# Well-posedness of path-dependent semilinear parabolic master equations

Shanjian Tang  S. Tang, School of Mathematical Sciences, Fudan University,
Handan Road 220, 200433, Shanghai, PRC<EMAIL_ADDRESS>and  Huilin Zhang
H. Zhang, Research Center for Mathematics and Interdisciplinary Sciences,
Shandong University, Binhai Road 72, 266237, Qingdao, PRC.
<EMAIL_ADDRESS>
###### Abstract.

Master equations are partial differential equations for measure-dependent
unknowns, and are introduced to describe asymptotic equilibrium of large scale
mean-field interacting systems, especially in games and control theory. In
this paper we introduce new semilinear master equations whose unknowns are
functionals of both paths and path measures. They include state-dependent
master equations, path-dependent partial differential equations (PPDEs),
history information dependent master equations and time inconsistent (e.g.
time-delayed) equations, which naturally arise in stochastic control theory
and games. We give a classical solution to the master equation by introducing
a new notation called strong vertical derivative (SVD) for path-dependent
functionals, inspired by Dupire’s vertical derivative [20], and applying
stochastic forward-backward system argument. Moreover, we consider a general
non-smooth case with a functional mollifying method.

###### Key words and phrases:

path-dependent, master equation, mean-field games, Itô-Dupire formula

###### 2010 Mathematics Subject Classification:

60G22, 60H10, 34C29

###### Contents

  1. 1 Introduction
  2. 2 Basic setup and Itô calculus for functionals of path and path-measure
    1. 2.1 The canonical setup
    2. 2.2 Strong vertical derivatives with respect to path and path-measure
    3. 2.3 Itô-Dupire formula
  3. 3 Differentiability of solutions for path-dependent mean-field BSDEs
    1. 3.1 First-order differentiability
    2. 3.2 Second-order differentiability
  4. 4 Solutions of semilinear path-dependent master equations
    1. 4.1 The decoupling field and its regularity
    2. 4.2 Classical solutions of path-dependent master equations
    3. 4.3 Some typical cases
    4. 4.4 The general case via functional mollifying
  5. 5 Appendix
    1. 5.1 Proof of Lemma 3.4
    2. 5.2 An extension of [41, Theorem 4.5] without assumption of local Lipschitz continuity in time variable

##  1\. Introduction

Denote by $\mathbb{C}_{T,d}$ the space of continuous functions on $[0,T]$ with
values in $\mathbb{R}^{d}$ and by $\mathcal{P}^{C}_{2}$ the totality of
probability measures on $\mathbb{C}_{T,d}$ with finite second order moments.
Given deterministic functions $(b_{1},\sigma_{1})$ and $(b_{2},\sigma_{2})$ on
$[0,T]$ with values in $\mathbb{R}^{d}\times\mathbb{R}^{d\times d}$, we study
the following path-dependent parabolic master equation,

(6)
$\displaystyle\left\\{\begin{array}[]{l}\partial_{t}u(t,\omega,\mu)+\frac{1}{2}\text{Tr}\
[\partial_{\omega}^{2}u(t,\omega,\mu)\sigma_{1}(t)\sigma_{1}(t)^{T}]+\partial_{\omega}u(t,\omega,\mu)b_{1}(t)\\\\[5.69054pt]
\quad+\frac{1}{2}\text{Tr}\
[\int_{\mathbb{C}_{T,d}}\partial_{\tilde{\omega}}\partial_{\mu}u(t,\omega,\mu,\tilde{\omega})\mu(d\tilde{\omega})\sigma_{2}(t)\sigma_{2}(t)^{T}]+\int_{\mathbb{C}_{T,d}}\partial_{\mu}u(t,\omega,\mu,\tilde{\omega})\mu(d\tilde{\omega})b_{2}(t)\\\\[5.69054pt]
\ \ \
+f(t,\omega,u(t,\omega,\mu),\sigma_{1}(t)\partial_{\omega}u(t,\omega,\mu),\mu,\mathcal{L}_{u(t,W^{\mu},\mu)})=0,\\\
\\\ u(T,\omega,\mu)=\Phi(\omega,\mu),\ \ \
(t,\omega,\mu)\in[0,T]\times\mathbb{C}_{T,d}\times\mathcal{P}^{C}_{2}.\end{array}\right.$

Here, (functional) derivatives $\partial_{\omega}$ and $\partial_{\mu}$ are
taken in the spirit of Dupire and Lions (see the subsequent definitions (17)
and (34)), respectively, and $W^{\mu}$ represents the canonical processes on
$\mathbb{C}_{T,d}$ under $\mu.$ Master equations (also called Hamiltonian-
Jacobi-Bellman (HJB) equations in a Wasserstein space when concerned with
control problems) naturally arise from mean-field games. The mean-field game
is introduced independently by Lasry and Lions [35] and Huang, Malhamé and
Caines [31] to the study of Nash equilibrium for systems consist of a large
number of “agents”. The classical mean-field theory, popular in statistical
physics (e.g. McKean-Vlasov and Boltzmann models), quantum mechanics and
quantum chemistry (e.g. Hartree-Fock type model), was developed in the last
century to study systems with large size of particles (see Kac [33], McKean
[37], Sznitman [45, 46, 47, 48], Bossy [3] and references therein). In the
last decade, the mean-field theory has been widely studied and applied from
theoretical areas including stochastic differential games, partial
differential equations (PDEs) and stochastic control, etc., to practical areas
such as engineering, economics and social science, just to mention a few
publications and references therein: [36], [15], [7], [11], [12], [27]. The
master equation is a PDE involving measure variables on a Wasserstein space of
infinite dimension and introduced by Lions in his lecture [36] for
differential games. It is a powerful analytic tool for the study of large
systems in physics and games with a mean-field interaction (see [14], [15]),
and includes interesting particular cases such as HJB and Fokker-Planck (FP)
equations in dynamical systems, stochastic control and mathematical finance of
mean-field type. Results are available in various frameworks: Bensoussan et
al. [5] consider the regular case when measure variables are restricted on
those measures of square integrable density functions, Cardaliaguet [15] gives
a viscosity solution for first-order HJB equations on a Wasserstein space,
Gomes and Saude [30] survey well-posedness of HJB-FP equations for reduced
mean-field games, Buckdahn et al. [9] and Chassagneux et al. [10] study
classical solutions for second order master equations through stochastic
differential equations (SDEs) and forward backward stochastic differential
equations (FBSDEs) respectively, Carmona and Delarue [13] consider the mean-
field games and corresponding master equation with common noise, Cardaliaguet
et al. [14] give an analytic approach for master equations, Pham and Wei [42]
study the dynamic programming principle for Bellman master equation, etc.
However, all these works consider the state-dependent case, where
$(\omega,\mu)$ in Equation (6) take values in
$\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R}^{k})$. Here,
$\mathcal{P}_{2}(\mathbb{R}^{k})$ is the set of probability measures on
$\mathbb{R}^{k}$ with finite second order moments. In practice, many problems
could be non-Markovian or path-dependent: to mention a few, optional pricing
for exotic options (e.g. Asian, chooser, lookback and barrier options [20],
[19], [32], [26]), stochastic game theory and stochastic control with delayed
information ([2], [28], [44], [51], [49]), rough volatility [29], [6], etc.

Dupire [20] introduces a functional Itô formula to incorporate the calculus of
path-dependent functionals, which is subsequently developed by Cont-Fournié
[16, 17] and references therein (on the other hand, see another approach to
path-dependent problems of Flandoli and Zanco [25] by lifting the primal
problem into a functional one in Banach spaces). In contrast to the classical
functional approach to the path-dependent stochastic analysis (see Ahn [1]),
Dupire’s approach is featured by the finite dimensional vertical derivative
(see the following definition (17)), and is admitted to solve non-Markovin
problems, in particular the one proposed by Peng in his ICM 2010 lecture [40]
that whether non-Markovian FBSDEs can be connected with path-dependent PDEs
(PPDEs). Concerning the well-posedness of PPDEs, Peng and Wang [41] consider
smooth solutions of parabolic PPDEs; Ekren et al. [21, 22, 23] study the
viscosity solution of quasilinear and fully nonlinear PPDEs; Cosso et al. [18]
treat PPDEs as the Hilbert space valued equations and build the viscosity
solution; Peng and Song [43] introduce a new path derivative and build Sobolev
solutions for corresponding parabolic fully-nonlinear PPDEs via $G-$BSDEs
[39]; Wu and Zhang [50] solve a master equation with solutions in a form of
$V(t,\mu)$, $\mu\in\mathcal{P}_{2}^{C}$. However, the path-dependent master
equation with solutions in the general form of $u(t,\omega,\mu)$,
$(\omega,\mu)\in\mathbb{C}_{T,d}\times\mathcal{P}^{C}_{2}$ still remains to be
studied.

In this article, we study the classical solution of path-dependent master
equation (6). In contrast to the state-dependent case [10], smooth solution of
equation (6) by FBSDEs meets with new issues. The first comes from the very
definition of vertical derivatives with respect to paths and measures on
$\mathbb{C}_{T,d}$ (see identities (17) and (34) for details). Dupire’s
vertical derivative [20] depends on the cut-off time for paths. In the same
spirit, the derivative with respect to measures on the path space (see also
[50]), also depends on the cut-off time. Note that in this case two different
flows are involved in the probabilistic representation of the path-dependent
field $u$. Application of the flow property of solutions for SDEs involves
vertical derivatives with respect to the path and the path measure at two
different times, while only one cut-off time is available in the field $u$.
Indeed, one of them should describe the smoothness of $u$ on “past”
information and is not defined in the sense of Dupire, which is only available
for non-anticipative (or adapted) functionals and describe the smoothness on
the “present” information. Secondly, the existence of the derivative with
respect to measure in Lions’ sense usually requires the separability of the
measurable space, which is the space of càdlàg functions here in view of
Dupire’s vertical derivative. However, FBSDEs are more consistent with the
uniform norm instead of Skorokhod norm, which leaves us without the general
existence result for derivatives. To handle the first issue, we propose a new
notation called “strong vertical derivative” (SVD) (see Definition 2.1) which
is derived from the vertical derivative of Dupire and restricts functionals to
be smooth before the cut-off time. By definition, non-anticipative functionals
with SVDs also have vertical derivatives. Moreover, the assumption of SVDs is
general enough to include all interesting continuously vertical differentiable
functionals (see Example 2.2). On the other hand, the SVD can be viewed as a
pathwise definition for the Malliavin derivative (see e.g. [38]) on the
canonical space (see subsequent Remark 2.3 for details). Based on the
differentiability with respect to paths and path measures, we build the Itô
formula and the partial Itô formula. Then we consider the non-Markovian FBSDE

(7)
$\left\\{\begin{array}[]{l}dX(r)=b_{1}(r)dr+\sigma_{1}(r)dB(r),\\\\[2.84526pt]
dX^{\prime}(r)=b_{2}(r)dr+\sigma_{2}(r)dB(r),\\\\[2.84526pt]
dY(r)=-f(X_{r},Y(r),Z(r),\mathcal{L}_{X^{\prime}_{r}},\mathcal{L}_{Y^{\eta_{t}}(r)})dr+Z(r)dB(r),\
\ \ r\geq t,\\\\[2.84526pt] X_{t}=\gamma_{t},\ \ X^{\prime}_{t}=\eta_{t},\ \
Y(T)=\Phi(X_{T},\mathcal{L}_{X^{\prime}_{T}}),\end{array}\right.$

where $\eta$ is a continuous process with law $\mu,$ and $Y^{\eta_{t}}$ solves
the associated mean-field FBSDE

(8)
$\mathcal{Y}(s)=\Phi(X_{T}^{\eta_{t}},\mathcal{L}_{X^{\prime}_{T}})+\int_{s}^{T}f(X_{r}^{\eta_{t}},\mathcal{Y}(r),\mathcal{Z}(r),\mathcal{L}_{X^{\prime}_{r}},\mathcal{L}_{\mathcal{Y}(r)})dr-\int_{s}^{T}\mathcal{Z}(r)dB(r),\quad
s\in[t,T],$

with $X_{r}^{\eta_{t}}:=X_{r}|_{\gamma=\eta}$ for $r\in[t,T].$ Here we denote
by $\omega(t)$ the value of path $\omega$ at time $t$ and by
$\omega_{t}(\cdot):=\omega(t\wedge\cdot)$ the path up to time $t.$ Assuming
that the functional generator $f$ and terminal value $\Phi$ have smooth SVDs,
we prove that the solution of corresponding FBSDE also has smooth SVDs.
Moreover, we construct all strong vertical derivatives of $u(t,\omega,\mu)$
via FBSDE, and give the required regularity to apply our Itô formula (see
Theorem 2.15 and Corollary 2.16)—which will benefit both numerical and
theoretical approximation of equilibrium by finite systems (see [24], [34]).
Furthermore, we also address some non-smooth case and connect it with
viscosity solutions via a functional mollifying argument, which is illustrated
with typical examples of the time-delayed case.

In summary, our main contribution is three-fold. Firstly, we propose the
general form of path-dependent master equation (6) and give the well-
posedness. Secondly, we introduce the notation of strong vertical
differentiability and build Itô and partial Itô formulas in this framework
which are fundamental tools in the study of path-dependent mean-field
problems. Thirdly, the argument of functional smoothing also seems new in the
path-dependent framework.

The rest of the article is organized as follows. In Section 2, we introduce
notations of SVD with respect to paths and measures on path space, and build
in the framework functional Itô calculus incorporating paths and path
measures. In Section 3, we show the differentiability and regularity of
associated FBSDE solutions. In Section 4, we prove the existence and
uniqueness of smooth solutions for path-dependent parabolic master equation.
Moreover, we extend our result to more general cases by a functional
mollifying argument.

Acknowledgements. This work is supported by NSF of China (Grant Numbers
12031009, 11901104) and Laboratory of Mathematics for Nonlinear Science, Fudan
University, Shanghai, China.

##  2\. Basic setup and Itô calculus for functionals of path and path-measure

###  2.1. The canonical setup

For any fixed $T>0$, we denote by $\mathbb{C}_{T,d}=C([0,T],\mathbb{R}^{d})$
the canonical space and equip it with the supreme norm $\|\cdot\|_{[0,T]}.$
$W$ is the canonical process and $\\{\mathcal{F}_{t}^{W}\\}_{0\leq t\leq T}$
is the natural filtration. For any $(t,\omega)\in[0,T]\times\mathbb{C}_{T,d},$
${\omega}_{t}$ is the cut-off path, meaning that
$\omega_{t}\in\mathbb{C}_{T,d}$ such that

(9) $\omega_{t}(r)=\omega(r)1_{[0,t)}(r)+\omega(t)1_{[t,T]}(r),\ \ r\in[0,T];$

and $\omega(t)$ is the state of $\omega$ at time $t$. Let
$\mathcal{P}^{C}_{2}$ be the set of probability measures on
$(\mathbb{C}_{T,d},\mathcal{F}_{T}^{W})$ with finite second order moments,
i.e. $\mu\in\mathcal{P}_{2}^{C}$ iff
$|||\mu|||^{2}:=\mathbb{E}^{\mu}[\|W\|_{[0,T]}^{2}]<\infty.$ For
$\mu\in\mathcal{P}^{C}_{2},$ $\mu_{t}\in\mathcal{P}^{C}_{2}$ is the
distribution of stopped process $W_{t}$ under $\mu.$ For any
$\mu,\nu\in\mathcal{P}^{C}_{2},$ we define the following classical
2-Wasserstein distance

(10)
$W_{2}(\mu,\nu)=\inf_{P\in\mathcal{P}(\mu,\nu)}\left(\int_{\mathbb{C}_{T,d}\times\mathbb{C}_{T,d}}\|u-v\|_{[0,T]}^{2}\
dP(u,v)\right)^{\frac{1}{2}},$

where $\mathcal{P}(\mu,\nu)$ is the set of all probability measures on
$(\mathbb{C}_{T,d}\times\mathbb{C}_{T,d},\mathcal{F}^{W}_{T}\times\mathcal{F}^{W}_{T})$
with marginal measures $\mu$ and $\nu.$ To introduce functional derivative in
the space of paths, we consider the space of càdlàg paths
$\mathbb{D}_{T,d}:=D([0,T],\mathbb{R}^{d}),$ which can be equipped with the
uniform topology $\|\cdot\|_{[0,T]},$ or the Skorohod topology

(11)
$d(\omega,\omega^{\prime}):=\inf_{\lambda\in\Lambda_{[0,T]}}\sup_{t\in[0,T]}(|t-\lambda(t)|+|\omega(t)-\omega^{\prime}(t)|),$

where $\Lambda_{[0,T]}$ is the set of all strictly increasing continuous
mappings on $[0,T]$ with $\lambda(0)=0$ and $\lambda(T)=T.$ In the following,
we equip $\mathbb{D}_{T,d}$ with the uniform topology unless stated otherwise.
With the space $\mathbb{C}_{T,d}$ being replaced with $\mathbb{D}_{T,d}$,
notations such as $\mathcal{P}^{D}_{2}$ and $W_{2}(\mu,\nu)$ are self-
explained.

Suppose that $(\Omega,\mathcal{F},P)$ is an atomless probability space
supporting a $d$-dimensional Brownian motion $B,$ and
$\\{\mathcal{F}_{t}\\}_{t\in[0,T]}$ is the natural augmented filtration. For
any $t\in[0,T]$ and $r\in[t,T],$ we define $\mathcal{F}_{r}^{t}$ as the
$\sigma$-algebra generated by $\\{B(s)-B(t);t\leq s\leq r\\}$ and completed
under $P$. For any (stopped up to time $t$) process $X_{t},$ we denote by
$\mathcal{L}_{X_{t}}$ the law of the process $X_{t}$ and $\mathcal{L}_{X(t)}$
the law of the random variable $X(t)$. In the following, we use notation
$\mathbb{M}_{2}^{C}$ ($\mathbb{M}_{2}^{D}$, resp.) as the collection of
measurable continuous processes (càdlàg processes, resp.) with laws in
$\mathcal{P}_{2}^{C}$ ($\mathcal{P}_{2}^{D}$, resp.). Since for any
$\mu\in\mathcal{P}_{2}^{D},$ we can always find an atomless probability space
$(\Omega,\mathcal{F}_{t},P)$ such that there exists a càdlàg process $\eta$ on
this probability space with law $\mu,$ we will always suppose for any
$\mu\in\mathcal{P}_{2}^{D},$ $(\Omega,\mathcal{F},P)$ is rich enough to
support a càdlàg process $\eta$ such that $\mathcal{L}_{\eta}=\mu.$ Moreover,
for any progressively measurable process $X$ and random variable $\xi$ on
$(\Omega,\mathcal{F},P)$, we define the following norms if they are finite:
for any $t\in[0,T],$ $p\in\mathbb{N^{+}},$

(12) $\|X\|_{\mathbb{S}^{p},[t,T]}^{p}:=\mathbb{E}^{P}[\|X\|^{p}_{[t,T]}],\ \
\|X\|_{\mathbb{H}^{p},[t,T]}^{p}:=\mathbb{E}^{P}[(\int_{t}^{T}|X(r)|^{2}dr)^{\frac{p}{2}}],\
\ \|\xi\|_{L^{p}}^{p}:=\mathbb{E}^{P}[|\xi|^{p}].$

We write $\mathbb{S}^{p}([t,T],\mathbb{R}^{k})$,
$\mathbb{H}^{p}([t,T],\mathbb{R}^{k})$ and
$L^{p}(\mathcal{F}_{T},\mathbb{R}^{k})$ for spaces of progressively measurable
processes on $[t,T]$ and random variables with values in $\mathbb{R}^{k}$ and
finite corresponding norms. Denote by $C^{n}(\mathbb{R}^{m},\mathbb{R}^{k})$
($C^{n}_{b}(\mathbb{R}^{m},\mathbb{R}^{k})$, resp.) the space of (bounded,
resp.) continuous functions from $\mathbb{R}^{m}$ to $\mathbb{R}^{k}$ with
(bounded, resp.) continuous derivatives up to order $n.$ Usually, we omit
$\mathbb{R}^{k}$ in
$\mathbb{S}^{p}([t,T],\mathbb{R}^{k}),\mathbb{H}^{p}([t,T],\mathbb{R}^{k}),L^{p}(\mathcal{F}_{T},\mathbb{R}^{k}),C(\mathbb{R}^{m},\mathbb{R}^{k})$
when $k=1,$ and also omit the time interval $[t,T]$ if no confusion raised.
Moreover, for
$(Y,Z)\in\mathbb{S}^{p}([t,T],\mathbb{R}^{m})\times\mathbb{H}^{p}([t,T],\mathbb{R}^{n}),$
we write

(13)
$\|(Y,Z)\|_{\mathbb{S}^{p}\times\mathbb{H}^{p}}:=\left(\|Y\|_{\mathbb{S}^{p}}^{p}+\|Z\|_{\mathbb{H}^{p}}^{p}\right)^{\frac{1}{p}}.$

###  2.2. Strong vertical derivatives with respect to path and path-measure

Denote by $\hat{\mathbb{D}}_{T,d}$ the product space
$[0,T]\times\mathbb{D}_{T,d}\times\mathcal{P}^{D}_{2}$ and by $\mathscr{D}$
the space of functionals on $\hat{\mathbb{D}}_{T,d}.$ A functional
$f\in\mathscr{D}$ is said to be non-anticipative if for any $(t,\omega,\mu),$
$f(t,\omega,\mu)=f(t,\omega_{t},\mu_{t})$. For non-anticipative
$f\in\mathscr{D},$ we call $f$ continuous on $\hat{\mathbb{D}}_{T,d}$ and
write $f\in\mathscr{C}(\hat{\mathbb{D}}_{T,d})$ if $f$ is continuous in the
product space $[0,T]\times\mathbb{D}_{T,d}\times\mathcal{P}^{D}_{2}$ equipped
with the premetric:

(14)
$d_{p}((t,\omega,\mu),(t^{\prime},\omega^{\prime},\mu^{\prime})):=|t-t^{\prime}|+\|\omega_{t}-\omega_{t^{\prime}}\|+W_{2}(\mu_{t},\mu_{t^{\prime}}).$

For any non-anticipative $f\in\mathscr{D},$ the horizontal derivative is
defined as

(15) $\partial_{t}f(t,\omega,\mu):=\lim_{h\rightarrow
0^{+}}\frac{1}{h}[f(t+h,\omega_{t},\mu_{t})-f(t,\omega_{t},\mu_{t})],\
\forall\ (t,\omega,\mu)\in\hat{\mathbb{D}}_{T,d}.$

For any $(t,x)\in[0,T]\times\mathbb{R}^{d},$ define
$\omega^{t,x}\in\mathbb{D}_{T,d}$ by

(16) $\omega^{t,x}:=\omega+x1_{[t,T]}.$

For any fixed $(t,\mu)\in[0,T]\times\mathcal{P}_{2}^{D},$
$f(t,\cdot,\mu):\mathbb{D}_{T,d}\to\mathbb{R}$ is called vertically
differentiable at $(t,\omega)$ (or $\omega_{t}$ for short), if
$f(t,\omega^{t,x},\mu)$ is differentiable at $x=0$, i.e. there exits
$\partial_{\omega}f(t,\omega,\mu)\in\mathbb{R}^{d}$ such that

(17)
$f(t,\omega+x1_{[t,T]},\mu)=f(t,\omega,\mu)+\partial_{\omega}f(t,\omega,\mu)x+o(|x|),\
\ \ \forall\ x\in\mathbb{R}^{d},$

and $\partial_{\omega}f(t,\omega,\mu)$ is then called the vertical derivative.
Now we extend the vertical derivative from non-anticipative functionals taken
at cut-off time to path functionals at any time before the cut-off time.

