# The Bismut-Elworthy-Li formula for semi-linear distribution-dependent SDEs
driven by fractional Brownian motion

Mahdieh<EMAIL_ADDRESS>[ Department of Applied
Mathematics, Tarbiat Modares University, P.O. Box 14115-134, Tehran,
Iran.presep=, ]e1

###### Abstract

In this work, we will show the existence, uniqueness, and weak
differentiability of the solution of semi-linear mean-field stochastic
differential equations driven by fractional Brownian motion. We prove an
extension of the Bismut-Elworthy-Li formula and show some applications in the
sensitivity analysis of variance swaps and the price of derivatives with
respect to the initial point.

60H10,

60G22,

Stochastic fractional integrals,

Malliavin calculus,

The Bismut-Elworthy-Li formula,

Mean-field stochastic differential equations,

###### keywords:

[class=MSC]

###### keywords:

,

##  1 Introduction

Mean-field stochastic differential equations (mean-field SDEs) are an
extension of stochastic differential equations where the coefficients are
allowed to depend on the law of the solution. Based on the works of Vlasov
[50], Kac [39] and McKean [43], mean-field SDEs, also referred to as McKean-
Vlasov equations or as distribution dependent equations, arisen from Boltzmann
equation in physics, is used to model weak interaction between particles in a
multi-particle system. Henceforth the applications of mean-field problem and
the study of dependent distribution SDEs attracted wide attention, (see for
instance, [31, 13, 33, 32]). In [41], Lasry and Lions have considered the
mean-field SDEs as an application in Economics and Finance, The mean-field
approach also has some application in systemic risk modeling, especially for
inter-bank lending and borrowing models, (see e.g., [24, 25, 15, 14]).
For the case where the mean-field SDEs have additive noise, Jourdain proves in
[38] the existence of a unique weak solution under the assumptions of a
bounded drift which is Lipschitz continuous in the law variable. In [42] and
[45] the existence of strong solutions of mean-field SDEs are shown in the
special form of drift which fulfills certain linear growth and Lipschitz
conditions. Recently, Bauer et. al. investigated the existence, uniqueness,
and regularity properties of (strong) solutions of one-dimensional mean-field
SDEs when the drift coefficient is allowed to be irregular in [8] and [7]. The
regularity of the solution of mean-field SDEs also discussed in many articles.
For instance, continuous differentiability of the solution $X_{t}^{x}$ with
respect to the initial point $x$, where the mean-field dependence is given via
an expectation functional of the form $\mathbb{E}(\phi(X_{t}^{x}))$, has been
shown in [12] under the assumption that the drift coeficient and
$\phi:\mathbb{R}\rightarrow\mathbb{R}$ are continuously differentiable with
bounded Lipschitz derivatives. Also, in [7], the authors deal with the
important case that the drift depends on the distribution function of the
solution for the indicator function $\phi(z)=1_{z\leq u}$. In this manuscript,
we are interested in the mean-field SDEs whose distribution dependence of the
solution is stated by an expectation functional form.
On the other hand, the Bismut-Elworthy-Li formula (BEL formula) gives a
representation of expressions of the form

$\frac{d}{dx}\mathbb{E}[\Phi(X_{t}^{x})]$

for $\Phi:\mathbb{R}^{d}\rightarrow\mathbb{R}$, $d\geq 1$ and for a desirable
stochastic process $X_{t}^{x}$, which doesn’t involve the derivative of
$\Phi$. The Bismut-Elworthy-Li formula, also known as the Bismut formula,
based on Malliavin calculus, is a very effective tool in the analysis of
distributional regularity for various stochastic models, with additive noise
and multiplicative noise (see e.g., [51, 34, 35]. The Bismut formula for
multi-dimensional mean-field SDEs with multiplicative noise and smooth drift
and volatility coefficients has been studied in [4]. Bauer et. al. in [7] have
derived a Bismut-Elworthy-Li type formula for mean-field SDEs under additive
noise where the dependence of the drift $b$ on the solution law is of the form

$dX_{t}=b(t,X_{t},\int_{0}^{t}\phi(t,X_{t},z)dP_{X_{t}}(z))dt+dB_{t},\quad
X_{0}=x\in\mathbb{R},$

for some $b,\phi:[0,T]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$
and $B_{t}$ is a Wiener process. Also, the Bismut formula was given for
distribution-path dependent SDEs with distribution-free noise in [5].
In this article, we are concerned with the equations driven by fractional
Brownian motions. In fractional literature, in 2017, Fan [22] investigated a
Bismut-Elworthy-Li type formula for the distribution-free SDEs with additive
noise driven by fractional Brownian motion, assuming the Lipschitz continuous
property on the drift. We also mention the interesting work [3], in 2018,
where Amine et. al. derived the Bismut-Elworthy-Li type formula for the
distribution-free SDEs with given additive fractional noise in rough path
sense with Hurst parameter $H<\frac{1}{2}$ and singular drift coefficients for
the modeling of regime switching effects in stock markets.
The mean field SDEs driven by fractional Brownian motion introduced by
Buckdhan and Jing in [11]. They have considered distribution-dependent SDEs of
the form

$dX_{t}=b(t,X_{t},\mathcal{L}_{X_{t}})dt+(\gamma_{s}X_{s}+\sigma(t,\mathcal{L}_{X_{t}})\\})dW_{t}^{H},\qquad
t\in[0,T],$ (1)

where $\mathcal{L}_{X_{t}}$ is the law of $X_{t}$, $H>\frac{1}{2}$ and
$W_{t}^{H}$ is $d$-dimensional fractional Brownian motion, under the Lipschitz
property on diffusion coefficient $\sigma$ and the drift coefficient $b$, with
Wasserstein measure $\mathbb{W}(.,.)$, as follows:

$|b(t,x,\mu)-b(t,y,\nu)|\leq
K(t)(|x-y|+\mathbb{W}(\mu,\nu)),\quad|\sigma(t,\mu)-\sigma(t,\nu)|\leq
K(t)\mathbb{W}(\mu,\nu),$ (2)

where $x,y\in\mathbb{R}^{d}$ and $\mu,\nu$ belongs to the space of probability
measures on $\mathbb{R}^{d}$ with finite $\theta$-th moment, $\theta\geq 1$,
also the stochastic integral is in the Skorohkod sense. They proved the
existence and uniqueness of the strong solution of that SDE in some special
Banach space, denoted by $L^{2,*}([0,T]\times\mathbb{R}^{d})$ (see Section 3).
In 2021, Fan et. al. [21] proved the BEL formula for distribution dependent
SDEs of the form (1) when the function $\gamma$ is zero.
We extend the result in [21] and [11] for the case, where the condition (2) is
not covered our assumption on $b$ in this paper and also we allow the
multiplicative noise driven by fractional Brownian motions with Hurst
parameter $H>\frac{1}{2}$.
In this article, we are interested in multiplicative stochastic differential
equations driven by fractional Brownian motion of the form

$dX_{t}=\\{b(t,\rho_{X_{t})})X_{t}+\beta(t,\rho_{X_{t}})\\}dt+\\{C_{t}X_{t}+a(t,\Gamma_{X_{t}})\\}dW_{t}^{H},\qquad
t\in[0,T],\quad X_{0}=x>0,$ (3)

where $\rho_{X_{t}}:=\mathbb{E}(\varphi(X_{t}))$ and
$\Gamma(X_{t})=\mathbb{E}(\psi(X_{t}))$, in which the functions
$\varphi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$,
$\psi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$, the functions $b,\beta$ and
$a$ from $[0,T]\times\mathbb{R}^{d}$ to $\mathbb{R}^{d}$ are measurable,
$C_{.}$ is a bounded deterministic function and $T\in\mathbb{R},T>0$. We
remark that the drift coefficient of (3) is not satisfies (2) and is an
extension of this type of dependent distribution SDEs. Also, we should study
the fractional integrals in the concept of Skorokhod integrals.
The first contribution to this manuscript is to establish existence,
uniqueness, and weak differentiability of strong solutions of this type of
mean-field SDEs and the second main contribution is to establish the BEL
formula in this multiplicative fractional distribution-dependent SDEs. To
achieve this purpose, usually one needs to show that the flow of the solution
is its Malliavin derivative in some direction. But for the solution of (3),
this relationship can not be covered unless one additional term appears in the
formula, (see Theorem 5.3). To handle this term in the BEL formula, we need
that its Malliavin derivative in some direction would be nonzero, however, the
solution can meet the zero during their path. To overcome this problem we
proceed on some stopping time to show that the fundamental solution of (3) can
be presented by the Mallivain derivative of the solution in some direction
which is in the domain of the Skorokhod operator. (see Theorem 4.5).
This article is organized as follows: Section 2 is devoted to recalling some
useful facts on fractional Brownian motion, fractional calculus and Girsanov
theorem and Malliavin calculus for stochastic fractional integrals. In Section
3, we prove the existence and uniqueness of the solutions of Mean-field SDEs
with multiplicative fractional noise when the drift coefficient is taken as a
product of the function of the law of the solution and the solution itself.
The regularity of the solution and obtaining a representation for the
Malliavin drivative are shown in Section 4. In Section 5, the existence of the
flow of the solution and a transfer principle between the flow and the
Malliavin derivative of the solution is derived. The Bismut formula is
established in section 6 and finally, we take some applications in finance in
the last section.

##  2 Preliminaries

In this section, we recall some concepts of fractional calculus, Girsanov
theorem, and Malliavin calculus for stochastic fractional integrals. For more
detailes, the reader can see the exhaustive references [10, 44].

###  2.1 Fractional operator

Let $a,b\in\mathbb{R}$ and $a<b$. For every $f\in L^{1}([a,b];\mathbb{R})$ and
$\alpha>0$, the left-sided and the right-sided fractional Riemann-Liouville
integral of $f$ of order $\alpha$ are defined as

$I_{a^{+}}^{\alpha}f(x)=\frac{1}{\Gamma(\alpha)}\int_{a}^{x}\frac{f(y)}{(x-y)^{1-\alpha}}dy,$
$I_{b^{-}}^{\alpha}f(x)=\frac{(-1)^{-\alpha}}{\Gamma(\alpha)}\int_{x}^{b}\frac{f(y)}{(x-y)^{1-\alpha}}dy,$

where $x\in(a,b)$ and $\Gamma$ denotes the Gamma function. For
$\alpha\in(0,1)$ and $p\geq 1$, fractional differentiation is shown as an
inverse operation. If $f\in I_{a^{+}}^{\alpha}(L^{p}([a,b];\mathbb{R}))$
(respectively, $f\in I_{b^{-}}^{\alpha}(L^{p}([a,b];\mathbb{R}))$), then there
exists a unique function $g$ such that $f=I_{a^{+}}^{\alpha}(g)$,
(respectively, $f=I_{b^{-}}^{\alpha}(g)$). Samko et al. [48] provide a
characterization of these functions coincides with the left-sided and the
right-sided fractional derivative, defined as

$D_{a^{+}}^{\alpha}f(x)=\frac{1}{\Gamma(1-\alpha)}\Big{(}\frac{f(x)}{x^{\alpha}}+\alpha\int_{a}^{x}\frac{f(x)-f(y)}{(x-y)^{1+\alpha}}dy\Big{)},$
$D_{b^{-}}^{\alpha}f(x)=\frac{1}{\Gamma(1-\alpha)}\Big{(}\frac{f(x)}{(b-x)^{\alpha}}+\alpha\int_{x}^{b}\frac{f(x)-f(y)}{(x-y)^{1+\alpha}}dy\Big{)}.$

###  2.2 Stochastic fractional integral

Let $(\Omega,\mathcal{F},P)$ be the Wiener space associated with fractional
Brownian motion $\\{W_{t}^{H};t\in[0,T]\\}$ with Hurst parameter
$H\in(\frac{1}{2},1)$ and consider $\\{\mathcal{F}_{t}\\}_{t\in[0,T]}$ as the
complete filteration generated by $W^{H}$.
Let $\mathcal{H}$ be the Hilbert space defined as the closure of the set of
step functions on $[0,T]$ with respect to the scalar product

$\langle 1_{[0,t]},1_{[0,s]}\rangle_{\mathcal{H}}=R_{H}(t,s)=\int_{0}^{t\wedge
s}K_{H}(t,r)K_{H}(r,s)dr,$ (4)

where $K_{H}(t,s)$ is the square integrable kernel given by

$K_{H}(t,s)=c_{H}s^{\frac{1}{2}-H}\int_{s}^{t}(u-s)^{H-\frac{3}{2}}u^{H-\frac{1}{2}}du,$

in which $c_{H}^{2}=\frac{H(2H-1)}{\beta(2-2H,H-\frac{1}{2})}$ and $t>s$.
Formula (4) provides the following explicit expression of $R_{H}(t,s)$

