# On the regularity of solutions of some linear parabolic path-dependent PDEs

Bruno Bouchard 111CEREMADE, Université Paris-Dauphine, PSL, CNRS.
<EMAIL_ADDRESS> Xiaolu Tan 222Department of Mathematics, The
Chinese University of Hong Kong<EMAIL_ADDRESS>
###### Abstract

We study a class of linear parabolic path-dependent PDEs (PPDEs) defined on
the space of càdlàg paths ${\rm x}\in D([0,T])$, in which the coefficient
functions at time $t$ depend on ${\rm x}(t)$ and $\int_{0}^{t}{\rm
x}(s)dA_{s}$, for some (deterministic) continuous function $A$ with bounded
variations. Under uniform ellipticity and Hölder regularity conditions on the
coefficients, together with some technical conditions on $A$, we obtain the
existence of a smooth solution to the PPDE by appealing to the notion of
Dupire’s derivatives. It provides a generalization to the existing literature
studying the case where $A_{t}=t$, and complements our recent work in [2] on
the regularity of approximate viscosity solutions for parabolic PPDEs. As a
by-product, we also obtain existence and uniqueness of weak solutions for a
class of path-dependent SDEs.

Keywords: Path-dependent PDE, degenerate parabolic PDE, Dupire’s functional
calculus.
MSC2020 subject classifications: 35K65,60H10.

##  1 Introduction

We consider linear parabolic path-dependent PDEs (PPDEs) of the form

$\displaystyle\partial_{t}{\rm v}~{}+~{}\bar{\mu}\partial_{{\rm x}}{\rm
v}~{}+~{}\frac{1}{2}\bar{\sigma}^{2}\partial^{2}_{{\rm x}}{\rm
v}~{}+~{}\bar{\ell}$ $\displaystyle=0,~{}\mbox{on}~{}[0,T)\times D([0,T])$
(1.1) $\displaystyle{\rm v}(T,\cdot)$
$\displaystyle=\bar{g}~{}\mbox{on}~{}D([0,T]).$

In the above, $D([0,T])$ denotes the space of all real-valued càdlàg path
${\rm x}=({\rm x}(t))_{t\in[0,T]}$ on $[0,T]$, the derivatives are taken in
the sense of Dupire [6, 3] (see Section 2.1 below), and the coefficient
functions $(\bar{\mu},\bar{\sigma},\bar{\ell},\bar{g}):[0,T]\times
D([0,T])\longrightarrow\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}$
are of the form

$\big{(}\bar{\mu}_{t},\bar{\sigma}_{t},\bar{\ell}_{t},\bar{g}\big{)}({\rm
x})~{}=~{}\big{(}\mu_{t},\sigma_{t},\ell_{t}\big{)}\big{(}{\rm
x}(t),I_{t}({\rm x})\big{)},\;\bar{g}({\rm x})=g({\rm x}_{T},I_{T}({\rm
x})),~{}~{}\mbox{with}~{}I_{t}({\rm x}):=\int_{0}^{t}{\rm x}(s)dA_{s},$

for some functions
$(\mu,\sigma,\ell,g):[0,T]\times\mathbb{R}^{2}\longrightarrow\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}$,
and a continuous process $A$ with bounded variations. When $A$ is absolutely
continuous, say simply $A_{t}=t$, the above can be written as a degenerate
parabolic PDE

$\partial_{t}v+\mu\partial_{x_{1}}v+x_{1}\partial_{x_{2}}v+\frac{1}{2}\sigma^{2}\partial^{2}_{x_{1}x_{1}}v+\ell=0~{}\mbox{on}~{}[0,T)\times\mathbb{R}^{2},\;v(T,\cdot)=g~{}\mbox{on}~{}\mathbb{R}^{2},$
(1.2)

in which the derivatives are now taken in the usual sense and

${\rm v}(t,{\rm x})~{}=~{}v\big{(}t,{\rm x}_{t},I_{t}({\rm x})\big{)}.$

Indeed, the Dupire’s horizontal derivative $\partial_{t}{\rm v}$ and vertical
derivatives $(\partial_{{\rm x}}{\rm v},\partial^{2}_{{\rm x}}{\rm v})$ are
related to the partial derivatives of $v$ through

$(\partial_{t},\partial_{{\rm x}},\partial^{2}_{{\rm x}}){\rm v}(t,{\rm
x})=\left(\partial_{t}+{\rm
x}(t)\partial_{x_{2}},\partial_{x_{1}},\partial^{2}_{x_{1}x_{1}}\right)v\big{(}t,{\rm
x}(t),I_{t}({\rm x})\big{)}.$

Various works are devoted to such equations, going back to [10], in more
complex multivariate frameworks, see e.g. [5, 8, 11, 13, 15] and the
references therein. The latter PDE may not admit a $C^{1,2}$-solution, in the
traditional sense, even when $\bar{\sigma}$ is uniformly elliptic:
$\partial_{t}v$ and $\partial_{x_{2}}v$ are in general not well-defined and
one needs to define $\partial_{t}v+x_{1}\partial_{x_{2}}v$ jointly, appealing
to the notion of Lie derivative, which amounts to considering Dupire’s
horizontal derivative when the PDE is seen as a PPDE.

The main novelty of this paper is that we do not assume anymore that
$(A_{t})_{t\in[0,T]}$ is absolutely continuous in $t$. In this case, the PDE
formulation (1.2) is not valide anymore, but the PPDE formulation (1.1) is
still adequate. We provide conditions under which (1.1) admits a solution that
is smooth in the sense of Dupire’s deviratives. It complements [2] in which
coefficients are assumed to be $C^{1+\alpha}$, which allows one to construct
the so-called approximate viscosity solutions of non-linear path-dependent
PDEs with first order Dupire’s vertical derivative enjoying some Hölder-type
regularity (see [2] for details). As shown in e.g. [1], in many situations,
this is already sufficient to derive a Feynman-Kac’s representation of the
solution by appealing to a version of Itô-Dupire’s stochastic calculus for
path-dependent functionals that are only vertically differentiable up to the
first order. In contrast to [2], we only assume here that the coefficients are
Hölder continuous, but require $\bar{\sigma}$ to be non-degenerate, so as to
expect the classical regularization effect to operate.

We rely on the parametrix approach, see e.g. [9, Chapter 1]. For this, we
perform a change of variables which allows us to reduce to a PDE of the form

$\partial_{t}u+\mu\langle(1,A),Du\rangle+\frac{1}{2}\sigma^{2}(1,A)D^{2}u(1,A)^{\top}=0,$

which can be written even if $A$ is not absolutely continuous. The above is
again degenerate and $(Du,D^{2}u)$ may not be well-defined. However, the
parametrix approach allows one to show that $\langle(1,A),Du\rangle$ and
$(1,A)D^{2}u(1,A)^{\top}$ are, which in turn implies that the vertical
derivatives $(\partial_{{\rm x}}{\rm v},\partial^{2}_{{\rm x}}{\rm v})$ of the
path-dependent functional ${\rm v}$ are.

As a by-product, we establish the existence and uniqueness of a weak solution
to the path-dependent stochastic differential equation (SDE)

$X_{t}=X_{0}+\int_{0}^{t}\mu_{s}(X_{s},I_{s})ds+\int_{0}^{t}\sigma_{t}(X_{s},I_{s})dW_{t},~{}~{}~{}I_{t}=\int_{0}^{t}X_{s}dA_{s},~{}~{}t\geq
0,$

and provide some first properties of the transition density of the Markov
process $(X,I)$, as well as the corresponding Feynman-Kac’s formula.

These results require structural conditions relating the Hölder regularity of
the coefficient $(\mu,\sigma)$ and the path behavior of $A$. If one knows a
priori that the above SDE admits a unique weak solution, then one can prove
under weaker conditions that the candidate solution to (1.1), deduced from a
formal application of the Feynman-Kac’s formula333Or more rigorously its
viscosity solution in the sense of [4, 16], see also e.g. [2, 7, 12] and the
references therein for an alternative definition., is already $C^{1}$ in
space, in the sense of Dupire. As mentioned above, this turns out to be enough
to deduce its Itô-Dupire’s semimartingale decomposition.

All over this paper, we stick to a one-dimensional setting for ease of
notations. Extensions to multivariate frameworks can be provided by using
similar techniques.

The rest of this paper is organized as follows. Section 2 states our main
results. Proofs are collected in Section 3.

In the following, the $i$-th component of a vector $x$ is denoted by $x_{i}$,
the $(i,j)$-component of a matrix $M$ is denoted by $M_{ij}$. Given
$\phi:(t,x)\in[0,T]\times\mathbb{R}^{2}\longrightarrow\phi(t,x)\in\mathbb{R}$,
we let $D\phi$ and $D^{2}\phi$ (or $D_{x}\phi$ and $D^{2}_{xx}\phi$) be the
gradient and the Hessian matrix with respect to $x$. The space partial
derivatives are denoted by $\partial_{x_{i}}\phi$,
$\partial^{2}_{x_{i}x_{j}}\phi$, and so on if we have to consider higher
orders.

##  2 Dupire’s regularity for linear PPDEs depending on the average of the
path

###  2.1 Notations and assumptions

Given $T>0$, let $D([0,T])$ denote the Skorokhod space of all
$\mathbb{R}$–valued càdlàg paths ${\rm x}=({\rm x}(t))_{t\in[0,T]}$ on
$[0,T]$, and let $C([0,T])$ denote the subspace of continuous paths. Let us
equipped $D([0,T])$ with the Skorokhod topology, and $C([0,T])$ with the
uniform convergence topology. Let $A=(A_{t})_{t\geq 0}$ be a deterministic
continuous process with finite variation, and
$(\mu,\sigma,\ell):[0,T]\times\mathbb{R}^{2}\longrightarrow\mathbb{R}\times\mathbb{R}\times\mathbb{R}$
be coefficient functions, from which we define path-dependent functionals
$(\bar{\mu},\bar{\sigma},\bar{\ell}):[0,T]\times
D([0,T])\longrightarrow\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ by

$\big{(}\bar{\mu}_{t},\bar{\sigma}_{t},\bar{\ell}_{t}\big{)}({\rm
x})~{}:=~{}\big{(}\mu_{t},\sigma_{t},\ell_{t}\big{)}\big{(}{\rm
x}(t),I_{t}({\rm x})\big{)},~{}~{}\mbox{with}~{}I_{t}({\rm
x}):=\int_{0}^{t}{\rm x}(s)dA_{s}.$

We study the following linear parabolic path-dependent PDE (PPDE):

$\displaystyle\partial_{t}{\rm v}~{}+~{}\bar{\mu}\partial_{{\rm x}}{\rm
v}~{}+~{}\frac{1}{2}\bar{\sigma}^{2}\partial^{2}_{{\rm x}}{\rm
v}~{}+~{}\bar{\ell}~{}=~{}0,~{}\mbox{on}~{}[0,T)\times D([0,T]),$ (2.1)

with terminal condition ${\rm v}(T,{\rm x})=\bar{g}({\rm x}):=g\big{(}{\rm
x}(T),I_{T}({\rm x})\big{)}$ for some function
$g:\mathbb{R}\times\mathbb{R}\longrightarrow\mathbb{R}$. In the above, the
derivatives are taken in the sense of Dupire.

##### Dupire’s derivatives for path-dependent functionals

To give a precise definition to the PPDE (2.1), let us recall Dupire’s [6, 3]
notion of horizontal derivative $\partial_{t}$ and vertical derivatives
$\partial_{{\rm x}}$ and $\partial^{2}_{{\rm x}}$ for path-dependent
functionals.

Let $F:[0,T]\times D([0,T])\longrightarrow\mathbb{R}$ be a path-dependent
functional, it is said to be non-anticipative if $F(s,{\rm x})=F(s,{\rm
x}(s\wedge\cdot))$ for all $(s,{\rm x})\in[0,T]\times D([0,T])$. For an non-
anticipative map $F$, its horizontal derivative $\partial_{s}F(s,{\rm x})$ at
$(s,{\rm x})\in[0,T)\times D([0,T])$ is defined as

$\partial_{s}F(s,{\rm x})~{}:=~{}\lim_{h\searrow 0}\frac{F(s+h,{\rm
x}(s\wedge\cdot))-F(s,{\rm x})}{h},$

and its vertical derivative $\partial_{{\rm x}}F(s,{\rm x})$ is defined as

$\partial_{{\rm x}}F(s,{\rm x})~{}:=~{}\lim_{y\to 0}\frac{F(s,{\rm
x}+y\mathbf{1}_{[s,T]})-F(s,{\rm x})}{y},$

whenever the limits exist. In the above, ${\rm x}+y\mathbf{1}_{[s,T]}$ denotes
the path taking value ${\rm x}_{t}+y\mathbf{1}_{[s,T]}(t)$ at time
$t\in[0,T]$. Similarly, one can define the second order vertical derivative
$\partial^{2}_{{\rm x}}F$ as the vertical derivative of $\partial_{{\rm x}}F$.
Given $t\in(0,T]$, we denote by ${\mathbb{C}}([0,t))$ the space of all
continuous non-anticipative functionals $F:[0,t)\times
D([0,T])\longrightarrow\mathbb{R}$, and we set

${\mathbb{C}}^{0,1}([0,t)):=\big{\\{}F\in{\mathbb{C}}([0,t))~{}:\partial_{{\rm
x}}F~{}\mbox{is well-defined and belongs to}~{}{\mathbb{C}}([0,t))\big{\\}},$

as well as

${\mathbb{C}}^{1,2}([0,t)):=\big{\\{}F\in{\mathbb{C}}^{0,1}([0,t))~{}:\partial_{s}F~{}\mbox{and}~{}\partial^{2}_{{\rm
x}}F~{}\mbox{are well-defined and belong to}~{}{\mathbb{C}}([0,t))\big{\\}}.$

##### Assumptions on the process $A$:

Recall that $A=(A_{t})_{t\in[0,T]}$ is a deterministic process with finite
variation. For $0\leq s<t\leq T$, let us define
$\overline{A}_{s,t}:=\frac{1}{t-s}\int_{s}^{t}A_{r}dr$ and

$m_{s,t}:=\frac{1}{t-s}\int_{s}^{t}\big{(}A_{r}-\overline{A}_{s,t}\big{)}^{2}dr,~{}~{}~{}\tilde{m}_{s,t}:=\frac{1}{t-s}\int_{s}^{t}(A_{r}-A_{s})^{2}dr.$

The above will play a major role in our analysis, as they will drive the
behavior of the parametrix density on small time intervals.

######  Assumption 2.1.

$\mathrm{(i)}$ There exist constants
$\beta_{0},\beta_{1},\beta_{2},\beta_{3}\geq 0$ and $C_{(\ref{eq : hyp vitesse
explo})},C_{(\ref{eq:order_m})}>0$ such that, for all $0\leq s<t\leq T$,

$\displaystyle\frac{1}{C_{(\ref{eq : hyp vitesse
explo})}}(t-s)^{-\beta_{1}}~{}\leq~{}$
$\displaystyle\frac{\tilde{m}_{s,t}}{m_{s,t}}~{}\leq~{}C_{(\ref{eq : hyp
vitesse explo})}(t-s)^{-\beta_{0}},$ (2.2)
$\displaystyle\frac{1}{C_{(\ref{eq:order_m})}}(t-s)^{-\beta_{2}}~{}\leq~{}$
$\displaystyle\frac{1}{m_{s,t}}~{}\leq~{}C_{(\ref{eq:order_m})}(t-s)^{-\beta_{3}}.$
(2.3)

$\mathrm{(ii)}$ There exist constants $\beta_{4}\geq 0$ and
$C_{\eqref{eq:holder A}}>0$ such that

$\displaystyle|A_{t}-A_{s}|~{}\leq~{}C_{\eqref{eq:holder
A}}(t-s)^{\beta_{4}},~{}~{}\mbox{for all}~{}0\leq s<t\leq T.$ (2.4)

######  Remark 2.2.

Notice that, $\lim_{t\downarrow s}\tilde{m}_{s,t}=0$ by continuity of $A$.
Without loss of generality, one can therefore assume that

$\beta_{1}\leq\beta_{0}\leq\beta_{3}\mbox{ and
}\beta_{1}\leq\beta_{2}\leq\beta_{3}.$

Moreover, one can always choose $\beta_{0}=\beta_{4}=0$ since
$m_{s,t}\leq\tilde{m}_{s,t}$ by their definitions.

Let us provide some typical examples.

######  Example 2.3.

$\mathrm{(i)}$ Let $A$ be defined by $A_{t}=\int_{0}^{t}\rho(s)ds$, $t\geq 0$,
with $\varepsilon\leq\rho\leq 1/\varepsilon$ a.e. for some $\varepsilon>0$.
Then, it is easy to check that Assumption 2.1 holds with

$\beta_{0}=0,~{}~{}\beta_{1}=0,~{}~{}\beta_{2}=2,~{}~{}\beta_{3}=2,~{}~{}\beta_{4}=1.$

In this setting, our main results are similar to those in [8], which studied a
multivariate version of the case where $\rho$ is constant.

$\mathrm{(ii)}$ Let $A_{t}=t^{\gamma}$ for some $\gamma\in(0,1)$. Then,

$m_{s,t}~{}=~{}\frac{t^{2\gamma+1}-s^{2\gamma+1}}{(2\gamma+1)(t-s)}-\frac{|t^{\gamma+1}-s^{\gamma+1}|^{2}}{(\gamma+1)^{2}(t-s)^{2}},$

and

$\tilde{m}_{s,t}~{}=~{}\frac{\frac{1}{2\gamma+1}(t^{2\gamma+1}-s^{2\gamma+1})-2\frac{1}{\gamma+1}(t^{\gamma+1}-s^{\gamma+1})s^{\gamma}+(t-s)s^{2\gamma}}{t-s}.$

In this setting, Assumption 2.1 holds true with

$\beta_{0}=0,~{}~{}\beta_{1}=0,~{}~{}\beta_{2}=2\gamma,~{}~{}\beta_{3}=2\gamma,~{}~{}\beta_{4}=\gamma.$

$\mathrm{(iii)}$ Assume that there exists $1\geq\gamma_{1}\geq\gamma_{2}>0$
and $C_{1},C_{2}>0$ such that

$C_{1}|t-s|^{\gamma_{1}}\leq A_{t}-A_{s}\leq C_{2}|t-s|^{\gamma_{2}},\mbox{
for all }s\leq t\leq T.$

Then, Assumption 2.1 holds with

$\beta_{0}=2(\gamma_{1}-\gamma_{2}),~{}~{}\beta_{1}=0,~{}~{}\beta_{2}=2\gamma_{2},~{}~{}\beta_{3}=2\gamma_{1},~{}~{}\beta_{4}=\gamma_{2}.$

Indeed, let us choose $t_{0}\in[0,T]$ such that
$A_{t_{0}}=\overline{A}_{s,t}$. Assume that $t-t_{0}\geq(t-s)/2$ (otherwise
$t_{0}-s\geq(t-s)/2$ and we can use similar computations), then

$\displaystyle{(t-s)m_{s,t}}~{}=~{}\int_{s}^{t}|A_{r}-A_{t_{0}}|^{2}$
$\displaystyle~{}\geq~{}\int_{t_{0}}^{t}C_{1}^{2}|{r}-{t_{0}}|^{2\gamma_{1}}dr~{}\geq~{}\frac{C_{1}^{2}}{2^{2\gamma_{1}+1}(2\gamma_{1}+1)}|t-s|^{2\gamma_{1}+1}.$

On the other hand,
${(t-s)m_{s,t}\leq}\int_{s}^{t}|A_{r}-A_{s}|^{2}\leq\frac{C_{2}^{2}}{2\gamma_{2}+1}|t-s|^{2\gamma_{2}+1}$.
Thus,

$1~{}\leq~{}\frac{\tilde{m}_{s,t}}{m_{s,t}}~{}\leq~{}2^{2\gamma_{1}+1}\frac{C_{2}^{2}(2\gamma_{1}+1)}{C_{1}^{2}(2\gamma_{2}+1)}|t-s|^{-2(\gamma_{1}-\gamma_{2})}.$

##### Assumptions on the coefficient functions $\mu$ and $\sigma$:

As in e.g. [8], the following Hölder regularity assumption on the coefficient
functions
$(\mu,\sigma):[0,T]\times\mathbb{R}^{2}\longrightarrow\mathbb{R}\times\mathbb{R}$
is calibrated to match with the explosion rate of the quadratic form entering
the parametrix. It does not impose smoothness conditions on $\mu$ and $\sigma$
as in e.g. [13], and facilitate the analysis, see Remark 2.6 below. Let us set

$\Theta~{}:=~{}\big{\\{}(s,x,t,y)\in[0,T]\times\mathbb{R}^{2}\times[0,T]\times\mathbb{R}^{2}~{}:~{}s<t\big{\\}},$

and, for $(s,x,t,y)\in\Theta$,

$\displaystyle{\rm
w}_{s,t}(x,y):=x-E_{s,t}(y)\in\mathbb{R}^{2},~{}~{}\mbox{with}~{}~{}E_{s,t}(y):=\left(\begin{array}[]{cc}1&0\\\
-(A_{t}-A_{s})&1\end{array}\right)y\in\mathbb{R}^{2}.$ (2.7)

######  Assumption 2.4.

Let $\beta_{0},\beta_{1},\beta_{2}\geq 0$ be the constants in Assumption 2.1.
Then,

$\mathrm{(i)}$ We have

$\beta^{\prime}_{1}:=\beta_{1}-\beta_{0}>-1,~{}~{}~{}~{}\beta^{\prime}_{2}:=\beta_{2}-\beta_{0}>-1.$

Moreover, the coefficients $\mu$ and $\sigma$ are continuous, and there exist
constants $(\underline{\mathfrak{a}},\bar{\mathfrak{a}})\in\mathbb{R}^{2}$,
$\mathfrak{b}\in\mathbb{R}$, $C_{(\ref{eq : hyp holder coeff bar sigma})}>0$
and $\alpha>0$ such that

$\displaystyle|\mu|\leq\mathfrak{b},~{}~{}0<\underline{\mathfrak{a}}\leq\sigma^{2}\leq\bar{\mathfrak{a}},\;\mbox{
on }[0,T]\times\mathbb{R}^{2},$ (2.8)

and

$\displaystyle\big{|}\sigma_{s}(x)-\sigma_{t}(y)\big{|}~{}\leq~{}C_{(\ref{eq :
hyp holder coeff bar sigma})}\Big{(}|t-s|^{\alpha}+\big{|}{\rm
w}_{s,t}(x,y)\big{|}^{\frac{2\alpha}{1+\beta^{\prime}_{1}}}+\big{|}{\rm
w}_{s,t}(x,y)\big{|}^{\frac{2\alpha}{1+\beta^{\prime}_{2}}}\Big{)},$ (2.9)

for all $(s,x,t,y)\in\Theta$.

