\varphi\left(t+\delta,X_{t+\delta}^{t,\mu;u}\right)\right]-\varphi\left(t,\xi\right)\right\\}\geq
0,\text{ a.s.}$

From the relation (64), we have
$\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\mathscr{Y}_{t}^{1,u}\geq 0,$
a.s., and furthermore, from Lemma 3.1, we have
$\inf_{u\in\mathcal{U}_{ad}^{2}\left[0,T\right]}\mathscr{Y}_{t}^{2,u}\geq-C\delta\rho_{1}\left(\delta\right).$
By means of Lemma 3.2, it follows

$\mathscr{Y}_{t}\geq-C\delta\rho_{1}\left(\delta\right).$

Therefore, we deduce that

$\Upsilon\left(t,\xi,0,0\right)=\inf_{u\in
U}\mathscr{F}\left(t,\xi,0,0,u\right)\geq 0.$

From the definition of $F$ we report that $\mathcal{V}$ is the viscosity
subsolution (resp., supersolution) of equation (55) with the terminal
condition (60), i.e., $\mathcal{V}$ is the viscosity solution.

Next we proceed the uniqueness of viscosity solution via a comparison
principle for viscosity solutions in infinite dimension to the lifted HJB
equation (58). Just like [45], we need check that all the assumptions of
Theorem 3.50 in [35] are fulfilled with the framework of the lifted
Hamiltonian $F$ defined in (57).

Observe first that there is no appearance of linear dissipative operator in
the equation, thus the HJB equation (58 is a bounded equation in the
terminology of [35]. Therefore, the notion of $B$-continuity reduces to the
standard notion of continuity in
$L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{R}^{n}\right)$,
namely, $B=I$ (the identity operator)888Let $A$ be a linear, densely defined,
closed operator in $H.$ Whenever $A=0,$ and if one sets $B=I$, it follows
immediately that $B$ satisfies the strong $B$-condition (Defintion 3.10 in
[35]). . The norm define in [35]

$\left|x\right|_{-1}^{2}=\left|x\right|^{2}=\left\langle
x,x\right\rangle,\text{ }x\in H,$

where $H$ is a real, separable Hilbert space with inner product
$\left\langle\cdot,\cdot\right\rangle.$ In particular, we set
$\left\\{e_{1},e_{2},\cdots\right\\}$ is an orthonormal basis of $H$, and for
$N>2$, $H_{N}=$span$\left\\{e_{1},e_{2},\cdots e_{N}\right\\}$, $P_{N}$ is the
orthogonal projection in $H$ onto $H_{N}$. We denote $S\left(H\right)$ the set
of bounded, self-adjoint operators on $H.$

We will check one by one the conditions in Theorem 3.50 in [35]. The
Hypothesis 3.44 holds from the uniform continuity of $b$, $\sigma$ and $f$ in
(A1)-(A2).

As for Hypothesis 3.45, invoking (A6), Hypothesis 3.45 is immediately
satisfied.

It is rather easy to check that the monotonicity condition in $P\in
L^{2}\left(\Omega^{1},\mathcal{G},\mathbb{P}^{1};\mathbb{S}^{n}\right)$) of
$F$ in Hypothesis 3.46 is satisfied since the generator $f$ is independent of
$P.$

Hypothesis 3.47 holds directly when dealing with bounded equations.

We now deal with Hypothesis 3.48. Let $X,Y\in S\left(H\right)$ such that
$X=P_{N}^{\ast}XP_{N},$ $Y=P_{N}^{\ast}YP_{N}$ for some $N$ and satisfying

$-\frac{3}{\varepsilon}\left(\begin{array}[]{cc}P_{N}&0\\\
0&P_{N}\end{array}\right)\leq\left(\begin{array}[]{cc}X&0\\\
0&-Y\end{array}\right)\leq\frac{1}{\varepsilon}\left(\begin{array}[]{cc}P_{N}&-P_{N}\\\
-P_{N}&P_{N}\end{array}\right).$ (72)

We proceed the following term

$\displaystyle
F\left(x,r,\frac{x-y}{\varepsilon},X\right)-F\left(y,r,\frac{x-y}{\varepsilon},Y\right)$
$\displaystyle=$ $\displaystyle\inf_{u\in
U}\mathbb{E}^{1}\Bigg{[}f\Big{(}x,\mathcal{L}\left(x\right),r,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(x,\mathcal{L}\left(\xi\right),u\right)\right],\mathcal{L}_{\left(r,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(x,\mathcal{L}\left(x\right),u\right)\right]\right)},u\Big{)}$
$\displaystyle+\left\langle\frac{\left(x-y\right)}{\varepsilon},b\left(x,\mathcal{L}\left(x\right),u\right)\right\rangle+\frac{1}{2}\left\langle
X\left(\sigma\left(x,\mathcal{L}\left(x\right),u\right)\right),\sigma\left(x,\mathcal{L}\left(x\right),u\right)\right\rangle\Bigg{]}$
$\displaystyle-\inf_{u\in
U}\mathbb{E}^{1}\Bigg{[}f\Big{(}y,\mathcal{L}\left(y\right),r,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(y,\mathcal{L}\left(y\right),u\right)\right],\mathcal{L}_{\left(r,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(y,\mathcal{L}\left(y\right),u\right)\right]\right)},u\Big{)}$
$\displaystyle+\left\langle\frac{\left(x-y\right)^{\top}}{\varepsilon},b\left(y,\mathcal{L}\left(y\right),u\right)\right\rangle+\frac{1}{2}\left\langle
Y\left(\sigma\left(y,\mathcal{L}\left(y\right),u\right)\right),\sigma\left(y,\mathcal{L}\left(y\right),u\right)\right\rangle\Bigg{]},$