$R_{H}(t,s)=\alpha_{H}\int_{0}^{t}\int_{0}^{s}\phi(r,u)drdu,$

where $\alpha_{H}=H(2H-1)$ and $\phi(t,s)=|t-s|^{2H-2}$.
The mapping $1_{[0,t]}\rightarrow W_{t}^{H}$ can be extended to an isometry
between $\mathcal{H}$ and a Gaussian space associated with $W^{H}$, denoted by
$\psi\rightarrow W^{H}(\psi)=\int_{0}^{T}\psi(t)dW_{t}^{H}$.
Besides, we define an operator $K_{H}$ on $\mathcal{H}$ as

$(K_{H}\psi)(s)=c_{H}\Gamma(H-\frac{1}{2})s^{\frac{1}{2}-H}I_{0^{+}}^{H-\frac{1}{2}}(u^{H-\frac{1}{2}}\psi(u))(s).$

This operator from $L^{2}([0,T];\mathbb{R}^{d})$ into
$I_{0^{+}}^{H+\frac{1}{2}}(L^{2}([0,T];\mathbb{R}^{d})$ is an isomorphism and
it’s inverse operator $K_{H}^{-1}$ is of the form

$((K_{H})^{-1}\psi)(s)=c_{H}\Gamma(H-\frac{1}{2})s^{H-\frac{1}{2}}D_{0^{+}}^{H-\frac{1}{2}}(u^{\frac{1}{2}-H}\psi^{\prime}(u))(s).$

Adjoint operator $K_{H}^{*}$ on $\mathcal{H}$ is

$(K_{H}^{*}\psi)(s)=c_{H}\Gamma(H-\frac{1}{2})s^{\frac{1}{2}-H}I_{T^{-}}^{H-\frac{1}{2}}(u^{H-\frac{1}{2}}\psi(u))(s).$

and its inverse operator is

$((K_{H}^{*})^{-1}\psi)(s)=\frac{1}{c_{H}\Gamma(H-\frac{1}{2})}s^{\frac{1}{2}-H}D_{T^{-}}^{H-\frac{1}{2}}(u^{H-\frac{1}{2}}\psi(u))(s).$

According to [2], for every $u,v\in\mathcal{H}$, $\langle
K_{H}^{*}u,K_{H}^{*}v\rangle_{L^{2}[0,T]}=\langle u,v\rangle_{\mathcal{H}}$
and the operator $K_{H}^{*}$ is an isometry between $\mathcal{H}$ and
$L^{2}([0,T];\mathbb{R}^{d})$ and then the injection $R_{H}=K_{H}oK_{H}^{*}$
embeds $\mathcal{H}$ densely into $\Omega$ such that for any
$\psi\in\Omega^{*}\subset\mathcal{H}$,
$\mathbb{E}\Big{(}\exp\\{iW^{H}(\psi)\\}\Big{)}=\exp\\{-\frac{1}{2}\|\psi\|_{\mathcal{H}}^{2}\\}$.
Consider the process $W_{t}=W_{t}^{H}((K_{H}^{*})^{-1}1_{[0,t]})$. It is a
Wiener process and the filteration generated by $W$ coincides with
$\\{\mathcal{F}_{t}\\}_{t\in[0,T]}$.

###  2.3 Girsanov transformation

Let $\gamma$ be a bounded function in $\mathcal{H}$. For any $w\in\Omega$ and
$t\in[0,T]$, we define the following operators

$\mathcal{T}_{t}(w)=w+\int_{0}^{t\wedge.}K_{H}^{*}(\gamma 1_{[0,t]})(s)ds,$
$\mathcal{A}_{t}(w)=w-\int_{0}^{t\wedge.}K_{H}^{*}(\gamma 1_{[0,t]})(s)ds.$

It is clear that
$\mathcal{T}_{t}(\mathcal{A}_{t})(w)=\mathcal{A}_{t}(\mathcal{T}_{t})(w)=w$.
Let

$\mathcal{E}_{t}(\gamma):=\exp\\{\int_{0}^{t}\gamma(s)dW_{s}^{H}-\frac{1}{2}\int_{0}^{t}(K_{H}^{*}(\gamma
1_{[0,t]}))^{2}(s)ds\\}.$

Hence

$\mathcal{E}_{t}^{-1}(\mathcal{T}_{t})(\gamma)=\exp\\{-\int_{0}^{t}\gamma(s)dW_{s}^{H}-\frac{1}{2}\int_{0}^{t}(K_{H}^{*}(\gamma
1_{[0,t]}))^{2}(s)ds\\}.$

Following a similar argument in Lemma 2.4 in [37], we conclude that for all
$p\geq 1$

$\mathbb{E}\Big{(}\sup_{0\leq t\leq
T}\mathcal{E}_{t}^{p}(\mathcal{T}_{t})\Big{)}<\infty\quad\mathbb{E}\Big{(}\sup_{0\leq
t\leq T}\mathcal{E}_{t}^{p}(\mathcal{T}_{t})\Big{)}<\infty.$ (5)

###  2.4 Malliavin calculus in fractional case

Denote by $\mathcal{S}$ the set of smooth functionals of the form

$F=f\left(W^{H}(u_{1}),W^{H}(u_{2}),\cdots W^{H}(u_{n})\right),$

where $f\in C^{\infty}_{b}(\mathbb{R}^{n})$ (f and all its derivatives are
bounded) and $u_{i}\in\mathcal{H},~{}i=1,2,\cdots,n$. For every
$F\in\mathcal{S}$, define

$D^{H}F=\sum_{i=0}^{n}\frac{\partial f}{\partial
x_{i}}\left(W^{H}(u_{1}),W^{H}(u_{2}),\cdots W^{H}(u_{n})\right)u_{i}.$

The derivative operator $D^{H}$ is a closable operator from $L^{p}(\Omega)$
into $L^{p}(\Omega,\mathcal{H})$ for every $p\geq 1$. We denote by
$\mathbb{D}^{1,p}$ the closure of $\mathcal{S}$ respect to the norm

$\|F\|_{1,p}^{p}=E|F|^{p}+E\|D^{H}F\|^{p}_{\mathcal{H}}.$

However, it is more convenient to use another Malliavin derivative for
fractional Brownian motions defined in [19].
Duncan et al. [19] has defined $\phi$-derivative of $F\in L^{p}$, $p\geq 1$,
as a directional derivative of the random variable $F$ and denoted by
$D_{s}^{\Phi}F$ in the following form

$\langle D^{\Phi}F(w),g\rangle_{L^{2}}=\lim_{\epsilon\rightarrow
0}\frac{1}{\epsilon}\Big{\\{}F(w+\int_{0}^{.}\Phi g(u)du)-F(w)\Big{\\}},$

where $\Phi g(u)=\int_{0}^{T}\phi(s,u)g(s)ds$ and $g\in\mathcal{H}$. For
instance, Equation (3.6) of [19] demonstrated that if
$f:\mathbb{R}\longrightarrow\mathbb{R}$ be a twice continuously differentiable
function with bounded second derivative, then

$D_{s}^{\phi}(\int_{0}^{\infty}f(u)dW_{u}^{H})=\int_{0}^{\infty}\phi(u,s)f(u)du.$

Recall $\mathcal{L}(0,T)$ as the family of Malliavin differentiable stochastic
processes $G$ such that
$\mathbb{E}\|G\|_{\mathcal{H}}+\mathbb{E}\|D^{\phi}G\|_{\mathcal{H}\otimes\mathcal{H}}<\infty$,
and for any sequence of partition $\pi:0=t^{n}_{0}\leq\cdots\leq t_{n}^{n}=T$
of $[0,T]$ such that $|\pi|\rightarrow 0$ as $n\rightarrow\infty$,

$\sum_{i=0}^{n}\mathbb{E}\Big{\\{}\int_{t_{i}^{n}}^{t_{i+1}^{n}}\int_{0}^{T}(D_{r}^{\phi}G^{\pi}_{t_{i}^{n}}-D_{r}^{\phi}G_{s})\phi(r,s)drds\Big{\\}}^{2}+\mathbb{E}\Big{\\{}\|G^{\pi}-G\|_{H}\Big{\\}}<\infty.$

The following proposition shows that how we can compute the $\phi$-derivative
of stochastic integral.

######  Proposition 2.1.

(Duncan [19], Theorem 4.2) If $(F_{t})_{t\in[0,T]}$ be a stochastic process in
$\mathcal{L}(0,T)$ and
$\sup\mathbb{E}(|\phi_{s}D_{s}^{\Phi}F_{s}|^{2})<\infty$, then
$\eta_{t}:=\int_{0}^{t}F(u)dW_{u}^{H}$ is $\phi$-differentiable and

$~{}D^{\phi}_{s}\eta_{t}=\int_{0}^{t}D_{s}^{\phi}F_{u}dW_{u}^{H}+\int_{0}^{t}F_{u}\phi(s,u)du,\qquad
a.s.$ (6)

These facts state Ito formula for the sochastic process

$\eta_{t}=\xi+\int_{0}^{t}G_{u}du+\int_{0}^{t}F_{u}dW_{u}^{H},\quad\xi\in\mathbb{R},t\in[0,\tau].$

######  Proposition 2.2.

(Duncan [19] Theorem 4.6) Let $(F_{s}^{i})_{s\in[0,T]}$, $i=0,...,n$,
satisfying the condition of Theorem 4.3 of [19] and
$\xi_{t}^{i}=\int_{0}^{t}F_{s}^{i}dB_{s}^{H}$. If
$f\in\mathbb{C}_{b}^{\frac{1}{2}}(\mathbb{R}_{+}\times\mathbb{R}^{n})$ and
$(f_{s_{k}}(s,\eta_{s})F_{s}^{k})_{s\in[0,T]}\in\mathcal{L}[0,T]$ then

$\displaystyle f(t,\xi_{+}^{1},...,\xi_{+}^{n})$
$\displaystyle=f(0,...,0)+\int_{0}^{t}f_{s}(u,\xi_{u}^{1},...,\xi_{u}^{n})du$
$\displaystyle~{}~{}+\Sigma_{k=1}^{n}\int_{0}^{t}f_{x_{k}}(u,\xi_{u}^{1},...,\xi_{u}^{n})F_{u}^{k}dW_{u}^{H}$
$\displaystyle~{}~{}+\Sigma_{k,l=1}^{n}\int
f_{x_{k}x_{l}}(u,\xi_{u}^{1},...,\xi_{u}^{n})F_{u}^{k}D^{\phi}_{u}\xi_{u}^{\ell}du,\quad
a.s.$

Consequently, from Section 5.7 in [29], we have

$D_{s}^{\phi}(F)=K_{H}K_{H}^{*}D_{s}^{H}F.$

Due to [9], for every stochastic process $F\in\mathbb{D}^{H}$,
$\phi$-derivative of $F$ is expressed as

$D_{s}^{\phi}(F)=\int_{0}^{T}D_{r}^{H}F\phi(r,s)dr.$ (7)

The adjoint operator of $D^{H}$, called $\delta^{H}(.)$, can be characterized
as a divergence operator associated to $D^{\phi}$ (See, for example Theorem
6.23 in [29], or Proposition 2.3 in [28]).

######  Definition 2.3.

For any measurable function $G\in L^{2}([0,T]\times\Omega)$, we called $G$ is
Skorohkod integrable if there exists $\delta^{H}(G)\in
L^{2}(\Omega;\mathcal{H})$ such that for any $F\in D^{1,2}$,

$E\left[F\delta^{H}(G)\right]=\int_{0}^{T}E[D_{t}^{\phi}FG(t)]dt=E\left\langle
D^{H}F,G\right\rangle_{\mathcal{H}},$

Let us finish this subsection by giving a relation between Malliavin
derivative and divergence operators of both processes $W^{H}$ and $W$ that are
needed later on.

######  Proposition 2.4.

([46] Proposition 5.2.1) For any $F\in D_{W}^{1,2}\cap D^{1,2}$,

$K_{H}^{*}D^{H}F=DF,$

where $D$ is the derivative operator with respect to the Wiener process $W$
and $D_{W}^{1,2}$ it’s domain in $L^{2}(\Omega)$

######  Proposition 2.5.

([46] Proposition 5.2.2) $Dom(\delta^{H})=(K_{h}^{*})^{-1}(Dom\delta_{W})$,
and for every $\mathcal{H}$-valued random variable $u\in dom(\delta_{W})$ we
have $\delta^{H}(u)=\delta(K_{H}^{*}u)$, where $\delta_{W}$ is the dual
operator of $D_{W}$.