$\mathrm{(ii)}$ There exists a constant $C_{(\ref{eq : hyp holder coeff bar
mu})}>0$ such that

$\big{|}\mu_{t}(x)-\mu_{t}(y)\big{|}\leq C_{(\ref{eq : hyp holder coeff bar
mu})}\Big{(}\big{|}x_{1}-y_{1}\big{|}^{\frac{2\alpha}{1+\beta^{\prime}_{1}}}+\big{|}x_{2}-y_{2}\big{|}^{\frac{2\alpha}{1+\beta^{\prime}_{2}}}\Big{)},~{}\mbox{for
all}~{}(t,x,y)\in[0,T]\times\mathbb{R}^{2}\times\mathbb{R}^{2}.$ (2.10)

######  Example 2.5.

The condition (2.9) holds for instance if $\sigma_{s}(x_{1},x_{2})$ depends
only on $(s,x_{1})$ and is Hölder with respect to $(s,x_{1})$. It would also
hold if it is of the form
$\sigma_{s}(x)=\tilde{\sigma}_{s}(x_{1},x_{2}-A_{s}x_{1})$ for some Hölder
continuous map $\tilde{\sigma}$. Indeed, one has

$\displaystyle\big{|}x_{2}-A_{s}x_{1}-(y_{2}-A_{t}y_{1})\big{|}$
$\displaystyle~{}=~{}\big{|}x_{2}-A_{s}x_{1}-(y_{2}-A_{t}y_{1})-A_{s}(y_{1}-x_{1})+A_{s}(y_{1}-x_{1})\big{|}$
$\displaystyle~{}\leq~{}\big{|}x_{2}-y_{2}+(A_{t}-A_{s})y_{1})\big{|}+|A_{s}|\big{|}y_{1}-x_{1}\big{|}$
$\displaystyle~{}\leq~{}\Big{(}1+\max_{[0,T]}|A|\Big{)}|{\rm w}_{s,t}(x,y)|.$

######  Remark 2.6.

The case where the coefficient $\sigma$ is Hölder in the classical sense, i.e.

$\big{|}\sigma_{s}(x)-\sigma_{t}(y)\big{|}~{}\leq~{}C\Big{(}|t-s|^{\alpha}+\big{|}x_{1}-y_{1}\big{|}^{\frac{2\alpha}{1+\beta^{\prime}_{1}}}+\big{|}x_{2}-y_{2}\big{|}^{\frac{2\alpha}{1+\beta^{\prime}_{2}}}\Big{)}$

can be tackled by combining the arguments below with those of e.g. [13]. This
will add additional exponentially growing terms in the estimates on
$\widetilde{\Phi}$ in Proposition 3.5 below, which can be handled, to the
price of adapted restrictions on the coefficients $(\beta_{i})_{0\leq i\leq
4}$. We chose the formulation of the conditions in (2.9) for sake of
simplicity.

###  2.2 Heuristic derivation using a change of variables and the parametrix
method

Let us consider the path-dependent SDE

$X_{t}=X_{0}+\int_{0}^{t}\mu_{s}(X_{s},I_{s})ds+\int_{0}^{t}\sigma_{t}(X_{s},I_{s})dW_{s},~{}~{}~{}I_{t}=\int_{0}^{t}X_{s}dA_{s},~{}~{}t\in[0,T],$
(2.11)

where $W$ is a Brownian motion. Assume that the above SDE has a solution $X$
such that $(X,I)$ is Markov. Then, to deduce a solution to the PPDE (2.1), it
suffices to find the transition probability (density) function $f(s,x;t,y)$ of
the Markov process $(X,I)$ from $(s,x)$ to $(t,y)$. When $t\longmapsto A_{t}$
is absolutely continuous, it is well-known that $(s,x)\longmapsto f(s,x;t,y)$
solves a Kolmogorov’s backward PDE and that $(t,y)\longmapsto f(s,x;t,y)$
solves a Kolmogorov’s forward PDE. One can then apply the classical parametrix
method as in [8, Section 4] to guess the expression of $f(s,x;t,y)$.

In our setting where $A$ is not necessarily absolutely continuous, it is no
more possible to write the Kolmogorov’s PDE for the transition probability
(density) function of $(X,I)$. We therefore perform a change of variable and
set

$\displaystyle\widetilde{X}_{t}:={\rm\bf
A}_{t}\left(\begin{array}[]{c}X_{t}\\\
I_{t}\end{array}\right),~{}\mbox{with}~{}{\rm\bf
A}_{t}:=\left(\begin{array}[]{cc}1&0\\\
A_{t}&-1\end{array}\right),~{}~{}t\in[0,T].$ (2.16)

Notice that ${\rm\bf A}_{t}^{-1}={\rm\bf A}_{t}$ and that $\widetilde{X}$ is a
diffusion process with dynamics

$\displaystyle\widetilde{X}_{t}$
$\displaystyle~{}=~{}\widetilde{X}_{0}+\int_{0}^{t}\tilde{\mu}_{s}(\widetilde{X}_{s})\overrightarrow{\rm
A}_{s}ds+\int_{0}^{t}\tilde{\sigma}_{s}(\widetilde{X}_{s})\overrightarrow{\rm
A}_{s}dW_{s},~{}~{}t\in[0,T],$ (2.17)

where $\overrightarrow{\rm A}$,
$\tilde{\mu}:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ and
$\tilde{\sigma}:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ are defined by

$\displaystyle\overrightarrow{\rm A}_{s}:=\left(\begin{array}[]{c}1\\\
A_{s}\end{array}\right),~{}\tilde{\mu}_{s}(x):=\mu_{s}({\rm\bf
A}_{s}x)~{}\mbox{and}~{}\tilde{\sigma}_{s}(x):=\sigma_{s}({\rm\bf
A}_{s}x),~{}(s,x)\in[0,T]\times\mathbb{R}^{2}.$ (2.20)

The generator $\widetilde{{\cal L}}$ of $\widetilde{X}$ is given by

$\widetilde{{\cal L}}\phi(s,x)~{}:=~{}\tilde{\mu}_{s}(x)~{}\overrightarrow{\rm
A}_{s}\cdot D\phi(s,x)+\frac{1}{2}\tilde{\sigma}_{s}(x)^{2}~{}{\rm
Tr}\left[\overrightarrow{\rm A}_{s}(\overrightarrow{\rm
A}_{s})^{\top}D^{2}\phi(s,x)\right],$

for smooth functions
$\phi:[0,T]\times\mathbb{R}^{2}\longrightarrow\mathbb{R}$.

Assume that the SDE (2.17) has a solution $\widetilde{X}$ which is Markovian
and has a smooth transition probability density function $\tilde{f}(s,x;t,y)$,
from $x$ at $s$ to $y$ at $t$, then $(s,x)\longmapsto\tilde{f}(s,x;t,y)$
solves the Kolmogorov backward equation

$\displaystyle\big{(}\partial_{s}+\widetilde{{\cal
L}}\big{)}\tilde{f}(s,x;t,y)=0,~{}~{}\mbox{for}~{}(s,x)\in[0,t)\times\mathbb{R}^{2}.$
(2.21)

Notice that, in the above, the operator $\widetilde{{\cal L}}$ acts on the
first two arguments $(s,x)$ of $\tilde{f}(s,x;t,y)$.

To construct the parametrix, we consider the following process, with
volatility frozen at $(r,z)\in[0,T]\times\mathbb{R}^{2}$,

$\widetilde{X}^{r,z}_{t}~{}:=~{}\tilde{\sigma}_{r}(z)\int_{0}^{t}\overrightarrow{\rm
A}_{s}dW_{s},~{}~{}t\in[0,T].$

The corresponding generator $\widetilde{{\cal L}}^{r,z}$ is then given by

$\widetilde{{\cal
L}}^{r,z}\phi(s,x)~{}:=~{}\frac{1}{2}\tilde{\sigma}_{r}(z)^{2}~{}{\rm
Tr}\left[\overrightarrow{\rm A}_{s}(\overrightarrow{\rm
A}_{s})^{\top}D^{2}\phi(s,x)\right],~{}\mbox{for smooth functions}~{}\phi.$

We further define $\tilde{f}_{r,z}(s,x;t,y)$ as the corresponding transition
probability function from $(s,x)$ to $(t,y)$, for $(s,x,t,y)\in\Theta$. Notice
that $\tilde{f}_{r,z}$ is explicitly given and that
$y\longmapsto\tilde{f}_{r,z}(s,x;t,y)$ is the density function of the Gaussian
random vector $x+\tilde{\sigma}_{r}(z)\int_{s}^{t}\overrightarrow{\rm
A}_{r}dW_{r}$. It satisfies

$\displaystyle\big{(}\partial_{s}+\widetilde{{\cal
L}}^{r,z}\big{)}\tilde{f}_{r,z}(s,x;t,y)=0,\;\mbox{ for
}(s,x)\in[0,t)\times\mathbb{R}^{2}.$ (2.22)

Now, we employ the machinery of the parametrix method (see e.g. [9, Chapter 1]
or [8]), taking $\tilde{f}_{t,y}(s,x;t,y)$ as parametrix, and expressing
$\tilde{f}(s,x;t,y)$ in the following form:

$\tilde{f}(s,x;t,y)=\tilde{f}_{t,y}(s,x;t,y)+\int_{s}^{t}\int_{\mathbb{R}^{2}}\tilde{f}_{r,z}(s,x;r,z)\widetilde{\Phi}(r,z,t,y)dzdr,$
(2.23)

for some function $\widetilde{\Phi}:\Theta\longrightarrow\mathbb{R}$. By
(2.21) and (2.22), one must have

$\displaystyle 0~{}=~{}$ $\displaystyle\big{(}\partial_{s}+\widetilde{{\cal
L}}\big{)}\tilde{f}(s,x;t,y)$ $\displaystyle~{}=~{}$
$\displaystyle\big{(}\partial_{s}+\widetilde{{\cal
L}}\big{)}\tilde{f}_{t,y}(s,x;t,y)+\big{(}\partial_{s}+\widetilde{{\cal
L}}\big{)}\int_{s}^{t}\int_{\mathbb{R}^{2}}\tilde{f}_{r,z}(s,x;r,z)\widetilde{\Phi}(r,z,t,y)dzdr$
$\displaystyle~{}=~{}$ $\displaystyle\big{(}\widetilde{{\cal
L}}-\widetilde{{\cal
L}}^{t,y}\big{)}\tilde{f}_{t,y}(s,x;t,y)-\widetilde{\Phi}(s,x;t,y)$
$\displaystyle+\int_{s}^{t}\int_{\mathbb{R}^{2}}\big{(}\widetilde{{\cal
L}}-\widetilde{{\cal
L}}^{t,z}\big{)}\tilde{f}_{r,z}(s,x;r,z)\widetilde{\Phi}(r,z;t,y)dzdr.$

Therefore, $\widetilde{\Phi}$ must satisfy

$\widetilde{\Phi}(s,x;t,y)~{}=~{}\big{(}\widetilde{{\cal L}}-\widetilde{{\cal
L}}^{t,y}\big{)}\tilde{f}_{t,y}(s,x;t,y)+\int_{s}^{t}\int_{\mathbb{R}^{2}}\big{(}\widetilde{{\cal
L}}-\widetilde{{\cal
L}}^{t,z}\big{)}\tilde{f}_{r,z}(s,x;r,z)\widetilde{\Phi}(r,z;t,y)dzdr.$ (2.24)

In view of (2.24), we obtain

$\widetilde{\Phi}(s,x;t,y)~{}:=~{}\sum_{k=0}^{\infty}\widetilde{\Delta}_{k}(s,x;t,y),$
(2.25)

where $\widetilde{\Delta}_{0}(s,x;t,y):=\big{(}\widetilde{{\cal
L}}-\widetilde{{\cal L}}^{t,y}\big{)}\tilde{f}_{t,y}(s,x;t,y)$, and

$\displaystyle\widetilde{\Delta}_{k+1}(s,x;t,y):=\int_{s}^{t}\int_{\mathbb{R}^{2}}\widetilde{\Delta}_{0}(s,x;r,z)\widetilde{\Delta}_{k}(r,z;t,y)dzdr,~{}~{}k\geq
0.$ (2.26)

Notice that $\widetilde{{\cal L}}$, $\widetilde{{\cal L}}^{t,y}$ and
$\tilde{f}_{t,y}$ have explicit expressions. The main strategy of the
classical parametrix method consists in checking that $\widetilde{\Phi}$ in
(2.25) is well-defined and solves the integral equation (2.24). Then, one
defines $\tilde{f}$ by (2.23), and check that it provides a solution to
(2.21). If $\tilde{f}$ is smooth, one can basically deduce that $\tilde{f}$ is
the transition probability density function of $\widetilde{X}$ in (2.17) by
using the Feynman-Kac’s formula.

The main difficulty here lies in the fact that $\tilde{f}$ is, in general, not
smooth enough. For smoothness properties, we will therefore turn back to the
initial coordinates $(X,I)$ and define the candidate transition probability
function $f$ of the process $(X,I)$ in (2.11) through (2.27)-(2.28) below, and
work on it directly.

###  2.3 Main results

Under some conditions on the constants $\alpha$ and $(\beta)_{i=0,\cdots,4}$
given in Assumptions 2.1 and 2.4, we will show that
$\widetilde{\Phi}:\Theta\longrightarrow\mathbb{R}$ is well-defined by
(2.25)-(2.26). For $(r,z)\in[0,T]\times\mathbb{R}^{2}$, we can then define
$f_{r,z}$ and $\Phi$ by inverting the change of variables in (2.16):

$f_{r,z}(s,x;t,y):=\tilde{f}_{r,{\rm\bf A}_{r}z}(s,{\rm\bf A}_{s}x;t,{\rm\bf
A}_{t}y),~{}~{}\Phi(s,x;t,y):=\widetilde{\Phi}(s,{\rm\bf A}_{s}x;t,{\rm\bf
A}_{t}y).$ (2.27)

The corresponding candidate transition density
$f:\Theta\longrightarrow\mathbb{R}$ for $(X,I)$ is therefore:

$f(s,x;t,y)~{}:=~{}f_{t,y}(s,x;t,y)+\int_{s}^{t}\int_{\mathbb{R}^{2}}f_{r,z}(s,x;r,z)\Phi(r,z;t,y)dzdr,$
(2.28)

for all $(s,x,t,y)\in\Theta$.

For any positive constant $a\in\mathbb{R}_{+}$ and $0\leq s<t\leq T$, let us
set

$\Sigma_{s,t}(a)~{}:=~{}a\left(\begin{array}[]{cc}t-s&-\int_{s}^{t}(A_{r}-A_{s})dr\\\
-\int_{s}^{t}(A_{r}-A_{s})dr&\int_{s}^{t}(A_{r}-A_{s})^{2}dr\end{array}\right).$

For $(r,z)\in[0,T]\times\mathbb{R}^{2}$, we write
$\Sigma_{s,t}(r,z):=\Sigma_{s,t}(\sigma^{2}_{r}(z))$ for simplicity.
Equivalently,

$\Sigma_{s,t}(r,z)~{}:=~{}\sigma^{2}_{r}\big{(}z\big{)}\left(\begin{array}[]{cc}t-s&-\int_{s}^{t}(A_{r}-A_{s})dr\\\
-\int_{s}^{t}(A_{r}-A_{s})dr&\int_{s}^{t}(A_{r}-A_{s})^{2}dr\end{array}\right).$
(2.29)

Then, it is easy to check that $y\longmapsto f_{r,z}(s,x;t,y)$ is the density
function of the Gaussian random vector

$\Big{(}x_{1}+\sigma_{r}(z)(W_{t}-W_{s}),~{}x_{2}+\int_{s}^{t}\big{(}x_{1}+\sigma_{r}(z)(W_{u}-W_{s})\big{)}dA_{u}\Big{)}^{\top},$

so that, with $w:={\rm w}_{s,t}(x,y)$ as in (2.7),

$f_{r,z}(s,x;t,y)~{}=~{}\frac{1}{2\pi~{}{\rm
det}\left(\Sigma_{s,t}(r,z)\right)^{\frac{1}{2}}}\exp\Big{(}-\frac{1}{2}\big{\langle}\Sigma^{-1}_{s,t}(r,z)w,w\big{\rangle}\Big{)}.$

Let us also define the Gaussian transition probability function
${f^{\circ}}:\Theta\longrightarrow\mathbb{R}$ by

$\displaystyle f^{\circ}(s,x;t,y)~{}:=~{}\frac{1}{2\pi~{}{\rm
det}\left(\Sigma_{s,t}(4\bar{\mathfrak{a}})\right)^{\frac{1}{2}}}\exp\Big{(}-\frac{1}{2}\big{\langle}\Sigma^{-1}_{s,t}(4\bar{\mathfrak{a}})w,w\big{\rangle}\Big{)},~{}~{}(s,x,t,y)\in\Theta.$
(2.30)

As a first main result, we show that $f$ is well-defined under some conditions
on the coefficients $\alpha$ and $\beta_{0}$, and then provide some first
regularity and bound estimates.

######  Theorem 2.7.

Let Assumption 2.1.$\mathrm{(i)}$ and Assumption 2.4.$\mathrm{(i)}$ hold true.