For any $\epsilon>0,$ there exists a measurable functions
$u^{\varepsilon}:\mathbb{R}^{n}\rightarrow U,$ such that

$\displaystyle\mathbb{E}^{1}\Bigg{[}f\Big{(}x,\mathcal{L}\left(x\right),r,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(x,\mathcal{L}\left(\xi\right),u^{\varepsilon}\right)\right],\mathcal{L}_{\left(r,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(x,\mathcal{L}\left(x\right),u^{\varepsilon}\right)\right]\right)},u^{\varepsilon}\Big{)}$
$\displaystyle+\left\langle\frac{\left(x-y\right)}{\varepsilon},b\left(x,\mathcal{L}\left(x\right),u^{\varepsilon}\right)\right\rangle+\frac{1}{2}\left\langle
P\left(\sigma\left(x,\mathcal{L}\left(x\right),u^{\varepsilon}\right)\right),\sigma\left(x,\mathcal{L}\left(x\right),u^{\varepsilon}\right)\right\rangle\Bigg{]}-\epsilon$
$\displaystyle\leq$ $\displaystyle\inf_{u\in
U}\mathbb{E}^{1}\Bigg{[}f\Big{(}x,\mathcal{L}\left(x\right),r,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(x,\mathcal{L}\left(\xi\right),u\right)\right],\mathcal{L}_{\left(r,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(x,\mathcal{L}\left(x\right),u\right)\right]\right)},u\Big{)}$
$\displaystyle+\left\langle\frac{\left(x-y\right)}{\varepsilon},b\left(x,\mathcal{L}\left(x\right),u\right)\right\rangle+\frac{1}{2}\left\langle
P\left(\sigma\left(x,\mathcal{L}\left(x\right),u\right)\right),\sigma\left(x,\mathcal{L}\left(x\right),u\right)\right\rangle\Bigg{]}.$

Immidiately, we have, from the condition of $b$ and $\sigma$ in (A1)-(A2), and
the uniform continuity condition on $f$ in (A3),

$\displaystyle
F\left(x,r,\frac{x-y}{\varepsilon},P\right)-F\left(y,r,\frac{x-y}{\varepsilon},\tilde{P}\right)$
$\displaystyle\geq$
$\displaystyle\mathbb{E}^{1}\Bigg{[}f\Big{(}x,\mathcal{L}\left(x\right),r,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(x,\mathcal{L}\left(\xi\right),u^{\varepsilon}\right)\right],\mathcal{L}_{\left(r,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(x,\mathcal{L}\left(x\right),u^{\varepsilon}\right)\right]\right)},u^{\varepsilon}\Big{)}$
$\displaystyle+\left\langle\frac{\left(x-y\right)}{\varepsilon},b\left(x,\mathcal{L}\left(x\right),u^{\varepsilon}\right)\right\rangle+\frac{1}{2}\left\langle
P\left(\sigma\left(x,\mathcal{L}\left(x\right),u^{\varepsilon}\right)\right),\sigma\left(x,\mathcal{L}\left(x\right),u^{\varepsilon}\right)\right\rangle\Bigg{]}-\varepsilon$
$\displaystyle-\mathbb{E}^{1}\Bigg{[}f\Big{(}y,\mathcal{L}\left(y\right),r,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(y,\mathcal{L}\left(y\right),u^{\varepsilon}\right)\right],\mathcal{L}_{\left(p,\mathbb{E}^{1}\left[\frac{\left(x-y\right)^{\top}}{\varepsilon}\sigma\left(y,\mathcal{L}\left(y\right),u^{\varepsilon}\right)\right]\right)},u^{\varepsilon}\Big{)}$
$\displaystyle-\left\langle\frac{\left(x-y\right)^{\top}}{\varepsilon},b\left(y,\mathcal{L}\left(y\right),u^{\varepsilon}\right)\right\rangle-\frac{1}{2}\left\langle\tilde{P}\left(\sigma\left(y,\mathcal{L}\left(y\right),u^{\varepsilon}\right)\right),\sigma\left(y,\mathcal{L}\left(y\right),u^{\varepsilon}\right)\right\rangle\Bigg{]}$
$\displaystyle\geq$
$\displaystyle-C\left(\left|x-y\right|\left(1+\frac{\left|x-y\right|}{\varepsilon}\right)\right)+\frac{1}{2}\left\langle
P\left(\sigma\left(x,\mathcal{L}\left(x\right),u^{\varepsilon}\right)\right),\sigma\left(x,\mathcal{L}\left(x\right),u^{\varepsilon}\right)\right\rangle$
$\displaystyle-\frac{1}{2}\left\langle\tilde{P}\left(\sigma\left(y,\mathcal{L}\left(y\right),u^{\varepsilon}\right)\right),\sigma\left(y,\mathcal{L}\left(y\right),u^{\varepsilon}\right)\right\rangle.$

We proceed the term (with