##  3 Fractional Mean-Field SDEs

In this section, we consider a mean-field type stochastic differential
equation (SDE) driven by a fractional Brownian motion with Hurst parameter
$H>\frac{1}{2}$ in the following form

$dX_{t}=\\{b(t,\rho_{X_{t})})X_{t}+\beta(t,\rho_{X_{t}})\\}dt+\\{C_{t}X_{t}+a(t,\Gamma_{X_{t}})\\}dW_{t}^{H},\qquad
t\in[0,T],\quad X_{0}=x>0,$ (8)

where $\rho_{X_{t}}:=\mathbb{E}(\varphi(X_{t}))$ and
$\Gamma(X_{t})=\mathbb{E}(\psi(X_{t}))$, in which the functions
$\varphi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$, and
$\psi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$, the functions $b,\beta$ and
$a$ from $[0,T]\times\mathbb{R}^{d}$ to $\mathbb{R}^{d}$ are measurable,
$C_{.}$ is a bounded deterministic function and $T\in\mathbb{R},T>0$. For
simplicity, we assume $d=1$ and we mention that a similar argument of all our
analysis techniques are hold true for $d>1$.
Denote $e_{t,X}=exp\Big{\\{}-\int_{0}^{t}b(s,\rho_{{X}_{s}})ds\Big{\\}}$ and
define

$\tilde{\beta}(t,{\rho}_{{X}_{t}})=\beta(t,{\rho}_{{X}_{t}})e_{t,X},\quad\quad\tilde{a}(t,{\Gamma}_{{X}_{t}})=a(t,{\Gamma}_{{X}_{t}})e_{t,X}.$

######  Assumption 3.1.

The functions $\varphi,\psi,b,a$ and $\beta$ satisfy the following conditions:

  1. 1.

they are continuously differentiable with bounded Lipschitz constants $K$,

  2. 2.

the function $\tilde{a}(.,.)$ is a bounded function on
$[0,T]\times\mathbb{R}$,

  3. 3.

the functions $\tilde{\beta}(.,.)$ has linear growth, i.e., for every
$t\in[0,T]$ and a stochastic process $X$, we have

$\displaystyle|\tilde{\beta}(t,\rho_{X_{t}})|\leq
K\Big{(}1+e_{t,X}\mathbb{E}{|X_{t}|}\Big{)},$

  4. 4.

the functions $\tilde{a}(.,.)$ and $\tilde{\beta}(.,.)$ are Lipschitz
functions, i.e., for every $t,s\in[0,T]$ and stochastic processes $X$ and $Z$,
we have

$\displaystyle|\tilde{\beta}(t,\rho_{X_{t}})-\tilde{\beta}(s,\rho_{Z_{s}})|$
$\displaystyle+|\tilde{a}(t,\Gamma_{X_{t}})-\tilde{a}(s,\Gamma_{Z_{s}})|$
$\displaystyle\leq
K\Big{(}|t-s|+\mathbb{E}\Big{|}e_{t,X}{X_{t}}-e_{s,Z}{Z_{s}}\Big{|}\Big{)}.$

We first show that under Assumption 3.1, the SDE (8) has a unique solution in
some Banach space. Let $\mathcal{E}_{t}:=\mathcal{E}_{t}(C)$ and denote by
$\mathcal{L}^{2,*}([0,T],\mathbb{R})$ the Banach space of
$\mathcal{F}_{t}$-adapted process $M_{t}$, $t\in[0,T]$, satisfying

$\sup_{0\leq t\leq
T}\mathbb{E}\Big{(}e_{t,M_{t}}^{2}M_{t}^{2}\mathcal{E}^{-1}_{t}\Big{)}<\infty,$

and ${L}^{2,*}([0,T],\mathbb{R})$ the Banach space of
$\mathcal{F}_{t}$-adapted process $M_{t}$, $t\in[0,T]$, satisfying

$\sup_{0\leq t\leq
T}\mathbb{E}\Big{(}M_{t}^{2}\mathcal{E}^{-1}_{t}\Big{)}<\infty.$

######  Definition 3.2.

A solution of Equation (8) is a stochastic process
$(X_{t})_{t\in[0,T]}\in\mathcal{L}^{2,*}([0,T],\mathbb{R})$ satisfying (8) and
$exp\Big{\\{}-\int_{0}^{.}b(s,\rho_{{X}_{s}})ds\Big{\\}}X1_{[0,t]}\in
Dom(\delta^{H})$, $t\in[0,T]$.

If, moreover, the function $b$ is uniformly bounded, the Cauchy-Schowartz
inequality implies that
$\mathcal{L}^{2,*}([0,T],\mathbb{R})\subset{L}^{2,*}([0,T],\mathbb{R})$, which
is exactly the case mentioned in [11].
Consider the following stochastic differential equation

$d\tilde{X}_{t}=\tilde{\beta}(t,{\rho}_{\tilde{X}_{t}})dt+\\{C_{t}\tilde{X}_{t}+\tilde{a}(t,{\Gamma}_{\tilde{X}_{t}})\\}dW_{t}^{H},\qquad
t\in[0,T],\quad\tilde{X}_{0}=x,$ (9)

From Ito formula (Proposition 2.2), the following lemma will be result,
obviously.

######  Lemma 3.3.

Under Assumption 3.1, for every solution $X_{t}\in\mathcal{L}^{2,*}$ of SDE
(8), the process $e_{t,X}X_{t}\in L^{2,*}$ is a solution of SDE (9).
Conversely, for every solution $\tilde{X}_{t}\in{L}^{2,*}$ of SDE (9), the
process $e_{t,\tilde{X}}\tilde{X}_{t}\in\mathcal{L}^{2,*}$ is a solution of
SDE (8).

Consequently, following a similar proof of Theorem 3.7 of [11], we verify that
SDE (8) has a unique solution in $X_{t}\in\mathcal{L}^{2,*}$ and also for
every two solutions $X_{t}$ and $Z_{t}$ of SDE (8) with initial conditions
$X_{0}=x$ and $Z_{0}=z$,

$\mathbb{E}\Big{(}\Big{|}e_{t,X}X_{t}-e_{t,Z}{Z}_{t}\Big{|}^{2}\mathcal{E}_{t}^{-1}\Big{)}\leq
c|x-z|^{2}+c\int_{0}^{t}\mathbb{E}\Big{(}\Big{|}e_{s,X}X_{s}-e_{s,Z}{Z}_{s}\Big{|}^{2}\mathcal{E}_{s}^{-1}\Big{)}ds,$

where $c$ is a constant. Therefore, for some positive constant $c_{0}$

$\mathbb{E}\Big{(}\Big{|}e_{t,X}X_{t}-e_{t,Z}{Z}_{t}\Big{|}^{2}\mathcal{E}_{t}^{-1}\Big{)}\leq
c_{0}|x-z|^{2}.$ (10)

These facts stated in the following theorem.

######  Theorem 3.4.

Under Assumption 3.1, SDE (8) has a unique solution
$X_{t}\in\mathcal{L}^{2,*}([0,T],\mathbb{R})$.

Thanks to Theorem 3.4 of [11], the following corollary is deduced immediately.

######  Corollary 3.5.

Under Assumption 3.1, the unique solution $X_{t}\in\mathcal{L}^{2,*}$ of SDE
(8) satisfies

$e_{t,X}X_{t}(\mathcal{T}_{t})\mathcal{E}_{t}^{-1}(\mathcal{T}_{t})=x+\int_{0}^{t}\tilde{\beta}(u,{\rho}_{{X}_{u}})\mathcal{E}_{u}^{-1}(\mathcal{T}_{u})du+\int_{0}^{t}\tilde{a}(u,{\Gamma}_{X_{u}})\mathcal{E}_{u}^{-1}(\mathcal{T}_{u})dW_{u}^{H}.$
(11)

Therefore, the unique solution of (8), for every $0\leq t\leq T$, is of the
form

$X_{t}(\mathcal{T}_{t})=e_{t,X}^{-1}\mathcal{E}_{t}(\mathcal{T}_{t})U_{t}^{(0)},$
(12)

where $U_{t}^{(0)}$ is denoted the rightside of Equation (11). This
representation show that $X_{t}$ is Malliavin differentiable, as the processes
$\mathcal{E}_{t}$ and $U_{t}^{(0)}(\mathcal{A}_{t})$ are differentiable.

######  Example.

(Linear fractional mean-field SDEs) Let the stochastic process
$\\{Y_{t}\\}_{t\in[0,T]}$ be the solution of the linear fractional mean-field
SDE

$dY_{t}=b(t,\rho_{Y_{t}})Y_{t}dt+C_{t}Y_{t}dW_{t}^{H},\qquad t\in[0,T],\qquad
Y_{0}=1,$ (13)

According to (11), the unique solution $Y_{t}$ satisfies

$exp\Big{\\{}-\int_{0}^{t}b(s,\rho_{{Y}_{s}})ds\Big{\\}}Y_{t}(\mathcal{T}_{t})=\mathcal{E}_{t}(\mathcal{T}_{t}).$
(14)

However, Theorem 2.5 of [30], thanks to Proposition 2.2, shows that this
solution is

$Y_{t}=exp\Big{\\{}\int_{0}^{t}C_{s}dW^{H}_{s}-\frac{1}{2}\int_{0}^{t}\int_{0}^{t}\phi(s,s^{\prime})C_{s}C_{s^{\prime}}dsds^{\prime}+\int_{0}^{t}b(s,\rho_{Y_{s}}^{1})ds\Big{\\}},$

which coincides with the expression (14), due to

$\langle K_{H}^{*}C1_{[0,t]},K_{H}^{*}C1_{[0,t]}\rangle_{L^{2}[0,T]}=\langle
1_{[0,t]},1_{[0,t]}\rangle_{\mathcal{H}},$

that results

$\mathcal{E}_{t}=exp\Big{\\{}\int_{0}^{t}C_{s}dW^{H}_{s}-\frac{1}{2}\int_{0}^{t}\int_{0}^{t}\phi(s,s^{\prime})C_{s}C_{s^{\prime}}dsds^{\prime}\Big{\\}}.$

The authors have also shown that $D^{H}_{r}Y_{t}=Y_{t}C_{r}1_{0\leq r\leq t}$,
equivalently $D_{r}^{\phi}Y_{t}=Y_{t}\int_{0}^{t}\phi(r,s)C_{s}ds$ (for
detailes, see Lemma 2.2 of [30] and the relation (7)). This expression, which
shows that the drift coefficient of SDE (13) does not appear in $D^{H}Y_{t}$
directly, in connection with Ito formula lead to conclude that the process
$Y_{t}^{-1}$ satisfying

$\displaystyle dY^{-1}_{t}$
$\displaystyle=\Big{\\{}-b(t,\rho_{Y_{t}}^{1})+2C_{t}\int_{0}^{t}\Phi(s,t)C_{s}ds\Big{\\}}Y^{-1}_{t}dt-
C_{t}Y^{-1}_{t}dW^{H}_{t}\quad t\in[0,T],$ $\displaystyle Y_{0}^{-1}$
$\displaystyle=1,$ (15)

is also differentiable and $D^{H}_{r}Y_{t}^{-1}=-Y_{t}^{-1}C_{r}1_{0\leq r\leq
t}$ and in other sense
$D^{\phi}_{r}Y_{t}^{-1}=-Y_{t}^{-1}\int_{0}^{t}\phi(r,s)C_{s}ds$.