######  Definition 2.1.

Suppose that $f:[0,T]\times\mathbb{D}_{T,d}\to\mathbb{R}.$ For any $\tau\leq
t,$ we call $f$ strongly vertically differentiable at $(\tau,t,\omega)$ (or
$\omega_{\tau}$ for short), if there exits
$\partial_{\omega_{\tau}}f(t,\omega)\in\mathbb{R}^{d}$ such that

(18)
$f(t,\omega+x1_{[\tau,T]})=f(t,\omega)+\partial_{\omega_{\tau}}f(t,\omega)x+o(|x|),\
\ \ \forall\ x\in\mathbb{R}^{d}.$

In this case, $\partial_{\omega_{\tau}}f(t,\omega)$ is called the strong
vertical derivative of $f$ at $(\tau,t,\omega)$. Moreover, if $f$ is strongly
vertically differentiable at $(\tau,t,\omega)$ for any $\tau\leq t,$ we call
$f$ strongly vertically differentiable at $(t,\omega)$ (or $\omega_{t}$ for
short).

Clearly, $f$ is strongly vertical differentiable at $\omega_{t}$ if and only
if the mapping $x\to f(t,\omega^{\tau,x})$ is differentiable at $x=0$ for any
$\tau\leq t$. In particular, if $f$ is non-anticipative and strongly
vertically differentiable, $f$ is vertically differentiable and its vertical
derivative at $(t,x)$ agrees with its strong vertical derivative at
$(t,t,\omega).$ For the SVD $\partial_{\omega_{\tau}}f(t,\omega),$ we can
further define its SVDs in the same spirit: for any $\tau^{\prime}\leq t,$
define $\partial_{\omega_{\tau^{\prime}}}\partial_{\omega_{\tau}}f(t,\omega)$
as the SVD of $\partial_{\omega_{\tau}}f(t,\omega)$ at
$(\tau^{\prime},t,\omega)$. In the following, we only need to consider the
case $\tau^{\prime}=\tau.$ We call $f$ has continuous SVDs or
$\partial_{\omega_{\tau}}f(t,\omega)$ is continuous if
$\partial_{\omega_{\tau}}f$ is continuous with respect to the metric: for any
$(\tau,t,\omega)$ and $(\tau^{\prime},t^{\prime},\omega^{\prime})$ with
$\tau\leq t,\ \tau^{\prime}\leq t^{\prime},$

(19)
$d_{sp}((\tau,t,\omega),(\tau^{\prime},t^{\prime},\omega^{\prime})):=|\tau-\tau^{\prime}|+|t-t^{\prime}|+\|\omega_{t}-\omega^{\prime}_{t^{\prime}}\|.$

Here are examples of strongly vertically differentiable functionals.

######  Example 2.2.

Let $f:[0,T]\times\mathbb{D}_{T,d}\longmapsto\mathbb{R}$ and
$(t,\omega)\in[0,T]\times\mathbb{D}_{T,d}.$

  * $(i)$

If $f(t,\omega)=F(t,\omega(t))$ for a function $F\in
C^{1,k}([0,T]\times\mathbb{R}^{d})$, then we have that for any
$\tau_{1},\tau_{2},\cdots,\tau_{j}\in[0,t],$ $j\leq k,$

(20)
$\partial_{t}f(t,\omega)=\partial_{t}F(t,\omega(t)),\quad\partial_{\omega_{\tau_{j}}}\cdots\partial_{\omega_{\tau_{1}}}f(t,\omega)=D_{x}^{j}F(t,\omega(t)),$

and thus $f$ has continuous strong vertical derivatives up to order $k$.

  * $(ii)$

Suppose that $f(t,\omega)=\int_{0}^{t}F(r,\omega(r))dr$ with $F\in
C^{1,k}([0,T]\times\mathbb{R}^{d})$. Then for any
$\tau_{1},\tau_{2},\cdots,\tau_{j}\in[0,t],$ $j\leq k,$

(21) $\partial_{t}f(t,\omega)=F(t,\omega(t)),\ \ \
\partial_{\omega_{\tau_{j}}}\cdots\partial_{\omega_{\tau_{1}}}f(t,\omega)=\int_{\tau}^{t}D_{x}^{j}F(r,\omega(r))dr,$

with $\tau=\max_{1\leq i\leq j}\\{\tau_{i}\\}.$ Thus $f$ has continuous SVDs
up to order $k$.

  * $(iii)$

For a partition $0=t_{0}<t_{1}<\cdots<t_{n}=T,$ and a continuously
differentiable function
$F:\underbrace{\mathbb{R}^{d}\times\mathbb{R}^{d}\times\cdots\mathbb{R}^{d}}_{n}\to\mathbb{R}$,
let

(22)
$f(T,\omega):=F(\omega(t_{1}),\omega(t_{2})-\omega(t_{1}),\cdots,\omega(T)-\omega(t_{n-1})).$

Then $f$ is strongly vertically differentiable at $(T,\omega)$ with

(23)
$\partial_{\omega_{t}}f(T,\omega)=\sum_{j=1}^{n}\partial_{x_{j}}F(\omega(t_{1}),\omega(t_{2})-\omega(t_{1}),\cdots,\omega(T)-\omega(t_{n-1}))1_{(t_{j-1},t_{j}]}(t),\
\ t>0.$

  * $(iv)$

For fixed $t_{0}\in(0,T)$ and $F\in C^{1}(\mathbb{R}^{d})$, define
$f(T,\omega):=F(\omega(t_{0})).$ Thus $f$ has SVDs

(24) $\partial_{\omega_{t}}f(T,\omega)=D_{x}F(\omega(t_{0}))1_{[0,t_{0}]}(t),$

which may not be continuous in $t\in[0,T]$. However, consider

(25)
$f_{\varepsilon}(T,\omega):=\int_{0}^{T}\rho_{\varepsilon}(t_{0}-s)F(\omega(s))ds,\
\ \omega\in\mathbb{D}_{T,d},\ \ \varepsilon>0,$

with $\rho_{\varepsilon}$ a standard mollifier on $\mathbb{R}.$ Then, for any
$\omega\in\mathbb{C}_{T,d},$ $\lim_{\varepsilon\rightarrow
0}f_{\varepsilon}(T,\omega)=f(T,\omega).$ Moreover, according to $(ii)$,
$f_{\varepsilon}(T,\omega)$ has continuous SVDs. Therefore, we can approximate
path-dependent master equations with non-smooth driven and terminal
functionals by those with smooth ones, see Example 4.11 for further details.

  * $(v)$

For a given partition of $[0,T]:$ $0=t_{0}<t_{1}<\cdots<t_{n}=T$ and smooth
functions $\\{f_{i}\\}_{i=0}^{n-1}$ on $\mathbb{R}^{d},$ consider

(26)
$f(t,\omega):=\sum_{i=0}^{n-1}f_{i}(\omega(t_{i}))1_{[t_{i},t_{i+1})}(t).$

Then $f$ is strongly vertically differentiable at $\omega_{t}$ with

(27)
$\partial_{\omega_{\tau}}f(t,\omega)=\sum_{i=0}^{n-1}Df_{i}(\omega(t_{i}))1_{[t_{i},t_{i+1})}(t)1_{[0,t_{i}]}(\tau),\
\ \forall\tau\leq t.$

######  Remark 2.3.

The relation between vertical derivative and Malliavin derivative is
considered in [17], where an equivalence is built through martingale
representation in both frameworks (see [17, Theorem 6.1]). However, according
to $(iii)$ of Example 2.2, there is a direct equivalence between SVDs and
Malliavin derivatives. Recall that $W$ is the canonical process, and then
$f(W):=F(W(t_{1}),W(t_{2})-W(t_{1}),\cdots,W(T)-W(t_{n-1}))$ gives a
cylindrical random variable under the Wiener measure. Then its Malliavin
derivative $\mathcal{D}_{r}f(W)$ agrees with its SVD at $(r,T,W)$ and then the
SVD can be viewed as a pathwise definition of Malliavin derivative without
involving any probability measure. Furthermore, we can consider SVDs with
respect to driven signals for integrals and equations by restricting the
domain of definition. Denote by $\mathbb{C}_{T,d}^{1}$ the subspace of
$\mathbb{C}_{T,d}$ with continuous derivatives. Consider functional
$g:[0,T]\times\mathbb{C}_{T,d}^{1}\to\mathbb{R}$ given by

(28)
$g(t,\omega):=\int_{0}^{t}G(\omega(r))d\omega(r),\quad(t,\omega)\in[0,T]\times\mathbb{C}_{T,d}^{1},$

with $G\in C_{b}^{1}(\mathbb{R}^{d},\mathbb{R}^{d}).$ Then, we have

(29)
$\partial_{\omega_{\tau}}g(t,\omega)=G(\omega(\tau))+\int_{\tau}^{t}DG(\omega(r))d\omega(r),\quad\tau\in[0,t).$

Similarly, consider $\phi:[0,T]\times\mathbb{C}_{T,d}^{1}\to\mathbb{R}$ given
by $\phi(t,\omega):=x(t)$ with

(30) $x(t)=\int_{0}^{t}H(x(r))d\omega(r),$

for a given function $H\in C_{b}^{2}(\mathbb{R},\mathbb{R}^{d})$. Then by a
nontrivial argument (note that ordinary differential equations (ODEs) are not
continuous with respect to the driven signal under the uniform norm), we have
that for any $(t,\omega)\in[0,T]\times\mathbb{C}_{T,d}^{1}$, $\phi$ is
strongly vertically differentiable at any $(t,\omega)$ and the derivative
$\partial_{\omega_{\tau}}\phi(t,\omega)$ at $(\tau,t,\omega)$ solves the
following linear ODE,

(31)
$\partial_{\omega_{\tau}}x(t)=H(x(\tau))+\int_{\tau}^{t}\partial_{\omega_{\tau}}x(r)H^{\prime}(x(r))d\omega(r),\quad\
t\geq\tau.$

The following lemma follows from Definition 2.1 directly.

######  Lemma 2.4.

Suppose that $f:[0,T]\times\mathbb{D}_{T,d}\to\mathbb{R}$ is strongly
vertically differentiable, and uniformly Lipschitz continuous in $\omega:$

(32) $|f(t,\omega)-f(t,\omega^{\prime})|\leq
C\|\omega_{t}-\omega^{\prime}_{t}\|,\quad\forall(t,\omega,\omega^{\prime})\in[0,T]\times\mathbb{D}_{T,d}\times\mathbb{D}_{T,d}.$

Then we have $|\partial_{\omega_{\tau}}f(t,\omega)|\leq C$ for any
$(t,\omega)\in[0,T]\times\mathbb{D}_{T,d}$ and $\tau\leq t.$

For a non-anticipative functional $f\in\mathscr{D}$, consider its lift
$\mathbf{f}:[0,T]\times\mathbb{D}_{T,d}\times\mathbb{M}^{D}_{2}\to\mathbb{R},$

(33) $\mathbf{f}(t,\omega,\eta):=f(t,\omega,\mathcal{L}_{\eta}).$

In the spirit of Lions [36] (also see [50] for derivative with respect to
measure on the path space), we call $f$ Fréchet (vertically) differentiable at
$(t,\mu)$ (or $\mu_{t}$ for short), if for any fixed $\omega,$ $\mathbf{f}$ is
Fréchet (vertically) differentiable at $(t,\eta)$ (or $\eta_{t}$ for short)
with $\mathcal{L}_{\eta}=\mu$ in the following sense: there exits
$D_{\eta}\mathbf{f}(t,\omega,\eta)\in
L^{2}_{P}(\mathcal{F}_{t},\mathbb{R}^{d})$ such that for any $\xi\in
L^{2}_{P}(\mathcal{F}_{t},\mathbb{R}^{d}),$

(34) $\mathbf{f}(t,\omega,\eta+\xi
1_{[t,T]})=\mathbf{f}(t,\omega,\eta)+\mathbb{E}^{P}[D_{\eta}\mathbf{f}(t,\omega,\eta)\xi]+o(\|\xi\|_{L^{2}}).$

In particular, it means that the following Gâteaux derivative exits

(35) $\lim_{h\rightarrow 0}\frac{1}{h}[\mathbf{f}(t,\omega,\eta+h\xi
1_{[t,T]})-\mathbf{f}(t,\omega,\eta)]=\mathbb{E}^{P}[D_{\eta}\mathbf{f}(t,\omega,\eta)\xi].$

Moreover, if there exists a non-anticipative jointly measurable functional
$\partial_{\mu}f:\hat{\mathbb{D}}_{T,d}\times\mathbb{D}_{T,d}\to\mathbb{R},$
such that

(36) $D_{\eta}\mathbf{f}(t,\omega,\eta)=\partial_{\mu}f(t,\omega,\mu,\eta),\ \
\ P\text{-}a.s.,$

we call $f$ vertically differentiable at $(t,\mu)$ and
$\partial_{\mu}f(t,\omega,\mu,\tilde{\omega})$ the vertical derivative of
$f(t,\omega,\cdot)$ at $(t,\mu)$ (or $\mu_{t}$).

######  Remark 2.5.

Here we give a crucial remark about the validity for notations of Fréchet and
Gâteaux differentiability. Denote by $\mathbf{f}$ the lift of
$f\in\mathscr{D}.$ For any $\xi\in L^{2}_{P}(\mathcal{F}_{t},\mathbb{R}^{d}),$
let $F(t,\omega,\eta,\xi):=\mathbf{f}(t,\omega,\eta+\xi 1_{[t,T]}).$ Then
$\mathbf{f}$ is Fréchet differentiable at $(t,\eta)$ in the above sense is
equivalent to that $F(t,\omega,\eta,\xi)$ is Fréchet differentiable at $\xi=0$
in the classical sense. Similar argument for Gâteaux differentiability also
holds.

######  Remark 2.6.

Concerning the existence of the derivative functional $\partial_{\mu}f$. If
the lift $\mathbf{f}(t,\omega,\eta)$ of $f(t,\omega,\mu)$ is Fréchet
differentiable at $\eta_{t},$ and the derivative
$D_{\eta}\mathbf{f}(t,\omega,\eta)$ is continuous in the sense that
$D_{\eta}\mathbf{f}(t,\omega,\eta^{n})\stackrel{{\scriptstyle
L^{2}}}{{\longrightarrow}}D_{\eta}\mathbf{f}(t,\omega,\eta)$ as
$\eta^{n}\stackrel{{\scriptstyle L^{2}}}{{\longrightarrow}}\eta$ under the
Skorohod topology (11), then according to [50, Theorem 2.2], $\partial_{\mu}f$
exists in the sense of (36). However, to build smooth solutions for (6), we
need our Itô formula (Theorem 2.15 and Corollary 2.16) to be applicable for
the larger class of functionals, which only need to be continuous with respect
to the uniform topology. Luckily, we can construct the derivative directly by
corresponding FBSDEs.

For the uniqueness of $\partial_{\mu}f(t,\omega,\mu,\cdot),$ in view of
identity (36), we see that it is unique $\mu$-a.s. in $\mathbb{D}_{T,d}$. Then
for any $\mu\in\mathcal{P}^{D}_{2}$ such that
$\text{supp}(\mu)=\mathbb{D}_{T,d},$ if
$\partial_{\mu}f(t,\omega,\mu,\tilde{\omega})$ is continuous in
$\tilde{\omega}\in\mathbb{D}_{T,d},$ $\partial_{\mu}f(t,\omega,\mu,\cdot)$ is
unique on $\mathbb{D}_{T,d}.$ Moreover, suppose that
$\partial_{\mu}f(t,\omega,\cdot,\cdot)$ is jointly continuous on
$\mathcal{P}_{2}^{D}\times\mathbb{D}_{T,d}.$ Then for any
$\mu_{0}\in\mathcal{P}_{2}^{D},$ $\partial_{\mu}f(t,\omega,\mu_{0},\cdot)$ is
unique on $\mathbb{D}_{T,d}$. Indeed, choose any $\eta\in\mathbb{M}^{D}_{2}$
with $\mathcal{L}_{\eta}=\mu_{0}\in\mathcal{P}_{2}^{D},$ and any
$\eta^{\prime}\in(\mathbb{M}^{D}_{2})^{\prime},$ which is independent of
$\eta,$ such that $\text{supp}(\mathcal{L}_{\eta^{\prime}})=\mathbb{D}_{T,d}.$
Then for any $\varepsilon>0,$ the functional
$\partial_{\mu}f(t,\omega,\mathcal{L}_{\eta+\varepsilon\eta^{\prime}},\cdot)$
is unique on $\mathbb{D}_{T,d}.$ It follows by continuity of
$\partial_{\mu}f(t,\omega,\cdot,\cdot)$ that
$\partial_{\mu}f(t,\omega,\mu_{0},\tilde{\omega})$ is unique as the limit of
$\partial_{\mu}f(t,\omega,\mathcal{L}_{\eta+\varepsilon\eta^{\prime}},\tilde{\omega})$
as $\varepsilon$ goes to zero. In conclusion, we have the following lemma.

######  Lemma 2.7.

Suppose that for any fixed $(t,\omega)\in[0,T]\times\mathbb{D}_{T,d},$ the
functional derivative $\partial_{\mu}f(t,\omega,\cdot,\cdot)$ is jointly
continuous in $\mathcal{P}_{2}^{D}\times\mathbb{D}_{T,d}.$ Then for any
$(t,\omega,\mu)\in\hat{\mathbb{D}}_{T,d},$
$\partial_{\mu}f(t,\omega,\mu,\cdot)$ is unique on $\mathbb{D}_{T,d}$.

######  Remark 2.8.

The definition of vertical derivative given by (34) and (35) has natural
extension for Banach space valued functionals. For any $t\in[0,T],$ suppose
that $f(t,\omega,\mu)$ takes values in a (stochastic) Banach space $E_{t}$
(e.g.
$\mathbb{S}^{2}({[t,T]}),\mathbb{H}^{2}({[t,T]}),L^{2}(\mathcal{F}_{t})$).
Indeed, $f(t,\omega,\mu)$ has the natural lift $\mathbf{f}(t,\omega,\eta)\in
E_{t}$ with $\mathcal{L}_{\eta}=\mu$. If the mapping from
$L^{2}(\mathcal{F}_{t})$ to $E_{t}$

$\begin{array}[]{lccl}\mathbf{f}(t,\omega,\eta+\cdot
1_{[t,T]}):&L^{2}(\mathcal{F}_{t})&\longrightarrow&E_{t}\\\
&\xi&&\mathbf{f}(t,\omega,\eta+\xi 1_{[t,T]})\end{array}$

is Fréchet (vertical) differentiable with derivative
$D_{\eta}\mathbf{f}(t,\omega,\eta)\in L(L^{2}(\mathcal{F}_{t}),E_{t})$ at
$\xi=0,$ we call $f(t,\omega,\cdot)$ Fréchet (vertically) differentiable at
$\mu_{t}$. Moreover, if there exists a jointly measurable functional
$U:\hat{\mathbb{D}}_{T,d}\times\mathbb{D}_{T,d}\to E_{t}$ such that for any
$\xi\in L^{2}(\mathcal{F}_{t})$,
$D_{\eta}\mathbf{f}(t,\omega,\eta)(\xi)=\mathbb{E}^{P}[U(t,\omega,\mu,\eta)\xi],$
we call $\partial_{\mu}f(t,\omega,\mu,\cdot):=U(t,\omega,\mu,\cdot)$ the
vertical derivative of $f(t,\omega,\cdot)$ at $\mu_{t}.$

Now we introduce SVDs with respect to path-measure.

######  Definition 2.9.

For any $\tau,t\in[0,T]$ with $\tau\leq t$ and $\mu\in\mathcal{P}_{2}^{D},$ we
call a non-anticipative functional
$f:[0,T]\times\mathcal{P}_{2}^{D}\to\mathbb{R}$ Fréchet (strongly vertically)
differentiable at $(\tau,t,\mu)$ if its lift $\mathbf{f}(t,\eta)$ with
$\mathcal{L}_{\eta}=\mu$ is Fréchet (strongly vertically) differentiable:
there exits $D_{\eta_{\tau}}\mathbf{f}(t,\eta)\in
L^{2}_{P}(\mathcal{F}_{t},\mathbb{R}^{d})$ such that for any $\xi\in
L^{2}_{P}(\mathcal{F}_{\tau},\mathbb{R}^{d}),$

(37) $\mathbf{f}(t,\eta+\xi
1_{[\tau,T]})=\mathbf{f}(t,\eta)+\mathbb{E}^{P}[D_{\eta_{\tau}}\mathbf{f}(t,\eta)\xi]+o(\|\xi\|_{L^{2}}).$

In particular, it means that the following Gâteaux derivative exits,

(38) $\lim_{h\rightarrow 0}\frac{1}{h}[\mathbf{f}(t,\eta+h\xi
1_{[\tau,T]})-\mathbf{f}(t,\eta)]=\mathbb{E}^{P}[D_{\eta_{\tau}}\mathbf{f}(t,\eta)\xi].$

We call $f$ strongly vertically differentiable at $(t,\mu)$ or $\mu_{t},$ if
it is Fréchet differentiable at $(\tau,t,\mu)$ for any $\tau\leq t$, and
moreover, there exists a jointly measurable non-anticipative functional
$\partial_{\mu_{\tau}}f:[0,T]\times\mathcal{P}_{2}^{D}\times\mathbb{D}_{T,d}\to\mathbb{R}^{d}$
such that

(39) $D_{\eta_{\tau}}\mathbf{f}(t,\eta)=\partial_{\mu_{\tau}}f(t,\mu,\eta),\ \
\ P\text{-}a.s..$

$\partial_{\mu_{\tau}}f(t,\mu,\cdot)$ is then called the strong vertical
derivative of $f(t,\cdot)$ at $(\tau,t,\mu).$

######  Remark 2.10.

For the existence and uniqueness of the SVD at $\mu_{\tau},$ we have similar
results as Remark 2.6 and Lemma 2.7 for vertical derivatives. In particular,
if for any $t\in[0,T],$ $\partial_{\mu_{\tau}}f(t,\cdot,\cdot)$ is jointly
continuous on $\mathcal{P}_{2}^{D}\times\mathbb{D}_{T,d}$, then the SVD is
unique. Moreover, we can extend SVDs in path-measure to the (stochastic)
Banach framework as Remark 2.8.