$\mathrm{(i)}$ Assume that

$\kappa_{0}~{}:=~{}\frac{1-\beta_{0}}{2}\wedge(\alpha-\beta_{0})~{}>~{}0.$
(2.31)

Then $\widetilde{\Phi}$ in (2.25)-(2.26) is well-defined, and so is
$f:\Theta\longrightarrow\mathbb{R}$ in (2.28). Moreover, $f$ is continuous on
$\Theta$, and there exists a constant $C>0$ such that

${\big{|}f(s,x;t,y)\big{|}}~{}\leq~{}Cf^{\circ}(s,x;t,y),~{}~{}\mbox{for
all}~{}(s,x,t,y)\in\Theta.$ (2.32)

$\mathrm{(ii)}$ Assume that (2.31) holds and that

$\displaystyle\kappa_{1}~{}:=~{}\kappa_{0}+\frac{1-\beta_{0}}{2}~{}=~{}(1-\beta_{0})\wedge(\frac{1}{2}+\alpha-\frac{3}{2}\beta_{0})~{}>~{}0.$
(2.33)

Then, the partial derivative
$(s,x;t,y)\in\Theta\mapsto\partial_{x_{1}}f(s,x;t,y)$ exists, is continuous on
$\Theta$, and, for some constant $C_{\eqref{eq: them estimation derive
fb}}>0$,

$\displaystyle\big{|}\partial_{x_{1}}f(s,x;t,y)\big{|}~{}\leq~{}\frac{C_{\eqref{eq:
them estimation derive
fb}}}{(t-s)^{1-\kappa_{1}}}f^{\circ}(s,x;t,y),~{}\mbox{for
all}~{}(s,x,t,y)\in\Theta.$ (2.34)

Under further conditions, we can obtain more regularity of $f$ and then check
that it is the transition probability function of the Markov process $(X,I)$.
To be more precise, let us rephrase this in terms of path-dependent
functionals. For $0\leq s<t\leq T$, ${\rm x}\in D([0,T])$ and
$y\in\mathbb{R}^{2}$, we set

$\displaystyle({\rm f},{\rm f}^{\circ})(s,{\rm
x};t,y)~{}:=~{}(f,f^{\circ})\big{(}s,{\rm x}(s),I_{s}({\rm
x});t,y\big{)},~{}\mbox{with}~{}I_{s}({\rm x}):=\int_{0}^{s}{\rm x}(r)dA_{r}.$
(2.35)

We now fix $\ell:[0,T]\times\mathbb{R}^{2}\longrightarrow\mathbb{R}$ and
$g:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ such that, for some constants
$C_{{\ell,g}}>0$ and $\alpha_{\ell}>0$,

$\big{|}\ell(t,x)\big{|}+\big{|}g(x)\big{|}~{}\leq~{}C_{{\ell,g}}\exp\big{(}C_{{\ell,g}}|x|\big{)},$
(2.36)

and

$\big{|}\ell(t,x)-\ell(t,x^{\prime})\big{|}~{}\leq~{}C_{{\ell,g}}\big{(}e^{C_{{\ell,g}}|x|}+e^{C_{{\ell,g}}|x^{\prime}|}\big{)}\Big{(}|x_{1}-x_{1}^{\prime}|^{\frac{2\alpha_{\ell}}{1+\beta_{1}^{\prime}}}+|x_{2}-x_{2}^{\prime}|^{\frac{2\alpha_{\ell}}{1+\beta_{2}^{\prime}}}\Big{)},$
(2.37)

for all $t\in[0,T]$ and $x,x^{\prime}\in\mathbb{R}^{2}$. In view of the upper-
bound estimate of $f$ in (2.32), we can then define

${\rm v}(s,{\rm
x})~{}:=\int_{s}^{T}\\!\\!\int_{\mathbb{R}^{2}}\\!\\!\ell(t,y){\rm f}(s,{\rm
x};t,y)dydt~{}+\int_{\mathbb{R}^{2}}\\!\\!g(y){\rm f}(s,{\rm
x};T,y)dy,~{}~{}(s,{\rm x})\in[0,T)\times D([0,T]).$ (2.38)

######  Remark 2.8.

By its definition in (2.35), it is straightforward to check that

$\partial_{{\rm x}}{\rm f}(s,{\rm x};t,y)~{}=~{}\partial_{x_{1}}f\big{(}s,{\rm
x}(s),I_{s}({\rm x});t,y\big{)}.$

Similarly, let us define, for $(r,z),(t,y)\in[0,T]\times\mathbb{R}^{d}$,

${\rm f}_{r,z}(s,{\rm x};t,y)~{}:=~{}f_{r,z}(s,{\rm x}(s),I_{t}({\rm
x});t,y),~{}(s,{\rm x})\in[0,t)\times D([0,T]).$

Then, the functional $(s,{\rm x})\longmapsto{\rm f}_{r,z}(s,{\rm x};t,y)$ is a
classical solution to the PPDE

$\partial_{s}{\rm f}_{r,z}(s,{\rm
x};t,y)+\frac{1}{2}\sigma_{r}(z)^{2}\partial^{2}_{{\rm x}{\rm x}}{\rm
f}_{r,z}(s,{\rm x};t,y)=0,~{}\mbox{for}~{}(s,{\rm x})\in[0,t)\times D([0,T]).$
(2.39)

######  Theorem 2.9.

Let Assumptions 2.1 and 2.4 hold true. Assume that (2.31), (2.33), (2.36) and
(2.37) hold, and that there exists $\alpha_{\Phi}\in\mathbb{R}$ such that

$0<\alpha_{\Phi}<\kappa_{0}\wedge\hat{\alpha}_{\Phi}\wedge\min\limits_{i=1,2}\frac{1+\beta^{\prime}_{i}}{2},~{}~{}\mbox{with}~{}~{}\hat{\alpha}_{\Phi}:=\frac{1}{2}-\beta_{0}-\frac{\widehat{\Delta\beta}}{2}-\frac{(\beta_{0}+1-2\alpha)^{+}}{2},$

where

$\displaystyle\widehat{\Delta\beta}:=\max\left\\{\beta_{0}-\beta_{1}\;,\;\beta_{3}-\beta_{2}\right\\},$
(2.40)

and

${\min\Big{(}\frac{2\beta_{4}+1+\beta^{\prime}_{1}}{1+\beta^{\prime}_{2}},1\Big{)}\min\\{\alpha_{\Phi},\alpha_{\ell},\alpha\\}-\beta_{0}>0.}$

$\mathrm{(i)}$ For each $(t,y)\in(0,T]\times\mathbb{R}^{2}$, the path-
dependent functional ${\rm f}(\cdot;t,y)$ belongs to
${\mathbb{C}}^{1,2}([0,t))$.

$\mathrm{{(ii)}}$ ${\rm v}\in{\mathbb{C}}^{1,2}([0,T))$ and it solves the PPDE
(2.1). Moreover, there exists $C>0$ such that, for all $(s,{\rm
x})\in[0,T)\times D([0,T])$,

$\big{|}\partial_{{\rm x}}{\rm v}(s,{\rm x})\big{|}~{}\leq~{}\frac{Ce^{C(|{\rm
x}_{s}|+|I_{s}({\rm
x})|)}}{(T-s)^{1-\kappa_{1}}},~{}\mbox{and}~{}\big{|}\partial_{t}{\rm
v}(s,{\rm x})\big{|}+\big{|}\partial^{2}_{{\rm x}}{\rm v}(s,{\rm
x})\big{|}~{}\leq~{}\frac{Ce^{C(|{\rm x}_{s}|+|I_{s}({\rm
x})|)}}{(T-s)^{1+\beta_{0}}}.$ (2.41)

If, in addition, $g:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ is continuous,
then ${\rm v}$ is the unique classical solution to the PPDE (2.1) satisfying

$\displaystyle\lim_{t\nearrow T}{\rm v}(t,{\rm x})=\bar{g}({\rm
x}){~{}~{}and~{}|{\rm v}(s,{\rm x})|\leq Ce^{C(|{\rm x}_{s}|+|I_{s}({\rm
x})|)}}$ (2.42)

for all $(s,{\rm x})\in[0,T]\times C([0,T])$, for some $C>0$.

$\mathrm{{(iii)}}$ The SDE (2.11) has a unique weak solution $X$. Moreover,
$(X,I)$ is a strong Markov process with transition probability given by $f$
and

${\rm v}(s,{\rm
x})~{}=~{}\mathbb{E}\Big{[}\int_{s}^{T}\ell(X_{t},I_{t})dt+g(X_{T},I_{T})\Big{|}X_{s}={\rm
x}(s),I_{s}=I_{s}({\rm x})\Big{]},\;(s,{\rm x})\in[0,T]\times D([0,T]).$
(2.43)

######  Remark 2.10.

To check the conditions on $\alpha$ and $\beta_{i},~{}i=0,\cdots,4$ in Theorem
2.9, let us stay in the setting of Example 2.3.

$\mathrm{(i)-(ii)}$ In these cases, $\kappa_{0}=\frac{1}{2}\wedge\alpha$,
$\kappa_{1}=1\wedge(\alpha+\frac{1}{2})$,
$\hat{\alpha}_{\Phi}=\frac{1}{2}-[\frac{1}{2}-\alpha]^{+}=\alpha\wedge\frac{1}{2}$
and we can choose $\alpha_{\Phi}\in(0,\frac{1}{2}\wedge\alpha)$.

$\mathrm{(iii)}$ In this case,
$\kappa_{0}=\frac{1-2(\gamma_{1}-\gamma_{2})}{2}\wedge(\alpha-2(\gamma_{1}-\gamma_{2}))$
which requires that $\gamma_{1}-\gamma_{2}<\frac{1}{2}\wedge\frac{\alpha}{2}$
to ensure that $\kappa_{0}>0$. Then, $\kappa_{1}>0$ and
$\hat{\alpha}_{\Phi}=\frac{1}{2}-3(\gamma_{1}-\gamma_{2})-[\gamma_{1}-\gamma_{2}+\frac{1}{2}-\alpha]^{+}$.
If $2\alpha/(1+\beta^{\prime}_{1})\leq 1$, then $\alpha\leq 1/2$ and therefore
$\gamma_{1}-\gamma_{2}+\frac{1}{2}-\alpha\geq 0$. In this case, we can choose
$\alpha_{\Phi}\in(0,\alpha-{4}(\gamma_{1}-\gamma_{2}))$ if
$\gamma_{1}-\gamma_{2}<\alpha/{4}$. If $2\alpha/(1+\beta^{\prime}_{1})>1$,
then $(\mu,\sigma)$ does no depend on its first argument, and the different
cases can also be treated explicitly, leading to a suitable $\alpha_{\Phi}$
when $\gamma_{1}-\gamma_{2}$ is small enough.

The conditions in Theorem 2.9 ensure that $f$ is smooth enough, so that one
can basically apply the Feynman-Kac’s formula to justify that it is the
transition probability function of a Markov process. It can then be used to
prove that the wellposedness (existence and uniqueness) of the SDE (2.11). If
one already knows that the SDE (2.11) has a unique a weak solution, then one
can rely on Theorem 2.11 below, which requires less technical conditions on
$A$ and $(\mu,\sigma)$, to check that $f$ is the corresponding transition
probability function. In this case, the path-dependent functional ${\rm v}$
defined above may only be ${\mathbb{C}}^{0,1}([0,T))$, but it is enough to
deduce that (2.43) holds, and obtain it’s Itô-Dupire’s decomposition, whenever
it satisfies for instance one of the conditions a. or b. of Theorem 2.11 .

######  Theorem 2.11.

Let Assumption 2.1.(i) and Assumption 2.4.(i) hold true, and assume that the
SDE (2.11) has a unique weak solution, so that the corresponding process
$(X,I)$ is a strong Markov process.

$\mathrm{(i)}$ Assume in addition that (2.31) holds true so that $f$ is well
defined. Then, $f$ is the transition probability function of $(X,I)$, and
(2.43) holds whenever (2.36) does.

$\mathrm{(ii)}$ Assume that (2.31), (2.33) and (2.36) hold. Then, ${\rm
v}\in{\mathbb{C}}^{0,1}([0,T))$. Suppose in addition that one of the following
holds:

  * (a)

there exists $C_{\eqref{eq: cond vr Lip}}>0$ such that

$\displaystyle\big{|}{\rm v}(t,{\rm x})-{\rm v}(t,{\rm
x}^{\prime})\big{|}~{}\leq~{}C_{\eqref{eq: cond vr Lip}}\int_{0}^{t}|{\rm
x}(r)-{\rm x}^{\prime}(r)|d|A|_{r},$ (2.44)

for all $t\in[0,T]$, ${\rm x},{\rm x}^{\prime}\in D([0,T])$ such that ${\rm
x}(t)={\rm x}^{\prime}(t)$, in which $|A|$ denotes the total variation of $A$.

  * (b)

$A$ is monotone and
$0<\frac{1+\beta_{2}-\beta_{0}}{2+4\beta_{4}}<1-\frac{\beta_{3}-\beta_{2}+\beta_{0}}{2}$.

Then,

$\displaystyle{\rm v}(t,X)={\rm v}(0,X)+\int_{0}^{t}\partial_{{\rm x}}{\rm
v}(s,X)\bar{\sigma}_{s}(X)dW_{s}-\int_{0}^{t}\bar{\ell}_{{s}}(X)ds,~{}~{}t\in[0,T].$
(2.45)

######  Remark 2.12.

When $b/\sigma$ is bounded, and $\sigma$ is Lipschitz in its space variable in
the sense that, for some constant $C_{\eqref{eq: coeff Lipschitz}}>0$,

$\big{|}\sigma_{s}(x)-\sigma_{s}(x^{\prime})\big{|}~{}\leq~{}C_{\eqref{eq:
coeff
Lipschitz}}|x-x^{\prime}|,~{}~{}s\in[0,T],\;x,x^{\prime}\in\mathbb{R}^{2},$
(2.46)

with $(\sigma_{s}(0))_{s\leq T}$ bounded, then the SDE (2.11) has a unique
weak solution.

######  Remark 2.13.

To check the conditions in Theorem 2.11.$\mathrm{(ii).(b)}$, let us consider
the situations of Example 2.3.

$\mathrm{(i)-(ii)}$ In these cases,
$\frac{1+\beta_{2}-\beta_{0}}{2+4\beta_{4}}=\frac{1}{2}$ and
$1-\frac{\beta_{3}-\beta_{2}+\beta_{0}}{2}=1$, so that the conditions
inTheorem 2.11.$\mathrm{(ii).(b)}$ hold true.

$\mathrm{(iii)}$ In this case,
$\frac{1+\beta_{2}-\beta_{0}}{2+4\beta_{4}}=\frac{1}{2}-\frac{\gamma_{1}-\gamma_{2}}{1+2\gamma_{2}}$
and $1-\frac{\beta_{3}-\beta_{2}+\beta_{0}}{2}=1-2(\gamma_{1}-\gamma_{2})$.
Therefore, the conditions in Theorem 2.11.$\mathrm{(ii).(b)}$ hold true when
$\gamma_{1}-\gamma_{2}$ is small enough.

##  3 Proofs

This section is devoted to the proof of Theorems 2.7, 2.9 and 2.11.

###  3.1 A priori estimates

Recall that, with $w:={\rm w}_{s,t}(x,y)$ (see (2.7)),

$f_{r,z}(s,x;t,y)~{}:=~{}\frac{1}{2\pi{\rm
det}\left(\Sigma_{s,t}(r,z)\right)^{\frac{1}{2}}}\exp\Big{(}-\frac{1}{2}\big{\langle}\Sigma^{-1}_{s,t}(r,z)w,w\big{\rangle}\Big{)},$

where

$\Sigma_{s,t}(r,z)~{}:=~{}\sigma^{2}_{r}\big{(}z\big{)}\left(\begin{array}[]{cc}t-s&-\int_{s}^{t}(A_{r}-A_{s})dr\\\
-\int_{s}^{t}(A_{r}-A_{s})dr&\int_{s}^{t}(A_{r}-A_{s})^{2}dr\end{array}\right).$

By direct computation, one has

${\rm
det}\left(\Sigma_{s,t}(r,z)\right)~{}=~{}\sigma^{4}_{r}\big{(}z\big{)}(t-s)^{2}m_{s,t},$
(3.1)

and hence

$\displaystyle\Sigma_{s,t}^{-1}(r,z)~{}=~{}\sigma^{-2}_{r}\big{(}z\big{)}\left(\begin{array}[]{cc}\frac{1}{t-s}\frac{\tilde{m}_{s,t}}{m_{s,t}}&\frac{\int_{s}^{t}(A_{r}-A_{s})dr}{(t-s)^{2}m_{s,t}}\\\
\frac{\int_{s}^{t}(A_{r}-A_{s})dr}{(t-s)^{2}m_{s,t}}&\frac{1}{(t-s)m_{s,t}}\end{array}\right).$
(3.4)

The following quantities will play an important role in our analysis. For
$i=1,2$, $(r,z)\in\mathbb{R}^{2}$ and $(s,x,t,y)\in\Theta$, with $w:={\rm
w}_{s,t}(x,y)$ (see (2.7)), we compute that

$\partial_{x_{i}}f_{r,z}(s,x;t,y)=f_{r,z}(s,x;t,y)\Big{(}-\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{i}\Big{)},$
(3.5)
$\partial^{2}_{x_{1}x_{i}}f_{r,z}(s,x;t,y)=f_{r,z}(s,x;t,y)\Big{(}\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{1}\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{i}-\big{(}\Sigma^{-1}_{s,t}(r,z)\big{)}_{1,i}\Big{)},$
(3.6)

and

$\displaystyle\frac{\partial^{3}_{x_{1}x_{1}x_{i}}f_{r,z}(s,x;t,y)}{f_{r,z}(s,x;t,y)}=~{}$
$\displaystyle
2\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{1}\big{(}\Sigma^{-1}_{s,t}(r,z)\big{)}_{1,i}+\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{i}\big{(}\Sigma^{-1}_{s,t}(r,z)\big{)}_{1,1}$
$\displaystyle-\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{1}^{2}\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{i}.$
(3.7)

Let us first provide some estimations in the following lemma.