Let us finish this part by giving a transformation to omit the term $a(.,.)$
in the SDE (8) and make a new stochastic process $K_{t}$ which are needed in
sequence. To this end, we need an additional condition in order to have a
unique solutionfor new constructed SDE.
H. Assume that $\frac{d}{dt}\Big{(}e_{t,X}a(t,\Gamma_{X_{t}})/{C_{t}}\Big{)}$
satisfy the parts 3 and 4 of Assumption 3.1.
Apply Ito formula (Proposition 2.2) for the process
$K_{t}=X_{t}+a(t,\Gamma_{X_{t}})/{C_{t}}$ to deduce

$\displaystyle dK_{t}$ $\displaystyle=dX_{t}+\frac{\partial}{\partial
t}\Big{(}\frac{a(t,\Gamma_{X_{t}})}{C_{t}}\Big{)}dt$
$\displaystyle=\Big{\\{}b(t,\rho_{X_{t}})X_{t}+\beta(t,\rho_{X_{t}})\Big{\\}}dt+\frac{\partial}{\partial
t}\Big{(}\frac{a(t,\Gamma_{X_{t}})}{C_{t}}\Big{)}dt+\Big{\\{}C_{t}X_{t}+a(t,\Gamma_{X_{t}})\Big{\\}}dW_{t}^{H}$
$\displaystyle=\Big{\\{}b(t,\rho_{X_{t}})K_{t}+\frac{\partial}{\partial
t}\Big{(}\frac{a(t,\Gamma_{X_{t}})}{C_{t}}\Big{)}-b(t,\rho_{X_{t}})\frac{a(t,\Gamma_{X_{t}})}{C_{t}}+\beta(t,\rho_{X_{t}})\Big{\\}}dt+C_{t}K_{t}dW_{t}^{H}$
$\displaystyle=:\Big{\\{}b(t,\rho_{X_{t}})K_{t}+\beta_{K}(t,\rho_{K_{t}},\Gamma_{K_{t}})\Big{\\}}dt+C_{t}K_{t}dW_{t}^{H}.$

Under condition (H), Corollary 3.5 and the relation (11) yield that

$\displaystyle e_{t,X}K_{t}(\mathcal{T}_{t})$
$\displaystyle=\mathcal{E}_{t}(\mathcal{T}_{t})\Big{(}K_{0}+\int_{0}^{t}\Big{\\{}e_{s,X}\beta_{K}(s,\rho_{K_{s}},\Gamma_{K_{s}})\Big{\\}}\mathcal{E}_{s}^{-1}(\mathcal{T}_{s})ds\Big{)},$
(16)

and since for every $s\leq t$,
$\mathcal{E}_{s}^{-1}\Big{(}\mathcal{T}_{s}(\mathcal{A}_{t})\Big{)}=\mathcal{E}_{s}^{-1}$,
it implies that

$e_{t,X}K_{t}=\mathcal{E}_{t}\Big{(}K_{0}+\int_{0}^{t}\Big{\\{}e_{s,X}\beta_{K}(s,\rho_{K_{s}},\Gamma_{K_{s}})\Big{\\}}\mathcal{E}_{s}^{-1}ds\Big{)}=:\mathcal{E}_{t}V_{t},$
(17)

and

$\sup_{0\leq t\leq T}\mathbb{E}\Big{(}V_{t}^{2}\Big{)}=\sup_{0\leq t\leq
T}\mathbb{E}\Big{(}e_{t}^{2}K_{t}^{2}(\mathcal{T}_{t})\mathcal{E}_{t}^{-2}(\mathcal{T}_{t})\Big{)}=\sup_{0\leq
t\leq
T}\mathbb{E}\Big{(}e_{t}^{2}K_{t}^{2}\mathcal{E}_{t}^{-1}\Big{)}<\infty.$ (18)

##  4 Malliavin’s derivative of $X_{t}$

We will give an expression for the Malliavin derivative of the solution $X$
and obtain a relation between the integral of its derivative and $\mathcal{E}$
in this section. To do this, according to the definition of $K_{t}$, it is
sufficient to find an exposition of $D^{H}K_{t}$ as a solution of some
stochastic differential equation.
Taking the derivative of both sides of (17) and substituting the derivative of
$\mathcal{E}^{-1}_{s}$ from Example Example in it to reach the following
expression for the Malliavin derivative of $K_{t}$.

$\displaystyle e_{t,X}D_{r}^{H}K_{t}$
$\displaystyle=\mathcal{E}_{t}C_{r}1_{[0,t]}(r)V_{t}-\mathcal{E}_{t}C_{r}1_{[0,t]}(r)\Big{(}\int_{r}^{t}\Big{\\{}e_{s,X}\beta_{K}(s,\rho_{K_{s}},\Gamma_{K_{s}})\Big{\\}}\mathcal{E}_{s}^{-1}ds\Big{)}$
$\displaystyle=\mathcal{E}_{t}C_{r}1_{[0,t]}(r)\Big{(}K_{0}+\int_{0}^{r}\Big{\\{}e_{s,X}\beta_{K}(s,\rho_{K_{s}},\Gamma_{K_{s}})\Big{\\}}\mathcal{E}_{s}^{-1}ds\Big{)}$
$\displaystyle=\mathcal{E}_{t}C_{r}1_{[0,t]}(r)e_{r,X}K_{r}\mathcal{E}^{-1}_{r}.$
(19)

In view of Corollary 3.5, we conclude the form of stochastic differential
equation associated with the weak derivative $D^{H}K_{t}$.

######  Proposition 4.1.

Under Assumption 3.1 and H, the Malliavin derivative of $K_{t}$ satisfies the
following equation

$D_{r}^{H}K_{t}=C_{r}K_{r}1_{[0,t]}(r)+\int_{r}^{t}b(s,\rho_{X_{s}})D_{r}^{H}K_{s}ds+\int_{r}^{t}C_{s}D_{r}^{H}K_{s}dW^{H}_{s},$
(20)

for every $0\leq r\leq t$. In addition, for every $h\in\mathcal{H}$

$\displaystyle<D^{H}K_{t},h>_{\mathcal{H}}$
$\displaystyle=<C_{.}K_{.}1_{[0,t]},h>_{\mathcal{H}}+\int_{0}^{t}b(s,\rho_{X_{s}})<D^{H}K_{s},h>_{\mathcal{H}}ds$
$\displaystyle+\int_{0}^{t}C_{s}<D^{H}K_{s},h>_{\mathcal{H}}dW^{H}_{s}.$

######  Remark 4.2.

For $h=1_{[0,t]}$, in view of (7), the process $D^{\phi}_{r}K_{t}$ is the
solution of the following SDE.

$D^{\phi}_{r}K_{t}=\int_{0}^{t}b(s,\rho_{X_{s}})D^{\phi}_{r}K_{s}ds+\int_{0}^{t}C_{u}K_{u}\phi(r,u)du+\int_{0}^{t}C(s)D^{\phi}_{r}K_{s}dW^{H}_{s},$

with $D^{\phi}_{r}K_{0}=0$ and for every $r\geq 0$.

Now, we are ready to establish a relation, that demonstrates the process
$\mathcal{E}$ as an inner product of the Malliavin derivative with other
Skorokhod integral processes needed in the Bismut formula.

######  Remark 4.3.

If $\beta_{K}(s,\rho_{K_{s}},\Gamma_{K_{s}})=0$ for every $s\in[0,T]$,
especially if $a(.,.)=\beta(.,.)=0$, then for $K_{0}\neq 0$ we have

$\int_{0}^{T}\frac{1}{K_{0}T}C_{r}^{-1}1_{[0,t]}(r)D_{r}^{H}K_{t}dr=e_{t,X}^{-1}\mathcal{E}_{t}.$
(21)

Also, if $K_{0}\beta_{K}(s,\rho_{K_{s}},\Gamma_{K_{s}})>0$, for every
$s\in[0,T]$, then

$\int_{0}^{T}\frac{1}{T}V_{r}^{-1}C_{r}^{-1}1_{[0,t]}(r)D_{r}^{H}K_{t}dr=e_{t,X}^{-1}\mathcal{E}_{t}.$

Nevertheless, in the case $\beta_{K}(s,\rho_{K_{s}},\Gamma_{K_{s}})\neq 0$,
the process $K_{r}$ would not be invertible almost everywhere, in general,
then the process $e_{t}^{-1}\mathcal{E}_{t}$ can not be obtain with respect to
the derivative of $K_{t}$ by taking inversion of (19) and integrate of that
with respect to $r$. So, we take a new trick to reach this goal in the
sequence of this section.
Let

$S=\inf\Big{\\{}t>0;\int_{0}^{t}1_{\\{V_{s}\neq 0\\}}ds>0\Big{\\}}\wedge T.$

It follows that for every $S\leq t$, there exists a set $A\subset(0,t]$ of
positive Lebesque measure such that for any $s\in A$,

$V_{s}>\lambda(s)>0.$

In sequence, in addition to Assumption 3.1, we need also the following
assumption.

######  Assumption 4.4.

The function $C$ is invertible and $C^{-1}$ is Hölder continuous of order
$\alpha_{0}\in(H-\frac{1}{2},1]$;

$|C^{-1}_{t}-C^{-1}_{s}|\leq K|t-s|^{\alpha_{0}},\quad t,s\in[0,T]$

Now, the following main result will establish on the set $\\{S<t\\}$.

######  Theorem 4.5.

Under Assumptions 3.1, 4.4 and H, for every $0<t\leq T$, on the set
$\\{S<t\\}$, there exists some random variable $V_{t}^{0}\in\mathcal{H}\cap
Dom(K_{H}^{*})$ satisfying

$\langle
D^{H}K_{t},V_{t}^{0}\rangle_{\mathcal{H}}=e_{t,X}^{-1}\mathcal{E}_{t},$

and $K_{H}^{*}V_{T}^{0}\in Dom(\delta_{W})$.

###### Proof.

According to the assumption $\\{S<t\\}$, there exists a set $A\subset(0,t]$ of
positive Lebesque measure such that for any $s\in A$,

$V_{s}\geq\lambda(s),$

where $\lambda(s)>0$. Therefore,

$\int_{0}^{t}V_{s}^{2}ds\geq\int_{A}\lambda(s)ds=:k>0.$

Multiply both sides of (19) in $V_{r}$ to deduce

$\displaystyle\int_{0}^{T}C_{r}^{-1}1_{[0,t]}(r)D_{r}^{H}K_{t}V_{r}dr$
$\displaystyle=e_{t,X}^{-1}\mathcal{E}_{t}\Big{(}\int_{0}^{t}V_{r}^{2}dr\Big{)}.$
(22)

Now, define

$V_{2,t}(r)=C_{r}^{-1}1_{[0,t]}(r)\frac{V_{r}}{\int_{0}^{t}V_{r}^{2}dr}.$

From Assumption 4.4, we see that $V_{2,t}\in L^{2}[0,T]$. Also, it is easily
verified that $V_{2,t}\in I_{0}^{H+\frac{1}{2}}(L^{2}[0,T];\mathbb{R})$. Thus
there exists some random variable $V_{t}^{0}\in Dom(K_{H}^{*})$ such that
$K_{H}oK_{H}^{*}V_{t}^{0}=\int_{0}^{.}V_{2,t}(r)dr$. To complete the proof, it
is sufficient to show that $K_{H}^{-1}(\int_{0}^{.}V_{2,t}(r)dr)\in
Dom(\delta_{W})$. Since ${D}_{W}^{1,2}\subset Dom(\delta_{W})$ and
$K_{H}^{*}V_{t}^{0}$ is an adapted process, (see Proposition 1.3.11 in [46]),
we will show that it belongs to $L^{2}([0,T]\times\Omega)$. To this end, as
$t\neq T$, for some constant $c$, we derive

$\displaystyle K_{H}^{-1}\Big{(}\int_{0}^{.}V_{2,t}(r)dr\Big{)}(u)$
$\displaystyle 1_{t<u\leq T}\int_{0}^{t}V_{s}^{2}ds$
$\displaystyle=(H-\frac{1}{2})u^{H-\frac{1}{2}}1_{t<u\leq
T}\int_{0}^{u}\frac{-s^{\frac{1}{2}-H}1_{[0,t]}(s)C_{s}^{-1}V_{s}}{(u-s)^{H+\frac{1}{2}}}ds$
$\displaystyle\leq(H-\frac{1}{2})u^{H-\frac{1}{2}}1_{t<u\leq
T}\Big{(}\int_{0}^{u}\frac{s^{1-2H}}{(u-s)^{2H+1}}ds\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{t}C_{s}^{-2}V_{s}^{2}ds\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq(H-\frac{1}{2})\frac{u^{H-\frac{1}{2}}}{(u-t)^{H+\frac{1}{2}}}1_{t<u\leq
T}\Big{(}\int_{0}^{u}s^{1-2H}ds\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{t}C_{s}^{-2}V_{s}^{2}ds\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq
c\frac{1}{2}(\frac{u}{u-t})^{H+\frac{1}{2}}u^{-1-H}1_{t<u\leq
T}\Big{(}\int_{0}^{t}V_{s}^{2}ds\Big{)}^{\frac{1}{2}},$

which implies that

$\displaystyle\mathbb{E}\Big{(}\int_{t}^{T}1_{S<t}\Big{[}K_{H}^{-1}\Big{(}\int_{0}^{.}V_{2,t}(r)dr\Big{)}(u)\Big{]}^{2}du\Big{)}\leq\frac{c}{\sqrt{k}}\mathbb{E}\Big{(}\int_{t}^{T}(\frac{u}{u-t})^{2H+1}u^{-2-2H}du\Big{)}<\infty.$