Given strongly vertically differentiable
$f:[0,T]\times\mathcal{P}_{2}^{D}\to\mathbb{R}$, for any
$(t,\mu,\tilde{\omega})\in[0,T]\times\mathcal{P}_{2}^{D}\times\mathbb{D}_{T,d}$
and $\tau\leq t,$ we can further consider SVDs of $\partial_{\mu_{\tau}}f$
with respect to $\mu_{t}$ and $\tilde{\omega}_{t}$: for any $\tau^{\prime}\leq
t,$ consider
$\partial_{\tilde{\omega}_{\tau^{\prime}}}\partial_{\mu_{\tau}}f(t,\mu,\tilde{\omega})$
as the SVD of $\partial_{\mu_{\tau}}f(t,\mu,\tilde{\omega})$ at
$(\tau^{\prime},t,\tilde{\omega})$;
$\partial_{\mu_{\tau^{\prime}}}\partial_{\mu_{\tau}}f(t,\mu,\tilde{\omega},\tilde{\omega}^{\prime})$
as the SVD of $\partial_{\mu_{\tau}}f(t,\mu,\tilde{\omega})$ at
$(\tau^{\prime},t,\mu).$ In the subsequent sections, we only need to consider
the case $\tau^{\prime}=\tau$ and the second order derivative
$\partial_{\tilde{\omega}_{\tau^{\prime}}}\partial_{\mu_{\tau}}f(t,\mu,\tilde{\omega})$.
Moreover, we call $f$ has continuous SVDs or
$\partial_{\mu_{\tau}}f(t,\mu,\tilde{\omega})$ is continuous if
$\partial_{\mu_{\tau}}f$ is continuous with respect to the following
premetric: for any $(t,\mu,\tilde{\omega})$ and
$(t^{\prime},\mu^{\prime},\tilde{\omega}^{\prime})$ with $\tau\leq
t,\tau^{\prime}\leq t^{\prime},$

(40)
$d_{sp}((\tau,t,\mu,\tilde{\omega}),(\tau^{\prime},t^{\prime},\mu^{\prime},\tilde{\omega}^{\prime})):=|\tau-\tau^{\prime}|+|t-t^{\prime}|+W_{2}(\mu_{t},\mu^{\prime}_{t^{\prime}})+\|\tilde{\omega}_{t}-\tilde{\omega}^{\prime}_{t^{\prime}}\|.$

$f$ is said to have continuous SVDs in path-measure up to order $2$, if both
$\partial_{\mu_{\tau}}f$ and
$\partial_{\tilde{\omega}_{\tau}}\partial_{\mu_{\tau}}f$ are continuous with
respect to the above topology.

######  Example 2.11.

Here we consider $f:[0,T]\times\mathcal{P}_{2}^{D}\to\mathbb{R}$ and
$(t,\mu)\in[0,T]\times\mathcal{P}_{2}^{D}.$

  * $(i)$

Suppose that $F\in C^{1,2}([0,T]\times\mathbb{R}^{d})$ with $|D_{x}^{2}F|$
being uniformly bounded, and $f(t,\mu):=\mathbb{E}^{\mu}[F(t,W(t))].$ Then we
have that

$\displaystyle\partial_{t}f(t,\mu)=\mathbb{E}^{\mu}[\partial_{t}F(t,W(t))],\ \
\ \partial_{{\mu_{\tau}}}f(t,\mu,\tilde{\omega})=D_{x}F(t,\tilde{\omega}(t)),$
$\displaystyle\quad\text{and}\quad\partial_{\tilde{\omega}_{\tau}}\partial_{{\mu_{\tau}}}f(t,\mu,\tilde{\omega})=D_{x}^{2}F(t,\tilde{\omega}(t)),\
\ \ \forall\tau\in[0,t].$

Thus $f$ has continuous SVDs up to order $2$.

  * $(ii)$

Let $F$ as defined in $(i)$ and
$f(t,\mu):=\mathbb{E}^{\mu}[\int_{0}^{t}F(r,W(r))dr].$ Then for any
$\tau\in[0,t],$

$\displaystyle\partial_{t}f(t,\mu)=\mathbb{E}^{\mu}[F(t,W(t))],\ \ \
\partial_{\mu_{\tau}}f(t,\mu,\tilde{\omega})=\int_{\tau}^{t}D_{x}F(r,\tilde{\omega}(r))dr,$
$\displaystyle\quad\text{and}\quad\partial_{\tilde{\omega}_{\tau}}\partial_{\mu_{\tau}}f(t,\mu,\tilde{\omega})=\int_{\tau}^{t}D^{2}_{x}F(r,\tilde{\omega}(r))dr.$

Therefore, the functional $f$ also has continuous SVDs up to order $2$.

  * $(iii)$

Let $F\in C^{1}(\mathbb{R}^{d})$ such that $|DF(x)|\leq C(1+|x|)$ for some
$C\geq 0.$ For fixed $t_{0}\in(0,T),$ consider
$\Phi(T,\mu):=\mathbb{E}^{\mu}[F(W(t_{0}))].$ Then the SVD at $\mu_{t}$ is
$\partial_{\mu_{t}}\Phi(T,\mu,\tilde{\omega}):=D_{x}F(\tilde{\omega}(t_{0}))1_{[0,t_{0}]}(t),$
which may not be continuous in $t\in[0,T]$. However, for any
$\mu\in\mathcal{P}_{2}^{D},$ consider

(41)
$\Phi_{\varepsilon}(T,\mu):=\mathbb{E}^{\mu}[\int_{0}^{T}\rho_{\varepsilon}(t_{0}-s)F(W(s))ds],$

with $\rho_{\varepsilon}$ being a standard mollifier on $\mathbb{R}.$ Then by
applying the dominated convergence theorem, we have

(42) $\lim_{\varepsilon\rightarrow 0}\Phi_{\varepsilon}(T,\mu)=\Phi(T,\mu),\ \
\forall\ \mu\in\mathcal{P}_{2}^{C}.$

Moreover, according to $(ii)$, $\Phi_{\varepsilon}(T,\mu)$ has continuous
SVDs. Therefore, we may approximate functionals with non-smooth SVDs by smooth
ones. See Example 4.11 for further application in path-dependent master
equations.

######  Example 2.12.

We consider non-anticipative functionals $f\in\mathscr{D}$ by combining
Example 2.2 and Example 2.11. For simplicity take $d=1$. Suppose that $F\in
C^{1,2}_{b}([0,T]\times\mathbb{R}^{5})$ and $f_{1},f_{2},f_{3},f_{5}\in
C^{2}_{b}(\mathbb{R})$. $f_{4}\in C^{2}_{b}(\mathbb{R}^{2})$. Consider the
following functional

$\displaystyle
f(t,\omega,\mu):=F\Big{(}t,\omega(t),\int_{0}^{t}f_{1}(\omega(r))dr,\mathbb{E}^{\mu}[f_{2}(W(t))],\mathbb{E}^{\mu}[\int_{0}^{t}f_{3}(W(r))dr],$
$\displaystyle\quad\quad\quad\quad
E^{\mu}[f_{4}(W(t),\int_{0}^{t}f_{5}(W(r))dr)]\Big{)},\quad\forall\
(t,\omega,\mu)\in\hat{\mathbb{D}}_{T,d}.$

Then we check that $f$ has continuous horizontal derivatives and twice
continuous SVDs in $\omega_{t}$ and $\mu_{t}.$ Indeed, for any $\tau\leq t,$

$\begin{split}&\partial_{t}f(t,\omega,\mu)=\partial_{t}F(t,x)+\partial_{x_{2}}F(t,x)f_{1}(\omega(t))+\partial_{x_{4}}F(t,x)\mathbb{E}^{\mu}\left[f_{3}(W(t))\right]\\\
&\ \ \ \ \quad\ \ \ \
\quad\quad+\partial_{x_{5}}F(t,x)\mathbb{E}^{\mu}\left[\partial_{y_{2}}f_{4}(Y)f_{5}(W(t))\right],\\\
&\partial_{\omega_{\tau}}f(t,\omega,\mu)=\partial_{x_{1}}F(t,x)+\partial_{x_{2}}F(t,x)\int_{\tau}^{t}f_{1}^{\prime}(\omega(r))dr,\\\
&\partial^{2}_{\omega_{\tau}}f(t,\omega,\mu)=\partial^{2}_{x_{1}}F(t,x)+\partial_{x_{2}}^{2}F(t,x)\left(\int_{\tau}^{t}f_{1}^{\prime}(\omega(r))dr\right)^{2}+\partial_{x_{2}}F(t,x)\int_{\tau}^{t}f_{1}^{(2)}(\omega(r))dr,\\\
&\partial_{\mu_{\tau}}f(t,\omega,\mu,\tilde{\omega})=\partial_{x_{3}}F(t,x)f^{\prime}_{2}(\tilde{\omega}(t))+\partial_{x_{4}}F(t,x)\int_{\tau}^{t}f_{3}^{\prime}(\tilde{\omega}(r))dr\\\
&\ \ \ \ \quad\ \ \ \quad\ \ \
\quad\quad+\partial_{x_{5}}F(t,x)\Big{[}\partial_{y_{1}}f_{4}(\tilde{y})+\partial_{y_{2}}f_{4}(\tilde{y})\int_{\tau}^{t}f_{5}^{\prime}(\tilde{\omega}(r))dr\Big{]},\quad\text{and}\\\
&\partial_{\tilde{\omega}_{\tau}}\partial_{\mu_{\tau}}f(t,\omega,\mu,\tilde{\omega})=\partial_{x_{3}}F(t,x)f^{(2)}_{2}(\tilde{\omega}(t))+\partial_{x_{4}}F(t,x)\int_{\tau}^{t}f_{3}^{(2)}(\tilde{\omega}(r))dr+\partial_{x_{5}}F(t,x)\Big{[}\partial^{2}_{y_{1}}f_{4}(\tilde{y}),\\\
&\quad\ \ \quad\quad\ \quad\quad\ \ \
\quad\quad+2\partial_{y_{2}}\partial_{y_{1}}f_{4}(\tilde{y})\int_{\tau}^{t}f_{5}^{\prime}(\tilde{\omega}(r))dr+\partial_{y_{2}}^{2}f_{4}(\tilde{y})(\int_{\tau}^{t}f_{5}^{\prime}(\tilde{\omega}(r))dr)^{2}\Big{]},\end{split}$

where

$\displaystyle(t,x)$
$\displaystyle=\left(t,\omega(t),\int_{0}^{t}f_{1}(\omega(r))dr,\mathbb{E}^{\mu}[f_{2}(W(t))],\mathbb{E}^{\mu}\Big{[}\int_{0}^{t}f_{3}(W(r))dr\Big{]},\
\mathbb{E}^{\mu}\Big{[}f_{4}(W(t),\int_{0}^{t}f_{5}(W(r))dr)\Big{]}\right),$
$\displaystyle Y$ $\displaystyle=\left(W(t),\int_{0}^{t}f_{5}(W(r))dr\right),\
\
\text{and}\quad\tilde{y}=\left(\tilde{\omega}(t),\int_{0}^{t}f_{5}(\tilde{\omega}(r))dr\right).$

In the following, for any $f\in\mathscr{D},$ we use generic notations
$(\partial_{\omega}f,\partial^{2}_{\omega}f)$
(($\partial_{\omega_{\tau}}f,\partial_{\omega_{\tau}}^{2}f)$, resp.) to denote
the vertical derivative (SVD, resp.) in path, and
$(\partial_{\mu}f,\partial_{\tilde{\omega}}\partial_{\mu}f)$
($(\partial_{\mu_{\tau}}f,\partial_{\tilde{\omega}_{\tau}}\partial_{\mu_{\tau}}f)$,
resp.) to denote the vertical derivative (SVD, resp.) in measure if there is
no confusion.

######  Definition 2.13.

Denote by $\mathscr{C}(\hat{\mathbb{D}}_{T,d})$ (or $\mathscr{C}$ when there
is no confusion), the subspace of $\mathscr{D}$ which consists of all non-
anticipative and continuous functionals. Furthermore,

  * (i)

$\mathscr{C}^{1,1,1}$ ($\mathscr{C}^{1,1,1}_{s}$, resp.) is the subset of
$\mathscr{C}$ whose element is continuously horizontally differentiable,
(strongly, resp.) vertically differentiable w.r.t. both path and measure, with
all derivatives being continuous;

  * (ii)

$\mathscr{C}^{1,2,1}$ ($\mathscr{C}^{1,2,1}_{s}$, resp.) is the subset of
$\mathscr{C}^{1,1,1}$ ( $\mathscr{C}^{1,1,1}_{s}$, resp.) whose element’s
derivative $\partial_{\omega}f(t,\cdot,\mu,\tilde{\omega})$ (
$\partial_{\omega_{\tau}}f(t,\cdot,\mu,\tilde{\omega})$, $\tau\leq t,$ resp.),
$(t,\omega,\mu,\tilde{\omega})\in\hat{\mathbb{D}}_{T,d}\times\mathbb{D}_{T,d}$,
is further vertically differentiable (strongly vertically differentiable at
$(\tau,t,\omega)$, resp.), with all derivatives being continuous;

  * (iii)

$\mathscr{C}^{1,2,1,1}$ ($\mathscr{C}^{1,2,1,1}_{s}$, resp.) is the subset of
$\mathscr{C}^{1,2,1}$ ($\mathscr{C}^{1,2,1}_{s}$, resp.) whose element’s
derivative functional $\partial_{\mu}f(t,\omega,\mu,\cdot)$ (
$\partial_{\mu_{\tau}}f(t,\omega,\mu,\cdot)$, $\tau\leq t$, resp.),
$(t,\omega,\mu,\tilde{\omega})\in\hat{\mathbb{D}}_{T,d}\times\mathbb{D}_{T,d}$,
is further vertically differentiable (strongly vertically differentiable at
$(\tau,t,\tilde{\omega})$, resp.), with all derivatives being continuous.

Moreover, denote by $\mathscr{C}^{1,1,1}_{p}$ the subset of
$\mathscr{C}^{1,1,1}$ such that the functional and all its first order
derivatives have at most polynomial growth in the path variable: there exists
$k\in\mathbb{Z}^{+}$, such that for $\phi=f,\partial_{t}f,\partial_{\omega}f,$
$\psi=\partial_{\mu}f$ and any $K>0,$

(43) $\begin{split}&|\phi(t,\omega,\mu)|\leq C_{K}(1+\|\omega_{t}\|^{k}),\ \ \
|\psi(t,\omega,\mu,\tilde{\omega})|\leq
C_{K}(1+\|\omega_{t}\|^{k}+\|\tilde{\omega}_{t}\|^{k}),\\\ &\ \ \ \
\forall(t,\omega,\mu,\tilde{\omega})\in\hat{\mathbb{D}}_{T,d}\times\mathbb{D}_{T,d}\
\text{such that }|||\mu_{t}|||\leq K,\end{split}$

for a constant $C_{K}$ depending only on $K.$ Notations such as
$\mathscr{C}_{p},$ $\mathscr{C}^{1,1,1}_{s,p}$ $\mathscr{C}^{0,1,1}$ and
$\mathscr{C}^{1,2,1,1}_{p}$ are defined similarly.

######  Remark 2.14.

Assume that $f\in\mathscr{D}$ is non-anticipative and has a state-dependent
structure: $f(t,\omega,\mu)=\tilde{f}(t,\omega(t),\mu(t))$ for some function
$\tilde{f}$ defined on
$[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R}^{d})$. Then the
horizontal differentiability and strongly vertical differentiability of $f$ is
reduced to the differentiability of $\tilde{f}$ on
$[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R}^{d}).$ Moreover,

(44)
$\displaystyle\partial_{t}f(t,\omega,\mu)=\partial_{t}\tilde{f}(t,\omega(t),\mu(t)),\
\
\partial_{\omega_{\tau}}f(t,\omega,\mu)=D_{x}f(t,\omega(t),\mu(t)),\quad\text{and}$
(45)
$\displaystyle\partial_{\mu_{\tau}}f(t,\omega,\mu,\tilde{\omega})=\partial_{\nu}\tilde{f}(t,\omega(t),\mu(t),\tilde{\omega}(t)),\
\forall(t,\omega,\mu)\in[0,T]\times\mathbb{D}_{T,d}\times\mathcal{P}_{2}^{D},\
\tau\leq t,$

where $\partial_{\nu}\tilde{f}$ is the Lions’ derivative (see e.g. [36]).

###  2.3. Itô-Dupire formula

Suppose that $(a,b)$ is a bounded progressively measurable process on
$(\Omega,\mathcal{F},P)$ with values in
$\mathbb{R}^{m}\times\mathbb{R}^{m\times d}.$ For any
$(t,\gamma)\in[0,T]\times\mathbb{D}_{T,d},$ $X$ is the solution of SDE

(46) $\left\\{\begin{array}[]{l}dX(r)=a(r)dr+b(r)dB(r),\\\ X_{t}=\gamma_{t},\
\ \ r\geq t.\end{array}\right.$

$(\Omega^{\prime},\mathcal{F}^{\prime},P^{\prime})$ is an atomless probability
space with a $k$-dimensional Brownian motion $B^{\prime}$ and $(c,d)$ is a
bounded progressively measurable process on
$(\Omega^{\prime},\mathcal{F}^{\prime},P^{\prime})$ with values in
$\mathbb{R}^{n}\times\mathbb{R}^{n\times k}.$ Given
$\eta\in(\mathbb{M}_{2}^{D})^{\prime}$, let $X^{\prime}$ defined by SDE

(47) $\left\\{\begin{array}[]{l}dX^{\prime}(r)=c(r)dr+d(r)dB^{\prime}(r),\\\
X^{\prime}_{t}=\eta_{t},\ \ \ r\geq t.\end{array}\right.$

Moreover, let
$({\tilde{X}}^{\prime},\tilde{c},\tilde{d},\tilde{B}^{\prime},{\tilde{\eta}})$
be an independent copy of $(X^{\prime},c,d,B^{\prime},\eta)$, which means that
$({\tilde{X}}^{\prime},\tilde{c},\tilde{d},\tilde{B}^{\prime},{\tilde{\eta}})$
is defined in an independent probability space
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{P})$ from $(\Omega,\mathcal{F},P)$
and $(\Omega^{\prime},\mathcal{F}^{\prime},P^{\prime})$, and it has the same
law as $(X^{\prime},c,d,B^{\prime},\eta)$. Then we have the following Itô-
Dupire formula.