######  Lemma 3.1.

Let Assumption 2.1.(i) hold. Then, there exists constants $C_{\eqref{eq: norne
Sigma-1 w 1}}$, $C_{\eqref{eq: norne Sigma-1 w 2}}$, $C_{{\eqref{eq: borne
Sigma -1 11}}}$, $C_{{\eqref{eq: borne Sigma -1 12}}}>0$, such that, for all
$(s,x,t,y)\in\Theta$ and $(r,z)\in[0,T]\times\mathbb{R}^{2}$, with $w:={\rm
w}_{s,t}(x,y)$ (see (2.7)), we have

$\displaystyle\Big{|}\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{1}\Big{|}$
$\displaystyle~{}=~{}\Big{|}\frac{\partial_{x_{1}}f_{r,z}(s,x;t,y)}{f_{r,z}(s,x;t,y)}\Big{|}~{}\leq~{}\frac{C_{\eqref{eq:
norne Sigma-1 w
1}}}{(t-s)^{\frac{1+\beta_{0}}{2}}}\sqrt{\big{\langle}\Sigma^{-1}_{s,t}(r,z)w,w\big{\rangle}},$
(3.8) $\displaystyle\Big{|}\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{2}\Big{|}$
$\displaystyle~{}\leq~{}\frac{C_{\eqref{eq: norne Sigma-1 w
2}}}{(t-s)^{\frac{1+\beta_{3}}{2}}}\sqrt{\big{\langle}\Sigma^{-1}_{s,t}(r,z)w,w\big{\rangle}},$
(3.9) $\displaystyle\Big{|}\big{(}\Sigma^{-1}_{s,t}(r,z)\big{)}_{1,1}\Big{|}$
$\displaystyle~{}\leq~{}\frac{C_{{\eqref{eq: borne Sigma -1
11}}}}{(t-s)^{1+\beta_{0}}},$ (3.10)
$\displaystyle\Big{|}\big{(}\Sigma^{-1}_{s,t}(r,z)\big{)}_{1,2}\Big{|}$
$\displaystyle~{}\leq~{}\frac{C_{{\eqref{eq: borne Sigma -1
12}}}}{(t-s)^{1+\frac{\beta_{0}+\beta_{3}}{2}}}.$ (3.11)

###### Proof.

The bound in (3.10) and (3.11) are immediate consequences of Assumption
2.1.(i) and (3.4), up to appealing to Cauchy-Schwarz’s inequality for the
latter. By direct computation, one has

$\displaystyle\sigma_{t}(z)^{2}\big{\langle}\Sigma^{-1}_{s,t}(r,z)w,w\big{\rangle}$
$\displaystyle=\frac{1}{(t-s)^{2}m_{s,t}}\int_{s}^{t}\Big{(}(A_{r}-A_{s})^{2}w_{1}^{2}+2(A_{r}-A_{s})w_{1}w_{2}+w_{2}^{2}\Big{)}dr$
$\displaystyle=\frac{1}{(t-s)^{2}m_{s,t}}\int_{s}^{t}\big{(}(A_{r}-A_{s})w_{1}+w_{2}\big{)}^{2}dr.$

Hence, using Assumption 2.4 and (2.2),

$\displaystyle\underline{\mathfrak{a}}\Big{|}\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{1}\Big{|}$
$\displaystyle\leq\sigma_{t}(z)^{2}\Big{|}\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{1}\Big{|}$
$\displaystyle=\frac{1}{(t-s)^{2}m_{s,t}}\Big{|}\int_{s}^{t}\big{(}A_{r}-A_{s}\big{)}\big{(}(A_{r}-A_{s})w_{1}+w_{2}\big{)}dr\Big{|}$
$\displaystyle\leq\sqrt{\frac{\tilde{m}_{s,t}}{(t-s)m_{s,t}}}\sqrt{\bar{\mathfrak{a}}\big{\langle}\Sigma^{-1}_{s,t}(r,z)w,w\big{\rangle}}\leq\frac{\sqrt{\bar{\mathfrak{a}}C_{\eqref{eq
: hyp vitesse
explo}}}}{(t-s)^{\frac{1+\beta_{0}}{2}}}\sqrt{\big{\langle}\Sigma^{-1}_{s,t}(r,z)w,w\big{\rangle}}.$

Similarly, using the above, Assumption 2.4 and Cauchy-Schwarz inequality,
implies that

$\displaystyle\underline{\mathfrak{a}}\Big{|}\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{2}\Big{|}$
$\displaystyle\leq\sigma_{t}(z)^{2}\Big{|}\big{(}\Sigma^{-1}_{s,t}(r,z)w\big{)}_{2}\Big{|}$
$\displaystyle=\Big{|}\frac{1}{(t-s)^{2}m_{s,t}}\int_{s}^{t}\big{(}(A_{r}-A_{s})w_{1}+w_{2}\big{)}dr\Big{|}$
$\displaystyle\leq\frac{\sqrt{\bar{\mathfrak{a}}C_{\eqref{eq:order_m}}}}{(t-s)^{\frac{1+\beta_{3}}{2}}}\sqrt{\big{\langle}\Sigma^{-1}_{s,t}(r,z)w,w\big{\rangle}}.$

∎

As usual, an important step consists in providing a suitable upper-bound on
the parametrix density. Recall that $y\longmapsto f^{\circ}(s,x;t,y)$ defined
in (2.30) is a Gaussian density function on $\mathbb{R}^{2}$.

######  Lemma 3.2.

Let Assumption 2.1.(i) hold. Then, there exists $C_{\eqref{eq: def
varomega}}>0$ such that, for all $(s,x,t,y)\in\Theta$ and
$(r,z)\in[0,T]\times\mathbb{R}^{2}$, we have

$\displaystyle
f_{r,z}(s,x;t,y)~{}\leq~{}\varpi(s,x;t,y)~{}f^{\circ}(s,x;t,y),$ (3.12)

in which $\varpi:=\varpi^{1}\varpi^{2}$ with

$\displaystyle\left\\{\begin{array}[]{l}\varpi^{1}(s,x;t,y)~{}:=~{}C_{\eqref{eq:
def varomega}}\exp\left(-\frac{1}{C_{\eqref{eq: def
varomega}}}\Big{(}\frac{|w_{1}|^{2}}{(t-s)^{1+\beta^{\prime}_{1}}}+\frac{|w_{2}|^{2}}{(t-s)^{1+\beta^{\prime}_{2}}}\Big{)}\right),\\\
\varpi^{2}(s,x;t,y)~{}:=~{}\exp\Big{(}-\frac{1}{2}\langle\Sigma^{-1}_{s,t}(4\bar{\mathfrak{a}})w,w\big{\rangle}\Big{)},\end{array}\right.$
(3.15)

where $w:={\rm w}_{s,t}(x,y)$ as defined in (2.7).

###### Proof.

Let us first observe that
$m_{s,t}=\tilde{m}_{s,t}-[(t-s)^{-1}\int_{s}^{t}(A_{r}-A_{s})ds]^{2}$, so that
the right-hand side of (2.2) is equivalent to

$\displaystyle\left(\frac{1}{t-s}\int_{s}^{t}(A_{r}-A_{s})ds\right)^{2}~{}\leq~{}\tilde{m}_{s,t}\left(1-\frac{(t-s)^{\beta_{0}}}{C_{\eqref{eq
: hyp vitesse explo}}}\right).$

Note that, upon changing the value of $C_{\eqref{eq : hyp vitesse explo}}$,
one can assume that $C_{\eqref{eq : hyp vitesse explo}}\geq 2T^{\beta_{0}}$.
Hence, using the inequality $2ab\leq a^{2}+b^{2}$ for $a,b\in\mathbb{R}$,

$\displaystyle
2\left|\frac{\int_{s}^{t}(A_{r}-A_{s})w_{1}w_{2}ds}{(t-s)^{2}m_{s,t}}\right|$
$\displaystyle\leq
2\left[\tilde{m}_{s,t}\left(1-\frac{(t-s)^{\beta_{0}}}{C_{\eqref{eq : hyp
vitesse explo}}}\right)\right]^{\frac{1}{2}}\frac{w_{1}w_{2}}{(t-s)m_{s,t}}$
$\displaystyle\leq\left(1-\frac{(t-s)^{\beta_{0}}}{C_{\eqref{eq : hyp vitesse
explo}}}\right)^{\frac{1}{2}}\left\\{\frac{\tilde{m}_{s,t}}{(t-s)m_{s,t}}|w_{1}|^{2}+\frac{1}{(t-s)m_{s,t}}|w_{2}|^{2}\right\\}.$

Combining the above with (2.2)-(2.3) and Assumption 2.4 implies that

$\displaystyle\big{\langle}\Sigma^{-1}_{s,t}(r,z)w,w\big{\rangle}$
$\displaystyle\geq\frac{1}{\sigma_{r}(z)^{2}}\left(\frac{\tilde{m}_{s,t}}{(t-s)m_{s,t}}|w_{1}|^{2}-2\left|\frac{\int_{s}^{t}(A_{r}-A_{s})w_{1}w_{2}ds}{(t-s)^{2}m_{s,t}}\right|+\frac{1}{(t-s)m_{s,t}}|w_{2}|^{2}\right)$
$\displaystyle\geq\frac{C}{\bar{\mathfrak{a}}}\left(\frac{|w_{1}|^{2}}{(t-s)^{1+\beta_{1}-\beta_{0}}}+\frac{|w_{2}|^{2}}{(t-s)^{1+\beta_{2}-\beta_{0}}}\right),$
(3.16)

for some $C>0$ that does not depend on $(s,x,t,y)$. The required result then
follows from obvious algebra and Assumption 2.4. ∎

######  Lemma 3.3.

Let Assumption 2.1.(i) hold. Let us define the transition density function
$f^{\circ,\frac{1}{2}}$ by, for $(s,x,t,y)\in\Theta$,

$\displaystyle f^{\circ,\frac{1}{2}}(s,x;t,y):=\frac{1}{2\pi~{}{\rm
det}\left(\Sigma_{s,t}(8\bar{\mathfrak{a}})\right)^{\frac{1}{2}}}\exp\Big{(}-\frac{1}{2}\big{\langle}\Sigma^{-1}_{s,t}(8\bar{\mathfrak{a}}){\rm
w}_{s,t}(x,y),{\rm w}_{s,t}(x,y)\big{\rangle}\Big{)}.$ (3.17)

Then, there exists $C_{[\ref{lem: fcirc 1/2}]}>0$ such that, for all
$(s,x,t,y)\in\Theta$ and $x^{\prime}\in\mathbb{R}^{2}$ satisfying

$\displaystyle|x_{1}-x_{1}^{\prime}|^{\frac{1}{1+\beta^{\prime}_{1}}}+|x_{2}-x_{2}^{\prime}|^{\frac{1}{1+\beta^{\prime}_{2}}}~{}\leq~{}(t-s)^{1/2},$
(3.18)

we have

$\displaystyle f^{\circ}(s,x^{\prime};t,y)\leq C_{[\ref{lem: fcirc
1/2}]}f^{\circ,\frac{1}{2}}(s,x;t,y)$ (3.19)

and

$\displaystyle\left(\frac{|{\rm
w}_{s,t}(x^{\prime},y)_{1}|^{2}}{(t-s)^{{1+\beta^{\prime}_{1}}}}+\frac{|{\rm
w}_{s,t}(x^{\prime},y)_{2}|^{2}}{(t-s)^{{1+\beta^{\prime}_{2}}}}\right)\varpi^{1}(s,x,t,y)\leq
C_{[\ref{lem: fcirc 1/2}]}.$ (3.20)

###### Proof.

Set $w:={\rm w}_{s,t}(x,y)$ and $w^{\prime}:={\rm w}_{s,t}(x^{\prime},y)$.
First observe that

$\left(\langle\Sigma^{-1}_{s,t}(\bar{\mathfrak{a}})w,w\rangle\right)^{\frac{1}{2}}\leq\left(\langle\Sigma^{-1}_{s,t}(\bar{\mathfrak{a}})(w-w^{\prime}),(w-w^{\prime})\rangle\right)^{\frac{1}{2}}+\left(\langle\Sigma^{-1}_{s,t}(\bar{\mathfrak{a}})w^{\prime},w^{\prime}\rangle\right)^{\frac{1}{2}}.$

Using that $2ab\leq 2a^{2}+2b^{2}$ for $a,b\geq 0$, we deduce that

$\displaystyle-\langle\Sigma^{-1}_{s,t}(\bar{\mathfrak{a}})w^{\prime},w^{\prime}\rangle$
$\displaystyle\leq-\frac{1}{2}\langle\Sigma^{-1}_{s,t}(\bar{\mathfrak{a}})w,w\rangle+\langle\Sigma^{-1}_{s,t}(\bar{\mathfrak{a}})(w-w^{\prime}),(w-w^{\prime})\rangle.$

Now, by the same arguments as in the proof of Lemma 3.2 and Assumption 2.4, we
have

$\displaystyle\big{\langle}\Sigma^{-1}_{s,t}(\bar{\mathfrak{a}})(w-w^{\prime}),(w-w^{\prime})\big{\rangle}$
$\displaystyle\leq
2\bar{\mathfrak{a}}\left(\frac{|x_{1}-x_{1}^{\prime}|^{2}}{(t-s)^{1+\beta^{\prime}_{1}}}+\frac{|x_{2}-x_{2}^{\prime}|^{2}}{(t-s)^{1+\beta^{\prime}_{2}}}\right)\leq
4\bar{\mathfrak{a}},$

in which we used (2.7) and our assumption (3.18). This proves (3.19). The
assertion (3.20) is proved similarly, upon interchanging the role of $x$ and
$x^{\prime}$. ∎

###  3.2 Wellposedness of $\widetilde{\Phi}$

In this section, we prove that $\widetilde{\Phi}$ in (2.25)-(2.26) is well
defined. Recall that $f^{\circ}$ is defined in (2.30), and let us define

$\displaystyle\tilde{f}^{\circ}(s,x;t,y)~{}:=~{}f^{\circ}(s,{\rm\bf
A}_{s}x;t,{\rm\bf A}_{t}y),~{}~{}(s,x,t,y)\in\Theta.$

Noticing that ${\rm\bf A}={\rm\bf A}^{-1}$, and recalling that

$f_{r,z}(s,x;t,y)~{}:=~{}\tilde{f}_{r,{\rm\bf A}_{r}z}(s,{\rm\bf
A}_{s}x;t,{\rm\bf A}_{t}y),$

it is straightforward to check that

$\displaystyle\partial_{x_{1}}f_{r,z}(s,x;t,y)$
$\displaystyle~{}=~{}\overrightarrow{\rm A}_{s}\cdot D_{x}\tilde{f}_{r,{\rm\bf
A}_{r}z}(s,{\rm\bf A}_{s}x;t,{\rm\bf A}_{t}y),$ (3.21)
$\displaystyle\partial^{2}_{x_{1}x_{1}}f_{r,z}(s,x;t,y)$
$\displaystyle~{}=~{}{\rm Tr}\Big{[}\overrightarrow{\rm
A}_{s}\big{(}\overrightarrow{\rm
A}_{s}\big{)}^{\top}D^{2}_{xx}\tilde{f}_{r,{\rm\bf A}_{r}z}(s,{\rm\bf
A}_{s}x;{\rm\bf A}_{t}y)\Big{]}.$ (3.22)

######  Lemma 3.4.

Let the conditions of Theorem 2.7.(i) hold. Then, there exist a constant
$C_{\eqref{eq: estimee L-Lfy}}>0$ such that

$\displaystyle\big{|}\big{(}\widetilde{{\cal L}}-\widetilde{{\cal
L}}^{t,\tilde{y}}\big{)}\tilde{f}_{t,\tilde{y}}(s,\tilde{x};t,\tilde{y})\big{|}$
$\displaystyle~{}\leq~{}\frac{C_{\eqref{eq: estimee
L-Lfy}}}{(t-s)^{1-\kappa_{0}}}\tilde{f}^{\circ}(s,\tilde{x};t,\tilde{y}),~{}\mbox{for
all}~{}(s,\tilde{x},t,\tilde{y})\in\Theta,$ (3.23)

in which $\kappa_{0}$ is defined in (2.31).

###### Proof.

For simplicity, we assume that $t-s\leq 1$, the case $t-s>1$ being trivially
handled. Let us denote

$x:={\rm\bf A}_{s}\tilde{x}~{}~{}\mbox{and}~{}y:={\rm\bf A}_{t}\tilde{y}.$
(3.24)

$\mathrm{(i)}$ Using (2.20) and (3.21), we first estimate

$\displaystyle I_{1}~{}:=~{}\tilde{\mu}_{s}(\tilde{x})~{}\overrightarrow{\rm
A}\cdot
D_{x}\tilde{f}_{t,\tilde{y}}(s,\tilde{x};t,\tilde{y})~{}=~{}\mu_{s}(x)~{}\partial_{x_{1}}f_{t,y}(s,x;t,y).$

Then, by Assumption 2.4, Lemmas 3.1 and 3.2, it follows that

$\displaystyle|I_{1}|~{}\leq~{}\frac{\mathfrak{b}C_{\eqref{eq: norne Sigma-1 w
1}}}{(t-s)^{\frac{1+\beta_{0}}{2}}}\sqrt{\big{\langle}\Sigma^{-1}_{s,t}(t,y){w},{w}\big{\rangle}}f_{t,y}(s,x;t,y)~{}\leq~{}\frac{\mathfrak{b}C_{\eqref{eq:
norne Sigma-1 w 1}}C_{\eqref{eq: def const
1}}}{(t-s)^{1-\frac{1-\beta_{0}}{2}}}f^{\circ}(s,x;t,y),$ (3.25)

in which $w:={\rm w}_{s,t}({x},{y})$ and, with $w^{\prime}:={\rm
w}_{{s^{\prime},t^{\prime}}}({x^{\prime}},{y^{\prime}})$,

$\displaystyle C_{\eqref{eq: def const
1}}~{}:=\sup_{(s^{\prime},x^{\prime},t^{\prime},y^{\prime},z^{\prime})\in\Theta\times\mathbb{R}^{2}}\sqrt{\big{\langle}\Sigma^{-1}_{s^{\prime},t^{\prime}}(t^{\prime},z^{\prime})w^{\prime},w^{\prime}\big{\rangle}}\varpi(s^{\prime},x^{\prime};t^{\prime},y^{\prime})~{}<~{}\infty.$
(3.26)

$\mathrm{(ii)}$ Using (2.20) and (3.22), we now estimate

$\displaystyle I_{2}$ $\displaystyle~{}:=~{}{\rm
Tr}\Big{[}\big{(}\tilde{\sigma}^{2}_{t}(\tilde{y})-\tilde{\sigma}^{2}_{s}(\tilde{x})\big{)}\overrightarrow{\rm
A}_{s}(\overrightarrow{\rm
A}_{s})^{\top}D^{2}_{xx}\tilde{f}_{t,\tilde{y}}(s,\tilde{x};t,\tilde{y})\Big{]}=\big{[}\sigma^{2}_{t}(y)-\sigma^{2}_{s}(x)\big{]}\partial^{2}_{x_{1}x_{1}}f_{y}(s,x;t,y).$

By (3.6) and Lemma 3.1, one obtains

$\displaystyle|I_{2}|$
$\displaystyle~{}\leq~{}\big{[}\sigma^{2}_{t}(y)-\sigma^{2}_{s}(x)\big{]}\frac{(C_{\eqref{eq:
norne Sigma-1 w 1}})^{2}\vee C_{(\ref{eq: borne Sigma -1
11})}}{(t-s)^{1+\beta_{0}}}\Big{(}\big{\langle}\overline{\Sigma}_{s,t}^{-1}(t,y)w,w\big{\rangle}+1\Big{)}f_{y}(s,x;t,y).$

Recalling (2.7), (2.9), (2.8) and Lemma 3.2, it follows that, for some $C>0$
that does not depend on $(s,x,t,y)$ and $z\in\mathbb{R}^{2}$,

$\displaystyle|I_{2}|$
$\displaystyle~{}\leq~{}C\frac{1}{(t-s)^{1+\beta_{0}-\alpha}}\Big{(}1+\big{|}{\rm
w}_{s,t}(x,y)\big{|}^{\frac{2\alpha}{1+\beta^{\prime}_{1}}}+\big{|}{\rm
w}_{s,t}(x,y)\big{|}^{\frac{2\alpha}{1+\beta^{\prime}_{2}}}\Big{)}(\varpi^{1}f^{\circ})(s,x;t,y),$

and we conclude by using the definition of $\varpi^{1}$ in (3.15). ∎

######  Proposition 3.5.

Let the conditions of Theorem 2.7.(i) hold. Then, the sum in (2.25) is well-
defined and there exists a constant $C_{\eqref{eq: estime vraie densite}}>0$
such that

$\displaystyle\big{|}\widetilde{\Phi}(s,x;t,y)\big{|}$
$\displaystyle~{}\leq~{}\frac{C_{\eqref{eq: estime vraie
densite}}}{(t-s)^{1-\kappa_{0}}}\tilde{f}^{\circ}(s,x;t,y),~{}\mbox{for
all}~{}(s,x,t,y)\in\Theta.$ (3.27)

Moreover, $\widetilde{\Phi}$ is continuous on $\Theta$ and satisfies

$\displaystyle\widetilde{\Phi}(s,x;t,y)$
$\displaystyle=\widetilde{\Delta}_{0}(s,x;t,y)+\int_{s}^{t}\int_{\mathbb{R}^{2}}\widetilde{\Delta}_{0}(s,x;r,z)\widetilde{\Phi}(r,z;t,y)dzdr,~{}\mbox{for
all}~{}(s,x,t,y)\in\Theta.$ (3.28)

###### Proof.

Let us recall that, if well-defined,

$\widetilde{\Phi}(s,x;t,y)~{}:=~{}\sum_{k=0}^{\infty}\widetilde{\Delta}_{k}(s,x;t,y),$

where $\widetilde{\Delta}_{0}(s,x;t,y):=\big{(}\widetilde{{\cal
L}}-\widetilde{{\cal L}}^{t,y}\big{)}\tilde{f}_{t,y}(s,x;t,y)$, and

$\displaystyle\widetilde{\Delta}_{k+1}(s,x;t,y):=\int_{s}^{t}\int_{\mathbb{R}^{2}}\widetilde{\Delta}_{0}(s,x;r,z)\widetilde{\Delta}_{k}(r,z;t,y)dzdr,~{}~{}k\geq
0.$

We already know from Lemma 3.4 that

$\displaystyle\big{|}\big{(}\widetilde{{\cal L}}-\widetilde{{\cal
L}}^{t,y}\big{)}\tilde{f}_{t,y}(s,x;t,y)\big{|}$
$\displaystyle~{}\leq~{}\frac{C_{\eqref{eq: estimee
L-Lfy}}}{(t-s)^{1-\kappa_{0}}}\tilde{f}^{\circ}(s,x;t,y),$

for all $(s,x,t,y)\in\Theta$. By the same induction argument as in [8, proof
of Proposition 4.1], together with (3.1) and (2.3), we then deduce that

$\displaystyle\big{|}\widetilde{\Delta}_{k}(s,x;t,y)\big{|}~{}\leq~{}\frac{M_{k}}{(t-s)^{1-k\kappa_{0}}}\tilde{f}^{\circ}(s,x;t,y)~{}\leq~{}CM_{k}(t-s)^{k\kappa_{0}-2-\frac{\beta_{3}}{2}},$
(3.29)

in which, $C>0$ does not depend on $(s,x,t,y)$ and $k$, and

$M_{k}~{}:=~{}\frac{\\{C_{\eqref{eq: estimee
L-Lfy}}\Gamma(\kappa_{0})\\}^{k}}{\Gamma(k\kappa_{0})},$

where $\Gamma$ denotes the Gamma function. By dominated convergence, each map
$\widetilde{\Delta}_{k}$ is continuous. Then, the well-posedness of
$\widetilde{\Phi}$ follows from the fact that the power series $\sum_{k\geq
0}M_{k}u^{k}$ has a radius of convergence equal to $\infty$. Continuity of
$\widetilde{\Phi}$ is a consequence of the absolute continuity of the series.