Now, if $0\leq u\leq t$, which also includes the case $t=T$, we know that

$\displaystyle K_{H}^{-1}\Big{(}\int_{0}^{.}V_{2,t}(r)dr\Big{)}(u)$
$\displaystyle 1_{[0,t]}(u)\int_{0}^{t}V_{s}^{2}ds$
$\displaystyle=\frac{1}{\Gamma(\frac{3}{2}-H)}\Big{(}u^{\frac{1}{2}-H}1_{[0,t]}(u)C_{u}^{-1}V_{u}$
$\displaystyle+(H-\frac{1}{2})C_{u}^{-1}V_{u}1_{[0,t]}(u)\int_{0}^{u}\frac{(1_{[0,t]}(u)-1_{[0,t]}(s))}{(u-s)^{H+\frac{1}{2}}}ds$
$\displaystyle+(H-\frac{1}{2})u^{H-\frac{1}{2}}C_{u}^{-1}V_{u}1_{[0,t]}(u)\int_{0}^{u}\frac{u^{\frac{1}{2}-H}-s^{\frac{1}{2}-H}}{(u-s)^{H+\frac{1}{2}}}1_{[0,t]}(s)ds$
$\displaystyle+(H-\frac{1}{2})u^{H-\frac{1}{2}}V_{u}1_{[0,t]}(u)\int_{0}^{u}\frac{s^{\frac{1}{2}-H}1_{[0,t]}(s)(C_{u}^{-1}-C^{-1}_{s})}{(u-s)^{H+\frac{1}{2}}}ds$
$\displaystyle+(H-\frac{1}{2})u^{H-\frac{1}{2}}1_{[0,t]}(u)\int_{0}^{u}\frac{s^{\frac{1}{2}-H}1_{[0,t]}(s)C^{-1}_{s}(V_{u}-V_{s})}{(u-s)^{H+\frac{1}{2}}}ds\Big{)}$
$\displaystyle=:\frac{1}{\Gamma(\frac{3}{2}-H)}\Big{(}I_{1}+I_{2}+I_{3}+I_{4}+I_{5}\Big{)}.$

Since $0\leq u\leq t$,

$u^{2H-1}\int_{0}^{u}\frac{u^{\frac{1}{2}-H}-s^{\frac{1}{2}-H}}{(u-s)^{H+\frac{1}{2}}}1_{[0,t]}(s)ds=\int_{0}^{1}\frac{r^{\frac{1}{2}-H}-1}{(1-r)^{\frac{1}{2}+H}}dr=:g_{0}<\infty,$
(23)

and

$\int_{0}^{u}\frac{(1_{[0,t]}(u)-1_{[0,t]}(s))}{(u-s)^{H+\frac{1}{2}}}ds=0.$

From Assumption 4.4, it follows that for some constant $c$

$u^{2H-1}1_{[0,t]}(u)\Big{(}\int_{0}^{u}\frac{s^{\frac{1}{2}-H}1_{[0,t]}(s)(C_{u}^{-1}-C^{-1}_{s})}{(u-s)^{H+\frac{1}{2}}}ds\Big{)}^{2}=cu^{2\alpha_{0}-2H+1}\leq
ct^{2\alpha_{0}-2H+1}.$

These imply that

$\mathbb{E}\Big{(}\int_{0}^{T}1_{S<t}\frac{I_{2}^{2}(u)+I_{3}^{2}(u)+I_{4}^{2}(u)}{\Big{(}\int_{0}^{t}V_{s}^{2}ds\Big{)}^{2}}du\Big{)}<\infty.$

Additionally, from (18)

$\mathbb{E}\Big{(}\int_{0}^{T}1_{S<t}\frac{I_{1}^{2}(u)}{\Big{(}\int_{0}^{t}V_{s}^{2}ds\Big{)}^{2}}du\Big{)}\leq\sup_{0\leq
s\leq
t}\mathbb{E}\Big{(}1_{S<t}\frac{V_{s}^{2}}{\Big{(}\int_{0}^{t}V_{s}^{2}ds\Big{)}^{2}}\Big{)}\int_{0}^{t}u^{1-2H}du<\infty.$

Now, to bound the last term $I_{5}$, we apply the definition of $V_{t}$ and
Hölder inequality. Therefore, for some constant $c$,

$\displaystyle\Big{(}\int_{0}^{u}\frac{s^{\frac{1}{2}-H}1_{[0,t]}(s)C^{-1}_{s}(V_{u}-V_{s})}{(u-s)^{H+\frac{1}{2}}}ds\Big{)}^{2}$
$\displaystyle\leq
c\int_{0}^{u}\frac{s^{1-2H}}{(u-s)^{2H}}(\int_{s}^{u}\mathcal{E}_{v}^{-2}dv)ds$
$\displaystyle\leq c\sup_{0\leq v\leq
t}\mathcal{E}_{v}^{-2}\int_{0}^{u}\frac{s^{1-2H}}{(u-s)^{2H-1}}ds$
$\displaystyle\leq\frac{2c}{2-2H}u^{2-2H}\sup_{0\leq v\leq
t}\mathcal{E}_{v}^{-2}.$

which yields finally

$\mathbb{E}\Big{(}\int_{0}^{T}1_{S<t}\frac{I_{5}^{2}(u)}{\Big{(}\int_{0}^{t}V_{s}^{2}ds\Big{)}^{2}}du\Big{)}\leq\frac{c}{2}t^{2}\mathbb{E}\Big{(}1_{S<t}\frac{\sup_{0\leq
s\leq
t}\mathcal{E}_{s}^{-2}}{\Big{(}\int_{0}^{t}V_{s}^{2}ds\Big{)}^{2}}\Big{)}<\infty,$

for some another constant $c>0$. These facts together, show that
$K_{H}^{-1}\Big{(}\int_{0}^{.}V_{2,t}(r)dr\Big{)}\in Dom(\delta_{W})$ and the
proof is comoleted. ∎

######  Remark 4.6.

In especial case, Remark 3.1, the stochastic process $V_{2,t}(r)$ will be in
the form

$V_{2,t}(r)=\frac{1}{T}V_{r}^{-1}C_{r}^{-1}1_{[0,t]}(r),$

Obviously, this process belongs to
$I_{0}^{H+\frac{1}{2}}(L^{2}[0,T];\mathbb{R})$ and also
$K_{H}^{-1}(\int_{0}^{.}C_{r}^{-1}1_{[0,t]}(r)dr)\in
Dom\delta_{W}\cap\mathcal{H}$.

##  5 Stochastic Flow of The Equation

This section is devoted to proving the existence of the stochastic flow of the
solution $K_{t}$ and establishing a relationship between this flow and the
Malliavin derivative of $K_{t}$, which will be essential in the remainder of
the paper.
Denote by $K_{t}^{u}$( and $X_{t}^{x}$ ) the solution of (9)( and (8)) with
initial condition $u$( and $x$). Also, let
$R(t,\rho_{X_{t}}):=\partial_{2}b(t,\rho_{X^{x}_{t}})\frac{\partial\rho_{X^{x}_{t}}}{\partial
x}$ and $r(t,f_{X_{t}}):=\frac{\partial}{\partial
x}\beta_{K}(t,\rho_{K_{t}},\Gamma_{K_{t}})$, where $\partial_{2}b(.,.)$ is the
derivative of $b$ with respect to the second component.
We need some additional conditions to show the existence of the stochastic
flow of the solution.

######  Assumption 5.1.

i) The function $e_{.,.}r(.,.)$ is a bounded measurable function and for every
$t,s\in[0,T]$ and stochastic processes $X$ and $Z$, we have

$\displaystyle|e_{t,X}r(t,f_{X_{t}})-e_{s,Z}r(s,f_{Z_{s}})|\leq
K\Big{(}|t-s|+\mathbb{E}\Big{|}e_{t,X}{X_{t}}-e_{s,Z}{Z_{s}}\Big{|}\Big{)},$

ii) The function $\frac{\partial}{\partial x}b(t,\rho_{X_{t}})$ is a bounded
measurable function and for every $t,s\in[0,T]$ and stochastic processes $X$
and $Z$, we have

$\displaystyle|R(t,\rho_{X_{t}})-R(s,\rho_{Z_{s}})|\leq
K^{\prime}|t-s|^{\alpha_{1}},$

where $K^{\prime}$ is a constant and $\alpha_{1}\in(H-\frac{1}{2},1]$.
iii) The function $e_{.,.}\frac{\partial}{\partial x}a(.,.)$ is a bounded
measurable function.

######  Theorem 5.2.

Under conditions (i) and (ii) of Assumption 5.1, the function $u\rightarrow
K_{t}^{u}$ is contiuously differentiable and $K_{t}^{K_{0}}$ admits a
stochastic flow satisfying the following SDE:

$\displaystyle d\frac{\partial K_{t}}{\partial x}$
$\displaystyle=\Big{\\{}b(t,\rho_{X_{t}^{x}})\frac{\partial K_{t}}{\partial
x}+R(t,\rho_{X_{t}^{x}})K_{t}+r(t,f_{X_{t}^{x}})\Big{\\}}dt+C_{t}\frac{\partial
K_{t}}{\partial{x}}dW^{H}_{t},$ (24)

with the initial condition $\frac{\partial}{\partial
x}K_{0}=1+\frac{1}{C_{0}}\frac{\partial a(0,\psi(x))}{\partial
x}\psi^{\prime}(x)=:k_{0}$,

###### Proof.

It follows from (17), part (i) of Assumption 5.1 and Hölder inequality that

$\displaystyle|e_{t,X^{x+h}}K_{t}^{K_{0}+h}-e_{t,X^{x}}K_{t}^{K_{0}}|^{2}$
$\displaystyle\leq\mathcal{E}_{t}^{2}\Big{(}h+\int_{0}^{t}|e_{s,X^{x+h}}r(s,f_{X_{s}^{x+h}})-e_{s,X^{x}}r(s,\rho_{X_{s}^{x}})|\mathcal{E}_{s}^{-1}ds\Big{)}^{2}$
$\displaystyle\leq\mathcal{E}_{t}^{2}h^{2}+\mathcal{E}_{t}^{2}\Big{(}\int_{0}^{t}\mathcal{E}_{s}^{-1}ds\Big{)}\int_{0}^{t}\mathbb{E}\Big{(}|e_{s,X^{x+h}}{X_{s}^{x+h}}-e_{s,X^{x}}{X_{s}^{x}}|^{2}\mathcal{E}_{s}^{-1}\Big{)}ds$
$\displaystyle\leq\mathcal{E}_{t}^{2}(1+c_{0}t\int_{0}^{t}\mathcal{E}_{s}^{-1}ds)h^{2},$

where we used (10) in the last inequality. Therefore, the function
$x\rightarrow e_{t,X^{x}}K_{t}^{K_{0}}$ is differentiable. This result in
connection with differentiability of $b(.,.)$ deduce that the function
$u\rightarrow K_{t}^{u}$ is continuously differentiable.
Now, take the derivative with respect to x from (17) to derive

$\displaystyle e_{t,X}\frac{\partial K_{t}}{\partial
x}+e_{t,X}K_{t}\int_{0}^{t}R(s,\rho_{X_{s}^{x}})ds$
$\displaystyle=\mathcal{E}_{t}\Big{[}\frac{\partial}{\partial
x}K_{0}+\int_{0}^{t}e_{s,X}r(s,f_{X_{s}^{x}})\mathcal{E}_{s}^{-1}ds$
$\displaystyle+\int_{0}^{t}e_{s,X}\Big{(}\int_{0}^{s}R(u,\rho_{X_{u}^{x}})du\Big{)}\beta_{K}(s,\rho_{K_{s}},\Gamma_{K_{s}})\mathcal{E}_{s}^{-1}ds\Big{]}.$
(25)

Integration by parts formula shows that

$\displaystyle\int_{0}^{t}e_{s,X}R(s,\rho_{X_{s}^{x}})K_{s}\mathcal{E}_{s}^{-1}ds$
$\displaystyle=e_{t,X}K_{t}\mathcal{E}_{t}^{-1}\int_{0}^{t}R(s,\rho_{X_{s}^{x}})ds$
$\displaystyle-\int_{0}^{t}e_{s,X}\Big{(}\int_{0}^{s}R(u,\rho_{X_{u}^{x}})du\Big{)}\beta_{K}(s,\rho_{K_{s}},\Gamma_{K_{s}})\mathcal{E}_{s}^{-1}ds.$
(26)

Substituting (26) into (25) results that

$\displaystyle e_{t,X}\frac{\partial K_{t}}{\partial x}$
$\displaystyle=\mathcal{E}_{t}\Big{[}\frac{\partial}{\partial
x}K_{0}+\int_{0}^{t}e_{s,X}r(s,f_{X_{s}^{x}})\mathcal{E}_{s}^{-1}ds+\int_{0}^{t}e_{s,X}R(s,\rho_{X_{s}^{x}})K_{s}\mathcal{E}_{s}^{-1}ds\Big{]}.$

Similar argument of obtaining $K_{t}$, and according to Corollary 3.5 in
connection with Assumption 5.1, state that $K_{t}$ is a solution of SDE (24).
∎