######  Theorem 2.15.

For any fixed
$(t,\gamma,\eta)\in[0,T]\times\mathbb{D}_{T,d}\times(\mathbb{M}^{D}_{2})^{\prime},$
$X$ and $X^{\prime}$ are diffusion processes defined by (46) and (47)
respectively. Suppose that
$f\in\mathscr{C}^{1,2,1,1}_{p}(\hat{\mathbb{D}}_{T,d})$, and then we have

$\displaystyle f(s,X,\mathcal{L}_{X^{\prime}})-f(t,\gamma,\mathcal{L}_{\eta})$
$\displaystyle\ \ \
=\int_{t}^{s}\partial_{r}f(r,X,\mathcal{L}_{X^{\prime}})dr+\int_{t}^{s}\partial_{\omega}f(r,X,\mathcal{L}_{X^{\prime}})dX(r)$
(48) $\displaystyle\ \ \ \ \ \ +\frac{1}{2}\int_{t}^{s}\text{Tr}\
[\partial_{\omega}^{2}f(r,X_{r},\mathcal{L}_{X^{\prime}})d\langle
X\rangle(r)]+\mathbb{E}^{\tilde{P}^{\prime}}[\int_{t}^{s}\partial_{\mu}f(r,X,\mathcal{L}_{X^{\prime}},\tilde{X}^{\prime})d\tilde{X}^{\prime}(r)]$
$\displaystyle\ \ \ \ \ \
+\frac{1}{2}\mathbb{E}^{\tilde{P}^{\prime}}\int_{t}^{s}\text{Tr}\
[\partial_{\tilde{\omega}}\partial_{\mu}f(r,X,\mathcal{L}_{X^{\prime}},\tilde{X}^{\prime})\tilde{d}(r)\tilde{d}(r)^{T}]dr,\quad\forall
s\geq t.$

###### Proof.

Without loss of generality, assume $d=k=m=n=1$ and $s=T.$ Since both sides of
identity (48) depend on $(X^{\prime},c,d,\eta)$ through its law, we assume
that $(\Omega^{\prime},\mathcal{F}^{\prime},P^{\prime})$ is independent from
$(\Omega,\mathcal{F},P)$ for simplicity of notations. Consider the following
discretization of $X$ and $X^{\prime}:$ for any $n\geq 1,$ take
$t=t_{0}<t_{1}<\cdots<t_{n}=T$ as any partition of $[0,T]$ with vanishing
modulus $\delta_{n}$. Define càdlàg processes $X^{n},{X^{\prime}}^{n}$ with
$X^{n}_{t}=\gamma_{t},\ {X^{\prime}}^{n}_{t}=\eta_{t}$ by

$\displaystyle
X^{n}(r):=\sum_{i=0}^{n-1}X(t_{i})1_{[t_{i},t_{i+1})}(r)+X(T)1_{\\{T\\}}(r),$
$\displaystyle{X^{\prime}}^{n}(r):=\sum_{i=0}^{n-1}X^{\prime}(t_{i})1_{[t_{i},t_{i+1})}(r)+X^{\prime}(T)1_{\\{T\\}}(r),\quad
r\geq t.$

Since $(a,b,c,d)$ is bounded, we see that for any $r\in[0,T],$

(49)
$\displaystyle\mathbb{E}\|X^{n}\|^{p}_{\mathbb{S}^{p}}\leq\mathbb{E}\|X\|^{p}_{\mathbb{S}^{p}}<\infty,\quad\lim_{n\rightarrow\infty}\|X^{n}_{t_{i}}-X_{r}\|=0,\
P\text{-}a.s.,$ (50)
$\displaystyle|||\mathcal{L}_{{X^{\prime}}^{n}}|||^{2}=\mathbb{E}\|{X^{\prime}}^{n}\|^{2}_{\mathbb{S}^{2}}\leq\mathbb{E}\|X^{\prime}\|^{2}_{\mathbb{S}^{2}}<\infty,\quad\lim_{n\rightarrow\infty}\|{X^{\prime}}^{n}_{t_{i}}-X^{\prime}_{r}\|=0,\
P^{\prime}\text{-}a.s.,$

where $i$ above satisfies $r\in[t_{i},t_{i+1}).$ It follows from (50) that

(51)
$\lim_{n\rightarrow\infty}W_{2}(\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}},\mathcal{L}_{X^{\prime}_{r}})=0.$

Then we have

(52)
$\begin{split}&f(T,X^{n}_{T},\mathcal{L}_{X^{{}^{\prime}n}_{T}})-f(t,\gamma_{t},\mathcal{L}_{\eta_{t}})\\\
&\ \ \
=\sum_{i=0}^{n-1}[f(t_{i+1},X_{t_{i+1}}^{n},\mathcal{L}_{(X^{{}^{\prime}n})_{t_{i+1}}})-f(t_{i},X_{t_{i}}^{n},\mathcal{L}_{(X^{{}^{\prime}n})_{t_{i}}})]\\\
&\ \ \
=\sum_{i=0}^{n-1}\Big{[}(f(t_{i+1},X^{n}_{t_{i}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})-f(t_{i},X^{n}_{t_{i}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}}))+(f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})\\\
&\ \ \ \ \ \
-f(t_{i+1},X^{n}_{t_{i}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}}))+(f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i+1}}})-f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}}))\Big{]}.\end{split}$

Since

(53)
$\begin{split}f(t_{i+1},X^{n}_{t_{i}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})-f(t_{i},X^{n}_{t_{i}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})&=\int_{t_{i}}^{t_{i+1}}\partial_{r}f(r,X^{n}_{t_{i}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})dr\\\
&=\int_{t}^{T}\partial_{r}f(r,X^{n}_{t_{i}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})1_{[t_{i},t_{i+1})}(r)dr,\end{split}$

in view of inequalities (49) and (51), applying the dominated convergence
theorem and passing to the limit for a subsequence, we have

(54)
$\lim_{n\rightarrow\infty}\sum_{i=0}^{n-1}\Big{(}f(t_{i+1},X^{n}_{t_{i}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})-f(t_{i},X^{n}_{t_{i}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})\Big{)}=\int_{t}^{T}\partial_{r}f(r,X,\mathcal{L}_{X^{\prime}})dr,\
\ \ P\text{-}a.s..$

For the second term on the right hand side of (52), since
$f\in\mathscr{C}^{1,2,1,1}_{p},$ we have that
$\phi_{i}(\theta):=f(t_{i+1},X^{n}_{t_{i}}+\theta
1_{[t_{i+1},T)},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})$ is twice continuously
differentiable in $\theta$, and moreover,

(55)
$\phi^{\prime}_{i}(\theta)=\partial_{\omega}f(t_{i+1},X^{n}_{t_{i}}+\theta
1_{[t_{i+1},T)},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}}),\ \
\phi^{\prime\prime}_{i}(\theta)=\partial_{\omega}^{2}f(t_{i+1},X^{n}_{t_{i}}+\theta
1_{[t_{i+1},T)},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}}).$

In the following, we will write $X_{i}:=X(t_{i})\equiv X^{n}(t_{i})$ and
$\delta X_{i}:=X_{i+1}-X_{i}.$ Similar notations such as $X^{{}^{\prime}}_{i}$
are self-explained. Note that
$X^{n}_{t_{i+1}}=X^{n}_{t_{i}}+(X_{i+1}-X_{i})1_{[t_{i+1},T)}.$ Using the Itô
formula to $\phi(X(r)-X_{i})$ on $r\in[t_{i},t_{i+1}],$ we have

$\displaystyle
f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})-f(t_{i+1},X^{n}_{t_{i}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})$
(56) $\displaystyle\ \ \
=\int_{t_{i}}^{t_{i+1}}\partial_{\omega}f(t_{i+1},X^{n}_{t_{i}}+(X(r)-X_{i})1_{[t_{i+1},T)},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})dX(r)$
$\displaystyle\ \ \ \ \ \
+\frac{1}{2}\int_{t_{i}}^{t_{i+1}}\partial_{\omega}^{2}f(t_{i+1},X^{n}_{t_{i}}+(X(r)-X_{i})1_{[t_{i+1},T]},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})d\langle
X\rangle(r).$

Since $\|X^{n}_{t_{i}}+(X(r)-X_{i})1_{[t_{i+1},T]}-X_{r}\|\rightarrow 0,\
P$-a.s. for any $r\in[t_{i},t_{i+1})$, according to inequality (56), passing
to the limit in a subsequence, we have

(57)
$\begin{split}&\lim_{n\rightarrow\infty}\sum_{i=0}^{n-1}\Big{(}f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})-f(t_{i+1},X^{n}_{t_{i}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})\Big{)}\\\
&\ \ \
=\int_{t}^{T}\partial_{\omega}f(r,X,\mathcal{L}_{X^{\prime}})dX(r)+\frac{1}{2}\int_{t}^{T}\partial_{\omega}^{2}f(r,X,\mathcal{L}_{X^{\prime}})d\langle
X\rangle(r),\ \ P\text{-}a.s..\end{split}$

For the last term in the decomposition (52), we have

$\displaystyle
f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i+1}}})-f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}})$
$\displaystyle\ \ \
=\int_{0}^{1}\mathbb{E}^{\prime}\Big{[}\partial_{\mu}f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}+\theta(\delta
X^{\prime}_{i})1_{[t_{i+1,T})}},X^{{}^{\prime}n}_{t_{i}}+\theta(\delta
X^{\prime}_{i})1_{[t_{i+1,T})})(\delta X^{\prime}_{i})\Big{]}d\theta$
$\displaystyle\ \ \
=\int_{0}^{1}\mathbb{E}^{\prime}\Big{[}\partial_{\mu}f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}+\theta(\delta
X^{\prime}_{i})1_{[t_{i+1,T})}},X^{{}^{\prime}n}_{t_{i}})(\delta
X^{\prime}_{i})\Big{]}d\theta$ $\displaystyle\ \ \ \ \ \
+\int_{0}^{1}\int_{0}^{1}\mathbb{E}^{\prime}\Big{[}\partial_{\tilde{\omega}}\partial_{\mu}f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}+\theta(\delta
X^{\prime}_{i})1_{[t_{i+1,T})}},X^{{}^{\prime}n}_{t_{i}}+\lambda\theta(\delta
X^{\prime}_{i})1_{[t_{i+1,T})})\theta(\delta X^{\prime}_{i})^{2}\Big{]}d\theta
d\lambda.$

Since $\|X^{{}^{\prime}n}_{t_{i}}+\theta(\delta
X^{\prime}_{i})1_{[t_{i+1},T)}-X^{\prime}_{r}\|\rightarrow 0,\
P^{\prime}$-a.s. for any $r\in[0,T]$ with $r\in[t_{i},t_{i+1}]$, we have

$\lim_{n\rightarrow\infty}W_{2}(\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}+\theta(\delta
X^{\prime}_{i})1_{[t_{i+1,T})}},\mathcal{L}_{X^{\prime}_{r}})=0.$

In view of (49), (51) and the dominated convergence theorem, we have

$\displaystyle\lim_{n\rightarrow\infty}\sum_{i=0}^{n-1}\int_{0}^{1}\Big{[}\partial_{\mu}f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}+\theta(\delta
X^{\prime}_{i})1_{[t_{i+1,T})}},X^{{}^{\prime}n}_{t_{i}})(\delta
X^{\prime}_{i})\Big{]}d\theta$ $\displaystyle\ \ \
=\int_{t}^{T}\partial_{\mu}f(r,X^{\gamma_{t}},\mathcal{L}_{X^{{}^{\prime}}},X^{{}^{\prime}})dX^{\prime}(r),\
P\times P^{\prime}\text{-}a.s..$

Then, according to Fubini’s theorem, we have

(58)
$\begin{split}&\lim_{n\rightarrow\infty}\sum_{i=0}^{n-1}\mathbb{E}^{\prime}\int_{0}^{1}\Big{[}\partial_{\mu}f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}+\theta(\delta
X^{\prime}_{i})1_{[t_{i+1,T})}},X^{{}^{\prime}n}_{t_{i}})(\delta
X^{\prime}_{i})\Big{]}d\theta\\\ &\ \ \
=\mathbb{E}^{\prime}[\int_{t}^{T}\partial_{\mu}f(r,X,\mathcal{L}_{X^{{}^{\prime}}},X^{{}^{\prime}})dX^{\prime}(r)],\
\ \ \ \ \ P\text{-}a.s..\end{split}$

By a similar argument as above, we have

(59)
$\begin{split}&\lim_{n\rightarrow\infty}\int_{0}^{1}\\!\\!\\!\int_{0}^{1}\mathbb{E}^{\prime}\Big{[}\partial_{\tilde{\omega}}\partial_{\mu}f(t_{i+1},X^{n}_{t_{i+1}},\mathcal{L}_{X^{{}^{\prime}n}_{t_{i}}+\theta(\delta
X^{\prime}_{i})1_{[t_{i+1,T})}},X^{{}^{\prime}n}_{t_{i}}+\lambda\theta(\delta
X^{\prime}_{i})1_{[t_{i+1,T})})\theta(\delta X^{\prime}_{i})^{2}\Big{]}d\theta
d\lambda\\\ &\ \ \
=\mathbb{E}^{\prime}[\int_{t}^{T}\partial_{\tilde{\omega}}\partial_{\mu}f(r,X,\mathcal{L}_{X^{\prime}},X^{\prime})dr],\
P\text{-}a.s..\end{split}$

In view of (54), (57), (58) and (59), taking $n\to\infty$ in (52), we obtain
the desired identity.

∎

Note that $(\omega_{s})_{\tau}=\omega_{s}$ and $(\mu_{s})_{\tau}=\mu_{s}$ for
any $\tau\geq s.$ In particular, if the non-anticipative functional $f$ is
strongly vertically differentiable, we have the following partial Itô-Dupire
formula.

######  Corollary 2.16.

Suppose that $(X,X^{\prime})$ is defined as in Theorem 2.15, and
$f\in\mathscr{C}^{0,2,1,1}_{s,p}(\hat{\mathbb{D}}_{T,d})$. Then we have that
for any $t\leq s\leq v\leq T,$

$\begin{split}&f(v,X_{s},\mathcal{L}_{X^{\prime}_{s}})-f(v,\gamma_{t},\mathcal{L}_{\eta_{t}})\\\
&\ \ \
=\int_{t}^{s}\partial_{\omega_{r}}f(v,X_{r},\mathcal{L}_{X^{\prime}_{r}})dX(r)+\frac{1}{2}\int_{t}^{s}\text{Tr}\
[\partial_{\omega_{r}}^{2}f(v,X_{r},\mathcal{L}_{X^{\prime}_{r}})d\langle
X\rangle(r)]\\\ &\ \ \ \ \ \
+\mathbb{E}^{\tilde{P}^{\prime}}[\int_{t}^{s}\partial_{\mu_{r}}f(v,X_{r},\mathcal{L}_{X^{\prime}},\tilde{X}^{\prime})d\tilde{X}^{\prime}(r)]+\frac{1}{2}\mathbb{E}^{\tilde{P}^{\prime}}\int_{t}^{s}\text{Tr}\
[\partial_{\tilde{\omega}_{r}}\partial_{\mu_{r}}f(v,X_{r},\mathcal{L}_{X^{\prime}_{r}},\tilde{X}^{\prime}_{r})\tilde{d}(r)\tilde{d}(r)^{T}]dr.\end{split}$

###### Proof.

Without loss of generality, assume $v=T.$ For any $r\in[t,s],$ let

(60) $\tilde{f}(r,\omega,\mu):=f(T,\omega_{r},\mu_{r}).$

Obviously, $\tilde{f}$ is non-anticipative, and moreover, we have that for any
$h\geq 0,$

$\tilde{f}(r+h,\omega_{r},\mu_{r})=f(T,(\omega_{r})_{r+h},(\mu_{r})_{r+h})=f(T,\omega_{r},\mu_{r})=\tilde{f}(r,\omega_{r},\mu_{r}),$

which implies $\partial_{r}\tilde{f}(r,\omega_{r},\mu_{r})=0.$ Furthermore, it
follows by definitions of vertical derivatives and strongly vertical
derivatives that

$\displaystyle\partial_{\omega}\tilde{f}(r,\omega,\mu)=\partial_{\omega_{r}}f(T,\omega_{r},\mu_{r}),\
\ \ $
$\displaystyle\partial_{\omega}^{2}\tilde{f}(r,\omega,\mu)=\partial_{\omega_{r}}^{2}f(T,\omega_{r},\mu_{r}),$
$\displaystyle\partial_{\mu}\tilde{f}(r,\omega,\mu,\tilde{\omega})=\partial_{\mu_{r}}f(T,\omega_{r},\mu_{r},\tilde{\omega}),\
\ $
$\displaystyle\text{and}\quad\partial_{\tilde{\omega}}\partial_{\mu}\tilde{f}(r,\omega,\mu,\tilde{\omega})=\partial_{\tilde{\omega}_{r}}\partial_{\mu_{r}}f(T,\omega_{r},\mu_{r},\tilde{\omega}_{r}).$

Applying Theorem 2.15 to $\tilde{f}(r,X,\mathcal{L}_{X^{\prime}})$ on
$r\in[t,s]$, and we obtain the desired formula.

∎

##  3\. Differentiability of solutions for path-dependent mean-field BSDEs

In the following, for any process $(X,Y,Z)$ on the probability space
$(\Omega,\mathcal{F},P)$, we denote by $(\tilde{X},\tilde{Y},\tilde{Z})$ an
independent copy of $(X,Y,Z)$, which means that
$(\tilde{X},\tilde{Y},\tilde{Z})$ is defined in an independent probability
space $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{P})$ and has the same law as
$(X,Y,Z).$ Recall that $B$ is a $d$-dimensional Brownian motion on
$(\Omega,\mathcal{F},P).$ The following linear mean-field BSDEs and estimates
are frequently used in subsequent discussions. Note that for a classical
linear BSDE, the generator has a linear growth in $Y$, which implies the
global well-posedness. Here in the mean-field case, things are different since
the expected evolutionary equation is an ODE. For simplicity, we only address
the one-dimensional case. Similar assertions in this section are still true in
the multi-dimensional case.

######  Lemma 3.1.

Let $\xi\in L^{2}(\mathcal{F}_{T})$ and $t\in[0,T)$. Suppose that
$(\alpha,\beta)\in\mathbb{H}^{2}([t,T],\mathbb{R}\times\mathbb{R}^{d})$ is
bounded, $c\in\mathbb{H}^{2}([t,T],\mathbb{R}^{k})$, and $h$ is a real valued
progressively measurable process such that $\int_{t}^{T}|h(r)|dr\in
L^{2}(\mathcal{F}_{T}).$ For any $(r,x)\in[t,T]\times\mathbb{R}^{k},$
$g(\cdot,x)\in\mathbb{H}^{2}([t,T])$ and $g(r,\cdot)$ is uniformly Lipschitz
continuous:

$\sup_{r\in[t,T]}|g(r,x)-g(r,y)|\leq L|x-y|,\quad\forall y\in\mathbb{R}^{k},\
P\text{-}a.s.$

for a constant $L.$ Then, the following linear mean-field BSDE

(61)
$Y(s)=\xi+\int_{s}^{T}\Big{(}\alpha(r)Y(r)+\beta(r)Z(r)+\tilde{\mathbb{E}}[g(r,\tilde{c}(r))\tilde{Y}(r)]+h(r)\Big{)}dr-\int_{s}^{T}Z(r)dB(r),s\in[t,T]$

with $(\tilde{c},\tilde{Y})$ being an independent copy of $(c,Y),$ has a
unique solution
$(Y,Z)\in\mathbb{S}^{2}([t,T])\times\mathbb{H}^{2}([t,T],\mathbb{R}^{d})$.
Moreover, we have

(62) $\|(Y,Z)\|_{\mathbb{S}^{2}\times\mathbb{H}^{2}}^{2}\leq
C(\|\xi\|_{L^{2}}^{2}+||\int_{t}^{T}|h(r)|dr||^{2}_{L^{2}})e^{C(\|c\|_{\mathbb{H}^{2}}+\|g(\cdot,0)\|_{\mathbb{H}^{2}})}$

for a constant $C$ depending on the bound of $\alpha,\beta$ and $L.$ In
particular, if $g$ is uniformly bounded, we have

(63) $\|(Y,Z)\|_{\mathbb{S}^{2}\times\mathbb{H}^{2}}^{2}\leq
C(\|\xi\|_{L^{2}}^{2}+||\int_{t}^{T}|h(r)|dr||^{2}_{L^{2}}).$

######  Remark 3.2.

Since neither $g(t,x)$ nor $g(r,c(r))$ is bounded or uniformly integrable for
any $c(r)\in\mathbb{H}^{2}([t,T],\mathbb{R}^{k})$, the well-posedness of the
mean-field BSDE is not an immediate consequence of existing works such as [8].

###### Proof.

For any $y\in\mathbb{H}^{2},$ consider the following classical linear BSDE

(64)
$Y(s)=\xi+\int_{s}^{T}\Big{(}\alpha(r)Y(r)+\beta(r)Z(r)+\tilde{\mathbb{E}}[g(r,\tilde{c}(r))\tilde{y}(r)]+h(r)\Big{)}dr-\int_{s}^{T}Z(r)dB(r),$

where $(\tilde{c},\tilde{y})$ is an independent copy of $(c,y)$. To prove that
it is well-posed on $[t,T]$, we only need to show

(65)
$\mathbb{E}\left[\int_{t}^{T}\Big{|}\tilde{\mathbb{E}}[g(r,\tilde{c}(r))\tilde{y}(r)]\Big{|}dr\right]^{2}<\infty.$

Indeed, by the uniformly Lipschitz continuity of $g$, we have

(66)
$\begin{split}&\mathbb{E}\Big{|}\int_{t}^{T}|\tilde{\mathbb{E}}[g(r,\tilde{c}(r))\tilde{y}(r)]|dr\Big{|}^{2}\leq
C\mathbb{E}\Big{[}\int_{t}^{T}|\tilde{\mathbb{E}}|g(r,0)\tilde{y}(r)|dr+\tilde{\mathbb{E}}|\tilde{c}(r)\tilde{y}(r)|dr\Big{]}^{2}\\\
&\ \ \leq
C\mathbb{E}\Big{[}\int_{t}^{T}|g(r,0)\tilde{\mathbb{E}}[\tilde{y}(r)]|dr\Big{]}^{2}+C\Big{[}\int_{t}^{T}\tilde{\mathbb{E}}[|\tilde{c}(r)\tilde{y}(r)|]dr\Big{]}^{2}\\\
&\ \ \leq
C\Big{[}(\mathbb{E}\int_{t}^{T}|g(r,0)|^{2}dr)(\int_{t}^{T}[\tilde{\mathbb{E}}|\tilde{y}(r)|]^{2}dr)+(\int_{t}^{T}\tilde{\mathbb{E}}|\tilde{c}(r)|^{2}dr)(\int_{t}^{T}\tilde{\mathbb{E}}|\tilde{y}(r)|^{2}dr)\Big{]}\\\
&\ \ \leq
C\Big{[}\|g(\cdot,0)\|_{\mathbb{H}^{2}}^{2}\|y\|_{\mathbb{H}^{2}}^{2}+\|c\|_{\mathbb{H}^{2}}^{2}\|y\|_{\mathbb{H}^{2}}^{2}\Big{]}\leq
C(\|g(\cdot,0)\|_{\mathbb{H}^{2}}^{2}+\|c\|_{\mathbb{H}^{2}}^{2})\|y\|_{\mathbb{H}^{2}}^{2},\end{split}$

where we apply the Hölder inequality to integrals in forms of $\int_{t}^{T}$
and $\int_{t}^{T}\tilde{\mathbb{E}}$ respectively in the third inequality
above. Then for any $y\in\mathbb{H}^{2},$ there exists a unique solution
$(Y,Z)\in\mathbb{H}^{2}\times\mathbb{H}^{2}$ of BSDE (64). The mapping $y\to
Y$, denoted by $\Phi$, is a transformation on $\mathbb{H}^{2}$ , and will be
shown to be a contraction under the following equivalent norm on
$\mathbb{H}^{2}$

(67)
$\|Y\|^{2}:=\mathbb{E}\int_{t}^{T}e^{As-\int_{s}^{T}(\|g(r,0)\|^{2}_{L^{2}}+\|c(r)\|^{2}_{L^{2}})dr}|Y(s)|^{2}ds$

with $A$ being a constant to be determined later. Take any
$y^{(1)},y^{(2)}\in\mathbb{H}^{2}$ and denote the corresponding solutions of
classical BSDE (64) by $(Y^{(1)},Z^{(1)}),(Y^{(2)},Z^{(2)})$. Set $(\Delta
Y,\Delta Z):=(Y^{(1)}-Y^{(2)},Z^{(1)}-Z^{(2)})$, $\Delta y:=y^{(1)}-y^{(2)}$,
and $f(r):=\|g(r,0)\|^{2}_{L^{2}}+\|c(r)\|^{2}_{L^{2}}$. Apply Itô’s formula
to $e^{As-\int_{s}^{T}f(r)dr}|\Delta Y(s)|^{2}$ on $s\in[t,T]$, and we have

$\begin{split}-e^{At-\int_{t}^{T}f(r)dr}|\Delta
Y(t)|^{2}&=\int_{t}^{T}(A+f(s))e^{As-\int_{s}^{T}f(r)dr}|\Delta Y(s)|^{2}ds\\\
&\ \ \ \ +2\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}\Delta Yd(\Delta
Y)+\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}|\Delta Z|^{2}ds.\end{split}$