It remains to prove (3.28). Note that, by the above,

$\widetilde{\Phi}(s,x;t,y)=\widetilde{\Delta}_{0}(s,x;t,y)+\sum_{k\geq
0}\int_{s}^{t}\int_{\mathbb{R}^{2}}\widetilde{\Delta}_{0}(s,x;r,z)\widetilde{\Delta}_{k}(r,z;t,y)dzdr$

and the family
$\\{(r,z)\in(s,t)\times\mathbb{R}^{2}\mapsto\sum_{k=0}^{n}\widetilde{\Delta}_{0}(s,x;r,z)\widetilde{\Delta}_{k}(r,z;t,y)$,
$n\geq 1\\}$ is uniformly integrable and converges to
$\widetilde{\Delta}_{0}(s,x;\cdot)\widetilde{\Phi}(\cdot;t,y)$. This implies
(3.28). ∎

Recall that

$\Phi(s,x;t,y)~{}:=~{}\widetilde{\Phi}(s,{\rm\bf A}_{s}x;t,{\rm\bf A}_{t}y).$

######  Proposition 3.6.

Let the conditions of Theorem 2.7.(i) hold. Then,
$f:\Theta\longrightarrow\mathbb{R}$ is well-defined in (2.28). Moreover, it is
continuous on $\Theta$ and, for some $C_{[\ref{prop:existence_fb_regul}]}>0$,

$\displaystyle\big{|}f(s,x;t,y)\big{|}~{}\leq~{}C_{[\ref{prop:existence_fb_regul}]}f^{\circ}(s,x;t,y),~{}\mbox{for
all}~{}(s,x,t,y)\in\Theta.$

###### Proof.

This is an immediate consequence of Proposition 3.5 and Lemma 3.2, recalling
that $f^{\circ}$ is a transition density and observing that
$\int_{s}^{t}(t-r)^{-1+\kappa_{0}}dr\leq CT^{\kappa_{0}}$. ∎

###  3.3 $C^{1}$-regularity

We now prove that $x=(x_{1},x_{2})\mapsto f(s,x;t,y)$ is $C^{1}$ in its first
space variable $x_{1}$, with partial derivative dominated by a Gaussian
density.

######  Lemma 3.7.

Let the conditions of Theorem 2.7 hold. Then, there exists $C_{[\ref{lem :
borne derive fbar}]}>0$ such that, for all $(r,z)\in[0,T]\times\mathbb{R}^{2}$
and $(s,x,t,y)\in\Theta$,

$\displaystyle\big{|}\partial_{x_{1}}f_{r,z}(s,x;t,y)\big{|}$
$\displaystyle~{}\leq~{}\frac{C_{[\ref{lem : borne derive
fbar}]}}{(t-s)^{\frac{\beta_{0}+1}{2}}}f^{\circ}(s,x;t,y).$

Moreover, let $h:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ be a (measurable)
function such that $\int_{\mathbb{R}^{2}}f^{\circ}(s,x;t,y)|h(y)|dy<\infty$,
and

$V(s,x;t)~{}:=~{}\int_{\mathbb{R}^{2}}f_{t,y}(s,x;t,y)h(y)dy,~{}~{}(s,x)\in[0,t)\times\mathbb{R}^{2},$

then $(s,x)\in[0,t)\times\mathbb{R}^{2}\mapsto V(s,x;t)$ is continuously
differentiable in its first space variable $x_{1}$ and satisfies

$\big{|}\partial_{x_{1}}V(s,x;t)\big{|}~{}\leq~{}\frac{C_{[\ref{lem : borne
derive
fbar}]}}{(t-s)^{\frac{\beta_{0}+1}{2}}}\int_{\mathbb{R}^{2}}f^{\circ}(s,x;t,y)~{}|h(y)|dy,$

in which $C_{[\ref{lem : borne derive fbar}]}>0$ does not depend on
$(s,x,t)\in[0,T]\times\mathbb{R}^{2}\times[0,T]$ with $s<t$.

###### Proof.

The first inequality follows immediately from Lemmas 3.1 and 3.2, as in the
proof of (3.25). The second one then follows by dominated convergence. ∎

For the following, we recall the defintion of $\kappa_{1}$ in (2.33).

######  Proposition 3.8.

Let the conditions of Theorem 2.7 hold. Then, for each
$(t,y)\in(0,T]\times\mathbb{R}^{2}$, the map
$(s,x)\in[0,t)\times\mathbb{R}^{2}\mapsto f(s,x;t,y)$ is continuously
differentiable in its first space variable $x_{1}$. Moreover, there exists
$C_{[\ref{prop : fb C1}]}>0$ such that

$|\partial_{x_{1}}f(s,x;t,y)|~{}\leq~{}\frac{C_{[\ref{prop : fb
C1}]}}{(t-s)^{1-\kappa_{1}}}f^{\circ}(s,x;t,y),~{}\mbox{for
all}~{}(s,x;t,y)\in\Theta.$

###### Proof.

Fix $z\in\mathbb{R}^{2}$. In view of the estimate in (3.27), together with
Lemma 3.7, we can find $C>0$, that does not depend on $(s,x,t,y)\in\Theta$,
such that

$\displaystyle\int_{\mathbb{R}^{2}}\Big{|}\partial_{x_{1}}f_{r,z}(s,x;r,z)\Phi(r,z;t,y)\Big{|}dz$
$\displaystyle\leq~{}$ $\displaystyle
C(r-s)^{\frac{-\beta_{0}-1}{2}}\int_{\mathbb{R}^{2}}f^{\circ}(s,x;r,z)\big{|}\Phi(r,z;t,y)\big{|}dz$
$\displaystyle\leq~{}$ $\displaystyle
C(t-r)^{-1+\kappa_{0}}(r-s)^{\frac{-\beta_{0}-1}{2}}\int_{\mathbb{R}^{2}}f^{\circ}(s,x;r,z)f^{\circ}(r,z;t,y)dz$
$\displaystyle=~{}$ $\displaystyle
C(t-r)^{-1+\kappa_{0}}(r-s)^{\frac{-\beta_{0}-1}{2}}f^{\circ}(s,x;t,y).$

Therefore, by the dominated convergence theorem,

$\partial_{x_{1}}\int_{s}^{t}\int_{\mathbb{R}^{2}}f_{r,z}(s,x;r,z)\Phi(r,z,t,y)dzdr$

is well-defined and continuous, and so is $\partial_{x_{1}}f(\cdot;t,y)$. The
latter is bounded from the above estimates by integrating over $r$ and using
the relation between the Euler-Gamma and the Beta functions. ∎

We conclude this section by a continuity property result on $f$, which allows
one to apply the $C^{1}$-Itô’s formula in the context of Theorem 2.11.

######  Proposition 3.9.

Let Assumptions 2.1 and 2.4.(i) hold true. Assume in addition that (2.31)
holds and that $\frac{\beta_{3}-\beta_{2}+\beta_{0}}{2}<1$, and let us fix
$\alpha^{\prime}\in\big{(}0,\frac{1+\beta^{\prime}_{2}}{2}\wedge(1-\frac{\beta_{3}-\beta_{2}+\beta_{0}}{2})\big{]}$.
Then, for all $\delta>0$, there exists $C_{[\ref{prop : holder f en x2}]}>0$
such that

$\displaystyle\big{|}f(s,x;t,y)-f(s,x^{\prime};t,y)\big{|}$
$\displaystyle~{}\leq~{}C_{[\ref{prop : holder f en
x2}]}|x_{2}-x^{\prime}_{2}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{2}}},$

for all $(s,x,t,y)\in\Theta$ and
$x^{\prime}=(x^{\prime}_{1},x^{\prime}_{2})\in\mathbb{R}^{2}$ such that
$t-s\geq\delta$ and $x_{1}=x^{\prime}_{1}$.

###### Proof.

Let $I:=\big{|}f_{r,z}(s,x;t,y)-f_{r,z}(s,x^{\prime};t,y)\big{|}$ and denote
by $C>0$ a generic constant that can change from line to line but does not
depend on $(s,x,x^{\prime},t,y,z)$. Then, by (3.5), Lemma 3.1 and Lemma 3.2,
one can find $x^{\prime\prime}_{2}$ in the interval formed by $x_{2}$ and
$x^{\prime}_{2}$ such that, with
$x^{\prime\prime}:=(x_{1},x^{\prime\prime}_{2})$,

$\displaystyle I$
$\displaystyle\leq|x_{2}-x^{\prime}_{2}|\;|\partial_{x_{2}}f_{r,z}(s,x^{\prime\prime};t,y)|\leq
C|x_{2}-x^{\prime}_{2}|\frac{1}{(t-s)^{\frac{1+\beta_{3}}{2}}}f^{\circ}(s,x^{\prime\prime};t,y).$

If $(t-s)^{\frac{1+\beta^{\prime}_{2}}{2}}/|x_{2}-x^{\prime}_{2}|\geq 1$, then

$\displaystyle I$ $\displaystyle\leq
C|x_{2}-x^{\prime}_{2}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{2}}}\frac{1}{(t-s)^{\frac{1+\beta_{3}}{2}-\frac{1+\beta^{\prime}_{2}}{2}+\alpha^{\prime}}}f^{\circ}(s,x^{\prime\prime};t,y)$
$\displaystyle=C|x_{2}-x^{\prime}_{2}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{2}}}\frac{1}{(t-s)^{\alpha^{\prime}+\frac{\beta_{3}-\beta_{2}+\beta_{0}}{2}}}f^{\circ}(s,x^{\prime\prime};t,y).$

Otherwise, by (3.12),

$I\leq|x_{2}-x^{\prime}_{2}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{2}}}\frac{1}{(t-s)^{\alpha^{\prime}}}\big{(}f^{\circ}(s,x;t,y)+f^{\circ}(s,x^{\prime};t,y)\big{)}.$

We conclude by using the fact that $\beta_{3}-\beta_{2}+\beta_{0}\geq 0$ and
by appealing to (3.27). ∎

###  3.4 $C^{2}$-regularity

We now prove that $f$ is $C^{2}$ in its first space variable $x_{1}$ and that
${\rm v}$ is a smooth solution of the path-dependent PDE (2.1).

####  3.4.1 Potential estimate and Hölder regularity of $\Phi$

Let $0\leq s<t\leq T$ and $x\in\mathbb{R}^{2}$,
$h:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ be a (measurable) function, we
first estimate the second order derivative of the following functional:

$V(s,x;t)~{}:=\int_{\mathbb{R}^{2}}f_{t,y}(s,x;t,y)h(y)dy.$

Let us also denote

$\displaystyle E^{-1}_{s,t}(x)~{}:=~{}\left(\begin{array}[]{cc}1&0\\\
A_{t}-A_{s}&1\end{array}\right)x.$ (3.32)

######  Lemma 3.10.

Let Assumption 2.1 and Assumption 2.4.(i) hold. Let
$h:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ and
$h_{\circ}:\mathbb{R}^{2}\longrightarrow\mathbb{R}_{+}$ be such that, for some
$\alpha_{h}>0$ and $C_{h}>0$,

$\displaystyle\big{|}h(y)-h(y^{\prime})\big{|}\leq
C_{h}\Big{(}|y_{1}-y^{\prime}_{1}|^{\frac{2\alpha_{h}}{1+\beta^{\prime}_{1}}}+|y_{2}-y^{\prime}_{2}|^{\frac{2\alpha_{h}}{1+\beta^{\prime}_{2}}}\Big{)}\big{(}h_{\circ}(y)+h_{\circ}(y^{\prime})\big{)},\mbox{
for all }y,y^{\prime}\in\mathbb{R}^{2},$

and

$\int_{\mathbb{R}^{2}}f^{\circ}(s,x;t,y)h_{\circ}(y)dy~{}<~{}\infty,~{}\mbox{for
all}~{}0\leq s<t\leq T,~{}~{}x\in\mathbb{R}^{2}.$

Assume that

$\kappa_{h}~{}:=~{}\min\Big{(}\frac{2\beta_{4}+1+\beta^{\prime}_{1}}{1+\beta^{\prime}_{2}},1\Big{)}\min\\{\alpha_{h},\alpha\\}-\beta_{0}~{}>~{}0.$

Then, $\partial^{2}_{x_{1}x_{1}}V(s,x;t)$ is well defined and continuous.
Moreover

$\displaystyle\partial^{2}_{x_{1}x_{1}}V(s,x;t)$
$\displaystyle=\int_{\mathbb{R}^{2}}\partial^{2}_{x_{1}x_{1}}f_{t,y}(s,x;t,y)h(y)dy,$

and there exists $C>0$, that does not depend on $C_{h}>0$, such that

$\displaystyle\big{|}\partial^{2}_{x_{1}x_{1}}V(s,x;t)\big{|}$
$\displaystyle\leq\frac{CC_{h}}{(t-s)^{1-\kappa_{h}}}\left(\big{|}h\big{(}E^{-1}_{s,t}(x)\big{)}\big{|}+{\big{|}h_{\circ}\big{(}E^{-1}_{s,t}(x)\big{)}\big{|}+\int_{\mathbb{R}^{2}}f^{\circ}(s,x;t,y)h_{\circ}(y)dy}\right),$

for all $0\leq s<t\leq T$ and $x\in\mathbb{R}^{2}$.

###### Proof.

For simplicity, we only consider the case $t-s\leq 1$. To estimate the second
order derivative, we decompose

$I:=\int_{\mathbb{R}^{2}}\partial_{x_{1}x_{1}}^{2}f_{t,y}(s,x;t,y)h(y)dy$

into the sum of the three following terms, with $\check{x}:=E^{-1}_{s,t}(x)$,

$\displaystyle I_{1}$
$\displaystyle:=\int_{\mathbb{R}^{2}}\partial_{x_{1}x_{1}}^{2}f_{t,y}(s,x;t,y)\big{[}h(y)-h\big{(}\check{x}\big{)}\big{]}dy,$
$\displaystyle I_{2}$
$\displaystyle:=h\big{(}\check{x}\big{)}\int_{\mathbb{R}^{2}}\left\\{\partial_{x_{1}x_{1}}^{2}f_{t,y}(s,x;t,y)-\partial_{x_{1}x_{1}}^{2}f_{t,\check{x}}(s,x;t,y)\right\\}dy,$
$\displaystyle I_{3}$
$\displaystyle:=h\big{(}\check{x}\big{)}\int_{\mathbb{R}^{2}}\partial_{x_{1}x_{1}}^{2}f_{t,\check{x}}(s,x;t,y)dy.$

All over this proof, $C>0$ denotes a generic constant that may change from
line to line but does not depend on $C_{h}$, $(s,x;t,y)\in\Theta$ and
$z\in\mathbb{R}^{2}$.

$\mathrm{(i)}$ We first estimate $I_{1}$. Set $w={\rm w}_{s,t}(x,y)$, recall
(2.7). By the Hölder regularity property of $h$ and the inequality
$(a+b)^{\gamma}\leq 2^{\gamma}(a^{\gamma}+b^{\gamma})$ for $a,b\geq 0$ and
$\gamma>0$, one has

$\displaystyle\Big{|}\partial_{x_{1}x_{1}}^{2}f_{t,y}(s,x;t,y)\big{[}h(y)-h\big{(}\check{x}\big{)}]\Big{|}$
$\displaystyle\leq
C_{h}\Big{|}\partial_{x_{1}x_{1}}^{2}f_{t,y}(s,x;t,y)\Big{|}\Big{(}|x_{1}-y_{1}|^{\frac{2\alpha_{h}}{1+\beta^{\prime}_{1}}}+|y_{2}-x_{2}-(A_{t}-A_{s})x_{1}|^{\frac{2\alpha_{h}}{1+\beta^{\prime}_{2}}}\Big{)}\big{(}h_{\circ}(y)+h_{\circ}(E^{-1}_{s,t}(x))\big{)}$
$\displaystyle\leq
CC_{h}\Big{|}\partial_{x_{1}x_{1}}^{2}f_{t,y}(s,x;t,y)\Big{|}\Big{(}|w_{1}|^{\frac{2\alpha_{h}}{1+\beta^{\prime}_{1}}}+|w_{2}|^{\frac{2\alpha_{h}}{1+\beta^{\prime}_{2}}}+\big{|}(A_{t}-A_{s})w_{1}\big{|}^{\frac{2\alpha_{h}}{1+\beta^{\prime}_{2}}}\Big{)}\big{(}h_{\circ}(y)+h_{\circ}(E^{-1}_{s,t}(x))\big{)}.$

Then, arguing as in the proof of Lemma 3.4 and using (2.4), we deduce that

$\displaystyle\big{|}I_{1}\big{|}\leq$ $\displaystyle
CC_{h}\Big{(}\frac{1}{(t-s)^{1+\beta_{0}-\alpha_{h}}}+\frac{1}{(t-s)^{1+\beta_{0}-\alpha_{h}\frac{2\beta_{4}+1+\beta^{\prime}_{1}}{1+\beta^{\prime}_{2}}}}\Big{)}\int_{\mathbb{R}^{2}}f^{\circ}(s,x;t,y)\big{(}h_{\circ}(y)+h_{\circ}(E^{-1}_{s,t}(x))\big{)}dy,$
$\displaystyle\leq~{}$
$\displaystyle\frac{CC_{h}}{(t-s)^{1-\kappa_{h}}}{\Big{(}\int_{\mathbb{R}^{2}}f^{\circ}(s,x;t,y)h_{\circ}(y)dy+h_{\circ}\big{(}E^{-1}_{s,t}(x)\big{)}\Big{)}.}$