To find a relationship between the flow of $K_{t}$ and its Malliavin
derivative, let us define the process $\mathcal{M}^{h}_{t}$ for every
$h\in\mathcal{H}$ as the solution of the following SDE

$d\mathcal{M}^{h}_{t}=b(t,\rho_{X_{t}^{x}})\mathcal{M}^{h}_{t}dt+r(t,f_{X_{t}^{x}})dt+C_{t}\mathcal{M}^{h}_{t}dW_{t}^{H},\qquad\mathcal{M}^{h}_{0}=k_{0}.$
(27)

According to Theorem 3.4 and Corollary 3.5, this equation has a unique
solution of the form

$\mathcal{M}^{h}_{t}=e_{t,X}^{-1}\mathcal{E}_{t}\Big{(}k_{0}+\int_{0}^{t}r(s)e_{s,X}\mathcal{E}_{s}^{-1}ds\Big{)},$

######  Theorem 5.3.

Under Assumptions 3.1, 4.4, 5.1 and H, there exists some $h\in\mathcal{H}\cap
Dom(\delta_{W})$ such that

$\frac{\partial K_{t}}{\partial
x}-<D^{H}K_{t},h>_{\mathcal{H}}=\mathcal{M}^{h}_{t}.$

###### Proof.

ۤFollowing a similar argument of Proposition 3.1 in [22], we verify that

$<C_{.}K_{.}1_{[0,t]},h>_{\mathcal{H}}=\int_{0}^{t}C_{s}K_{s}d(R_{H}h)(s),\quad{for~{}all}~{}h\in\mathcal{H}.$

Take $\mathcal{N}^{h}_{t}:=\frac{\partial K_{t}}{\partial
x}-<D^{H}K_{t},h>_{\mathcal{H}}$ and use Ito formula to derive

$\displaystyle
d\mathcal{N}^{h}_{t}=b(t,\rho_{X_{t}^{x}})\mathcal{N}^{h}_{t}dt+R(t,\rho_{X_{t}^{x}})K_{t}dt-C(t)K_{t}d(R_{H}h)(t)+r(t,f_{X_{t}^{x}})dt+C_{t}\mathcal{N}^{h}_{t}dW_{t}^{H}.$

The function $C_{.}^{-1}R(.,.)\in
I_{0}^{H+\frac{1}{2}}L^{2}([0,T];\mathbb{R})$. So, one can choose a function
$h\in\mathcal{H}$ satisfying

$R_{H}h(t)=\int_{0}^{t}C_{s}^{-1}R(s,\rho_{X_{s}^{x}})ds.$ (28)

In this case, from the uniqueness of the solution of (27), we conclude that
$\mathcal{N}^{h}_{t}=\mathcal{M}^{h}_{t}$ in $L^{2,*}([0,T]\times\Omega)$. To
complete the assertion, it is sufficient to show that $K_{H}^{-1}(R_{H}h)\in
Dom(\delta_{W})$. The definition of $K_{H}^{-1}$ follows that

$\displaystyle
K_{H}^{-1}\Big{(}\int_{0}^{.}C_{r}^{-1}R(r,\rho_{X_{r}^{x}})dr\Big{)}(u)$
$\displaystyle=\frac{1}{\Gamma(\frac{3}{2}-H)}\Big{(}u^{\frac{1}{2}-H}C_{u}^{-1}R(u,\rho_{X_{u}^{x}})$
$\displaystyle+(H-\frac{1}{2})u^{H-\frac{1}{2}}C_{u}^{-1}R(u,\rho_{X_{u}^{x}})\int_{0}^{u}\frac{u^{\frac{1}{2}-H}-s^{\frac{1}{2}-H}}{(u-s)^{H+\frac{1}{2}}}ds$
$\displaystyle+(H-\frac{1}{2})u^{H-\frac{1}{2}}R(u,\rho_{X_{u}^{x}})\int_{0}^{u}\frac{s^{\frac{1}{2}-H}(C_{u}^{-1}-C^{-1}_{s})}{(u-s)^{H+\frac{1}{2}}}ds$
$\displaystyle+(H-\frac{1}{2})u^{H-\frac{1}{2}}\int_{0}^{u}\frac{s^{\frac{1}{2}-H}C^{-1}_{s}(R(u,\rho_{X_{u}^{x}})-R(s,\rho_{X_{s}^{x}}))}{(u-s)^{H+\frac{1}{2}}}ds\Big{)}.$

Similar to the proof of Theorem 4.5 and notice to Assumption 5.1, we verify
that

$\mathbb{E}\Big{(}\int_{0}^{T}\Big{(}K_{H}^{-1}\Big{(}\int_{0}^{.}C_{r}^{-1}R(r,\rho_{X_{r}^{x}})dr\Big{)}(u)\Big{)}^{2}du\Big{)}<\infty.$

which completes the proof. ∎

Now, define the stochastic process

$\mathcal{W_{HT}}(t):=\frac{\mathcal{M}^{h}_{t}}{<D^{H}K_{t},V_{t}^{0}>_{\mathcal{H}}},$

where $V_{t}^{0}\in\mathcal{H}\cap Dom(K^{*}_{H})$, defined in Theorem 4.5.

######  Corollary 5.4.

For the stochastic process $V_{t}^{0}\in\mathcal{H}\cap Dom(K^{*}_{H})$, we
have

$\mathcal{W_{HT}}(t)=k_{0}+\int_{0}^{t}e_{s,X}r(s,f_{X_{s}^{x}})\mathcal{E}_{s}^{-1}ds\in
Dom(\delta_{W}).$

##  6 Bismut-Elworthy-Li formula

In this section, we find a Bismut-Elworthy-Li type formula with the help of
Theorems 4.5 and 5.3. The following lemma prepares some requirements to reach
this main result.

######  Lemma 6.1.

Assume that $F,G\in D_{W}^{1,2}\cap D^{1,2}$, and continuously differentiable
bounded function $\phi$ is given. For every $v\in\mathcal{H}\cap
Dom(K^{*}_{H})$ such that $<D^{H}F,v>_{\mathcal{H}}\neq 0$ and every random
variable $G$ such that $K_{H}^{*}v\frac{G}{<D^{H}F,v>_{\mathcal{H}}}\in
Dom(\delta_{W})$, it holds that

$\mathbb{E}\Big{(}\phi^{\prime}(F)G\Big{)}=\mathbb{E}\left(\phi(F)\delta^{H}\Big{(}(K_{H}^{*})^{-1}\frac{GK_{H}^{*}v}{<D^{H}F,v>_{\mathcal{H}}}\Big{)}\right).$

###### Proof.

Due to Proposition 2.5 and Proposition 2.4, for $u=K_{H}^{*}v$ we derive

$\displaystyle\mathbb{E}\Big{(}\phi^{\prime}(F)G\Big{)}$
$\displaystyle=\mathbb{E}\left(\phi(F)\delta\Big{(}\frac{Gu}{<DF,u>_{L^{2}[0,T]}}\Big{)}\right)$
$\displaystyle=\mathbb{E}\left(\phi(F)\delta\Big{(}\frac{GK_{H}^{*}v}{<K_{H}^{*}D^{H}F,K_{H}^{*}v>_{L^{2}[0,T]}}\Big{)}\right)$
$\displaystyle=\mathbb{E}\left(\phi(F)\delta\Big{(}\frac{GK_{H}^{*}v}{<D^{H}F,v>_{\mathcal{H}}}\Big{)}\right)$
$\displaystyle=\mathbb{E}\left(\phi(F)\delta^{H}\Big{(}(K_{H}^{*})^{-1}\frac{GK_{H}^{*}v}{<D^{H}F,v>_{\mathcal{H}}}\Big{)}\right).$

∎

######  Theorem 6.2.

(Bismut-Elworthy-Li formula I) Under Assumption 5.1, for every measurabe
function $\phi:\mathbb{R}^{d}\longrightarrow\mathbb{R}$ such that
$\phi(K_{t}^{K_{0}})\in{L}^{2}(\Omega)$ and $\phi\in C_{b}^{1}(\mathbb{R})$
(the space of bounded and continuously differentiable functions on
$\mathbb{R}$), sensitivity of the payoff $\phi$ with respect to the initial
value $x$ is given by the following equation.

$\displaystyle\frac{\partial}{\partial
x}\mathbb{E}\left(\phi(K_{t}^{K_{0}})\right)$
$\displaystyle=\mathbb{E}\left(\phi(K_{t}^{K_{0}})\delta^{H}\Big{(}1_{S\leq
t}h+(K_{H}^{*})^{-1}\mathcal{W_{HT}}(t)1_{S\leq
t}K_{H}^{-1}w_{t}\Big{)}\right)$
$\displaystyle=\mathbb{E}\left(\phi(K_{t}^{K_{0}})\delta\Big{(}1_{S\leq
t}K_{H}^{-1}\\{\int_{0}^{.}C_{s}^{-1}1_{[0,t]}(s)R(s,\rho_{X_{s}^{x}})ds\\}\Big{)}\right)$
$\displaystyle+\mathbb{E}\left(\phi(K_{t}^{K_{0}})\delta\Big{(}1_{S\leq
t}\mathcal{W_{HT}}(t)K_{H}^{-1}w_{t}\Big{)}\right),$

for $h$ satisfying (28). Especially, as
$\beta_{K}(t,\rho_{K_{t}},\Gamma_{K_{t}})=0$,

$\frac{\partial}{\partial
x}\mathbb{E}\Big{(}\phi(K_{t}^{K_{0}})\Big{)}=\mathbb{E}\left(\phi(K_{t}^{K_{0}})\delta\Big{(}1_{S\leq
t}K_{H}^{-1}\Big{\\{}\int_{0}^{.}C_{s}^{-1}1_{[0,t]}(s)[R(s,\rho_{X_{s}^{x}})+\frac{1}{K_{0}T}]ds\Big{\\}}\Big{)}\right).$

###### Proof.

For simplicity we use $K_{t}$ instead of $K_{t}^{K_{0}}$ and note that on the
set $\\{S>t\\}$, the process $K_{t}$ is zero. From Theorem 5.3, Lemma 6.1 and
Theorem 4.5, for $V_{t}^{0}\in\mathcal{H}\cap Dom(K^{*}_{H})$, we result

$\displaystyle\frac{\partial}{\partial x}\mathbb{E}\Big{(}\phi(K_{t})\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}1_{S>t}\phi^{\prime}(K_{t})\frac{\partial
K_{t}}{\partial x}\Big{)}+\mathbb{E}\Big{(}1_{S\leq
t}\phi^{\prime}(K_{t})\frac{\partial K_{t}}{\partial x}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\phi^{\prime}(K_{t})1_{S\leq
t}<D^{H}K_{t},h>_{\mathcal{H}}\Big{)}+\mathbb{E}\Big{(}\phi^{\prime}(K_{t})1_{S\leq
t}\mathcal{M}^{h}_{t}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\phi(K_{t})\delta^{H}(1_{S\leq
t}h)\Big{)}+\mathbb{E}\Big{(}\phi(K_{t})\delta^{H}\Big{(}(K_{H}^{*})^{-1}\frac{1_{S\leq
t}\mathcal{M}^{h}_{t}K_{H}^{-1}(\int_{0}^{.}V_{2,t}(s)ds)}{<D^{H}K_{t},V_{t}^{0}>_{\mathcal{H}}}\Big{)}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\phi(K_{t})\delta^{H}(1_{S\leq
t}h)\Big{)}+\mathbb{E}\Big{(}\phi(K_{t})\delta^{H}\Big{(}(K_{H}^{*})^{-1}(1_{S\leq
t}\mathcal{W_{HT}}(t)K_{H}^{-1}w_{t})\Big{)}\Big{)},$

where $w_{t}=\int_{0}^{.}V_{2,t}(s)ds$. It implies the first part of the
claim. Especially, from Equation (21),
$w_{t}=\int_{0}^{.}\frac{1}{K_{0}T}C_{s}^{-1}1_{[0,t]}(s)ds$ and the second
part of the claim is held exactly in the same way. ∎

We are actually able to rewrite the above theorem for the stochastic process
$X$ by considering the relation between two processes $K_{t}$ and $X_{t}$.