Therefore,

$\displaystyle e^{At-\int_{t}^{T}f(r)dr}|\Delta
Y(t)|^{2}+\int_{t}^{T}(A+f(s))e^{As-\int_{s}^{T}f(r)dr}|\Delta
Y(s)|^{2}dr+\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}|\Delta Z(s)|^{2}dr$
$\displaystyle\quad=2\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}\Delta
Y[\alpha\Delta Y+\beta\Delta
Z+\tilde{\mathbb{E}}[g(s,\tilde{c}(s))\Delta\tilde{y}]]ds-2\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}\Delta
Y\Delta ZdW(s)$ $\displaystyle\quad\leq
C\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}|\Delta
Y|^{2}ds+C\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}|\Delta
Y|^{2}ds+\frac{1}{2}\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}|\Delta Z|^{2}ds$
$\displaystyle\ \ \ \ \ \ +2\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}|\Delta
Y||g(s,0)|\|\Delta\tilde{y}\|_{L^{2}}ds+2\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}|\Delta
Y|\|c\|_{L^{2}}\|\Delta\tilde{y}\|_{L^{2}}ds$ $\displaystyle\ \ \ \ \ \
-2\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}\Delta Y\Delta ZdB(s)$
$\displaystyle\quad\leq C\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}|\Delta
Y|^{2}ds+\frac{1}{2}\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}|\Delta
Z|^{2}ds-2\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}\Delta Y\Delta ZdB(s)$
$\displaystyle\ \ \ \ \ \ +2\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}(|\Delta
Y||g(s,0)|\|\Delta\tilde{y}\|_{L^{2}})ds+\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}(|\Delta
Y|^{2}\|c\|^{2}_{L^{2}}+\|\Delta\tilde{y}\|^{2}_{L^{2}})ds.$

Taking expectation on both sides of the above inequality, we have

$\displaystyle\int_{t}^{T}(A-C+f(s))e^{As-\int_{s}^{T}f(r)dr}\|\Delta
Y(s)\|_{L^{2}}^{2}dr$ $\displaystyle\quad\leq
2\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}\mathbb{E}[|\Delta
Y||g(s,0)|]\|\Delta\tilde{y}\|_{L^{2}}ds+\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}(\|\Delta
Y\|_{L^{2}}^{2}\|c\|^{2}_{L^{2}}+\|\Delta\tilde{y}\|^{2}_{L^{2}})ds$
$\displaystyle\quad\leq\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}\Big{[}\Big{(}\mathbb{E}[|\Delta
Y||g(s,0)|]\Big{)}^{2}+\|\Delta\tilde{y}\|_{L^{2}}^{2}\Big{]}ds+\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}(\|\Delta
Y\|_{L^{2}}^{2}\|c\|^{2}_{L^{2}}+\|\Delta\tilde{y}\|^{2}_{L^{2}})ds$
$\displaystyle\quad\leq\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}\Big{(}\|g(s,0)\|_{L^{2}}^{2}+\|c\|^{2}_{L^{2}}\Big{)}\|\Delta
Y\|_{L^{2}}^{2}ds+\int_{t}^{T}e^{As-\int_{s}^{T}f(r)dr}\|\Delta\tilde{y}\|_{L^{2}}^{2}ds.$

Therefore, choosing a sufficiently large number $A$ such that $A-C>1$, we
obtain a contraction and then the well-posedness of (61).

Now BSDE (61) can be written as the following classical BSDE

(68)
$Y(s)=\xi+\int_{s}^{T}\Big{(}\alpha(r)Y(r)+\beta(r)Z(r)+h^{\prime}(r)\Big{)}dr-\int_{s}^{T}Z(r)dB(r),$

with $h^{\prime}(r)=\tilde{\mathbb{E}}[g(r,\tilde{c}(r))\tilde{Y}(r)]+h(r).$
Thus it is standard that

(69) $\begin{split}\|Y\|_{\mathbb{S}^{2}}^{2}+\|Z\|_{\mathbb{H}^{2}}^{2}&\leq
C(\|\xi\|_{L^{2}}^{2}+\|\int_{t}^{T}|h^{\prime}(r)|dr\|^{2}_{L^{2}})\\\ &\leq
C(\|\xi\|_{L^{2}}^{2}+\|\int_{t}^{T}|h(r)|dr\|^{2}_{L^{2}}+\|\int_{t}^{T}|\tilde{\mathbb{E}}[g(r,\tilde{c}(r))\tilde{Y}(r)]|dr\|^{2}_{L^{2}}).\end{split}$

Furthermore, similar to the proof of inequality (66), we have

(70)
$\|\int_{t}^{T}|\tilde{\mathbb{E}}[g(r,\tilde{c}(r))\tilde{Y}(r)]|dr\|^{2}_{L^{2}}\leq
C[\int_{t}^{T}\|g(r,0)\|_{L^{2}}^{2}\|Y\|^{2}_{\mathbb{S}^{2},[t,r]}dr+\int_{t}^{T}\|c(r)\|_{L^{2}}^{2}\|Y\|^{2}_{\mathbb{S}^{2},[t,r]}dr].$

Then, using Gronwall’s inequality, we obtain the desired estimate (62).

∎

We consider the following path-dependent master equation

(76)
$\displaystyle\left\\{\begin{array}[]{l}\partial_{t}u(t,\gamma,\mu)+\frac{1}{2}\text{Tr}\left[\partial_{\omega}^{2}u(t,\gamma,\mu)\sigma_{1}(t)\sigma_{1}(t)^{T}\right]+\partial_{\omega}u(t,\gamma,\mu)b_{1}(t)\\\\[5.69054pt]
\quad+\frac{1}{2}\text{Tr}\left[\mathbb{E}^{P}[\partial_{\tilde{\omega}}\partial_{\mu}u(t,\gamma,\mu,\eta)]\sigma_{2}(t)\sigma_{2}(t)^{T}\right]+\mathbb{E}^{P}[\partial_{\mu}u(t,\gamma,\mu,\eta)]b_{2}(t)\\\\[5.69054pt]
\quad+f(t,\gamma,u(t,\gamma,\mu),\sigma_{1}(t)\partial_{\omega}u(t,\gamma,\mu),\mu,\mathcal{L}_{u(t,\eta,\mu)})=0,\\\
\\\ u(T,\gamma,\mu)=\Phi(\gamma_{T},\mu_{T}),\ \ \
(t,\gamma,\mu)\in[0,T]\times\mathbb{C}_{T,d}\times\mathcal{P}_{2}^{C},\end{array}\right.$

where $(b_{1},\sigma_{1},b_{2},\sigma_{2})$ are continuous functions and
$\eta\in\mathbb{M}^{C}_{2}$ with law $\mu$. For simplicity, we take
$(b_{1},\sigma_{1})=(b_{2},\sigma_{2})=(0,I)$ and refer to Remark 4.9 for the
above form. In the following, we write
$f(\omega_{t},\mu_{t}):=f(t,\omega,\mu)$ for simplicity since $f$ is non-
anticipative. Moreover, for any $\gamma,\omega\in\mathbb{D}_{T,d},$ define
$\omega^{\gamma_{t}}\in\mathbb{D}_{T,d}$ as the following

(77)
$\omega^{\gamma_{t}}(\cdot):=\gamma_{t}(\cdot)+(\omega(\cdot)-\omega(t))1_{[t,T]}(\cdot).$

To give a classical solution through FBSDEs, for any
$(t,\eta)\in[0,T]\times\mathbb{M}^{D}_{2}$, we denote by
$(Y^{\eta_{t}},Z^{\eta_{t}})$ the solution of the following path-dependent
mean-field BSDE

(78)
$Y(s)=\Phi(B_{T}^{\eta_{t}},\mathcal{L}_{B_{T}^{\eta_{t}}})+\int_{s}^{T}f(B_{r}^{\eta_{t}},Y(r),Z(r),\mathcal{L}_{B_{r}^{\eta_{t}}},\mathcal{L}_{Y(r)})dr-\int_{s}^{T}Z(r)dB(r),\quad
s\in[t,T].$

On the other hand, for any $\gamma\in\mathbb{D}_{T,d},$ let
$(Y^{\gamma_{t},\eta_{t}},Z^{\gamma_{t},\eta_{t}})$ solve the associated path-
dependent BSDE

(79)
$\mathcal{Y}{(s)}=\Phi(B_{T}^{\gamma_{t}},\mathcal{L}_{B_{T}^{\eta_{t}}})+\int_{s}^{T}f(B_{r}^{\gamma_{t}},\mathcal{Y}(r),\mathcal{Z}(r),\mathcal{L}_{B_{r}^{\eta_{t}}},\mathcal{L}_{Y^{\eta_{t}}(r)})dr-\int_{s}^{T}\mathcal{Z}(r)dB(r),\quad
s\in[t,T].$

######  Definition 3.3.

We write $\Phi\in\mathscr{C}_{T}(\hat{\mathbb{D}}_{T,d})$ (or
$\mathscr{C}_{T}$ if no confusion raised) if
$\Phi:\mathbb{D}_{T,d}\times\mathcal{P}_{2}^{D}\to\mathbb{R}$ is continuous on
$\mathbb{D}_{T,d}\times\mathcal{P}_{2}^{D}.$ Furthermore, we write

  * (i)

$\Phi\in\mathscr{C}_{T,lip}$ if it is uniformly Lipschitz continuous on
$\mathbb{D}_{T,d}\times\mathcal{P}_{2}^{D}:$

$|\Phi(\omega_{T},\mu_{T})-\Phi(\omega^{\prime}_{T},\mu^{\prime}_{T})|\leq
C(\|\omega_{T}-\omega^{\prime}_{T}\|+W_{2}(\mu_{T},\mu^{\prime}_{T})),\
\forall(\omega,\mu),(\omega^{\prime},\mu^{\prime})\in\mathbb{D}_{T,d}\times\mathcal{P}_{2}^{D},$

for some constant $C$;

  * (ii)

$\Phi\in\mathscr{C}^{1,1}_{T,lip}$ if $\Phi\in\mathscr{C}_{T,lip}$ and $\Phi$
is continuously strongly vertically differentiable in path and measure.
Furthermore, for any $\tau\in[0,T),$ SVDs $\partial_{\omega_{\tau}}\Phi$ and
$\partial_{\mu_{\tau}}\Phi$ are uniformly Lipschitz continuous in
$(\omega,\mu)\in\mathbb{D}_{T,d}\times\mathcal{P}_{2}^{D}$ and
$(\omega,\mu,\tilde{\omega})\in\mathbb{D}_{T,d}\times\mathcal{P}_{2}^{D}\times\mathbb{D}_{T,d}$,
respectively;

  * (iii)

$\Phi\in\mathscr{C}^{2,1,1}_{T,lip}$ if $\Phi\in\mathscr{C}^{1,1}_{T,lip}$ and
for any
$(\tau,\omega,\mu,\tilde{\omega})\in\hat{\mathbb{D}}_{T,d}\times\mathbb{D}_{T,d},$
its SVDs $\partial_{\omega_{\tau}}\Phi(\cdot,\mu_{T})$ and
$\partial_{\mu_{\tau}}\Phi(\omega_{T},\mu_{T},\cdot)$ are continuously
strongly vertically differentiable at $(\tau,T,\omega)$ and
$(\tau,T,\tilde{\omega})$ respectively. Moreover, all second-order derivatives
are uniformly Lipschitz continuous.

To obtain the well-posedness and estimates of BSDEs (78) and (79), we assume
that

  * (H0)

$(i)$ The functional $\Phi\in\mathscr{C}_{T,lip}(\hat{\mathbb{D}}_{T,d})$ ;
$(ii)$ $f$ is a non-anticipative continuous function on
$[0,T]\times\mathbb{D}_{T,d}\times\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{P}_{2}^{D}\times\mathcal{P}_{2}(\mathbb{R})$,
and for any
$(t,\omega,\mu)\in[0,T]\times\mathbb{D}_{T,d}\times\mathcal{P}_{2}^{D},$
$f(t,\omega,\cdot,\cdot,\mu,\cdot)$ is continuously differentiable on
$\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R})$. Moreover,
for any $t\in[0,T]$, $f(t,\cdot,\cdot,\cdot,\cdot,\cdot)$ and
$\partial_{\nu}f(t,\cdot,\cdot,\cdot,\cdot,\cdot,\cdot)$ are uniformly
Lipschitz continuous.

Note that under Assumption (H0), the functional

(80)
$\hat{f}(r,y,z,\nu):=f(B_{r}^{\eta_{t}},y,z,\mathcal{L}_{B_{r}^{\eta_{t}}},\nu),\quad(r,y,z,\nu)\in[t,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R}),$

is uniformly Lipschitz continuous in $(y,z)\in\mathbb{R}\times\mathbb{R}^{d}$.
According to [11, Theorem 4.23], BSDE (78) is well posed with
$(Y^{\eta_{t}},Z^{\eta_{t}},\mathcal{L}_{Y^{\eta_{t}}})\in\mathbb{S}^{2}\times\mathbb{H}^{2}\times\mathcal{P}_{2}(\mathbb{R}).$
Then (79) is a well-defined classical BSDE with
$(Y^{\gamma_{t},\eta_{t}},Z^{\gamma_{t},\eta_{t}})\in\mathbb{S}^{p}\times\mathbb{H}^{p}$
for any $p\geq 1.$ In the following BSDEs, we write
$\Theta^{\eta_{t}}_{r}:=(B_{r}^{\eta_{t}},Y^{\eta_{t}}(r),Z^{\eta_{t}}(r)),\Theta^{\gamma_{t},\eta_{t}}_{r}:=(B_{r}^{\gamma_{t}},Y^{\gamma_{t},\eta_{t}}(r),Z^{\gamma_{t},\eta_{t}}(r))$,
$\mathcal{L}_{\Theta^{\eta_{t}}_{r}}:=(\mathcal{L}_{B_{r}^{\eta_{t}}},\mathcal{L}_{Y^{\eta_{t}}(r)})$
and $(Y,Z):=(Y(t),Z(t))$ if no confusion is raised. Then we have the following
basic estimates for BSDEs (78) and (79).

######  Lemma 3.4.

Assume that $(\Phi,f)$ satisfies (H0). For any $K>0$ and
$(\gamma,\eta),(\gamma^{\prime},\eta^{\prime})\in\mathbb{D}_{T,d}\times\mathbb{M}_{2}^{D}$
such that
$|||\mathcal{L}_{\eta_{t}}|||,|||\mathcal{L}_{\eta^{\prime}_{t}}|||\leq K,$ we
have

(81)
$\displaystyle\|(Y^{\eta_{t}},Z^{\eta_{t}})\|_{\mathbb{S}^{2}\times\mathbb{H}^{2}}\leq
C(1+\|\eta_{t}\|_{\mathbb{S}^{2}}),$ (82)
$\displaystyle\|(Y^{\gamma_{t},\eta_{t}},Z^{\gamma_{t},\eta_{t}})\|_{\mathbb{S}^{p}\times\mathbb{H}^{p}}\leq
C_{p}(1+\|\gamma_{t}\|+\|\eta_{t}\|_{\mathbb{S}^{2}}),\ \ \forall p\geq 1,$
(83)
$\displaystyle\|(Y^{\eta_{t}}-Y^{\eta^{\prime}_{t}},Z^{\eta_{t}}-Z^{\eta^{\prime}_{t}})\|_{\mathbb{S}^{2}\times\mathbb{H}^{2}}\leq
C_{K}\|\eta_{t}-\eta^{\prime}_{t}\|_{\mathbb{S}^{2}},\quad\quad\text{and }$
(84)
$\displaystyle\|(Y^{\gamma_{t},\eta_{t}}-Y^{\gamma^{\prime}_{t},\eta^{\prime}_{t}},Z^{\gamma_{t},\eta_{t}}-Z^{\gamma^{\prime}_{t},\eta^{\prime}_{t}})\|_{\mathbb{S}^{p}\times\mathbb{H}^{p}}\leq
C_{K,p}(\|\gamma_{t}-\gamma^{\prime}_{t}\|+W_{2}(\mathcal{L}_{\eta_{t}},\mathcal{L}_{\eta^{\prime}_{t}})),\
\ \forall p\geq 1,$

where $(C,C_{p})$ does not depend on $(\gamma,\eta)$, and $(C_{K},C_{K,p})$
does not depend on $(\gamma,\gamma^{\prime})$.

######  Remark 3.5.

According to inequality (84),
$(Y^{\gamma_{t},\eta_{t}},Z^{\gamma_{t},\eta_{t}})$ and
$(Y^{\gamma_{t},\eta^{\prime}_{t}},Z^{\gamma_{t},\eta^{\prime}_{t}})$ are
indistinguishable if $\mathcal{L}_{\eta_{t}}=\mathcal{L}_{\eta^{\prime}_{t}},$
which implies the following definition is well-posed

(85)
$(Y^{\gamma_{t},\mathcal{L}_{\eta_{t}}},Z^{\gamma_{t},\mathcal{L}_{\eta_{t}}}):=(Y^{\gamma_{t},\eta_{t}},Z^{\gamma_{t},\eta_{t}}).$

The proof of Lemma 3.4 is rather standard with an application of Lemma 3.1,
and is left in the appendix.

###  3.1. First-order differentiability

For any $(\gamma,\eta)\in[0,T]\times\mathbb{M}^{D}_{2},$ in the following, we
consider the first order differentiability of
$Y^{{\gamma_{t},\eta_{t}}}=Y^{\gamma_{t},\mathcal{L}_{\eta_{t}}}$ with respect
to $\gamma_{t}$ and $\mathcal{L}_{\eta_{t}}.$ For the differentiability in
$\gamma_{t},$ let

(86)
$\begin{split}&\hat{f}(\omega_{s},y,z):=f(\omega_{s},y,z,\mathcal{L}_{B_{s}^{\eta_{t}}},\mathcal{L}_{Y^{\eta_{t}}(s)}),\\\
&\hat{\Phi}(\omega_{T}):=\Phi(\omega_{T},\mathcal{L}_{B_{T}^{\eta_{t}}}),\
\forall(s,\omega,y,z)\in[t,T]\times\mathbb{D}_{T,d}\times\mathbb{R}\times\mathbb{R}^{d},\end{split}$

and then the solution $Y^{\gamma_{t},\eta_{t}}{(s)}$ to equation (79) solves
the following path-dependent BSDE

(87)
$\hat{Y}{(s)}=\hat{\Phi}(B_{T}^{\gamma_{t}})+\int_{s}^{T}\hat{f}(B^{\gamma_{t}}_{r},\hat{Y}(r),\hat{Z}(r))dr-\int_{s}^{T}\hat{Z}(r)dB(r).$

Define $\hat{u}_{\eta_{t}}(t,{\gamma}):=Y^{\gamma_{t},\eta_{t}}(t).$ If $f$
and $\Phi$ are regular enough, according to [41, Theorem 4.5] ,
$\hat{u}_{\eta_{t}}(t,{\gamma})$ is twice vertically differentiable at
$(t,\gamma),$ and moreover for any $s\geq t,$

(88) $\hat{u}_{\eta_{t}}(s,B^{\gamma_{t}})=Y^{\gamma_{t},\eta_{t}}(s),\ \ \
\partial_{\gamma_{t}}\hat{u}_{\eta_{t}}(s,B^{\gamma_{t}})=Z^{\gamma_{t},\eta_{t}}(s).$

Furthermore, $\hat{u}_{\eta_{t}}(t,\gamma)$ is the unique solution to the
following semilinear PPDE

(89)
$\left\\{\begin{array}[]{l}\partial_{t}\hat{u}_{\eta_{t}}(t,\gamma)+\frac{1}{2}\text{Tr}\left[\partial_{\omega}^{2}\hat{u}_{\eta_{t}}(t,\gamma)\right]+\hat{f}(\gamma_{t},\hat{u}_{\eta_{t}}(t,{\gamma}),\partial_{\omega}\hat{u}_{\eta_{t}}(t,{\gamma}))=0,\\\\[5.69054pt]
\hat{u}_{\eta_{t}}(T,\gamma)=\hat{\Phi}(\gamma),\ \ \
(t,\gamma)\in[0,T]\times\mathbb{C}_{T,d}.\end{array}\right.$

In the following, we denote by
$\partial_{(t,\omega,y,z,\mu,\nu,{\omega_{\tau}},{\mu_{\tau}})}f$ the
derivative vector

$(\partial_{t}f,\partial_{\omega}f,\partial_{y}f,\partial_{z}f,\partial_{\mu}f,\partial_{\nu}f,\partial_{\omega_{\tau}}f,\partial_{\mu_{\tau}}f).$

In this subsection, we assume that

  * (H1)

$(i)$ The functional
$\Phi\in\mathscr{C}^{1,1}_{T,lip}(\hat{\mathbb{D}}_{T,d})$; $(ii)$ $f$ is a
non-anticipative continuous function on
$[0,T]\times\mathbb{D}_{T,d}\times\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{P}_{2}^{D}\times\mathcal{P}_{2}(\mathbb{R})$,
and for any
$(t,\omega,\mu)\in[0,T]\times\mathbb{D}_{T,d}\times\mathcal{P}_{2}^{D},$
$f(t,\omega,\cdot,\cdot,\mu,\cdot)$ is differentiable on
$\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R})$ with bounded
derivatives. For any
$(y,z,\nu)\in\mathbb{R}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R}),$
$f(t,\omega,y,z,\cdot,\nu)$ is strongly vertically differentiable at $\mu_{t}$
and $f(t,\cdot,y,z,\mu,\nu)$ is strongly vertically differentiable at
$\omega_{t}$. Moreover, $\partial_{(y,z,\nu,{\omega_{\tau}},{\mu_{\tau}})}f$
is continuous, and for any $\tau\leq t,$
$(I,\partial_{(y,z,\nu,{\omega_{\tau}},{\mu_{\tau}})})f(t,\cdot)$ is uniformly
Lipschitz continuous.