$\mathrm{(ii)}$ We now consider $I_{2}$. By (3.6) and Lemma 3.1,

$\displaystyle\big{|}\partial_{x_{1}x_{1}}^{2}f_{t,y}(s,x;t,y)-\partial_{x_{1}x_{1}}^{2}f_{t,\check{x}}(s,x;t,y)\big{|}$
$\displaystyle~{}\leq~{}$
$\displaystyle\big{|}f_{t,y}(s,x;t,y)-f_{t,\check{x}}(s,x;t,y)\big{|}\Big{|}\big{(}\Sigma^{-1}_{s,t}(t,y)w\big{)}_{1}^{2}-\big{(}\Sigma^{-1}_{s,t}(t,y)\big{)}_{1,1}\Big{|}$
$\displaystyle+f_{t,\check{x}}(s,x;t,y)\Big{|}\big{(}\Sigma^{-1}_{s,t}(t,y)w\big{)}_{1}^{2}-\big{(}\Sigma^{-1}_{s,t}(t,\check{x})w\big{)}_{1}^{2}\Big{|}$
$\displaystyle+f_{t,\check{x}}(s,x;t,y)\Big{|}\big{(}\Sigma^{-1}_{s,t}(t,y)\big{)}_{1,1}-\big{(}\Sigma^{-1}_{s,t}(t,\check{x})\big{)}_{1,1}\Big{|}$
$\displaystyle~{}=~{}$
$\displaystyle\big{|}f_{t,y}(s,x;t,y)-f_{t,\check{x}}(s,x;t,y)\big{|}\Big{|}\big{(}\Sigma^{-1}_{s,t}(t,y)w\big{)}_{1}^{2}-\big{(}\Sigma^{-1}_{s,t}(t,y)\big{)}_{1,1}\Big{|}$
$\displaystyle+f_{t,\check{x}}(s,x;t,y)\left|\sigma_{t}(y)^{-4}-\sigma_{t}(\check{x})^{-4}\right|\Big{|}\big{(}\Sigma^{-1}_{s,t}(1)w\big{)}_{1}\Big{|}^{2}$
$\displaystyle+f_{t,\check{x}}(s,x;t,y)\left|\sigma_{t}(y)^{-2}-\sigma_{t}(\check{x})^{-2}\right|\Big{|}\big{(}\Sigma^{-1}_{s,t}(1)\big{)}_{1,1}\Big{|},$

in which, by (2.9),

$\displaystyle\left|\sigma_{t}(y)-\sigma_{t}(\check{x})\right|$
$\displaystyle\leq C_{(\ref{eq : hyp holder coeff bar
sigma})}\Big{(}\big{|}y_{1}-x_{1}\big{|}^{\frac{2\alpha}{1+\beta^{\prime}_{1}}}+\big{|}y_{2}-x_{1}(A_{t}-A_{s})-x_{2}\big{|}^{\frac{2\alpha}{1+\beta^{\prime}_{2}}}\Big{)}$
$\displaystyle\leq
C\Big{(}|w_{1}|^{\frac{2\alpha_{g}}{1+\beta^{\prime}_{1}}}+|w_{2}|^{\frac{2\alpha_{g}}{1+\beta^{\prime}_{2}}}+\big{|}(A_{t}-A_{s})w_{1}\big{|}^{\frac{2\alpha_{g}}{1+\beta^{\prime}_{2}}}\Big{)}$

by the same arguments as in in step 1. Using Lemma 3.12 below, (3.8), (3.10)
and (2.8), it follows that

$\big{|}I_{2}\big{|}~{}\leq~{}\frac{C|h\big{(}\check{x}\big{)}|}{(t-s)^{1+\beta_{0}-\alpha\frac{2\beta_{4}+1+\beta^{\prime}_{1}}{1+\beta^{\prime}_{2}}}}\int_{\mathbb{R}^{2}}f^{\circ}(s,x;t,y)dy\leq\frac{C|h\big{(}E^{-1}_{s,t}(x)\big{)}|}{(t-s)^{1-\kappa_{h}}}.$

$\mathrm{(iii)}$ We finally consider $I_{3}$. Notice that $y\mapsto
f_{t,\check{x}}(s,x;t,y)$ is a Gaussian density function, so that

$\int_{\mathbb{R}^{2}}\big{|}\partial^{2}_{x_{1}x_{1}}f_{t,\check{x}}(s,x;t,y)\big{|}dy~{}<~{}\infty.$

Moreover, by the definition of $f_{t,\check{x}}(s,x;t,y)$, one has

$D_{y}f_{t,\check{x}}(s,x;t,y)=-\left(\begin{array}[]{cc}1&0\\\
-(A_{t}-A_{s})&1\end{array}\right)^{\top}D_{x}f_{t,\check{x}}(s,x;t,y),$

so that

$\displaystyle\partial_{x_{1}}f_{t,\check{x}}(s,x;t,y)=-\partial_{y_{1}}f_{t,\check{x}}(s,x;t,y)-(A_{t}-A_{s})\partial_{y_{2}}f_{t,\check{x}}(s,x;t,y),$

which implies

$\int_{\mathbb{R}}\int_{\mathbb{R}}\partial_{x_{1}}f_{t,\check{x}}(s,x;t,y)dy_{1}dy_{2}~{}=~{}0,$

and therefore $I_{3}=0$.

Finally, we can apply the Leibniz integral rule to interchange the derivative
and the integral, and hence to conclude the proof. ∎

######  Remark 3.11.

Let us consider $V(s,x;t)$ as a path-dependent functional:

$\overline{V}(s,{\rm x};t)~{}:=~{}V(s,{\rm x}(s),I_{s}({\rm
x});t)~{}=~{}\int_{\mathbb{R}^{2}}{\rm f}_{t,y}(s,{\rm x};t,y)h(y)dy.$

In view of Remark 2.8, the above results implies that, in the context of Lemma
3.10, the second order vertical derivative $\partial^{2}_{{\rm x}{\rm
x}}\overline{V}(s,{\rm x};t)$ is well defined. Moreover, by (2.39), one has

$\partial_{s}{\rm f}_{t,y}(s,{\rm
x};t,y)~{}=~{}-\frac{1}{2}\sigma(t,y)^{2}\partial^{2}_{{\rm x}{\rm x}}{\rm
f}_{t,y}(s,{\rm x};t,y).$ (3.33)

Then, by the same technique as in Lemma 3.10, we can deduce that the
horizontal derivative $\partial_{s}\overline{V}(s,{\rm x};t)$ is also well
defined.

We now provide the following easy estimate which is used in the proof of Lemma
3.10.

######  Lemma 3.12.

Let Assumption 2.1.(i) and Assumption 2.4.(i) hold. Then, there exists
$C_{[\ref{lem : derive f par rapport vol}]}>0$ such that, for all
$(s,x,t,y)\in\Theta$ and $z,z^{\prime}\in\mathbb{R}^{2}$,

$\displaystyle\big{|}f_{t,z}(s,x;t,y)-f_{t,z^{\prime}}(s,x;t,y)\big{|}$
$\displaystyle\leq C_{[\ref{lem : derive f par rapport
vol}]}\left(|z_{1}-z_{1}^{\prime}|^{\frac{2\alpha}{1+\beta^{\prime}_{1}}}+|z_{2}-z_{2}^{\prime}|^{\frac{2\alpha}{1+\beta^{\prime}_{2}}}\right)\left[1+\langle\Sigma^{-1}_{s,t}(1)w,w\rangle\right](\varpi
f^{\circ})(s,x;t,y)$

in which $w:={\rm w}_{s,t}(x,y)$.

###### Proof.

Let us write $f_{[a]}$ for $f_{t,z}$ if $\sigma^{2}_{t}(z)=a$, and let
$\partial_{a}f_{[a]}$ denote its derivative with respect to this parameter
$a$. Then,

$\partial_{a}f_{[a]}(s,x;t,y)=\left[-\frac{1}{a}+\frac{\sigma_{t}(0)^{2}}{2a^{2}}\langle\Sigma^{-1}_{s,t}(0)w,w\rangle\right]f_{[a]}(s,x;t,y)$

in which $w={\rm w}_{s,t}(x,y)$ is as in (2.7). In view of Assumption 2.4 and
Lemma 3.2, it follows that

$|\partial_{a}f_{[a]}(s,x;t,y)|\leq
C\left[1+\langle\Sigma^{-1}_{s,t}(0)w,w\rangle\right]\varpi(s,x;t,y)f^{\circ}(s,x;t,y),$

for some $C>0$ that does not depend on $a,(s,x,t,y)$. We conclude by appealing
to (2.9). ∎

In order to apply Lemma 3.10 to (2.23), we need to prove that the function
$\Phi(s,x;t,y)$ defined by (2.25) and (2.27) is Hölder in $x$. Recall the
definition of $\widehat{\Delta\beta}$ in (2.40), of $\kappa_{0}$ in (2.31) and
of $f^{\circ,\frac{1}{2}}$ in (3.17).

######  Lemma 3.13.

Let the conditions of Theorem 2.9 hold. Fix
$\alpha_{\Phi}\in\big{(}0,\hat{\alpha}_{\Phi}\wedge\kappa_{0}\wedge\min\limits_{i=1,2}\frac{1+\beta^{\prime}_{i}}{2}\big{)}$.
Then, there exists $C_{\alpha_{\Phi}}>0$ such that,

$\displaystyle|\Phi(s,x;t,y)-\Phi(s,x^{\prime};t,y)|$ $\displaystyle\leq~{}$
$\displaystyle
C_{\alpha_{\Phi}}\frac{|x_{1}-x_{1}^{\prime}|^{\frac{2\alpha_{\Phi}}{1+\beta^{\prime}_{1}}}+|x_{2}-x_{2}^{\prime}|^{\frac{2\alpha_{\Phi}}{1+\beta^{\prime}_{2}}}}{(t-s)^{1-\eta_{\Phi}}}\Big{(}f^{\circ,\frac{1}{2}}(s,x;t,y)+f^{\circ,\frac{1}{2}}(s,x^{\prime};t,y)\Big{)},$

for all $(s,x,t,y)\in\Theta$, in which

$\eta_{\Phi}~{}:=~{}\hat{\alpha}_{\Phi}\wedge\kappa_{0}-\alpha_{\Phi}~{}>~{}0.$

###### Proof.

In all this proof, $C>0$ denotes a generic constant, whose value can change
from line to line, but which does not depend on $(s,x,t,y)\in\Theta$. We set
$\Delta_{k}(s,x;t,y):=\widetilde{\Delta}_{k}(s,{\rm\bf A}_{s}x;t,{\rm\bf
A}_{t}y)$ and recall that $\Phi(s,x;t,y):=\widetilde{\Phi}(s,{\rm\bf
A}_{s}x;t,{\rm\bf A}_{t}y)$, $(s,x,t,y)\in\Theta$.

$\mathrm{(i)}$ Let us first consider

$\displaystyle I~{}:=~{}$
$\displaystyle\Delta_{0}(s,x;t,y)-\Delta_{0}(s,x^{\prime};t,y)$
$\displaystyle=~{}$
$\displaystyle\mu_{s}(x)\partial_{x_{1}}f_{t,y}(s,x;t,y)+\frac{1}{2}\big{(}\sigma^{2}_{s}(x)-\sigma^{2}_{t}(y)\big{)}\partial^{2}_{x_{1}x_{1}}f_{t,y}(s,x;t,y)$
$\displaystyle-\Big{(}\mu_{s}(x^{\prime})\partial_{x_{1}}f_{t,y}(s,x^{\prime};t,y)+\frac{1}{2}\big{(}\sigma^{2}_{s}(x^{\prime})-\sigma^{2}_{t}(y)\big{)}\partial^{2}_{x_{1}x_{1}}f_{t,y}(s,x^{\prime};t,y)\Big{)}.$
(3.34)

$\mathrm{(i.1)}$ In the case where

$|x_{1}-x_{1}^{\prime}|^{\frac{1}{1+\beta^{\prime}_{1}}}+|x_{2}-x_{2}^{\prime}|^{\frac{1}{1+\beta^{\prime}_{2}}}~{}>~{}(t-s)^{1/2},$

Lemma 3.4 implies that, for $\alpha^{\prime}\in(0,\kappa_{0})$,

$\displaystyle\Big{|}\Delta_{0}(s,x;t,y)-\Delta_{0}(s,x^{\prime};t,y)\Big{|}$
$\displaystyle\leq~{}$ $\displaystyle C_{\eqref{eq: estimee
L-Lfy}}\frac{1}{(t-s)^{1-\kappa_{0}}}\big{(}f^{\circ}(s,x;t,y)+f^{\circ}(s,x^{\prime};t,y)\big{)}$
$\displaystyle\leq~{}$ $\displaystyle C_{\eqref{eq: estimee
L-Lfy}}\frac{|x_{1}-x_{1}^{\prime}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{1}}}+|x_{2}-x_{2}^{\prime}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{2}}}}{(t-s)^{1-\kappa_{0}+\alpha^{\prime}}}\big{(}f^{\circ}(s,x;t,y)+f^{\circ}(s,x^{\prime};t,y)\big{)}.$

$\mathrm{(i.2)}$ We next consider the case where

$|x_{1}-x_{1}^{\prime}|^{\frac{1}{1+\beta^{\prime}_{1}}}+|x_{2}-x_{2}^{\prime}|^{\frac{1}{1+\beta^{\prime}_{2}}}~{}\leq~{}(t-s)^{1/2}.$
(3.35)

Let us write

$I~{}:=~{}\Delta_{0}(s,x;t,y)-\Delta_{0}(s,x^{\prime};t,y)~{}=~{}I_{1}+I_{2}+I_{3}+I_{4},$

where

$I_{1}~{}:=~{}\big{(}\mu_{s}(x)-\mu_{s}(x^{\prime})\big{)}\partial_{x_{1}}f_{t,y}(s,x;t,y),$
$I_{2}~{}:=~{}\mu_{s}(x^{\prime})\big{(}\partial_{x_{1}}f_{t,y}(s,x;t,y)-\partial_{x_{1}}f_{t,y}(s,x^{\prime};t,y)\big{)},$
$I_{3}~{}:=~{}\frac{1}{2}\big{(}\sigma^{2}_{s}(x)-\sigma^{2}_{s}(x^{\prime})\big{)}\partial^{2}_{x_{1}x_{1}}f_{t,y}(s,x^{\prime};t,y),$

and

$I_{4}~{}:=~{}\frac{1}{2}\big{(}\sigma^{2}_{s}(x)-\sigma^{2}_{t}(y)\big{)}\Big{(}\partial^{2}_{x_{1}x_{1}}f_{t,y}(s,x;t,y)-\partial^{2}_{x_{1}x_{1}}f_{t,y}(s,x^{\prime};t,y)\Big{)}.$

For $I_{1}$, we use the Hölder continuity property of $\mu$ in (2.10), Lemma
3.1 and Lemma 3.2 to obtain that

$\displaystyle\big{|}I_{1}\big{|}$
$\displaystyle~{}\leq~{}C\frac{|x_{1}-x_{1}^{\prime}|^{\frac{2\alpha}{1+\beta^{\prime}_{1}}}+|x_{2}-x_{2}^{\prime}|^{\frac{2\alpha}{1+\beta^{\prime}_{2}}}}{(t-s)^{\frac{1+\beta_{0}}{2}}}f^{\circ}(s,x;t,y).$
(3.36)

For $I_{2}$, let us fix $\rho\in[0,1]$ and $x^{\prime\prime}=\rho
x+(1-\rho)x^{\prime}$ so that, using Assumption 2.4,

$\big{|}I_{2}\big{|}~{}\leq~{}\mathfrak{b}\Big{(}\big{|}\partial^{2}_{x_{1}x_{1}}f_{t,y}(s,x^{\prime\prime};t,y)\big{|}\big{|}x_{1}-x_{1}^{\prime}\big{|}+\big{|}\partial^{2}_{x_{1}x_{2}}f_{t,y}(s,x^{\prime\prime};t,y)\big{|}\big{|}x_{2}-x_{2}^{\prime}\big{|}\Big{)}.$

Using (3.6), (3.6), Lemma 3.1, Lemma 3.2 and the fact that
$\beta_{0}\leq\beta_{3}$, it follows that

$\displaystyle\big{|}I_{2}\big{|}$ $\displaystyle\leq
C\Big{(}\frac{|x_{1}-x_{1}^{\prime}|}{(t-s)^{1+\beta_{0}}}+\frac{|x_{2}-x_{2}^{\prime}|}{(t-s)^{1+\frac{\beta_{0}+\beta_{3}}{2}}}\Big{)}f^{\circ}(s,x^{\prime\prime};t,y).$

Since $x^{\prime\prime}$ lies in the interval formed by $x$ and $x^{\prime}$,
Lemma 3.3 and (3.35) imply that

$\displaystyle\big{|}I_{2}\big{|}$ $\displaystyle\leq
C\Big{(}\frac{|x_{1}-x_{1}^{\prime}|}{(t-s)^{1+\beta_{0}}}+\frac{|x_{2}-x_{2}^{\prime}|}{(t-s)^{1+\frac{\beta_{0}+\beta_{3}}{2}}}\Big{)}f^{\circ,\frac{1}{2}}(s,x;t,y).$
(3.37)

Next, using the Hölder property of $\sigma$ in (2.9), Lemma 3.1 and Lemma 3.2,
it follows that

$\displaystyle\big{|}I_{3}\big{|}~{}\leq~{}\frac{C}{(t-s)^{1+\beta_{0}}}\Big{(}|x_{1}-x_{1}^{\prime}|^{\frac{2\alpha}{1+\beta^{\prime}_{1}}}+|x_{2}-x_{2}^{\prime}|^{\frac{2\alpha}{1+\beta^{\prime}_{2}}}\Big{)}f^{\circ}(s,x;t,y).$
(3.38)

Finally, $I_{4}$ is tackled as $I_{2}$. Namely, we can find
$\tilde{x}^{\prime\prime}=\tilde{\rho}x+(1-\tilde{\rho})x^{\prime}$ with
$\tilde{\rho}\in[0,1]$ such that

$\displaystyle\Big{|}\partial^{2}_{x_{1}x_{1}}f_{t,y}(s,x;t,y)-\partial^{2}_{x_{1}x_{1}}f_{t,y}(s,x^{\prime};t,y)\Big{|}$
$\displaystyle\leq~{}$
$\displaystyle\big{|}\partial^{3}_{x_{1}x_{1}x_{1}}f_{t,y}(s,\tilde{x}^{\prime\prime};t,y)\big{|}\big{|}x_{1}-x_{1}^{\prime}\big{|}+\big{|}\partial^{3}_{x_{1}x_{1}x_{2}}f_{t,y}(s,\tilde{x}^{\prime\prime};t,y)\big{|}\big{|}x_{2}-x_{2}^{\prime}\big{|}$
$\displaystyle\leq~{}$ $\displaystyle
C\left(\frac{\big{|}x_{1}-x_{1}^{\prime}\big{|}}{(t-s)^{\frac{3}{2}(1+\beta_{0})}}+\frac{\big{|}x_{2}-x_{2}^{\prime}\big{|}}{(t-s)^{\frac{3}{2}+\beta_{0}+\frac{\beta_{3}}{2}}}\right)(\varpi^{1}f^{\circ})(s,\tilde{x}^{\prime\prime};t,y),$

in which we used Lemma 3.1 and Lemma 3.2 again. Next, we appeal to (2.9) to
deduce that

$\displaystyle\big{|}\sigma^{2}_{s}(x)-\sigma^{2}_{t}(y)\big{|}$
$\displaystyle\leq C_{\eqref{eq : hyp holder coeff bar
sigma}}\Big{(}|t-s|^{\alpha}+|w_{1}|^{\frac{2\alpha}{1+\beta^{\prime}_{1}}}+|w_{2}|^{\frac{2\alpha}{1+\beta^{\prime}_{2}}}\Big{)}.$

Using that $\beta_{3}\geq\beta_{0}$, the condition (3.35) together with Lemma
3.3 and the fact that $\tilde{x}^{\prime\prime}$ lies on the interval formed
by $x$ and $x^{\prime}$ implies that

$\displaystyle|I_{4}|$
$\displaystyle~{}\leq~{}C|t-s|^{\alpha}\left(\frac{\big{|}x_{1}-x^{\prime}_{1}\big{|}}{(t-s)^{\frac{3}{2}(1+\beta_{0})}}+\frac{\big{|}x_{2}-x_{2}^{\prime}\big{|}}{(t-s)^{\frac{3}{2}+\beta_{0}+\frac{\beta_{3}}{2}}}\right)f^{\circ,\frac{1}{2}}(s,x;t,y).$
(3.39)

Note that there exists $C>0$, that does not depend on $(s,x,t,y)$ such that

$f^{\circ}(s,x;t,y)~{}\leq~{}Cf^{\circ,\frac{1}{2}}(s,x;t,y).$

Thus, combining (3.36)-(3.39) and recalling (3.34) and (3.19) leads to a upper
bound for

$J:=\frac{|I|}{C\big{(}f^{\circ,\frac{1}{2}}(s,x;t,y)+f^{\circ,\frac{1}{2}}(s,x^{\prime};t,y)\big{)}}.$