######  Theorem 6.3.

(Bismut-Elworthy-Li formula) Assume that the measurabe function
$\phi:\mathbb{R}^{d}\longrightarrow\mathbb{R}$ belongs to
$C_{b}^{1}(\mathbb{R})$ and $\phi(X_{T}^{x})\in L^{2}(\Omega)$. Then under
Assumption 5.1,

$\displaystyle\frac{\partial}{\partial
x}\mathbb{E}\Big{(}\phi(X_{T}^{x})\Big{)}$
$\displaystyle=\mathbb{E}\left(\phi(X_{T}^{x})\delta^{H}\Big{(}h+(K_{H}^{*})^{-1}\\{U(T)K_{H}^{-1}w_{T}\\}\Big{)}\right)$
$\displaystyle=\mathbb{E}\left(\phi(X_{T}^{x})\delta\Big{(}K_{H}^{-1}\Big{\\{}\int_{0}^{.}C_{s}^{-1}1_{[0,t]}(s)R(s,\rho_{X_{s}^{x}})ds\Big{\\}}+U(T)K_{H}^{-1}w_{T}\Big{)}\right).$

where $U(T)=\mathcal{W_{HT}}(T)-\frac{\frac{\partial}{\partial
x}a(T,\Gamma_{X^{x}_{T}})}{C_{T}}e_{T}\mathcal{E}_{T}^{-1}$ and $h$ satisfying
(28).

###### Proof.

From Theorem 6.2 and Lemma 6.1 as $G=1$, in connection with the fact
$D^{H}K_{t}=D^{H}X_{t}$,

$\displaystyle\frac{\partial}{\partial
x}\mathbb{E}\Big{(}\phi(X_{T}^{x})\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\phi^{\prime}(K_{T}^{k_{0}}-\frac{a(T,\Gamma_{X_{T}^{x}})}{C_{T}})\left[\frac{\partial
K_{T}}{\partial x}-\frac{\frac{\partial}{\partial
x}a(T,\Gamma_{X_{T}^{x}})}{C_{T}}\right]\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\phi({X_{T}^{x}})\delta^{H}(h)\Big{)}+\mathbb{E}\Big{(}\phi(X_{T}^{x})\delta^{H}\Big{(}(K_{H}^{*})^{-1}\\{\mathcal{W_{HT}}(T)K_{H}^{-1}w_{T}\\}\Big{)}\Big{)}$
$\displaystyle-\mathbb{E}\Big{(}\phi(X_{T}^{x})\delta^{H}\Big{(}(K_{H}^{*})^{-1}\frac{\frac{\partial}{\partial
x}a(T,\Gamma_{X_{T}^{x}})}{C_{T}}\frac{K_{H}^{*}V_{T}^{0}}{<D^{H}K_{T},V_{T}^{0}>_{\mathcal{H}}}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\phi(X_{T}^{x})\delta^{H}(h)\Big{)}+\mathbb{E}\Big{(}\phi(X_{T}^{x})\delta^{H}\Big{(}U(T)K_{H}^{-1}w_{T}\Big{)}\Big{)}.$

∎

##  7 Application

The first fractional volatility model was presented in [17], which is based on
the fractional Ornstein-Uhlenbeck process with a Hurst parameter $H>1/2$ to
describe the slow flattening of smiles and skews as the time to maturity
increases. The volatility model around the fractional literature has
subsequently developed in many articles, for instance, see [18, 16, 1, 6].
Long memory in the volatility process becomes much needed in order to justify
the long-standing conundrums such as steep volatility smiles in long-term
options and co-movements between implied and realized volatility. (see also
[52]). A fractional Ornstein–Uhlenbeck process is defined as the solution of
the SDE

$d\sigma_{t}=(\nu-\kappa\sigma_{t})dt+udW_{t}^{H},$ (29)

where $\nu,\kappa$, and $u$ are positive parameters and the stochastic
integral is a pathwise Riemann-Stieltjes integral. Gatheral [26] have
considered rough stochastic fractional volatility model on the time interval
$[0,T]$, as the volatility $x_{t}$ is the form $x_{t}=e^{\sigma_{t}}$ for
$\sigma_{t}$ satisfying (29). The volatility model is essential for investors
in predicting their portfolio [23], forming suitable hedging strategies [20,
47], pricing volatility indices and other derivative products [36], etc.
Analysis of variance and volatility swaps, a forward contract on realized
stock variance and volatility, are developed with the underlying stochastic
volatility models. For example, the authors in [49] choose a drift-adjusted
version of the Heston model, and in [40] the authors choose the log-normal
SABR model with fractional stochastic volatility to price variance and
volatility swaps. It is reasonable to develop the model as a mean-field
stochastic differential equation and consider the sensitivity of the price of
swaps, without achieving a closed-form exact formula for the price of the
swaps, based on the Malliavin derivative of the volatility. The model
considered in the next example is inspired of the Example 4.1 in [4] in
fractional literature.

######  Example.

Consider the stochastic volatility dynamics $\\{\sigma_{t}\\}_{t\in[0,t]}$ of
an asset with initial volatility $x>0$ in Block-Scholes model with coninuous
dividend payments represented by the following SDE

$d\sigma_{t}=(\mu-q\rho_{t}^{x})\sigma_{t}dt+\alpha\sigma_{t}dW_{t}^{H},\qquad\rho_{t}^{x}=\mathbb{E}(\sigma_{t}^{x}),\quad
t\in[0,T],$ (30)

where $\mu,q,\alpha\in\mathbb{R^{+}}$, $\mu>0$. It is worth mention that if
the parameter $q$ in (30) would be zero, our model coincides with the models
we refereed them.
A variance swap is a forward contract on realized stock variance
$\sigma^{2}_{R}(x)=\frac{1}{T}\int_{0}^{T}\sigma_{s}^{2}ds$, with payoff
function $N(\sigma_{R}^{2}-K_{var})$ in which $N$ is the notional amount and
$K_{var}$ is strike of the swap. The price of a variance forward contract is
the discounted value of the payoff in a risk-neutral world with a risk-less
rate $r>0$.

$P_{var}(x)=\mathbb{E^{Q}}\Big{(}e^{-rT}(\sigma_{\mathbb{R}}^{2}(x)-K_{var})\Big{)},$

where $Q$ is a minimal martingale measure, i.e.;

$\forall
t\in[0,T]\quad{M}_{t}^{x}:=\frac{d\mathbb{Q}}{dp}|_{\mathcal{F}_{t}}=exp\Big{\\{}\int_{0}^{t}\Theta_{u}^{x}dW_{u}^{H}-\frac{1}{2}\int_{0}^{t}\int_{0}^{t}\Theta_{\nu}^{x}\Theta_{u}^{x}\phi(\nu,u)dud\nu\Big{\\}},$

for some $\Theta(.)$ satisfying
$\mu-q\rho_{t}^{x}-\alpha\int_{0}^{t}\phi(t,u)\Theta_{u}du=r$, (see also
[27]).
To consider the sensitivity of the price with respect to initial volatility
$x$, written by $\frac{1}{T}\int_{0}^{T}\frac{\partial}{\partial
x}\mathbb{E}(\sigma_{s}^{2})ds$, we need to find an expression for
$\frac{\partial}{\partial x}\mathbb{E}(\sigma_{s}^{2})$. To this end, we
follow the technical approach described in this manuscript.
One can easily solves a Riccati’s equation to obtain $\rho_{t}^{x}=\frac{x\mu
e^{\mu t}}{qxe^{\mu t}+\mu-qx}$. Therefore, $\frac{\partial}{\partial
x}\rho_{t}^{x}=\frac{e^{-\mu t}}{x^{2}}(\rho_{t}^{x})^{2}$,

$\int_{0}^{t}\phi(t,s)\frac{\partial\Theta_{s}}{\partial x}ds=-\frac{qe^{-\mu
t}}{\alpha x^{2}}(\rho_{t}^{x})^{2},$

and

$\frac{\partial}{\partial
x}M_{t}^{x}=M_{t}^{x}\Big{[}\int_{0}^{t}\frac{\partial\Theta_{u}^{x}}{\partial
x}dW_{u}^{H}-\int_{0}^{t}\int_{0}^{t}\frac{\partial\Theta_{u}^{x}}{\partial
x}\Theta_{\nu}^{x}\phi(\nu,u)dud\nu\Big{]}.$

Assume that $\mu>qx>0$. Theorem 5.3 shows that $\frac{\partial}{\partial
x}\sigma_{t}=\langle
D^{H}\sigma_{t},h\rangle_{\mathcal{H}}+e_{t,X}^{-1}\mathcal{E}_{t}$ and

$R_{H}h(t)=-\frac{q}{x^{2}}\alpha^{-1}\int_{0}^{t}e^{-\mu
s}(\rho_{s}^{x})^{2}ds.$

In addition, $w_{t}(u)=\frac{1}{K_{0}T\alpha}(t\wedge u)$ and therefore from
(23),

$(K_{H}^{-1}w_{t})(u)=\frac{1}{K_{0}T\alpha}u^{\frac{1}{2}-H}\Big{(}1_{[0,t]}(u)+g_{0}\Big{)}.$

Now, applying Lemma 6.1 for $v=V_{t}^{0}$ and then Theorem 4.5 to reult

$\displaystyle\frac{\partial}{\partial x}\mathbb{E}^{Q}(\sigma_{t}^{2})$
$\displaystyle=\mathbb{E}(\sigma_{t}^{2}\frac{\partial}{\partial
x}{M}_{t}^{x})+2\mathbb{E}(M_{t}^{x}\sigma_{t}\frac{\partial}{\partial
x}\sigma_{t})$
$\displaystyle=\mathbb{E}(\sigma_{t}^{2}\frac{\partial}{\partial
x}{M}_{t}^{x})+\mathbb{E}\Big{(}M_{t}^{x}\langle
D^{H}\sigma_{t}^{2},h\rangle_{\mathcal{H}}\Big{)}+\mathbb{E}\Big{(}2\sigma_{t}M_{t}^{x}e_{t,X}^{-1}\mathcal{E}_{t}\Big{)}$
$\displaystyle=\mathbb{E}(\sigma_{t}^{2}\frac{\partial}{\partial
x}{M}_{t}^{x})+\mathbb{E}\Big{(}\sigma_{t}^{2}M_{t}^{x}\delta^{H}(h)\Big{)}-\mathbb{E}\Big{(}\sigma_{t}^{2}\langle
D^{H}M_{t}^{x},h\rangle_{\mathcal{H}}\Big{)}+\mathbb{E}\Big{(}\sigma_{t}^{2}\int_{0}^{t}M_{t}^{x}K_{H}^{-1}(w_{t})(s)dW_{s}\Big{)}$
$\displaystyle=\mathbb{E}(\sigma_{t}^{2}\frac{\partial}{\partial
x}{M}_{t}^{x})-\frac{q}{x^{2}}\mathbb{E}\Big{(}\sigma_{t}^{2}M_{t}^{x}\int_{0}^{t}K_{H}^{-1}\Big{\\{}\int_{0}^{.}e^{-\mu
u}(\rho_{u}^{x})^{2}du\Big{\\}}(s)dW_{s}\Big{)}$
$\displaystyle-\mathbb{E}^{Q}(\sigma_{t}^{2})\int_{0}^{t}\int_{0}^{s}\phi(s,u)\frac{\partial\Theta_{u}^{x}}{\partial
x}duds+\frac{1+g_{0}}{K_{0}T\alpha}\mathbb{E}\Big{(}\sigma_{t}^{2}\int_{0}^{t}M_{t}^{x}s^{\frac{1}{2}-H}dW_{s}\Big{)}.$

######  Remark 7.1.

We note that if $\mathbb{E}(\sigma_{t})=\rho_{t}^{x}=0$ or $q=0$, then

$\frac{\partial}{\partial
x}\mathbb{E}(\sigma_{t}^{2})=\mathbb{E}\Big{(}\sigma_{t}^{2}\Big{[}\frac{\partial}{\partial
x}{M}_{t}^{x}+\frac{1+g_{0}}{K_{0}T\alpha}\int_{0}^{t}M_{t}^{x}s^{\frac{1}{2}-H}dW_{s}\Big{]}\Big{)}.$

######  Example.

Consider the following stochastic volatility model

$\left\\{\begin{array}[]{lr}S_{t}^{x_{1},x_{2}}=x_{1}+\int_{0}^{t}\mu
S_{u}^{x_{1},x_{2}}du+\int_{0}^{t}g(\sigma_{u}^{x_{2}})S_{u}^{x_{1},x_{2}}dW_{u},\\\
\sigma_{t}^{x_{2}}=x_{2}+\int_{0}^{t}(\mu-q\rho_{t}^{x_{2}})\sigma_{t}^{x_{2}}du+\int_{0}^{t}\alpha\sigma_{u}^{x_{2}}dW_{u}^{H},\quad
t\in[0,T],\end{array}\right.$

where $\mu,q,\alpha\in\mathbb{R^{+}}$, $\mu>qx>0$ and
$g:\mathbb{R}\rightarrow\mathbb{R}$ is a bounded measurable function with
Lipschitz continuity property. For simplicity, denote by $S_{t}$ and
$\sigma_{t}$, respectively, the solution of the above equation. We would like
to investigate the sensitivity of the payoff functions of the form with
respect to $x_{2}$. To do this, it follows from above example that