######  Remark 3.6.

Assume that $\Phi:\mathbb{D}_{T,d}\to\mathbb{R}$ is twice continuously
strongly vertically differentiable and satisfies the following locally
Lipschitz continuous condition: for any $t\in[0,T]$ and
$\phi=\Phi,\partial_{\omega_{t}}\Phi,\partial_{\omega_{t}}^{2}\Phi$,

(90) $|\phi(\omega_{T})-\phi(\omega^{\prime}_{T})|\leq
C(1+\|\omega_{T}|^{k}+\|\omega^{\prime}_{T}\|^{k})\|\omega_{T}-\omega^{\prime}_{T}\|,\quad\forall\
(\omega,\omega^{\prime})\in\mathbb{D}_{T,d}^{2}.$

Then, the main result [41, Theorem 4.5] is still true. For the reader’s
convenience, the proof is sketched in the appendix, using our partial Itô-
Dupire formula.

######  Lemma 3.7.

Let $(f,\Phi)$ satisfy Assumption (H1). Then
$(Y^{\gamma_{t},\eta_{t}},Z^{\gamma_{t},\eta_{t}})$ is almost surely
vertically differentiable at $(t,\gamma).$ The derivative
$(\partial_{\omega_{t}}Y^{{\gamma_{t},\eta_{t}}},\partial_{\omega_{t}}Z^{{\gamma_{t},\eta_{t}}})\in\mathbb{S}^{p}([t,T],\mathbb{R}^{d})\times\mathbb{H}^{p}([t,T],\mathbb{R}^{d\times
d}),$ for any $p\geq 1$, is the unique solution to BSDE

(91) $\begin{split}\mathcal{Y}(s)=&\
\partial_{\omega_{t}}\Phi(B^{\gamma_{t}},\mathcal{L}_{B^{\eta_{t}}})+\int_{s}^{T}\partial_{\omega_{t}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})dr+\int_{s}^{T}\partial_{y}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\mathcal{Y}(r)dr\\\
&+\int_{s}^{T}\mathcal{Z}(r)\partial_{z}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})dr-\int_{s}^{T}\mathcal{Z}(r)dB(r),\
\ s\in[t,T].\end{split}$

Furthermore, since
$(\partial_{\omega_{t}}Y^{{\gamma_{t},\eta_{t}}},\partial_{\omega_{t}}Z^{{\gamma_{t},\eta_{t}}})$
is independent of $\mathcal{F}_{t}$, we have that for any $K>0,$ there are
positive constants $C_{p}$ and $C_{K,p}$ such that

(92)
$\begin{split}&\|(\partial_{\omega_{t}}Y^{{\gamma_{t},\eta_{t}}},\partial_{\omega_{t}}Z^{{\gamma_{t},\eta_{t}}})\|_{\mathbb{S}^{p}\times\mathbb{H}^{p}}<C_{p},\\\
&\|(\partial_{\omega_{t}}Y^{{\gamma_{t},\eta_{t}}}-\partial_{\omega_{t}}Y^{\gamma^{\prime}_{t},\eta^{\prime}_{t}},\partial_{\omega_{t}}Z^{{\gamma_{t},\eta_{t}}}-\partial_{\omega_{t}}Z^{\gamma^{\prime}_{t},\eta^{\prime}_{t}})\|_{\mathbb{S}^{p}\times\mathbb{H}^{p}}<C_{K,p}(\|\gamma_{t}-\gamma^{\prime}_{t}\|+W_{2}(\mathcal{L}_{\eta_{t}},\mathcal{L}_{\eta^{\prime}_{t}})),\\\
&\ \ \ \ \ \ \forall\
(\gamma,\eta),(\gamma^{\prime},\eta^{\prime})\in\mathbb{D}_{T,d}\times\mathbb{M}_{2}^{D}\text{
such that
}|||\mathcal{L}_{\eta_{t}}|||,|||\mathcal{L}_{\eta^{\prime}_{t}}|||\leq
K.\end{split}$

###### Proof.

According to the preceding remark and [41, Lemma 3.8], we have the first two
assertions.

In view of the standard estimate for linear BSDEs, Lipschitz continuity of
$(\Phi,f)$ and Lemma 2.4, we have

$\displaystyle\|\partial_{\omega_{t}}Y^{{\gamma_{t},\eta_{t}}}\|_{\mathbb{S}^{p}}+\|\partial_{\omega_{t}}Z^{{\gamma_{t},\eta_{t}}}\|_{\mathbb{H}^{p}}$
$\displaystyle\ \ \ \leq
C(\|\partial_{\omega_{t}}\Phi(B^{\gamma_{t}},\mathcal{L}_{B^{\eta_{t}}})\|_{L^{p}}+\|\int_{t}^{T}|\partial_{\omega_{t}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})|dr\|_{L^{p}}),$

and thus the first inequality of (92). The last inequality is proved in a
similar way to Lemma 3.4. ∎

Furthermore, we have

######  Proposition 3.8.

Let $(f,\Phi)$ satisfy Assumption (H1). Then for any $\tau\leq t,$
$(Y^{{\gamma_{t},\eta_{t}}}(s),Z^{{\gamma_{t},\eta_{t}}}(s))$ is strongly
vertically differentiable at $(\tau,t,\gamma)$. Moreover, the derivative
$(\partial_{\omega_{\tau}}Y^{{\gamma_{t},\eta_{t}}},\partial_{\omega_{\tau}}Z^{{\gamma_{t},\eta_{t}}})\in\mathbb{S}^{p}([t,T],\mathbb{R}^{d})\times\mathbb{H}^{p}([t,T],\mathbb{R}^{d\times
d}),\ \forall\ p\geq 1$, is the unique solution to BSDE

(93) $\begin{split}\mathcal{Y}(s)=\
&\partial_{\omega_{\tau}}\Phi(B^{\gamma_{t}},\mathcal{L}_{B^{\eta_{t}}})+\int_{s}^{T}\partial_{\omega_{\tau}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})dr+\int_{s}^{T}\partial_{y}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\mathcal{Y}(r)dr\\\
&+\int_{s}^{T}\partial_{z}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\mathcal{Z}(r)dr-\int_{s}^{T}\mathcal{Z}(r)dB(r),\
\ s\in[t,T].\end{split}$

Furthermore, since
$(\partial_{\omega_{\tau}}Y^{{\gamma_{t},\eta_{t}}},\partial_{\omega_{\tau}}Z^{{\gamma_{t},\eta_{t}}})$
is independent of $\mathcal{F}_{t}$, we have that for any $K>0,$

(94)
$\begin{split}&\|(\partial_{\omega_{\tau}}Y^{{\gamma_{t},\eta_{t}}},\partial_{\omega_{\tau}}Z^{{\gamma_{t},\eta_{t}}})\|_{\mathbb{S}^{p}\times\mathbb{H}^{p}}<C_{p},\\\
&\|(\partial_{\omega_{\tau}}Y^{{\gamma_{t},\eta_{t}}}-\partial_{\omega_{\tau}}Y^{\gamma^{\prime}_{t},\eta^{\prime}_{t}},\partial_{\omega_{\tau}}Z^{{\gamma_{t},\eta_{t}}}-\partial_{\omega_{\tau}}Z^{\gamma^{\prime}_{t},\eta^{\prime}_{t}})\|_{\mathbb{S}^{p}\times\mathbb{H}^{p}}<C_{K,p}(\|\gamma_{t}-\gamma^{\prime}_{t}\|+W_{2}(\mathcal{L}_{\eta_{t}},\mathcal{L}_{\eta^{\prime}_{t}})),\\\
&\ \ \ \ \ \ \forall\
(\gamma,\eta),(\gamma^{\prime},\eta^{\prime})\in\mathbb{D}_{T,d}\times\mathbb{M}_{2}^{D}\text{
such that
}|||\mathcal{L}_{\eta_{t}}|||,|||\mathcal{L}_{\eta^{\prime}_{t}}|||\leq
K,\end{split}$

for some positive constants $C_{p}$ and $C_{K,p}$.

###### Proof.

In view of Assumption (H1) and Lemma 3.1, we see that equation (93) has a
unique solution
$(\partial_{\omega_{\tau}}Y,\partial_{\omega_{\tau}}Z)\in\mathbb{S}^{p}\times\mathbb{H}^{p},\
\forall p\geq 1.$ Here, we consider the one-dimensional case for simplicity.
For any $h>0,$ recall that $\gamma^{\tau,h}=\gamma+h1_{[\tau,T]}$. Set

(95)
$\begin{split}&\gamma^{\prime}:=\gamma^{\tau,h},\quad\Delta_{h}Y:=\frac{1}{h}(Y^{\prime}-Y):=\frac{1}{h}(Y^{\gamma^{\tau,h}_{t},\eta_{t}}-Y^{{\gamma_{t},\eta_{t}}}),\quad\quad\text{and}\\\
&\quad\Delta_{h}Z:=\frac{1}{h}(Z^{\prime}-Z):=\frac{1}{h}(Z^{\gamma^{\tau,h}_{t},\eta_{t}}-Z^{{\gamma_{t},\eta_{t}}}).\end{split}$

Then we know that $(\Delta_{h}Y,\Delta_{h}Z)$ solves the following BSDE

$\begin{split}\Delta_{h}Y(s)&=\frac{1}{h}(\Phi^{\prime}-\Phi)+\frac{1}{h}\int_{s}^{T}[f(\Theta^{\gamma^{\prime},\eta}_{r},\mathcal{L}_{\Theta^{\eta}_{r}})-f(\Theta^{\gamma,\eta}_{r},\mathcal{L}_{\Theta^{\eta}_{r}})]dr-\int_{s}^{T}\Delta_{h}Z(r)dB(r)\\\
&=:\Delta_{h}\Phi+\int_{s}^{T}\Big{(}a_{r}\Delta_{h}Y(r)+b_{r}\Delta_{h}Z(r)+\Delta_{h}f\Big{)}dr-\int_{s}^{T}\Delta_{h}Z(r)dB(r),\end{split}$

where

$\displaystyle\Phi^{\prime}:=\Phi(B^{\gamma^{\prime}},\mathcal{L}_{B^{\eta}}),\
\ \Phi:=\Phi(B^{\gamma},\mathcal{L}_{B^{\eta}}),\ \
\Delta_{h}\Phi:=\int_{0}^{1}\partial_{\gamma_{\tau}}\Phi(B^{\gamma^{\tau,h\theta}},\mathcal{L}_{B^{\eta}})\
d\theta,$ $\displaystyle
a_{r}:=\int_{0}^{1}\partial_{y}{f}(B^{\gamma^{\prime}}_{r},Y+\theta(Y^{\prime}-Y),Z^{\prime},\mathcal{L}_{\Theta^{\eta}_{r}})\
d\theta,\ \
b_{r}:=\int_{0}^{1}\partial_{z}{f}(B^{\gamma^{\prime}}_{r},Y,Z+\theta(Z^{\prime}-Z),\mathcal{L}_{\Theta^{\eta}_{r}})\
d\theta,$
$\displaystyle\quad\text{and}\quad\Delta_{h}f:=\frac{1}{h}f(B^{\omega}_{r},Y,Z,\mathcal{L}_{B^{\eta}_{r}},\mathcal{L}_{Y})\Big{|}_{\omega=\gamma}^{\omega=\gamma^{\prime}}=\int_{0}^{1}\partial_{\omega_{\tau}}f(B^{\gamma^{\tau,h\theta}},Y,Z,\mathcal{L}_{B^{\eta}_{r}},\mathcal{L}_{Y})\
d\theta.$

Then $(\delta Y,\delta
Z):=(\Delta_{h}Y-\partial_{\omega_{\tau}}Y,\Delta_{h}Z-\partial_{\omega_{\tau}}Z)$
satisfies BSDE

$\begin{split}\delta Y(s)=&\
(\Delta_{h}\Phi-\partial_{\omega_{\tau}}\Phi)+\int_{s}^{T}\left(a_{r}\delta
Y+b_{r}\delta
Z+(\Delta_{h}f-\partial_{\omega_{\tau}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}}))\right)dr\\\
\ \
&+\int_{s}^{T}[(a_{r}-\partial_{y}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}}))\partial_{\omega_{\tau}}Y+(b_{r}-\partial_{z}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}}))\partial_{\omega_{\tau}}Z]dr-\int_{s}^{T}\delta
ZdB(r).\end{split}$

According to standard estimate for BSDEs (or Lemma 3.1 for $p=2$) and Lemma
3.4, we have

$\begin{split}\|\delta Y\|_{\mathbb{S}^{p}}^{p}+\|\delta
Z\|_{\mathbb{H}^{p}}^{p}&\leq
C\|\Delta_{h}\Phi-\partial_{\omega_{\tau}}\Phi\|_{L^{p}}^{p}+\|\int_{t}^{T}|\Delta_{h}f-\partial_{\omega_{\tau}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})|dr\|_{L^{p}}^{p}+O(|h|)\\\
&\leq O(|h|),\end{split}$

and thus the strongly vertical differentiability.

∎

To show the differentiability of $Y^{\gamma_{t},\eta_{t}}$ with respect to
$\eta_{t},$ we follow a similar argument as in the state-dependent case for
SDEs made in [9]. Firstly we show that $Y^{\gamma_{t},\eta_{t}}$ is Gâteaux
differentiable in $\eta_{t}$ in the sense of (35) and Remark 2.8. To this end,
we need to prove that for any $\xi\in L^{2}(\mathcal{F}_{t},\mathbb{R}^{d})$
and $\eta_{t}^{\lambda\xi}:=\eta_{t}+\lambda\xi 1_{[t,T]}$, $\lambda>0,$ the
following limit exits in $\mathbb{S}^{2}([t,T],\mathbb{R}^{d}),$

(96) $\partial_{\eta}Y^{\gamma_{t},\eta_{t}}(\xi):=\lim_{\lambda\rightarrow
0}\frac{1}{\lambda}(Y^{\gamma_{t},\eta_{t}^{\lambda\xi}}-Y^{\gamma_{t},\eta_{t}}).$

Then we show that
$\partial_{\eta}Y^{\gamma_{t},\eta_{t}}(\cdot):L^{2}(\mathcal{F}_{t},\mathbb{R}^{d})\to\mathbb{S}^{2}([t,T],\mathbb{R}^{d})$
is a bounded linear operator, and moreover, it is continuous in the following
sense: for any $\zeta\in L^{2}(\mathcal{F}_{t},\mathbb{R}^{d}),$
$\partial_{\eta}Y^{\gamma_{t},\eta_{t}+\zeta 1_{[t,T]}}$ converges to
$\partial_{\eta}Y^{\gamma_{t},\eta_{t}}$ in the sense of operators as $\zeta$
goes to zero. In view of Remark 2.5, we see that $Y^{\gamma_{t},\eta_{t}}$ is
Fréchet (vertically) differentiable in the sense of (34) and Remark 2.8.

To this end, consider the following linear BSDE

(97)
$\begin{split}\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}(s)&=\tilde{\mathbb{E}}[\partial_{\mu_{t}}\Phi(B^{\gamma_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}]+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\mu_{t}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{y}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}(r)dr+\int_{s}^{T}\partial_{z}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\mathcal{Z}^{{\gamma_{t},\eta_{t}},\xi}(r)dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})(\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}\tilde{\xi}+\tilde{\mathcal{Y}}^{\tilde{\eta}_{t},\tilde{\xi}})(r)]dr\\\
&\ \ \ -\int_{s}^{T}\mathcal{Z}^{{\gamma_{t},\eta_{t}},\xi}(r)dB(r),\quad
s\in[t,T].\end{split}$

Here,
$(\tilde{B},\tilde{\eta},\tilde{\xi},\tilde{Y}^{\tilde{\eta}},\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}},\tilde{\mathcal{Y}}^{\tilde{\eta}_{t},\tilde{\xi}})$
is an independent copy of
$(B,\eta,\xi,Y^{\eta_{t}},\partial_{\omega_{t}}Y^{\gamma_{t},\mathcal{L}_{\eta_{t}}}|_{\gamma=\eta},\mathcal{Y}^{\eta_{t},\xi})$,
and $\mathcal{Y}^{\eta_{t},\xi}$ satisfies the following linear mean-field
BSDE

(98)
$\begin{split}\mathcal{Y}^{\eta_{t},\xi}(s)&=\tilde{\mathbb{E}}[\partial_{\mu_{t}}\Phi(B^{\eta_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}]+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\mu_{t}}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}}_{r})\tilde{\xi}]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{y}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\mathcal{Y}^{{\eta_{t}},\xi}(r)dr+\int_{s}^{T}\partial_{z}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\mathcal{Z}^{{\eta_{t}},\xi}(r)dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})(\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}\tilde{\xi}+\tilde{\mathcal{Y}}^{\tilde{\eta}_{t},\tilde{\xi}})(r)]dr\\\
&\ \ \ -\int_{s}^{T}\mathcal{Z}^{{\eta_{t}},\xi}(r)dB(r),\quad
s\in[t,T].\end{split}$

######  Lemma 3.9.

For any $\xi\in L^{2}(\mathcal{F}_{t},\mathbb{R}^{d}),$ there exits a unique
solution
$(\mathcal{Y}^{\eta_{t},\xi},\mathcal{Z}^{\eta_{t},\xi})\in\mathbb{S}^{2}([t,T])\times\mathbb{H}^{2}([t,T],\mathbb{R}^{d})$
for BSDE (98). Moreover,
$(\mathcal{Y}^{\eta_{t},\xi},\mathcal{Z}^{\eta_{t},\xi})$ is linear in $\xi$,
and we have

(99)
$\|(\mathcal{Y}^{\eta_{t},\xi},\mathcal{Z}^{\eta_{t},\xi})\|_{\mathbb{S}^{2}\times\mathbb{H}^{2}}\leq
C\|\xi\|_{L^{2}}$

for some constant $C$.

###### Proof.

By Lipschitz continuity of $(\partial_{\mu_{t}}\Phi,\partial_{\mu_{t}}f),$ we
have

$\tilde{\mathbb{E}}[\partial_{\mu_{t}}\Phi(B_{T}^{\eta_{t}},\mathcal{L}_{B_{T}^{\eta_{t}}},\tilde{B}_{T}^{\tilde{\eta}_{t}})\tilde{\xi}]\in
L^{2}(\mathcal{F}_{T}),\ \ \
\tilde{\mathbb{E}}[\partial_{\mu_{t}}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}}_{r})\tilde{\xi}]\in
L^{2}(\mathcal{F}_{r}).$

Since $f$ is uniformly Lipschitz continuous in $(y,z),$
$\partial_{(y,z)}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})$
is uniformly bounded. Set
$g(r,x):=\partial_{\nu}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},x)$.
In view of Lemma 3.4 and Assumption (H1), we see that
$g(\cdot,0)\in\mathbb{H}^{2}.$ Then by Lemma 3.1, to show the well-posedness
of linear mean-field BSDE (98), we only need to check the following

$\int_{t}^{T}\left|\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})(\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}\tilde{\xi})]\right|dr\in
L^{2}(\mathcal{F}_{T}).$

Let

(100)
$F_{2}(t,x,y,z,\mu,\nu):=\tilde{\mathbb{E}}[\partial_{\nu}f(t,x,y,z,\mu,\nu,\tilde{Y}^{\tilde{\eta}_{t}}(r))(\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}(r)\tilde{\xi})].$

Then by Lipschitz continuity of $\partial_{\nu}f$ and Lemma 3.7, we have

$\displaystyle F_{2}(t,x,y,z,\mu,\nu)$ $\displaystyle\ \
=\tilde{\mathbb{E}}\Big{[}\tilde{\mathbb{E}}_{\tilde{\mathcal{F}}_{t}}[\partial_{\nu}f(t,x,y,z,\mu,\nu,\tilde{Y}^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(r))(\partial_{\omega_{t}}\tilde{Y}^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(r))]\Big{|}_{\gamma_{t}=\tilde{\eta}_{t}}\tilde{\xi}\Big{]}$
$\displaystyle\ \ \leq
C\tilde{\mathbb{E}}\Big{[}\tilde{\mathbb{E}}_{\tilde{\mathcal{F}}_{t}}[|\tilde{Y}^{\gamma_{t},\mathcal{L}_{\eta_{t}}}||\partial_{\omega_{t}}\tilde{Y}^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(r)|]\left|{}_{\gamma_{t}=\tilde{\eta}_{t}}\right.\tilde{\xi}\Big{]}+\partial_{\nu}f(t,x,y,z,\mu,\nu,0)\tilde{\mathbb{E}}\Big{[}\tilde{\mathbb{E}}_{\tilde{\mathcal{F}}_{t}}[\tilde{Y}^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(r)|]\left|{}_{\gamma_{t}=\tilde{\eta}_{t}}\right.\tilde{\xi}\Big{]}$
$\displaystyle\ \ \leq
C\tilde{\mathbb{E}}\left[(\tilde{\mathbb{E}}_{\tilde{\mathcal{F}}_{t}}[|\tilde{Y}^{\gamma_{t},\mathcal{L}_{\eta_{t}}}|^{2}]^{\frac{1}{2}})(\tilde{\mathbb{E}}_{\tilde{\mathcal{F}}_{t}}[|\partial_{\omega_{t}}\tilde{Y}^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(r)|^{2}]^{\frac{1}{2}})\Big{|}_{\gamma_{t}=\tilde{\eta}_{t}}\tilde{\xi}\right]+C\partial_{\nu}f(t,x,y,z,\mu,\nu,0)$
$\displaystyle\ \ \leq
C\tilde{\mathbb{E}}\left[\tilde{\mathbb{E}}_{\tilde{\mathcal{F}}_{t}}[(1+\|\gamma_{t}\|)]\Big{|}_{\gamma_{t}=\tilde{\eta}_{t}}\tilde{\xi}\right]+C\partial_{\nu}f(t,x,y,z,\mu,\nu,0)$
$\displaystyle\ \ \leq C+C\partial_{\nu}f(t,x,y,z,\mu,\nu,0),$

where we applied Lemmas 3.4 and 3.7 in the second last inequality. Then
according to Lemma 3.4, we have
$\partial_{\nu}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta^{\eta_{t}}_{r}},0)\in\mathbb{H}^{2},$
and thus the well-posedness of (98). For inequality (99), similar to the proof
of Lemma 3.4, we have

$\displaystyle\|\mathcal{Y}^{\eta_{t},\xi}\|_{\mathbb{S}^{2}}^{2}+\|\mathcal{Z}^{\eta_{t},\xi}\|^{2}_{\mathbb{H}^{2}}$
$\displaystyle\ \ \leq
C\mathbb{E}\Big{[}|\tilde{\mathbb{E}}[\partial_{\mu_{t}}\Phi(B^{\eta_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}]|^{2}+\int_{s}^{T}|\tilde{\mathbb{E}}[\partial_{\mu_{t}}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}]|^{2}dr$
$\displaystyle\ \ \ \ \ \
+\int_{s}^{T}|\tilde{\mathbb{E}}\partial_{\nu}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})(\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}\tilde{\xi})|^{2}dr\Big{]}$
$\displaystyle\ \ \leq
C\Big{(}(\tilde{\mathbb{E}}\|\tilde{B}^{\tilde{\eta}_{t}}\|\
|\tilde{\xi}|)^{2}+\|\xi\|_{L^{2}}^{2}\mathbb{E}|\partial_{\mu_{t}}\Phi(B^{\eta_{t}},\mathcal{L}_{B^{\eta_{t}}},0)|^{2}$
$\displaystyle\ \ \ \ \ \
+\|\xi\|_{L^{2}}^{2}\mathbb{E}\int_{s}^{T}|\partial_{\mu_{t}}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},0)|^{2}dr+\mathbb{E}\int_{s}^{T}|\tilde{\mathbb{E}}[\tilde{Y}^{\tilde{\eta}_{t}}\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}\tilde{\xi}]|^{2}dr$
$\displaystyle\ \ \ \ \ \
+\|\xi\|_{L^{2}}^{2}\mathbb{E}\int_{s}^{T}|\partial_{\nu}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},0)|^{2}dr\Big{)}\leq
C\|\xi\|_{L^{2}}^{2}.$

∎

Since BSDE (98) is well-posed, we see that BSDE (97) is also well-posed. In
conclusion, we have

######  Corollary 3.10.

There exits a unique solution
$(\mathcal{Y}^{\gamma_{t},\eta_{t},\xi},\mathcal{Z}^{\gamma_{t},\eta_{t},\xi})\in\mathbb{S}^{2}([t,T])\times\mathbb{H}^{2}([t,T],\mathbb{R}^{d})$
to BSDE (97). Moreover,

(101)
$(\mathcal{Y}^{\eta_{t},\xi},\mathcal{Z}^{\eta_{t},\xi})=(\mathcal{Y}^{\gamma_{t},\eta_{t},\xi},\mathcal{Z}^{\gamma_{t},\eta_{t},\xi})|_{\gamma=\eta}.$

######  Lemma 3.11.

The map $\xi\to\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}$ is a bounded linear
operator from $L^{2}({\mathcal{F}_{t}},\mathbb{R}^{d})$ to
$\mathbb{S}^{2}([t,T])$. Moreover, it is the Gâteaux derivative of
$Y^{\gamma_{t},\eta_{t}}$ with respect to $\eta_{t}$ in the following sense

(102) $\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}=\lim_{\lambda\rightarrow
0}\frac{1}{\lambda}(Y^{\gamma_{t},\eta_{t}^{\lambda\xi}}-Y^{\gamma_{t},\eta_{t}})\quad\text{strongly
in $\mathbb{S}^{2}([t,T])$}.$

In particular, $\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}(s)$ is the Gâteaux
derivative of $Y^{\gamma_{t},\eta_{t}}(s)$ in the sense of (35).