Namely,

$\displaystyle J~{}\leq~{}$
$\displaystyle\frac{|x_{1}-x_{1}^{\prime}|^{\frac{2\alpha}{1+\beta^{\prime}_{1}}}+|x_{2}-x_{2}^{\prime}|^{\frac{2\alpha}{1+\beta^{\prime}_{2}}}}{(t-s)^{1+\beta_{0}}}$
$\displaystyle+~{}|x_{1}-x_{1}^{\prime}|\left(\frac{1}{(t-s)^{1+\beta_{0}}}+\frac{1}{(t-s)^{\frac{3}{2}(1+\beta_{0})-\alpha}}\right)$
$\displaystyle+~{}|x_{2}-x_{2}^{\prime}|\left(\frac{1}{(t-s)^{1+\frac{\beta_{0}+\beta_{3}}{2}}}+\frac{1}{(t-s)^{\frac{3}{2}+\beta_{0}+\frac{\beta_{3}}{2}-\alpha}}\right).$

We then use that
$(t-s)^{\frac{1+\beta^{\prime}_{i}}{2}}/|x_{i}-x^{\prime}_{i}|\geq 1$, for
$i=1,2$, to deduce that, for
$0<\alpha^{\prime}\leq\alpha\wedge\min\limits_{i=1,2}\frac{1+\beta^{\prime}_{i}}{2}$,

$\displaystyle J~{}\leq~{}$
$\displaystyle\frac{|x_{1}-x_{1}^{\prime}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{1}}}+|x_{2}-x_{2}^{\prime}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{2}}}}{(t-s)^{1+\beta_{0}+\alpha^{\prime}-\alpha}}$
$\displaystyle+~{}\frac{|x_{1}-x_{1}^{\prime}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{1}}}}{(t-s)^{(1+\beta_{0})\vee(\frac{3}{2}(1+\beta_{0})-\alpha)-\frac{1+\beta^{\prime}_{1}}{2}+\alpha^{\prime}}}+\frac{|x_{2}-x_{2}^{\prime}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{2}}}}{(t-s)^{(1+\frac{\beta_{0}+\beta_{3}}{2})\vee(\frac{3}{2}+\beta_{0}+\frac{\beta_{3}}{2}-\alpha)-\frac{1+\beta^{\prime}_{2}}{2}+\alpha^{\prime}}}.$

Since $\beta_{0}\geq\beta_{1}$ and $\beta_{3}\geq\beta_{2}$,

$\displaystyle J$
$\displaystyle\leq\frac{|x_{1}-x_{1}^{\prime}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{1}}}}{(t-s)^{(\frac{1}{2}+\beta_{0}+\frac{\beta_{0}-\beta_{1}}{2})\vee(1+\frac{3}{2}\beta_{0}+\frac{\beta_{0}-\beta_{1}}{2}-\alpha)+\alpha^{\prime}}}+\frac{|x_{2}-x_{2}^{\prime}|^{\frac{2\alpha^{\prime}}{1+\beta^{\prime}_{2}}}}{(t-s)^{(\frac{1}{2}+\beta_{0}+\frac{\beta_{3}-\beta_{2}}{2})\vee(1+\frac{3}{2}\beta_{0}+\frac{\beta_{3}-\beta_{2}}{2}-\alpha)+\alpha^{\prime}}}.$

$\mathrm{(i.3)}$ We now combine the results of steps $\mathrm{(i.1)}$ and
$\mathrm{(i.2)}$ to deduce that, when
$\alpha^{\prime}=\alpha_{\Phi}\in(0,\hat{\alpha}_{\Phi}\wedge\kappa_{0})$,

$\displaystyle|I|$
$\displaystyle~{}\leq~{}C\frac{|x_{1}-x_{1}^{\prime}|^{\frac{2\alpha_{\Phi}}{1+\beta^{\prime}_{1}}}+|x_{2}-x_{2}^{\prime}|^{\frac{2\alpha_{\Phi}}{1+\beta^{\prime}_{2}}}}{(t-s)^{1-\eta_{\Phi}}}\left(f^{\circ,\frac{1}{2}}(s,x;t,y)+f^{\circ,\frac{1}{2}}(s,x^{\prime};t,y)\right).$

$\mathrm{(ii)}$ To conclude, it remains to use an induction argument as in the
end of the proof of Proposition 3.5. ∎

####  3.4.2 Smoothness of the transition density and Feynman-Kac’s
representation

Recall that ${\rm f}(s,{\rm x};t,y)$ is defined in (2.35).

######  Proposition 3.14.

Let the conditions of Theorem 2.9 hold. Then, the vertical derivative
$\partial^{2}_{{\rm x}{\rm x}}{\rm f}(s,{\rm x};t,y)$ and horizontal
derivative $\partial_{s}{\rm f}(s,{\rm x};t,y)$ are well-defined for all
$0\leq s<t\leq T$, ${\rm x}\in D([0,T])$ and $y\in\mathbb{R}^{2}$. Moreover,
for all $(t,y)\in[0,T]\times\mathbb{R}^{2}$, $\partial^{2}_{{\rm x}{\rm
x}}{\rm f}(\cdot;t,y)$ and $\partial_{s}{\rm f}(\cdot;t,y)$ are continuous on
$[0,t)\times C([0,T])$.

###### Proof.

We denote by $C>0$ a generic constant that does not depend on $(s,x,t,y)$. Let
us fix $t_{0}\in(s,t)$, then by (2.28) and (2.35),

$\displaystyle{\rm f}(s,{\rm x};t,y)~{}:=~{}$ $\displaystyle{\rm
f}_{t,y}(s,{\rm x};t,y)+\int_{s}^{t_{0}}\\!\\!\int_{\mathbb{R}^{2}}{\rm
f}_{r,z}(s,{\rm x};r,z)\Phi(r,z;t,y)dzdr$
$\displaystyle+\int_{t_{0}}^{t}\int_{\mathbb{R}^{2}}{\rm f}_{r,z}(s,{\rm
x};r,z)\Phi(r,z;t,y)dzdr$ $\displaystyle~{}=:~{}$ $\displaystyle{\rm
f}_{t,y}(s,{\rm x};t,y)~{}+~{}{\rm f}_{1}(s,{\rm x};t,y)~{}+~{}{\rm
f}_{2}(s,{\rm x};t,y).$

First, the existence and continuity of the vertical derivative and horizontal
derivative of ${\rm f}_{t,y}(s,{\rm x};t,y)$ is trivial.

For ${\rm f}_{1}(s,{\rm x};t,y)$, we can use Lemmas 3.10 and 3.13, Proposition
3.5, together with (3.19), to obtain that

$\int_{s}^{t_{0}}\\!\\!\left|\int_{\mathbb{R}^{2}}\partial^{2}_{{\rm x}{\rm
x}}{\rm f}_{r,z}(s,{\rm
x};r,z)\Phi(r,z;t,y)dz\right|dr~{}\leq~{}C\int_{s}^{t_{0}}\frac{I_{1}(s,{\rm
x};r;t,y)+I_{2}(s,{\rm x};r;t,y)}{(r-s)^{1-\kappa_{\Phi}}}dr,$

where

$\kappa_{\Phi}~{}:=~{}\min\Big{(}\frac{2\beta_{4}+1+\beta^{\prime}_{1}}{1+\beta^{\prime}_{2}},1\Big{)}\min\\{\alpha_{\Phi},\alpha\\}-\beta_{0}~{}>~{}0,$

and, with $x:=({\rm x}(s),I_{s}({\rm x}))$,

$\displaystyle I_{1}(s,{\rm x};r;t,y)~{}:=~{}$
$\displaystyle\int_{\mathbb{R}^{2}}f^{\circ,\frac{1}{2}}(s,x;r,z)\big{(}f^{\circ,\frac{1}{2}}(r,E^{-1}_{s,r}(x);t,y)+f^{\circ,\frac{1}{2}}(r,z;t,y)\big{)}dz$
$\displaystyle~{}=~{}$ $\displaystyle
f^{\circ,\frac{1}{2}}(r,E^{-1}_{s,r}(x);t,y)+f^{\circ,\frac{1}{2}}(s,x;t,y)$
$\displaystyle I_{2}(s,{\rm x};r;t,y)~{}:=~{}$ $\displaystyle
f^{\circ,\frac{1}{2}}(r,E^{-1}_{s,r}(x);t,y).$

Since $t_{0}<t$, we can then easily obtain the existence and continuity of
$\partial^{2}_{{\rm x}{\rm x}}{\rm f}_{1}(\cdot;t,y)$ by dominated
convergence. Further, in view of Remark 3.11 and in particular (3.33), we can
also deduce the existence and continuity of the horizontal derivative
$\partial_{s}{\rm f}_{1}(\cdot;t,y)$.

For ${\rm f}_{2}(s,{\rm x};t,y)$, we notice that

$\big{|}\partial_{s}{\rm f}_{r,z}(s,{\rm
x};r,z)\big{|}+\big{|}\partial^{2}_{{\rm x}{\rm x}}{\rm f}_{r,z}(s,{\rm
x};r,z)\big{|}~{}\leq~{}C{\rm f}^{\circ}(s,{\rm x};r,z),~{}\mbox{for}~{}r\geq
t_{0}>s,~{}z\in\mathbb{R}^{2}.$

Together with the estimate on $\Phi(r,z;t,y)$ in Proposition 3.5, it follows
the existence and continuity of the vertical derivative $\partial^{2}_{{\rm
x}{\rm x}}{\rm f}_{2}(\cdot;t,y)$ and the horizontal derivative
$\partial_{s}{\rm f}_{2}(\cdot;t,y)$. ∎

Recall the growth condition (2.36) on $\ell$ and $g$, and the Hölder
continuity condition (2.37) on $\ell$. Let

$v(s,x)~{}:=~{}\int_{s}^{T}\\!\\!\int_{\mathbb{R}^{2}}\ell(t,y)f(s,x;t,y)dydt+\int_{\mathbb{R}^{2}}g(y)f(s,x;T,y)dy,~{}~{}(s,x)\in[0,T)\times\mathbb{R}^{2}.$

Then, with ${\rm v}$ defined in (2.38), one has, for $x=({\rm x}(s),I_{s}({\rm
x}))$,

${\rm v}(s,{\rm x})=v(s,x),~{}\partial_{{\rm x}}{\rm v}(s,{\rm
x})=\partial_{x_{1}}v(s,x)~{}\mbox{and}~{}\partial^{2}_{{\rm x}{\rm x}}{\rm
v}(s,{\rm x})=\partial^{2}_{x_{1}x_{1}}v(s,x).$

######  Proposition 3.15.

Let the conditions of Theorem 2.9 hold. Then:

$\mathrm{(i)}$ ${\rm v}\in{\mathbb{C}}^{1,2}([0,T))$ and the bound estimates
in (2.41) hold true.

$\mathrm{(ii)}$ The function ${\rm v}$ is a classical solution to the PPDE
(2.1). If in addition $g$ is continuous, then ${\rm v}$ is the unique
classical solution of (2.1) satisfying (2.42).

###### Proof.

$\mathrm{(i)}$ Let us define, for $(r,z)\in[0,T)\times\mathbb{R}^{2}$,

$\displaystyle
v_{\Phi}(r,z)~{}:=\int_{r}^{T}\int_{\mathbb{R}^{2}}\Phi(r,z;t,y)\ell(t,y)dydt~{}+\int_{\mathbb{R}^{2}}\Phi(r,z;T,y)g(y)dy,$
(3.40)

so that

$\displaystyle v(s,x)~{}=$
$\displaystyle\int_{\mathbb{R}^{2}}f_{T,y}(s,x;T,y)g(y)dy~{}+\int_{s}^{T}\int_{\mathbb{R}^{2}}f_{r,z}(s,x;r,z)\big{(}v_{\Phi}(r,z)+\ell(r,z)\big{)}dzdr.$
(3.41)

Then, it follows from Lemma 3.13 that

$\displaystyle|v_{\Phi}(r,z)-v_{\Phi}(r,z^{\prime})|\leq
C_{\alpha_{\Phi}}\frac{|z_{1}-z_{1}^{\prime}|^{\frac{2\alpha_{\Phi}}{1+\beta^{\prime}_{1}}}+|z_{2}-z_{2}^{\prime}|^{\frac{2\alpha_{\Phi}}{1+\beta^{\prime}_{2}}}}{(T-r)^{1-\eta_{\Phi}}}\big{(}v^{\circ,\frac{1}{2}}(r,z)+v^{\circ,\frac{1}{2}}(r,z^{\prime})\big{)},$

in which

$v^{\circ,\frac{1}{2}}(r,z)~{}:=~{}\int_{r}^{T}\int_{\mathbb{R}^{2}}f^{\circ,\frac{1}{2}}(r,z;T,y)\big{|}\ell(t,y)\big{|}dydt+\int_{\mathbb{R}^{2}}f^{\circ,\frac{1}{2}}(r,z;T,y)\big{|}g(y)\big{|}dy.$

Together with the Hölder continuity condition on $\ell$ in (2.37), we can then
apply Lemma 3.10 to deduce that $\partial^{2}_{x_{1}x_{1}}v(s,x)$ exists and

$\displaystyle~{}\partial^{2}_{x_{1}x_{1}}v(s,x)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2}}\\!\partial^{2}_{x_{1}x_{1}}f_{T,y}(s,x;T,y)g(y)dy+\\!\int_{s}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\\!\partial^{2}_{x_{1}x_{1}}f_{r,z}(s,x;r,z)\big{(}v_{\Phi}(r,z)+\ell(r.z)\big{)}dzdr.$

Then, using (3.6) and Lemma 3.1, we deduce that, for some constant $C>0$,

$\int_{\mathbb{R}^{2}}\\!\Big{|}\partial^{2}_{x_{1}x_{1}}f_{T,y}(s,x;T,y)g(y)\Big{|}dy~{}\leq~{}\frac{C}{(T-s)^{1+\beta_{0}}}\int_{\mathbb{R}^{2}}f^{\circ}(s,x;T,y)|g(y)|dy~{}\leq~{}\frac{Ce^{C|x|}}{(T-s)^{1+\beta_{0}}}.$

By Lemma 3.10, one can choose $C>0$ such that,

$\displaystyle\left|\int_{s}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\\!\partial^{2}_{x_{1}x_{1}}f_{r,z}(s,x;r,z)\ell(r.z)dzdr\right|$
$\displaystyle\leq~{}$
$\displaystyle\int_{s}^{T}\frac{C}{(r-s)^{1-\kappa_{\ell}}}\Big{(}\big{|}\ell(E^{-1}_{s,r}(x))\big{|}+Ce^{|E^{-1}_{s,r}(x)|}+\int_{\mathbb{R}^{2}}f^{\circ}(s,x;r,z)e^{C|z|}dz\Big{)}dr~{}\leq~{}Ce^{C|x|},$

in which

$\kappa_{\ell}~{}:=~{}\min\Big{(}\frac{2\beta_{4}+1+\beta^{\prime}_{1}}{1+\beta^{\prime}_{2}},1\Big{)}\min\\{\alpha_{\ell},\alpha\\}-\beta_{0}~{}>~{}0,$

and

$\displaystyle\left|\int_{s}^{T}\\!\\!\\!\int_{\mathbb{R}^{2}}\\!\partial^{2}_{x_{1}x_{1}}f_{r,z}(s,x;r,z)v_{\Phi}(r.z)dzdr\right|\leq\int_{s}^{T}\frac{C}{(r-s)^{1-\kappa_{\Phi}}(T-r)^{1-\eta_{\Phi}}}e^{C|x|}dr~{}\leq~{}Ce^{C|x|}.$

This proves the bound estimate on $\partial^{2}_{x_{1}x_{1}}v(s,x)$ (or
equivalently $\partial^{2}_{{\rm x}{\rm x}}{\rm v}(s,{\rm x})$) in (2.41). In
view of (3.33), one can obtain the same bound on $\partial_{s}{\rm v}(s,{\rm
x})$ in (2.41). Finally, $\partial_{{\rm x}}{\rm v}(s,{\rm x})$ is estimated
by appealing to Proposition 3.8 and (2.36). The bound on the right-hand side
of (2.42) is proved similarly.

$\mathrm{(ii)}$ Recall that

$\displaystyle{\rm f}(s,{\rm x};t,y)~{}=~{}{\rm f}_{t,y}(s,{\rm
x};t,y)~{}+\int_{s}^{t}\int_{\mathbb{R}^{2}}{\rm f}_{r,z}(s,{\rm
x};r,z)\Phi(r,z,t,y)dzdr,$

and that $(s,{\rm x})\in[0,t)\times D([0,T])\mapsto{\rm f}_{t,y}(s,{\rm
x};t,y)$ solves

$\displaystyle{\rm L}_{t,y}{\rm f}_{t,y}(\cdot;t,y)=0\mbox{ on $[s,t)\times
C([0,T])$,}$ (3.42)

where

${\rm
L}_{t,y}~{}:=~{}\partial_{s}+\frac{1}{2}\sigma_{t}(y)^{2}\partial^{2}_{{\rm
x}{\rm x}}.$

Let

${\rm L}\phi(s,{\rm x})~{}:=~{}\partial_{s}\phi(s,{\rm x})+\mu_{s}({\rm
x})\partial_{{\rm x}}\phi(s,{\rm x})+\frac{1}{2}\sigma_{s}^{2}({\rm
x})\partial^{2}_{{\rm x}}\phi(s,{\rm x}),$

for $\phi\in{\mathbb{C}}^{1,2}([0,T))$. Recalling the definition of $v_{\Phi}$
in (3.40) and using (3.41), we obtain that, with $x:=({\rm x}(s),I_{s}({\rm
x}))$,

$\displaystyle{\rm L}{\rm v}(s,{\rm x})=$
$\displaystyle\int_{\mathbb{R}^{2}}{\rm L}{\rm f}_{{T},y}(s,{\rm
x};T,y)g(y)dy-v_{\Phi}(s,x){-\ell(s,x)}$
$\displaystyle+\int_{s}^{T}\\!\\!\int_{\mathbb{R}^{2}}{\rm L}{\rm
f}_{r,z}(s,{\rm x};r,z)\big{(}v_{\Phi}(r,z){+\ell(r,z)}\big{)}dzdr.$ (3.43)

At the same time, as a consequence of (3.28) and (3.21)-(3.22), we observe
that

$\displaystyle\Phi(s,x;T,y)=~{}$ $\displaystyle({\rm L}-{\rm L}_{t,y}){\rm
f}_{t,y}(s,{\rm x};t,y)$
$\displaystyle+\int_{s}^{t}\\!\int_{\mathbb{R}^{2}}({\rm L}-{\rm
L}_{r,z}){{\rm f}_{r,z}}(s,{\rm x};r,z)\Phi(r,z;t,y)dzdr.$

Hence, recalling Lemma 3.4 and Proposition 3.5, it follows by (3.40) that

$\displaystyle v_{\Phi}(s,x)=$ $\displaystyle\int_{\mathbb{R}^{2}}({\rm
L}-{\rm L}_{T,y}){\rm f}_{{T,}y}(s,{\rm x};T,y)g(y)dy$
$\displaystyle+\int_{s}^{T}\\!\\!\int_{\mathbb{R}^{2}}({\rm L}-{\rm
L}_{r,z}){\rm f}_{{r,}z}(s,{\rm
x};r,z)\big{(}v_{\Phi}(r,z){+\ell(r,z)}\big{)}dzdr.$

We then use (3.42) to obtain

$\displaystyle v_{\Phi}(s,x)~{}=$ $\displaystyle\int_{\mathbb{R}^{2}}{\rm
L}{\rm f}_{T,y}(s,{\rm
x};T,y)g(y)dy+\int_{s}^{T}\\!\\!\int_{\mathbb{R}^{2}}{\rm L}{\rm
f}_{r,z}(s,{\rm x};r,z)\big{(}v_{\Phi}(r,z){+\ell(r,z)}\big{)}dzdr.$

It follows then by (3.43) that ${\rm v}$ is a classical solution to the PPDE
(2.1).