$\frac{\partial}{\partial x_{2}}\sigma_{t}=\langle
D^{H}\sigma_{t},h\rangle_{\mathcal{H}}+e_{t,X}^{-1}\mathcal{E}_{t},$

where $R_{H}h(t)=-\frac{q}{x_{2}^{2}}\alpha^{-1}\int_{0}^{t}e^{-\mu
s}(\rho_{s}^{x_{2}})^{2}ds$. Also, from Theorem 4.5, we know $\langle
D^{H}\sigma_{t},V_{t}^{0}\rangle_{\mathcal{H}}=e_{t,X}^{-1}\mathcal{E}_{t}$.
Thus,

$\frac{\partial}{\partial x_{2}}\sigma_{t}=\langle
D^{H}\sigma_{t},h+V_{t}^{0}\rangle_{\mathcal{H}}.$

On the other hand,

$\frac{\partial}{\partial x_{2}}S_{t}=\int_{0}^{t}\mu\frac{\partial}{\partial
x_{2}}S_{s}ds+\int_{0}^{t}\\{g(\sigma_{s})\frac{\partial}{\partial
x_{2}}S_{s}+Sg^{\prime}(\sigma_{s})\frac{\partial}{\partial
x_{2}}\sigma_{s}\\}dW_{s},\\\ $

and

$\displaystyle D_{r}^{H}S_{t}$ $\displaystyle=\int_{0}^{t}\mu
D_{r}^{H}S_{s}ds+\int_{0}^{t}\\{g(\sigma_{s})D_{r}^{H}S_{s}+S_{s}g^{\prime}(\sigma_{s})D_{r}^{H}\sigma_{s}\\}dW_{s},$

Then, since the solution is unique, $\frac{\partial}{\partial
x_{2}}S_{t}=\langle D^{H}S_{t},h+V_{t}^{0}\rangle_{\mathcal{H}}$. Therefore,
it follows for $X_{t}=(S_{t},\sigma_{t})$ that

$\displaystyle\frac{\partial}{\partial
x_{2}}\mathbb{E}\Big{(}\Phi_{s}^{\prime}(X_{t})\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\Phi_{s}^{\prime}(X_{t})\frac{\partial}{\partial
x_{2}}S_{t}\Big{)}+\mathbb{E}\Big{(}\Phi^{\prime}_{\sigma}(X_{t})\frac{\partial}{\partial
x_{2}}\sigma_{t}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\Phi_{s}^{\prime}(X_{t})\langle
D^{H}S_{t},h+V_{t}^{0}\rangle_{\mathcal{H}}\Big{)}+\mathbb{E}\Big{(}\Phi_{\sigma}^{\prime}(X_{t})\langle
D^{H}\sigma_{t},h+V_{t}^{0}\rangle_{\mathcal{H}}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\langle
D^{H}(\Phi(X_{t})),h+V_{t}^{0}\rangle_{\mathcal{H}}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\Phi(X_{t})\delta(-\frac{q}{x^{2}}K_{H}^{-1}\Big{\\{}\int_{0}^{.}e^{-\mu
u}(\rho_{u}^{x})^{2}du\Big{\\}}+K_{H}^{-1}w_{t})\Big{)}.$

##  8 Conclusions

In this work, we proved the Malliavin differentiability of the solution on
mean-field stochastic differential equations whose dependence is the form of
expection functional of the solution itself. We formulated the Bismut formula
for the solution by changing the solution by another process whose Malliavin
derivative and flow can be easily modified each other. We showed the
application of the Bismut formula in the sensitivity analysis of variance
swaps with distribution-dependent models.

## References

  * [1] E. Alós and Y. Yang, (2017). A fractional Heston model with H ¿ 1/2, Stochastics, 89(1), 384-399.
  * [2] E. Alós, O. Mazet and D. Nualart, (2001). Stochastic calculus with respect to Gaussian processes, Ann. Probab. 29, 766-801.
  * [3] O. Amine, E. Coffie, F. Harang and F. Proske, A Bismut-Elworthy-Li Formula for Singular SDEs Driven by a Fractional Brownian Motion and Applications to Rough Volatility Modeling, submitted 2018, https://doi.org/10.48550/arXiv.1805.11435.
  * [4] D. Baños, (2018). The Bismut-Elworthy-Li formula for mean-field stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 54(1), 220-233.
  * [5] J. Bao, P. Ren and F.-Y. Wang, (2021). Bismut formula for Lions derivative of distribution-path dependent SDEs, J. Differential Equations, 282, 285-329.
  * [6] C. Bayer, P. Friz, P. Gassiat, J. Martin, and B. Stemper, (2020). A regularity structure for rough volatility, Math. Finance, 30 (3), 782-832.
  * [7] M. Bauer and T. Meyer-Brandis. Strong Solutions of Mean-Field SDEs with irregular expectation functional in the drift, submitted 2019, https://doi.org/10.48550/arXiv.1912.06534.
  * [8] M. Bauer and T. Meyer-Brandis and F. Proske, Strong solutions of mean-field stochastic differential equations with irregular drift, submitted 2018, https://doi.org/10.48550/arXiv.1806.11451.
  * [9] F. Biagini, Y., Hu, B. oksendal, and A. Sulem, (2002). A stochastic maximum principle for processes driven by fractional Brownian motion, Stochastic Process. Appl. 100, 233-253.
  * [10] F. Biagini, Y. Hu, B. Oksendal and T. Zhang, (2008). Stochastic Calculus for Fractional Brownian Motion and Applications, Springer.
  * [11] R. Buckdahn and S. Jing, (2017). Mean-field SDE driven by a fractional Brownian motion and related stochastic control problem, SIAM J. Control Optim. 55, 1500-1533.
  * [12] R. Buckdahn, J. Li, S. Peng and C. Rainer, (2017). Mean-field stochastic differential equations and associated PDEs, Ann. Probab. 45(2), 824-878.
  * [13] R. Carmona and F. Delarue, (2013). Probabilistic analysis of mean-field games, SIAM J. Control Optim. 51, 2705-2734.
  * [14] R. Carmona, J. P. Fouque, S. M. Mousavi, and L. H.Sun, (2018). Systemic Risk and Stochastic Games with Delay. Journal of Optimization Theory and Applications, 179(2), 366-399.
  * [15] R. Carmona, J. P. Fouque, and L. H. Sun, (2015). Mean field games and systemic risk. Communications in Mathematical Sciences, 13(4), 911-933.
  * [16] F. Comte, L. Coutin, and É. Renault, (2012). Affine fractional stochastic volatility models. Ann. Finance, 8(2-3), 337-378.
  * [17] F. Comte, E. Renault, (1998). Long memory in continuous-time stochastic volatility models. Mathematical Finance. Int. J. Math. Stat. Financ. Econ. 8, 291-323.
  * [18] P. Cheridito, H. Kawaguchi, and M. Maejima, (2003). Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab, 8(3), 1-14.
  * [19] E. Duncan, Y. Hu, B. Pasik-Duncan, (2000). Stochastic calculus for fractional Brownian motion I. Theory, SIAM J. Control Optim. 38(2), 582-612.
  * [20] O. E. Euch, M. Rosenbaum, (2018). Perfect hedging in rough Heston models. Annals of Probability, 28(6), 3813-3856.
  * [21] X. L. Fan, X. Huang, Y. Suo and C. Yuan, (2022). Distribution dependent SDEs driven by fractional Brownian motions, Stochastic Process. Appl. 151, 23-67.
  * [22] X. L. Fan and Y. Ren, (2017). Bismut formulae and applications for stochastic (functional) differential equations driven by fractional Brownian motions, Stoch. Dyn. 17, 1750028.
  * [23] J. P. Fouque, R. Hu, (2018). Optimal portfolio under fast mean-reverting fractional stochastic environment. SIAM Journal of Financial Mathematics, 9(2), 564-601.
  * [24] J. P. Fouque and T. Ichiba, (2013). Stability in a model of inter-bank lending. SIAM Journal on Financial Mathematics, 4, 784-803.
  * [25] J. Garnier, G. Papanicolaou, and T. W. Yang, (2013). Large deviations for a mean field model of systemic risk. SIAM Journal on Financial Mathematics, 4(1), 151-184.
  * [26] J. Gatheral, T. Jaisson, and M. Rosenbaum, (2018). Volatility is rough, Quantitative Finance, 18(6), 933-949.
  * [27] S. Habtemicael and I. SenGupta, (2016). Pricing variance and volatility swaps for Barndorff-Nielsen and Shephard process driven financial markets, International Journal of Financial Engineering, 3(4), 1650027\.
  * [28] Y. Han, Y. Hu and J. Song, (2013). Maximum principle for general controlled systems driven by fractional Brownian motions. Appl. Math. Optim., 67, 279-322.
  * [29] Y. Hu, ( 2005).Integral Transformations and Anticipative Calculus for Fractional Brownian Motions, Mem. Amer. Math. Soc.
  * [30] Y. Hu, X. Y. Zhou, (2005). Stochastic control for linear systems driven by fractional noises. SIAM J. Control and Optim. 43(6), 2245-2277.
  * [31] M. Huang, R. Malhamé and P. Caines, (2006). Large population stochastic dynamic games: closedloop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst. 6, 221–251.
  * [32] X. Huang, M. Röckner and F.-Y. Wang, (2019). Nonlinear Fokker-Planck equations for probability measures on path space and path-distribution dependent SDEs, Discrete Contin. Dyn. Syst. 39, 3017-3035
  * [33] X. Huang and F.-Y. Wang, (2019). Distribution dependent SDEs with singular coefficients, Stochastic Process. Appl. 129, 4747-4770.
  * [34] X. Huang and F. Y. Wang, (2022). Singular McKean-Vlasov (reflecting) SDEs with distribution dependent noise, J. mathematical Analysis and Applications, 514 (1), 126301.
  * [35] X. Huang and F. Y. Wang, Log-Harnack Inequality and Bismut Formula for Singular McKean-Vlasov SDEs, submitted 2022, https://doi.org/10.48550/arXiv.2207.11536.
  * [36] T. Jaisson, M. Rosenbaum, (2016). Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes. Annals of Applied Probability, 26(5), 2860-2882.
  * [37] S. Jing and J. A. León, (2011). Semilinear backward doubly stochastic differential equations and SPDEs driven by fractional Brownian motion with Hurst parameter in (0, 1/2), Bull. Sci. Math. 135, 896-935.
  * [38] B. Jourdain, (1997). Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burger equations. ESAIM: Probability and Statistics, 1, 339-355.
  * [39] M. Kac, Foundations of Kinetic Theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics, pages 171-197, Berkeley, Calif., 1956. University of California Press.
  * [40] S-W. Kim and J-H. Kim, (2020). Volatility and variance swaps and options in the fractional SABR model, The European Journal of Finance, 26(17), 1725-1745.
  * [41] J. M. Lasry and P. L. Lions, (2007). Mean field games, Japanese Journal of Mathematics, 2(1), 229-260.
  * [42] J. Li and H. Min, (2016). Weak Solutions of Mean-Field Stochastic Differential Equations and Application to Zero-Sum Stochastic Differential Games, SIAM Journal on Control and Optimization, 54(3),1826-1858.
  * [43] H. P. McKean, (1966). A class of Markov processes associated with nonlinear parabolic equations. Proc Natl Acad Sci USA, 56(6),1907-1911.
  * [44] Y. S. Mishura, (2008). Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer.
  * [45] Y. S. Mishura and A. Y. Veretennikov, (2021). Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations, Theory of Probability and Mathematical Statistics, 103, 59-101.
  * [46] D. Nualart, (2010). Malliavin Calculus and Related Topics, 2nd ed., Springer, New York.
  * [47] N. Salmon, (2021). Fractional Barndorff-Nielsen and Shephard model: applications in variance and volatility swaps, and hedging, Annals of Finance, 17, 529-558.
  * [48] S. G. Samko, A. A. Kilbas and O. I. Marichev, (1993).Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yvendon.
  * [49] A. Swishchuk, N. Vadori, (2014). Smiling for the delayed volatility swaps, Wilmott, 2014(74), 62-73.
  * [50] A. A. Vlasov, (1968). The vibrational properties of an electron gas, Soviet Physics Uspekhi, 10(6), 721.
  * [51] F. Y. Wang, (2018). Distribution dependent SDEs for Landau type equations, Stochastic Processes and their Applications, 128(2), 595-621.
  * [52] X. Wang, W. Xiao, J. Yu, (2021). Modeling and forecasting realized volatility with the fractional Ornstein-Uhlenbeck Process, . Journal of Econometrics, 1-27, Research Collection School Of Economics.