###### Proof.

Since $\mathcal{Y}^{\eta_{t},\xi}$ is linear in $\xi,$ we see that
$(\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi},\
\mathcal{Z}^{{\gamma_{t},\eta_{t}},\xi})$ is also linear in $\xi$. Moreover,
we have the following estimate

(103)
$\|(\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi},\mathcal{Z}^{{\gamma_{t},\eta_{t}},\xi})\|_{\mathbb{S}^{2}\times\mathbb{H}^{2}}\leq
C\|\xi\|_{L^{2}}.$

Therefore, we have the first assertion.

In the following, we omit the fixed subscript $t$ and write
$(Y,Z):=(Y(r),Z(r))$ if no confusion raised. Besides, the constant $C$ may
change from line to line. Set

(104)
$\begin{split}&\Delta_{\lambda}Y:=\frac{1}{\lambda}(Y^{\gamma,\eta^{\lambda\xi}}-Y^{\gamma,\eta}),\
\
\Delta_{\lambda}Z:=\frac{1}{\lambda}(Z^{\gamma,\eta^{\lambda\xi}}-Z^{\gamma,\eta}),\quad\quad\text{and
}\\\
&\quad\Delta_{\lambda}\Phi:=\frac{1}{\lambda}[\Phi(B_{T}^{\gamma},\mathcal{L}_{B_{T}^{\eta^{\lambda\xi}}})-\Phi(B_{T}^{\gamma},\mathcal{L}_{B_{T}^{\eta}})].\end{split}$

Then according to Lemma 3.4, we have

(105)
$\|\Delta_{\lambda}Y\|_{\mathbb{S}^{2}}+\|\Delta_{\lambda}Z\|_{\mathbb{H}^{2}}\leq
C\frac{1}{\lambda}\|\eta_{t}^{\lambda\xi}-\eta_{t}\|_{\mathbb{S}^{2}}\leq
C\|\xi\|_{L^{2}}.$

In view of BSDE (79), we see that $(\Delta_{\lambda}Y,\Delta_{\lambda}Z)$
satisfies the following linear mean-field BSDE

(106)
$\begin{split}\Delta_{\lambda}Y&=\Delta_{\lambda}\Phi+\int_{s}^{T}\left[\alpha(r)\Delta_{\lambda}Y+\beta(r)\Delta_{\lambda}Z+\tilde{\mathbb{E}}[\tilde{g}(r)\frac{1}{\lambda}(\tilde{Y}^{\tilde{\eta}^{\lambda\tilde{\xi}}}-\tilde{Y}^{\tilde{\eta}})]+\Delta_{\lambda}f\right]dr\\\
&\quad-\int_{s}^{T}\Delta_{\lambda}ZdB(r),\end{split}$

where

$\displaystyle\alpha(r):=\int_{0}^{1}\partial_{y}{f}(B^{\gamma}_{r},Y^{\gamma,\eta}+\theta(Y^{\gamma,\eta^{\lambda\xi}}-Y^{\gamma,\eta}),Z^{\gamma,\eta^{\lambda\xi}},\mathcal{L}_{\Theta^{\eta^{\lambda\xi}}_{r}})d\theta,$
$\displaystyle\beta(r):=\int_{0}^{1}\partial_{z}{f}(B^{\gamma}_{r},Y^{\gamma,\eta},Z^{\gamma,\eta}+\theta(Z^{\gamma,\eta^{\lambda\xi}}-Z^{\gamma,\eta}),\mathcal{L}_{\Theta^{\eta^{\lambda\xi}}_{r}})d\theta,$
$\displaystyle\tilde{g}(r):=\int_{0}^{1}\partial_{\nu}f(\Theta^{\gamma,\eta},\mathcal{L}_{B^{\eta^{\lambda\xi}}},\mathcal{L}_{Y^{\eta}+\theta(Y^{\eta^{\lambda\xi}}-Y^{\eta})},\tilde{Y}^{\tilde{\eta}}+\theta(\tilde{Y}^{\tilde{\eta}^{\lambda\tilde{\xi}}}-\tilde{Y}^{\tilde{\eta}}))d\theta,\quad\quad\text{and}$
$\displaystyle\Delta_{\lambda}f(r):=\frac{1}{\lambda}[f(\Theta^{\gamma,\eta},\mathcal{L}_{B^{\eta^{\lambda\xi}}_{r}},\mathcal{L}_{Y^{\eta}})-f(\Theta^{\gamma,\eta},\mathcal{L}_{B^{\eta}_{r}},\mathcal{L}_{Y^{\eta}})].$

According to estimate (83) in Lemma 3.4, we have

(107)
$\|\Delta_{\lambda}\tilde{Y}^{\tilde{\eta}}\|_{\mathbb{S}^{2}}:=\|\frac{1}{\lambda}(\tilde{Y}^{\tilde{\eta}^{\lambda\tilde{\xi}}}-\tilde{Y}^{\tilde{\eta}})\|_{\mathbb{S}^{2}}\leq
C\frac{1}{\lambda}\|\eta_{t}^{\lambda\xi}-\eta_{t}\|_{\mathbb{S}^{2}}\leq
C\|\xi\|_{L^{2}}.$

Then, in view of Assumption (H1), we have

(108)
$\|\int_{t}^{T}\tilde{\mathbb{E}}[\tilde{g}(r)\Delta_{\lambda}\tilde{Y}^{\tilde{\eta}}]dr\|_{L^{2}}+\|\int_{t}^{T}\Delta_{\lambda}fdr\|_{L^{2}}\leq
C\|\xi\|_{L^{2}}.$

Thus BSDE (106) has a unique solution $(\Delta_{\lambda}Y,\Delta_{\lambda}Z)$,
and therefore,
$(\Delta_{\lambda}Y-\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi},\Delta_{\lambda}Z-\mathcal{Z}^{{\gamma_{t},\eta_{t}},\xi})$
is the unique solution of the following BSDE

$\displaystyle Y(s)=$ $\displaystyle\
(\Delta_{\lambda}\Phi-\tilde{\mathbb{E}}[\partial_{\mu_{t}}\Phi(B^{\gamma_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}])+\int_{s}^{T}\partial_{y}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})Ydr$
$\displaystyle\
+\int_{s}^{T}\partial_{z}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})Z(r)dr+\int_{s}^{T}(\Delta_{\lambda}f-\tilde{\mathbb{E}}[\partial_{\mu_{t}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}])dr$
$\displaystyle\
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})(\Delta_{\lambda}\tilde{Y}^{\tilde{\eta}}-\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}\tilde{\xi}-\tilde{\mathcal{Y}}^{\tilde{\eta}_{t},\tilde{\xi}})]dr$
$\displaystyle\ +\int_{t}^{T}R_{1}(r)dr-\int_{s}^{T}ZdB(r)$

with

$\displaystyle R_{1}(r):=$
$\displaystyle\left(\alpha(r)-\partial_{y}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\right)\Delta_{\lambda}Y+\left(\beta(r)-\partial_{z}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\right)\Delta_{\lambda}Z$
$\displaystyle+\tilde{\mathbb{E}}\left[(\tilde{g}(r)-\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}}))\Delta_{\lambda}\tilde{Y}^{\tilde{\eta}}\right].$

Since $\partial_{(y,z)}f$ is bounded, from the standard estimate for solutions
of BSDEs, we have

(109)
$\|\Delta_{\lambda}Y-\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}\|^{2}_{\mathbb{S}^{2}}\leq
C(\|A_{1}\|^{2}_{L^{2}}+\|A_{2}\|^{2}_{L^{2}}+\|A_{3}\|^{2}_{L^{2}}+\|A_{4}\|^{2}_{L^{2}})$

with

$\displaystyle A_{1}$
$\displaystyle:=\Delta_{\lambda}\Phi-\tilde{\mathbb{E}}[\partial_{\mu_{t}}\Phi(B^{\gamma_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}],\quad
A_{2}:=\int_{t}^{T}|R_{1}(r)|dr,$ $\displaystyle A_{3}$
$\displaystyle:=\int_{t}^{T}\left|(\Delta_{\lambda}f-\tilde{\mathbb{E}}[\partial_{\mu_{t}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}])\right|dr,\quad\text{and}$
$\displaystyle A_{4}$
$\displaystyle:=\int_{t}^{T}\left|\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})(\Delta_{\lambda}\tilde{Y}^{\tilde{\eta}}-\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}\tilde{\xi}-\tilde{\mathcal{Y}}^{\tilde{\eta}_{t},\tilde{\xi}})(r)]\right|dr.$

For $A_{1},$ according to the Lipschitz continuity of
$\partial_{\mu_{t}}\Phi,$ we have

(110) $\begin{split}\mathbb{E}|A_{1}|^{2}=&\
\mathbb{E}\Big{|}\int_{0}^{1}\tilde{\mathbb{E}}[\partial_{\mu_{t}}\Phi(B^{\gamma},\mathcal{L}_{B^{\eta}+\theta(B^{\eta^{\lambda\xi}}-B^{\eta})},\tilde{B}^{\tilde{\eta}}+\theta(\tilde{B}^{\tilde{\eta}^{\lambda\tilde{\xi}}}-\tilde{B}^{\tilde{\eta}}))\tilde{\xi}\\\
&\
-\partial_{\mu_{t}}\Phi(B^{\gamma_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}]d\theta\Big{|}^{2}\\\
\leq&\
C\left([\bar{E}\|\bar{B}^{\bar{\eta}^{\lambda\bar{\xi}}}-\bar{B}^{\bar{\eta}}\|^{2}]^{\frac{1}{2}}\|\xi\|_{L^{2}}+\tilde{\mathbb{E}}[\|\tilde{B}^{\tilde{\eta}^{\lambda\tilde{\xi}}}-\tilde{B}^{\tilde{\eta}}\||\tilde{\xi}|]\right)^{2}\leq
C\lambda^{2}\|\xi\|_{L^{2}}^{4},\end{split}$

for a constant $C$ independent of $\gamma$ and $\eta.$

Term $A_{2}$ is estimated as follows:

(111) $\begin{split}|A_{2}|^{2}\leq&\
C\Big{|}\int_{t}^{T}(\alpha(r)-\partial_{y}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}}))\Delta_{\lambda}Ydr\Big{|}^{2}\\\
&+C\Big{|}\int_{t}^{T}\left(\beta(r)-\partial_{z}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\right)\Delta_{\lambda}Zdr\Big{|}^{2}\\\
&+C\Big{|}\int_{t}^{T}\tilde{\mathbb{E}}[(\tilde{g}(r)-\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}}))\Delta_{\lambda}\tilde{Y}^{\tilde{\eta}}]dr\Big{|}^{2}.\end{split}$

For the first two terms on the right hand side of the above inequality, by the
Lipschitz continuity of $\partial_{(y,z)}f$ and inequality (105), we obtain

$\displaystyle\mathbb{E}\Big{|}\int_{t}^{T}\left(\alpha(r)-\partial_{y}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\right)\Delta_{\lambda}Ydr\Big{|}^{2}+\Big{|}\int_{t}^{T}\left(\beta(r)-\partial_{z}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\right)\Delta_{\lambda}Zdr\Big{|}^{2}\leq
C\lambda^{2}\|\xi\|_{L^{2}}^{4}.$

For the third term, we claim that

(112)
$\mathbb{E}\Big{|}\int_{t}^{T}\tilde{\mathbb{E}}\left[(\tilde{g}(r)-\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}}))\Delta_{\lambda}\tilde{Y}^{\tilde{\eta}}\right]dr\Big{|}^{2}\leq
C\lambda^{2}\|\xi\|_{L^{2}}^{4},$

with $C$ depending only on $\|\eta_{t}\|_{\mathbb{S}^{2}},$ and therefore we
have

(113) $\mathbb{E}|A_{2}|^{2}\leq C\lambda^{2}\|\xi\|_{L^{2}}^{4},$

in view of (111) and above estimates. Indeed, by the Hölder inequality and
estimate (107), we have

$\displaystyle\mathbb{E}\Big{|}\int_{t}^{T}\tilde{\mathbb{E}}[(\tilde{g}(r)-\partial_{\nu}f)\Delta_{\lambda}\tilde{Y}^{\tilde{\eta}}]dr\Big{|}^{2}\leq\mathbb{E}[\int_{t}^{T}\tilde{\mathbb{E}}|\tilde{g}-\partial_{\nu}f|^{2}dr]\int_{t}^{T}\tilde{\mathbb{E}}|\Delta_{\lambda}\tilde{Y}^{\tilde{\eta}}|^{2}dr$
$\displaystyle\quad\leq
C\|\xi\|_{L^{2}}^{2}\mathbb{E}\int_{t}^{T}\tilde{\mathbb{E}}\Big{|}\int_{0}^{1}(\partial_{\nu}f(\Theta^{\gamma,\eta},\mathcal{L}_{B^{\eta^{\lambda\xi}}},\mathcal{L}_{Y^{\eta}+\theta(Y^{\eta^{\lambda\xi}}-Y^{\eta})},\tilde{Y}^{\tilde{\eta}}+\theta(\tilde{Y}^{\tilde{\eta}^{\lambda\tilde{\xi}}}-\tilde{Y}^{\tilde{\eta}}))$
$\displaystyle\quad\quad-\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}}))d\theta\Big{|}^{2}dr$
$\displaystyle\quad\leq
C\|\xi\|_{L^{2}}^{2}(\|\tilde{Y}^{\tilde{\eta}^{\lambda\tilde{\xi}}}-\tilde{Y}^{\tilde{\eta}}\|_{\mathbb{S}^{2}}+\|B^{\eta^{\lambda\xi}}-B^{\eta}\|_{\mathbb{S}^{2}})^{2}\leq
C\lambda^{2}\|\xi\|_{L^{2}}^{4}.$

For $A_{3},$ from Lipschitz continuity of $\partial_{\mu_{t}}f$ in
$(\mu,\nu,\tilde{\omega})$, we have

(114) $\begin{split}\mathbb{E}|A_{3}|^{2}=&\
\mathbb{E}\big{|}\int_{t}^{T}\int_{0}^{1}\tilde{\mathbb{E}}[\partial_{\mu_{t}}f(\Theta^{{\gamma_{t},\eta_{t}}},\mathcal{L}_{B^{\eta+\theta(\eta^{\lambda\xi}-\eta)}},\mathcal{L}_{Y^{\eta}},\tilde{B}^{\tilde{\eta}+\theta(\tilde{\eta}^{\lambda\tilde{\xi}}-\tilde{\eta})})\\\
&\
-\partial_{\mu_{t}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})]\tilde{\xi}d\theta
dr\big{|}^{2}\\\ \leq&\
C\left|\tilde{\mathbb{E}}[(\|\eta^{\lambda\xi}_{t}-\eta_{t}\|_{\mathbb{S}^{2}}+\|\tilde{\eta}^{\lambda\tilde{\xi}}-\tilde{\eta}\|)|\tilde{\xi}|]\right|^{2}\\\
\leq&\
C\left|\tilde{\mathbb{E}}[\lambda\|\xi\|_{L^{2}}\tilde{\xi}+\lambda|\tilde{\xi}|^{2}]\right|^{2}\leq
C\lambda^{2}\|\xi\|_{L^{2}}^{4}.\end{split}$

We now estimate $A_{4}$. Since

(115)
$\Delta_{\lambda}\tilde{Y}^{\tilde{\eta}}-\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}\tilde{\xi}-\tilde{\mathcal{Y}}^{\tilde{\eta}_{t},\tilde{\xi}}=\
A_{41}+A_{42}$

with

(116) $\begin{split}A_{41}&:=\
[\frac{1}{\lambda}(\tilde{Y}^{\tilde{\eta}^{\lambda\tilde{\xi}},\mathcal{L}_{\tilde{\eta}^{\lambda\tilde{\xi}}}}-\tilde{Y}^{\tilde{\eta},\mathcal{L}_{\tilde{\eta}^{\lambda\tilde{\xi}}}})-\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}\tilde{\xi}],\quad\quad\text{and}\\\
A_{42}&:=[\frac{1}{\lambda}(\tilde{Y}^{\tilde{\eta},\mathcal{L}_{\tilde{\eta}^{\lambda\tilde{\xi}}}}-\tilde{Y}^{\tilde{\eta},\mathcal{L}_{\tilde{\eta}}})-\tilde{\mathcal{Y}}^{\tilde{\eta}_{t},\tilde{\xi}}],\end{split}$

then, from boundedness of $\partial_{\nu}f,$ we have

(117)
$\begin{split}\mathbb{E}|A_{4}|^{2}&=\mathbb{E}\Big{|}\int_{t}^{T}\tilde{\mathbb{E}}\Big{[}\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})(A_{41}(r)+A_{42}(r))\Big{]}dr\Big{|}^{2}\\\
&\leq
C\Big{(}\Big{|}\int_{t}^{T}\tilde{\mathbb{E}}[A_{41}]dr\Big{|}^{2}+\Big{|}\int_{t}^{T}\tilde{\mathbb{E}}[A_{42}]dr\Big{|}^{2}\Big{)}.\end{split}$

From Lemma 3.7, we have

(118)
$\begin{split}\left|\int_{t}^{T}\tilde{\mathbb{E}}[A_{41}]dr\right|^{2}&\leq
C\int_{t}^{T}\left[\tilde{\mathbb{E}}\int_{0}^{1}|(\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}^{\lambda\theta\tilde{\xi}},\mathcal{L}_{\eta^{\lambda\xi}}}-\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta},\mathcal{L}_{\eta}})\tilde{\xi}|d\theta\right]^{2}dr\\\
&\leq
C\|\xi\|_{L^{2}}^{2}\int_{t}^{T}\int_{0}^{1}\tilde{\mathbb{E}}|\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta}^{\lambda\theta\tilde{\xi}},\mathcal{L}_{\eta^{\lambda\xi}}}-\partial_{\omega_{t}}\tilde{Y}^{\tilde{\eta},\mathcal{L}_{\eta}}|^{2}d\theta
dr\\\ &\leq
C\|\xi\|_{L^{2}}^{2}\int_{t}^{T}\int_{0}^{1}\tilde{\mathbb{E}}(\|\tilde{\eta}_{t}^{\lambda\theta\tilde{\xi}}-\tilde{\eta}_{t}\|)^{2}d\theta
dr\leq C\lambda^{2}\|\xi\|_{L^{2}}^{4}\end{split}$

for a constant $C$ only depending on $\|\eta_{t}\|_{\mathbb{S}^{2}}.$ Since

(119)
$\displaystyle\left|\int_{t}^{T}\tilde{\mathbb{E}}[A_{42}]dr\right|^{2}\leq\int_{t}^{T}\tilde{\mathbb{E}}|A_{42}|^{2}dr\leq
C\sup_{\gamma_{t}}\int_{t}^{T}\tilde{\mathbb{E}}|\Delta_{\lambda}Y-\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}|^{2}dr,$

for a constant $C$ independent of $(\gamma,\eta),$ we have

(120) $\mathbb{E}|A_{4}|^{2}\leq
C\left(\lambda^{2}\|\xi\|_{L^{2}}^{4}+\sup_{\gamma_{t}}\int_{t}^{T}\tilde{\mathbb{E}}|\Delta_{\lambda}Y-\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}|^{2}dr\right).$

Finally, in view of inequalities (110), (113), (114), (120) and (109), we have

$\displaystyle\|\Delta_{\lambda}Y-\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}\|^{2}_{\mathbb{S}^{2}}\leq
C(\lambda^{2}\|\xi\|_{L^{2}}^{4}+\sup_{\gamma_{t}}\int_{t}^{T}\|\Delta_{\lambda}Y-\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}\|_{\mathbb{S}^{2}}^{2}dr),$

where $C$ only depends on $\|\eta_{t}\|_{\mathbb{S}^{2}}$. Then, using
Gronwall’s inequality, we have

(121)
$\|\Delta_{\lambda}Y-\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}\|^{2}_{\mathbb{S}^{2}}\leq
C\lambda^{2}\|\xi\|_{L^{2}}^{4}\rightarrow 0,\ \ \text{ as }\lambda\rightarrow
0.$

∎

In view of our Assumption (H1), the solution
$(Y^{\gamma_{t},\eta_{t}},Z^{\gamma_{t},\eta_{t}})$ of BSDE (79) is indeed
strongly vertically differentiable in $\eta_{t}.$ According to Definition 2.9,
for any $\tau\leq t$ and $\xi\in L^{2}(\mathcal{F}_{\tau},\mathbb{R}^{d}),$
let $\eta_{t}^{\tau,\lambda\xi}:=\eta_{t}+\lambda\xi 1_{[\tau,T]}$. Similar as
the vertical differentiable case, we firstly need to show the following limit
exits in $\mathbb{S}^{2}([t,T]),$

(122)
$\partial_{\eta_{\tau}}Y^{\gamma_{t},\eta_{t},\xi}:=\lim_{\lambda\rightarrow
0}\frac{1}{\lambda}(Y^{\gamma_{t},\eta_{t}^{\tau,\lambda\xi}}-Y^{\gamma_{t},\eta_{t}}).$

Indeed, $\partial_{\eta_{\tau}}Y^{\gamma_{t},\eta_{t},\xi}$ is the unique
solution of the following BSDE

(123)
$\begin{split}\partial_{\eta_{\tau}}Y^{{\gamma_{t},\eta_{t}},\xi}(s)&=\tilde{\mathbb{E}}[\partial_{\mu_{\tau}}\Phi(B^{\gamma_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}]+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\mu_{\tau}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{y}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\partial_{\eta_{\tau}}Y^{{\gamma_{t},\eta_{t}},\xi}(r)dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})(\partial_{\omega_{\tau}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}\tilde{\xi}+\partial_{\eta_{\tau}}\tilde{Y}^{\tilde{\eta}_{t},\tilde{\xi}})(r)]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{z}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\partial_{\eta_{\tau}}Z^{{\gamma_{t},\eta_{t}},\xi}(r)dr\\\
&\ \ \
-\int_{s}^{T}\partial_{\eta_{\tau}}Z^{{\gamma_{t},\eta_{t}},\xi}(r)dB(r),\
s\in[t,T],\end{split}$

where $\partial_{\eta_{\tau}}{Y}^{\eta_{t},\xi}$ solves the following mean-
field BSDE

(124)
$\begin{split}\partial_{\eta_{\tau}}Y^{\eta_{t},\xi}(s)&=\tilde{\mathbb{E}}[\partial_{\mu_{\tau}}\Phi(B^{\eta_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}]+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\mu_{\tau}}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{\tilde{\eta}_{t}})\tilde{\xi}]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{y}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\partial_{\eta_{\tau}}Y^{{\eta_{t}},\xi}(r)dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})(\partial_{\omega_{\tau}}\tilde{Y}^{\tilde{\eta}_{t},\mathcal{L}_{\eta_{t}}}\tilde{\xi}+\partial_{\eta_{\tau}}\tilde{Y}^{\tilde{\eta}_{t},\tilde{\xi}})(r)]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{z}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\partial_{\eta_{\tau}}Z^{{\eta_{t}},\xi}(r)dr-\int_{s}^{T}\partial_{\eta_{\tau}}Z^{{\eta_{t}},\xi}(r)dB(r),\
s\in[t,T].\end{split}$

According to Assumption (H1), we see that BSDEs (124) and (123) are well-
posed. Moreover, following a similar argument as in Lemma 3.11, for the
Gâteaux strong vertical differentiability, we have

######  Lemma 3.12.

$\partial_{\eta_{\tau}}Y^{{\gamma_{t},\eta_{t}},\cdot}$ is a bounded linear
operator from $L^{2}({\mathcal{F}_{\tau}},\mathbb{R}^{d})$ to
$\mathbb{S}^{2}([t,T])$. Moreover,
$\partial_{\eta_{\tau}}Y^{{\gamma_{t},\eta_{t}},\xi}$ is the Gâteaux strong
vertical derivative of $Y^{\gamma_{t},\eta_{t}}$ at $(\tau,t,\eta)$:

(125)
$\partial_{\eta_{\tau}}Y^{{\gamma_{t},\eta_{t}},\xi}=\lim_{\lambda\rightarrow
0}\frac{1}{\lambda}(Y^{\gamma_{t},\eta_{t}^{\tau,\lambda\xi}}-Y^{\gamma_{t},\eta_{t}}),\quad\text{strongly
in $\mathbb{S}^{2}([t,T])$}.$

In particular, $\partial_{\eta_{\tau}}Y^{{\gamma_{t},\eta_{t}},\cdot}(s)$ is
the Gâteaux derivative of $Y^{\gamma_{t},\eta_{t}}(s)$ at $(\tau,t,\eta)$ in
the sense of (38).

To give an explicit representation of the vertical derivative
$Y^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(\cdot)$ with respect to
$\mathcal{L}_{\eta_{t}}$ in view of (36), we need to find out a measurable
random field
$U^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(\cdot):\mathbb{D}_{T,d}\to\mathbb{S}^{2}([t,T],\mathbb{R}^{d})$,
such that for any $s\geq t$ and $\xi\in
L^{2}(\mathcal{F}_{t},\mathbb{R}^{d}),$

(126)
$\mathcal{Y}^{{\gamma_{t},\eta_{t}},\xi}(s)=\bar{E}[U^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(\bar{\eta}_{t})(s)\bar{\xi}],$

where $(\bar{\eta},\bar{\xi})$ is an independent copy of $(\eta,\xi).$ If
(126) holds and moreover we show that $Y^{\gamma_{t},{\eta_{t}}}$ is Fréchet
differentiable with respect to $\eta_{t}$ in the sense of (34) and Remark 2.8,
we have that

(127)
$\partial_{\mu_{t}}Y^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(x_{t}):=U^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(x_{t}),\
\ \forall\ x\in\mathbb{D}_{T,d},$

is the vertical derivative of $Y^{\gamma_{t},\mathcal{L}_{\eta_{t}}}$ at
$\mathcal{L}_{\eta_{t}}$. Here and in the following, we write $\partial_{\mu}$
instead of $\partial_{\mathcal{L}_{\eta}}$. In view of (97) and (126), we
formally deduce that
$(U^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(x_{t}),V^{\gamma_{t},\mathcal{L}_{\eta_{t}}}(x_{t}))$
solves the following BSDE

(128)
$\begin{split}U^{{\gamma_{t},\eta_{t}},x_{t}}(s)&=\tilde{\mathbb{E}}[\partial_{\mu_{t}}\Phi(B^{\gamma_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{x_{t}})]+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\mu_{t}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{x_{t}})]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{y}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})U^{{\gamma_{t},\eta_{t}},x_{t}}(r)dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{x_{t},\mathcal{L}_{\eta_{t}}})\partial_{\omega_{t}}\tilde{Y}^{x_{t},\mathcal{L}_{\eta_{t}}}(r)]dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})\tilde{U}^{\tilde{\eta}_{t},x_{t}}(r)]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{z}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})V^{{\gamma_{t},\eta_{t}},x_{t}}(r)dr-\int_{s}^{T}V^{{\gamma_{t},\eta_{t}},x_{t}}(r)dB(r),\quad
s\in[t,T],\end{split}$

where ${U}^{\eta_{t},x_{t}}$ solves the associated mean-field BSDE:

(129)
$\begin{split}U^{\eta_{t},x_{t}}(s)&=\tilde{\mathbb{E}}[\partial_{\mu_{t}}\Phi(B^{\eta_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{x_{t}})]+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\mu_{t}}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{x_{t}})]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{y}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})U^{\eta_{t},x_{t}}(r)dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{x_{t},\mathcal{L}_{\eta_{t}}})\partial_{\omega_{t}}\tilde{Y}^{x_{t},\mathcal{L}_{\eta_{t}}}(r)]dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})\tilde{U}^{\tilde{\eta}_{t},x_{t}})(r)]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{z}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})V^{\eta_{t},x_{t}}(r)dr-\int_{s}^{T}V^{\eta_{t},x_{t}}(r)dB(r).\end{split}$

According to Lemma 3.1, we see that mean-field BSDE (129) is well posed with
$(U^{\eta_{t},x_{t}},V^{\eta_{t},x_{t}})\in\mathbb{S}^{2}([t,T],\mathbb{R}^{d})\times\mathbb{H}^{2}([t,T],\mathbb{R}^{d\times
d}).$ Then BSDE (128) also has a unique solution
$(U^{{\gamma_{t},\eta_{t}},x_{t}},V^{{\gamma_{t},\eta_{t}},x_{t}})\in\mathbb{S}^{2}([t,T],\mathbb{R}^{d})\times\mathbb{H}^{2}([t,T],\mathbb{R}^{d\times
d})$. Moreover, according to the uniqueness of solutions for BSDEs (129), we
see $U^{\eta_{t},x_{t}}=U^{\gamma_{t},{\eta_{t}},x_{t}}|_{\gamma=\eta}.$
Concerning the regularity of $U^{{\gamma_{t},\eta_{t}},x_{t}}$ and
$U^{\eta_{t},x_{t}}$ with respect to $(\gamma,\eta,x)$, we have

######  Lemma 3.13.

For any $x,x^{\prime},\gamma,\gamma^{\prime}\in\mathbb{D}_{T,d},$ and
$\eta,\eta^{\prime}\in\mathbb{M}_{2}^{D},$ we have

(130)
$\displaystyle\|U^{\eta_{t},x_{t}}-U^{\eta^{\prime}_{t},x^{\prime}_{t}}\|_{\mathbb{S}^{2}}\leq
C(\|\eta_{t}-\eta^{\prime}_{t}\|_{\mathbb{S}^{2}}+\|x_{t}-x^{\prime}_{t}\|),\quad\text{and}$
(131)
$\displaystyle\|U^{{\gamma_{t},\eta_{t}},x_{t}}-U^{\gamma^{\prime}_{t},\eta^{\prime}_{t},x^{\prime}_{t}}\|_{\mathbb{S}^{2}}\leq
C(\|\gamma_{t}-\gamma^{\prime}_{t}\|+W_{2}(\mathcal{L}_{\eta_{t}},\mathcal{L}_{\eta^{\prime}_{t}})+\|x_{t}-x^{\prime}_{t}\|),$

with $C$ only depending on
$\|\eta_{t}\|_{\mathbb{S}^{2}}+\|\eta^{\prime}_{t}\|_{\mathbb{S}^{2}}$.

######  Remark 3.14.

Similar to Lemma 3.4, according to estimate (131),
$U^{\gamma_{t},\mathcal{L}_{\eta_{t}},x_{t}}:=U^{{\gamma_{t},\eta_{t}},x_{t}}$
is well-defined.

###### Proof.

In the following we omit the subscript $t$ and write
$(U,V,Y,Z):=(U(r),V(r),Y(r),Z(r))$. Moreover, we only show the proof for (130)
since (131) follows by (130) and similar argument. Denote

$\displaystyle(\Delta U,\Delta
V):=(U^{\eta,x}-U^{\eta^{\prime},x^{\prime}},V^{\eta,x}-V^{\eta^{\prime},x^{\prime}}),$
$\displaystyle\Delta\partial_{\mu}\Phi:=\partial_{\mu_{t}}\Phi(B^{\eta},\mathcal{L}_{B^{\eta}},B^{x})-\partial_{\mu_{t}}\Phi(B^{\eta^{\prime}},\mathcal{L}_{B^{\eta^{\prime}}},B^{x^{\prime}}),$
$\displaystyle\Delta\partial_{\mu}f:=\partial_{\mu_{t}}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta}},\tilde{B}^{x_{t}})-\partial_{\mu_{t}}f(\Theta^{\eta^{\prime}}_{r},\mathcal{L}_{\Theta_{r}^{\eta^{\prime}}},\tilde{B}^{x^{\prime}_{t}}),$
$\displaystyle\Delta\partial_{\nu}f^{(1)}:=\partial_{\nu}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta}},\tilde{Y}^{x_{t},\mathcal{L}_{\eta_{t}}})-\partial_{\nu}f(\Theta^{\eta^{\prime}}_{r},\mathcal{L}_{\Theta_{r}^{\eta^{\prime}}},\tilde{Y}^{x^{\prime}_{t},\mathcal{L}_{\eta^{\prime}_{t}}}),$
$\displaystyle\Delta\partial_{\nu}f^{(2)}:=\partial_{\nu}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta}},\tilde{Y}^{\tilde{\eta}_{t}})-\partial_{\nu}f(\Theta^{\eta^{\prime}}_{r},\mathcal{L}_{\Theta_{r}^{\eta^{\prime}}},\tilde{Y}^{\tilde{\eta}^{\prime}_{t}}),$
$\displaystyle\Delta\partial_{(y,z)}f:=\partial_{(y,z)}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta}})-\partial_{(y,z)}f(\Theta^{\eta^{\prime}}_{r},\mathcal{L}_{\Theta_{r}^{\eta^{\prime}}}),\quad\text{and}\quad$
$\displaystyle\Delta\partial_{\omega}\tilde{Y}:=\partial_{\omega_{t}}\tilde{Y}^{x_{t},\mathcal{L}_{\eta_{t}}}-\partial_{\omega_{t}}\tilde{Y}^{x^{\prime}_{t},\mathcal{L}_{\eta^{\prime}_{t}}},$

and we see that $(\Delta U,\Delta V)$ is the unique solution of BSDE

(132) $\begin{split}\Delta
U(s)&=\Delta\partial_{\mu}\Phi+\int_{s}^{T}\tilde{\mathbb{E}}[\Delta\partial_{\mu}f]dr+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta}},\tilde{Y}^{\tilde{\eta}})\Delta\tilde{U}]dr\\\
&\ \ \
+\int_{s}^{T}(\partial_{y}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta}})\Delta
U+\partial_{z}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta}})\Delta
V)dr-\int_{s}^{T}\Delta VdB(r)\\\ &\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[(\Delta\partial_{\nu}f^{(1)})\partial_{\omega}\tilde{Y}^{x^{\prime},\tilde{\eta}^{\prime}}+(\Delta\partial_{\nu}f^{(2)})\tilde{U}^{\tilde{\eta}^{\prime},x^{\prime}}]dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta}},\tilde{Y}^{x,\mathcal{L}_{\eta}})\Delta\partial_{\omega}\tilde{Y}^{x,\mathcal{L}_{\eta}}]dr\\\
&\ \ \
+\int_{s}^{T}\left((\Delta\partial_{y}f)U^{\eta^{\prime},x^{\prime}}+(\Delta\partial_{z}f)V^{\eta^{\prime},x^{\prime}}\right)dr.\end{split}$

By Lipschitz continuity of $(\partial_{(\mu,\nu,y,z)}f,\partial_{\mu}\Phi),$
and boudnedness of $\partial_{(y,z)}f,$ we see that

$\displaystyle\|\Delta\partial_{\mu}\Phi\|^{2}_{L^{2}}+\|\int_{t}^{T}\tilde{\mathbb{E}}[\Delta\partial_{\mu}f]dr\|^{2}_{L^{2}}+\mathbb{E}[\int_{t}^{T}\tilde{\mathbb{E}}(|\Delta\partial_{\nu}f^{(1)}|^{2}+|\Delta\partial_{\nu}f^{(2)}|^{2})dr]$
(133) $\displaystyle\ \ \ \ \
+\mathbb{E}[\int_{t}^{T}|\Delta\partial_{y}f|^{2}dr]+\mathbb{E}[\int_{t}^{T}|\Delta\partial_{z}f|^{2}dr]$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \leq
C(\|\eta_{t}-\eta^{\prime}_{t}\|^{2}_{\mathbb{S}^{2}}+\|x_{t}-x^{\prime}_{t}\|^{2}).$

Moreover, since
$\partial_{\omega}\tilde{Y}^{x^{\prime},\tilde{\eta}^{\prime}},\tilde{U}^{\eta^{\prime},x^{\prime}},\tilde{Y}^{x,\mathcal{L}_{\eta}},\tilde{Y}^{\tilde{\eta}}\in\mathbb{S}^{2}$,
and $V^{\eta^{\prime},x^{\prime}}\in\mathbb{H}^{2},$ from the above estimate
and the Cauchy-Schwartz inequality, we obtain

(134)
$\begin{split}&\|\int_{s}^{T}\tilde{\mathbb{E}}\left[(\Delta\partial_{\nu}f^{(1)})\partial_{\omega}\tilde{Y}^{x^{\prime},\tilde{\eta}^{\prime}}+(\Delta\partial_{\nu}f^{(2)})\tilde{U}^{\tilde{\eta}^{\prime},x^{\prime}}\right]dr\|_{L^{2}}\\\
&\ \ \ \ \
+\|\int_{s}^{T}\left((\Delta\partial_{y}f)U^{\eta^{\prime},x^{\prime}}+(\Delta\partial_{z}f)V^{\eta^{\prime},x^{\prime}}\right)dr\|_{L^{2}}\\\
&\ \ \ \ \ \ \ \ \leq
C(\|\eta_{t}-\eta^{\prime}_{t}\|_{\mathbb{S}^{2}}+\|x_{t}-x^{\prime}_{t}\|).\end{split}$

According to estimates given in Lemma 3.7 and boundedness of
$\partial_{\nu}f,$ we check that

(135)
$\|\int_{s}^{T}|\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta}},\tilde{Y}^{x,\mathcal{L}_{\eta}})\Delta\partial_{\omega}\tilde{Y}^{x,\mathcal{L}_{\eta}}]|dr\|_{L^{2}}\leq
C(\|\eta_{t}-\eta^{\prime}_{t}\|_{\mathbb{S}^{2}}+\|x_{t}-x^{\prime}_{t}\|).$

Then in view of Lemma 3.1, inequalities (133), (134) and (135), we obtain the
desired inequality (130).

∎

Concerning the SVD $\partial_{\mu_{\tau}}Y^{{\gamma_{t},\eta_{t}},\cdot}$ of
$Y^{\gamma_{t},\mathcal{L}_{\eta_{t}}}$ at $(\tau,t,\mathcal{L}_{\eta})$,
$\tau\leq t,$ in view of Definition 2.9 and BSDE (123), we deduce that it is
the unique solution of the following BSDE: for any $x\in\mathbb{D}_{T,d},$

(136)
$\begin{split}\partial_{\mu_{\tau}}Y^{{\gamma_{t},\eta_{t}},x_{t}}(s)&=\tilde{\mathbb{E}}[\partial_{\mu_{\tau}}\Phi(B^{\gamma_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{x_{t}})]+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\mu_{\tau}}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{x_{t}})]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{y}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\partial_{\mu_{\tau}}Y^{{\gamma_{t},\eta_{t}},x_{t}}(r)dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})\partial_{\mu_{\tau}}\tilde{Y}^{\tilde{\eta}_{t},x_{t}}(r)]dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{x_{t},\mathcal{L}_{\eta_{t}}})\partial_{\omega_{\tau}}\tilde{Y}^{x_{t},\mathcal{L}_{\eta_{t}}}(r)]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{z}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\partial_{\mu_{\tau}}Z^{{\gamma_{t},\eta_{t}},x_{t}}(r)dr\\\
&\ \ \
-\int_{s}^{T}\partial_{\mu_{\tau}}Z^{{\gamma_{t},\eta_{t}},x_{t}}(r)dB(r),\
s\in[t,T],\end{split}$

where $\partial_{\mu_{\tau}}Y^{\eta_{t},x_{t}}$ sloves the mean-field BSDE
below

(137)
$\begin{split}\partial_{\mu_{\tau}}Y^{\eta_{t},x_{t}}(s)&=\tilde{\mathbb{E}}[\partial_{\mu_{\tau}}\Phi(B^{\eta_{t}},\mathcal{L}_{B^{\eta_{t}}},\tilde{B}^{x_{t}})]+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\mu_{\tau}}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{B}^{x_{t}})]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{y}f(\Theta^{\eta}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\partial_{\mu_{\tau}}Y^{\eta_{t},x_{t}}(r)dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{x_{t},\mathcal{L}_{\eta_{t}}})\partial_{\omega_{\tau}}\tilde{Y}^{x_{t},\mathcal{L}_{\eta_{t}}}(r)]dr\\\
&\ \ \
+\int_{s}^{T}\tilde{\mathbb{E}}[\partial_{\nu}f(\Theta^{\gamma_{t},\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}},\tilde{Y}^{\tilde{\eta}_{t}})\partial_{\mu_{\tau}}\tilde{Y}^{\tilde{\eta}_{t},x_{t}})(r)]dr\\\
&\ \ \
+\int_{s}^{T}\partial_{z}f(\Theta^{\eta_{t}}_{r},\mathcal{L}_{\Theta_{r}^{\eta_{t}}})\partial_{\mu_{\tau}}Z^{\eta_{t},x_{t}}(r)dr-\int_{s}^{T}\partial_{\mu_{\tau}}Z^{\eta_{t},x_{t}}(r)dB(r),\
s\in[t,T].\end{split}$

Thanks to Lemma 3.1 again, mean-field BSDE (137) has a unique solution
$(\partial_{\mu_{\tau}}Y^{\eta_{t},x_{t}},\partial_{\mu_{\tau}}Z^{\eta_{t},x_{t}})\in\mathbb{S}^{2}\times\mathbb{H}^{2}.$
Then the well-posedness of equation (136) follows similarly. Moreover we have
that if $\tau=t,$

(138)
$\partial_{\mu_{t}}Y^{{\gamma_{t},\eta_{t}},x_{t}}=U^{{\gamma_{t},\eta_{t}},x_{t}},\
\ \ \partial_{\mu_{t}}Y^{\eta_{t},x_{t}}=U^{\eta_{t},x_{t}},$

and
$\partial_{\mu_{\tau}}Y^{\eta_{t},x_{t}}=\partial_{\mu_{\tau}}Y^{\gamma_{t},\mathcal{L}_{\eta_{t}},x_{t}}|_{\gamma=\eta}.$
Thus the following lemma follows similarly as that of Lemma 3.13.

######  Lemma 3.15.

For any $x,x^{\prime},\gamma,\gamma^{\prime}\in\mathbb{D}_{T,d},$ and