$\mathrm{(iii)}$ We now prove that $\lim_{s\nearrow T}{\rm v}(s,{\rm
x})=g({\rm x}_{T},I_{T}({\rm x}))$, or equivalently $\lim_{s\nearrow
T}v(s,x)=g(x)$, whenever $g$ is continuous. In view of the estimates in (3.12)
and (3.27), and Proposition 3.6, one has

$\displaystyle\lim_{s\nearrow T}v(s,x)~{}=~{}$ $\displaystyle\lim_{s\nearrow
T}\int_{\mathbb{R}^{2}}f(s,x;T,y)g(y)dy~{}=~{}\lim_{s\nearrow
T}\int_{\mathbb{R}^{2}}f_{T,y}(s,x;T,y)g(y)dy$ $\displaystyle~{}=~{}$
$\displaystyle\lim_{M\to\infty}\lim_{s\nearrow
T}\int_{D^{M}_{s,T}}\\!\\!f_{T,y}(s,x;T,y)g(y)dy~{}=\lim_{M\to\infty}\lim_{s\nearrow
T}\int_{D^{M}_{s,T}}\\!\\!f_{T,x}(s,x;T,y)g(y)dy$ $\displaystyle~{}=~{}$
$\displaystyle\lim_{s\nearrow
T}\int_{\mathbb{R}^{2}}f_{T,x}(s,x;T,y)g(y)dy~{}=~{}g(x),$

in which

$D^{M}_{s,T}:=\Big{[}x_{1}-M\sqrt{T-s},~{}x_{1}+M\sqrt{T-s}\Big{]}\times\Big{[}x_{2}-M\sqrt{(T-s)\tilde{m}_{s,t}},~{}x_{2}+M\sqrt{(T-s)\tilde{m}_{s,t}}\Big{]},$

so that third and fifth equalities are true since both $f_{T,y}(s,x;T,y)$ and
$f_{T,x}(s,x;T,y)$ are dominated by $Cf^{\circ}(s,x;T,y)$ in which the
covariance matrix in $f^{\circ}$ is given by
$\Sigma_{s,T}(4\bar{\mathfrak{a}})$, and the fourth equality follows by the
fact that, for every fixed $M>0$,

$\lim_{s\nearrow T}\sup_{y\in
D^{M}_{s,T}}\left|\frac{f_{T,y}(s,x;T,y)}{f_{T,x}(s,x;T,y)}-1\right|=0.$

$\mathrm{(iv)}$ The fact that ${\rm v}$ is the unique solution of (2.1)
satisfying (2.42) holds true follows easily by a verification argument based
on Itô-Dupire’s formula, see [3], whenever $g$ is continuous. ∎

###  3.5 Proofs of Theorems 2.7, 2.9 and 2.11

###### Proof of Theorem 2.7.

$\mathrm{(i)}$ First, the well-posedness of $\widetilde{\Phi}$ in
(2.25)-(2.26) is proved in Proposition 3.5. Further, the well-posedness of $f$
in (2.28) as well as its continuity and growth property is proved in
Proposition 3.6.

$\mathrm{(ii)}$ Under further conditions, the existence of
$\partial_{x_{1}}f(s,x;t,y)$ as well as its continuity and growth property is
proved in Proposition 3.8. ∎

###### Proof of Theorem 2.9.

$\mathrm{(i)}$ The fact that ${\rm f}(\cdot;t,y)\in{\mathbb{C}}^{1,2}([0,t))$
is proved in Proposition 3.14.

$\mathrm{(ii)}$ The fact that ${\rm v}$ provides a classical solution to the
PPDE, as well as the estimation on the derivatives are proved in Proposition
3.15.

$\mathrm{(iii)}$ We now use the PPDE results in Item $\mathrm{(ii)}$ to study
the path-dependent SDE (2.11). To study the weak solution of the SDE (2.11),
we consider the martingale problem on the canonical space $C([0,T])$ of all
$\mathbb{R}^{2}$-valued continuous paths on $[0,T]$. By abuse of notation, we
denote by $(X_{t},I_{t})_{t\in[0,T]}$ the canonical process, which generates
the canonical filtration $\mathbb{F}$. Then, given an initial condition
$(t,x)\in[0,T]\times\mathbb{R}^{2}$, a solution to the corresponding
martingale problem is a probability measure $\mathbb{P}$ on $C([0,T])$ such
that $\mathbb{P}[(X_{s},I_{s})=x=(x_{1},x_{2}),~{}s\in[0,t]]=1$,
$\mathbb{P}[I_{s}=x_{2}+\int_{t}^{s}X_{r}dA_{r},~{}s\in[t,T]]=1$ and the
process

$\varphi(X_{s})-\int_{t}^{s}\Big{(}\bar{\mu}_{r}(X)D\varphi(X_{r})+\frac{1}{2}\bar{\sigma}_{r}(X)D^{2}\varphi(X_{r})\Big{)}dr,~{}s\in[t,T],$

is a $(\mathbb{P},\mathbb{F})$-martingale for all bounded smooth functions
$\varphi:\mathbb{R}\longrightarrow\mathbb{R}$. Let us denote, for all
$(t,x)\in[0,T]\times\mathbb{R}^{2}$,

${\cal P}(t,x)~{}:=~{}\big{\\{}\mathbb{P}~{}:\mathbb{P}~{}\mbox{is solution to
the martingale problem with initial condition}~{}(t,x)\big{\\}}.$

Notice that $\bar{\mu}$ and $\bar{\sigma}$ are both bounded continuous, it is
then classical to know that ${\cal P}(t,x)$ is a nonempty compact set (see
e.g. Stroock and Varadhan [14, Chapter VI]).

We next apply the classical Markovian selection technique (see e.g. [14,
Chapter 12.2]) to construct a weak solution to the SDE (2.11) such that
$(X_{t},I_{t})_{t\in[0,T]}$ is a strong Markov process. Let $(\phi_{n})_{n\geq
1}$ be a sequence of bounded continuous functions from
$[0,T]\times\mathbb{R}^{2}\longrightarrow\mathbb{R}$ such that it is a measure
determining sequence in the sense that the sequence

$\Big{\\{}\mathbb{E}^{\mathbb{P}}\Big{[}\int_{0}^{T}\phi_{n}(t,X_{t},I_{t})dt\Big{]}\Big{\\}}_{n\geq
1}$

can determinate the probability measure $\mathbb{P}$ on $C([0,T])$. For each
$(t,x)\in[0,T]\times\mathbb{R}^{2}$, let ${\cal P}^{+}_{0}(t,x):={\cal
P}(t,x)$, and then define, for each $n\geq 0$,

${\cal P}^{+}_{n+1}(t,x)=\Big{\\{}\mathbb{P}\in{\cal
P}^{+}_{n}(t,x)~{}:\mathbb{E}^{\mathbb{P}}\Big{[}\int_{0}^{T}\phi_{n}(t,X_{t},I_{t})dt\Big{]}=\max_{\mathbb{P}^{\prime}\in{\cal
P}^{+}_{n}(t,x)}\mathbb{E}^{\mathbb{P}^{\prime}}\Big{[}\int_{0}^{T}\phi_{n}(t,X_{t},I_{t})dt\Big{]}\Big{\\}}.$

It is easy to see that each ${\cal P}^{+}_{n}(t,x)$ is a non-empty compact
set, so that ${\cal P}^{+}(t,x):=\cap_{n\geq 1}{\cal P}^{+}_{n}(t,x)$ is also
non-empty compact, as the sequence is non-increasing. Moreover, since any two
probability measures in ${\cal P}^{+}(t,x)$ has the same value by evaluating
w.r.t. any $\phi_{n}$, this implies that ${\cal P}^{+}(t,x)$ contains exactly
one probability measure denoted by $\mathbb{P}^{+}_{t,x}$. By the dynamic
programming principle for the optimal control problem in the definition of
${\cal P}^{+}_{n+1}$, it follows that
$(X,I,(\mathbb{P}^{+}_{t,x})_{(t,x)\in[0,T]\times\mathbb{R}^{2}})$ provides a
Markov process solution to SDE (2.11) such that $(X,I)$ is a strong Markov
process.

At the same time, one can apply the above Markovian selection argument to
construct another Markov process
$(X,I,(\mathbb{P}^{-}_{t,x})_{(t,x)\in[0,T]\times\mathbb{R}^{2}}$ by replacing
“$\max$” by “$\min$” in the definition of ${\cal P}^{+}_{n+1}(t,x)$. If the
class of all martingale solutions ${\cal P}(t,x)$ is not unique, then
$\mathbb{P}^{+}_{t,x}\neq\mathbb{P}^{-}_{t,x}$ as $(\phi_{n})_{n\geq 1}$ is
measure determining.

At the same time, by the results in Item $\mathrm{(ii)}$ and the Feynman-Kac’s
formula in the case $g\equiv 0$, one has

$\mathbb{E}^{\mathbb{P}^{+}_{s,x}}\Big{[}\int_{s}^{T}\ell_{t}(X_{t},I_{t})dt\Big{]}=\mathbb{E}^{\mathbb{P}^{-}_{s,x}}\Big{[}\int_{s}^{T}\ell_{t}(X_{t},I_{t})dt\Big{]}=\int_{s}^{T}\int_{\mathbb{R}^{2}}f(s,x;t,y)\ell_{t}(y)dydt.$

Since $\ell$ could be an arbitrary bounded continuous function, this implies
that $\mathbb{P}^{+}_{s,x}=\mathbb{P}^{-}_{s,x}$ for all
$(s,x)\in[0,T]\times\mathbb{R}^{2}$. Therefore, for all initial condition
$(t,x)$, there exists a unique solution to the martingale problem, i.e. a
unique weak solution. Moreover the (unique) solution process $(X,I)$ is a
strong Markov process, and the transition probability function is given by
$f$. ∎

###### Proof of Theorem 2.11..

When the SDE (2.11) admits weak uniqueness, the above Markovian selection
argument shows that the only solution $(X,I)$ is a strong Markov process.

$\mathrm{(i)}$ Let $W^{\perp}$ be a Brownian motion independent of $W$,
$(\varepsilon_{n})_{n\geq 1}$ be a sequence of positive constants such that
$\varepsilon_{n}\longrightarrow 0$. For each $n>1$, let us define
$\widetilde{X}^{n}=(\widetilde{X}^{n,1},\widetilde{X}^{n,2})$ as the unique
(Markovian) solution to the SDE

$\displaystyle d\widetilde{X}^{n,1}_{t}$
$\displaystyle=\tilde{\mu}_{t}(\widetilde{X}^{n}_{t})dt+\tilde{\sigma}_{t}(\widetilde{X}^{n}_{t})dW_{t},~{}~{}~{}d\widetilde{X}^{n,2}_{t}=\tilde{\mu}_{t}(\widetilde{X}^{n}_{t})A_{t}dt+\tilde{\sigma}_{t}(\widetilde{X}^{n}_{t})A_{t}dW_{t}+\varepsilon_{n}dW^{\perp}_{t}.$

By stability of weak solutions of SDEs, it is clear that, by using the same
initial condition for the above SDE as that in (2.17) for $\widetilde{X}$, one
has $\widetilde{X}^{n}\longrightarrow\widetilde{X}$ weakly.

At the same time, it follows from e.g. [8] that, for each $t\in(s,T]$,
$\widetilde{X}^{n}_{t}$ has a density $\tilde{f}^{n}(s,x;t,\cdot)$ whenever
$\widetilde{X}^{n}_{s}=x$. Moreover, $\tilde{f}^{n}$ can be defined in the
form

$\tilde{f}^{n}(s,x;t,y)=\tilde{f}^{n}_{{t},y}(s,x;t,y)+\int_{s}^{t}\int_{\mathbb{R}^{2}}\tilde{f}^{n}_{{r},z}(s,x;r,z)\widetilde{\Phi}^{n}(r,z;t,y)dzdr$

in which $\tilde{f}^{n}_{{t},y}$ is defined as $\tilde{f}_{{t},y}$ but with

$\Sigma^{n}_{s,t}(r,z)=\sigma^{2}_{r}(z)\left(\begin{array}[]{cc}t-s&-\int_{s}^{t}(A_{u}-A_{s})du\\\
-\int_{s}^{t}(A_{u}-A_{s})du&\int_{s}^{t}\big{[}(A_{u}-A_{s})^{2}+\varepsilon^{2}_{{n}}\sigma_{r}(z)^{-2}\big{]}du\end{array}\right)$

in place of $\Sigma_{s,t}(r,z)$, and

$\widetilde{\Phi}^{n}(s,x;t,y)~{}:=~{}\sum_{k=0}^{\infty}\widetilde{\Delta}^{n}_{k}(s,x;t,y),$

where $\widetilde{\Delta}^{n}_{0}(s,x;t,y):=\big{(}\widetilde{{\cal
L}}^{n}_{s}-\widetilde{{\cal
L}}^{n,t,y}_{s}\big{)}\tilde{f}^{n}_{t,y}(s,x;t,y)$,

$\displaystyle\widetilde{\Delta}^{n}_{k+1}(s,x;t,y):=\int_{s}^{t}\int_{\mathbb{R}^{2}}\widetilde{\Delta}^{n}_{0}(s,x;r,z)\widetilde{\Delta}^{n}_{k}(r,z;t,y)dzdr,~{}~{}k\geq
0.$

In the above, $\widetilde{{\cal L}}^{n}$ is the generator of
$\widetilde{X}^{n}$ and $\widetilde{{\cal L}}^{n,t,y}$ is defined from
$\widetilde{{\cal L}}^{n}$ as $\widetilde{{\cal L}}^{t,y}$ is defined from
$\widetilde{{\cal L}}$ by freezing $\sigma$ to $\sigma_{t}(y)$ and erasing the
drift term. Then, we define $f^{n}_{r,z}$ from $\tilde{f}^{n}_{\cdot}$ as
$f_{r,z}$ is defined from $\tilde{f}_{\cdot}$ in Section 2.3.

It is straightforward to check that the estimates in (3.29) hold for
$(\widetilde{\Delta}^{n}_{k})_{k\geq 0}$ in place of
$(\widetilde{\Delta}_{k})_{k\geq 0}$, uniformly in $n>0$. Then, an induction
argument combined with the fact that
$\tilde{f}^{n}_{t,y}(s,x;t,y)\to\tilde{f}_{t,y}(s,x;t,y)$ as $n\to\infty$, for
all $(s,x,t,y)\in\Theta$, implies that
$f^{n}(s,x;t,y):=\tilde{f}^{n}(s,{\rm\bf A}_{s}x;t,{\rm\bf A}_{t}y)$ converges
to $f(s,x;t,y)$ as $n\longrightarrow\infty$, for all $(s,x,t,y)\in\Theta$. By
the weak convergence of the sequence of processes $(\widetilde{X}^{n})_{n\geq
1}$ to $X$, this shows that $f$ is the transition probability function of
$(X,I)$.

$\mathrm{(ii)}$ As shown in Theorem 2.7, one has ${\rm
v}\in{\mathbb{C}}^{0,1}([0,T))$ and the vertical derivative $\partial_{{\rm
x}}{\rm v}$ is locally bounded. Let $(X,I)$ be the solution of SDE (2.11),
then by Feynman-Kac’s formula, the process

${\rm v}(t,X)+\int_{0}^{t}\bar{\ell}(s,X)ds,~{}t\in[0,T],~{}\mbox{is a local
martingale}.$

One can further apply the $C^{1}$-Itô formula for path-dependent functionals
in [1] to prove (2.45). Indeed, when (2.44) holds true, one can directly apply
[1, Proposition 2.11 and Theorem 2.5].

Otherwise, when $A$ is monotone and
$0<\frac{1+\beta_{2}-\beta_{0}}{2+4\beta_{4}}<1-\frac{\beta_{3}-\beta_{2}+\beta_{0}}{2}$,
we can fix
$\alpha^{\prime}\in(\frac{1+\beta_{2}-\beta_{0}}{2+4\beta_{4}},1-\frac{\beta_{3}-\beta_{2}+\beta_{0}}{2})$,
and, by Proposition 3.9, there exists a constant $C>0$ such that, for all
$\varepsilon>0$,

$\displaystyle\mathbb{E}\Big{[}\Big{|}{\rm
v}\big{(}s+\varepsilon,X\big{)}-{\rm
v}\big{(}s+\varepsilon,X_{s\wedge\cdot}\oplus_{s+\varepsilon}(X_{s+\varepsilon}-X_{s})\big{)}\Big{|}^{2}\Big{]}$
$\displaystyle~{}\leq~{}$ $\displaystyle C\mathbb{E}\Big{[}\Big{(}\sup_{s\leq
t\leq
s+\varepsilon}\big{|}X_{t}-X_{s}\big{|}~{}\varepsilon^{\beta_{4}}\Big{)}^{\frac{4\alpha^{\prime}}{1+\beta_{2}^{\prime}}}\Big{]}~{}\leq~{}C\varepsilon^{\frac{\alpha^{\prime}(2+4\beta_{4})}{1+\beta_{2}^{\prime}}},$

where
$\big{(}X_{s\wedge\cdot}\oplus_{s+\varepsilon}(X_{s+\varepsilon}-X_{s})\big{)}_{t}:=\mathbf{1}_{[0,s+\varepsilon)}(t)X_{s\wedge
t}+\mathbf{1}_{[s+\varepsilon,T]}(t)X_{s+\varepsilon}$ for all $t\in[0,T]$.
Since $\frac{\alpha^{\prime}(2+4\beta_{4})}{1+\beta_{2}^{\prime}}>1$, it
follows that

$\lim_{\varepsilon\searrow 0}\frac{1}{\varepsilon}\mathbb{E}\Big{[}\Big{|}{\rm
v}\big{(}s+\varepsilon,X\big{)}-{\rm
v}\big{(}X_{s\wedge\cdot}\oplus_{s+\varepsilon}(X_{s+\varepsilon}-X_{s})\big{)}\Big{|}^{2}\Big{]}~{}=~{}0.$

Finally, we can apply [1, Proposition 2.6 and Theorem 2.5] to deduce (2.45). ∎

## References

  * [1] Bruno Bouchard, Grégoire Loeper, and Xiaolu Tan.  A $C^{0,1}$ -functional Itô’s formula and its applications in mathematical finance.  Stochastic Processes and their Applications, 148:1299–323, 2022\.
  * [2] Bruno Bouchard, Grégoire Loeper, and Xiaolu Tan.  Approximate viscosity solutions of path-dependent pdes and dupire’s vertical differentiability.  Annals of Applied Probability, to appear.
  * [3] Rama Cont and David-Antoine Fournié.  Functional Itô calculus and stochastic integral representation of martingales.  The Annals of Probability, 41(1):109–133, 2013.
  * [4] Andrea Cosso, Fausto Gozzi, Mauro Rosestolato, and Francesco Russo.  Path-dependent Hamilton-Jacobi-Bellman equation: Uniqueness of Crandall-Lions viscosity solutions.  arXiv preprint arXiv:2107.05959, 2021.
  * [5] François Delarue and Stéphane Menozzi.  Density estimates for a random noise propagating through a chain of differential equations.  Journal of functional analysis, 259(6):1577–1630, 2010.
  * [6] Bruno Dupire.  Functional Itô calculus.  Portfolio Research Paper, 04, 2009.
  * [7] Ibrahim Ekren, Christian Keller, Nizar Touzi, and Jianfeng Zhang.  On viscosity solutions of path dependent pdes.  The Annals of Probability, 42(1):204–236, 2014.
  * [8] Marco Di Francesco and Andrea Pascucci.  On a class of degenerate parabolic equations of Kolmogorov type.  Applied Mathematics Research eXpress, 2005(3):77–116, 2005.
  * [9] Avner Friedman.  Partial differential equations of parabolic type.  Courier Dover Publications, 2008.
  * [10] Andreï N. Kolmogorov.  Zufällige Bewegungen (zur Theorie der Brownschen Bewegung).  Ann Math, 35:116–117, 1934.
  * [11] Ermanno Lanconelli, Andrea Pascucci, and Sergio Polidoro.  Linear and nonlinear ultraparabolic equations of kolmogorov type arising in diffusion theory and in finance.  Nonlinear problems in mathematical physics and related topics, II, 2:243–265, 2002.
  * [12] Zhenjie Ren, Nizar Touzi, and Jianfeng Zhang.  Comparison of viscosity solutions of semi-linear path-dependent PDEs.  SIAM Journal on Control and Optimization, 58(1):277-302, 2020.
  * [13] Isaac M. Sonin.  On a class of degenerate diffusion processes.  Theory of Probability & Its Applications, 12(3):490–496, 1967\.
  * [14] Daniel W. Stroock and Srinivasa R. Varadhan.  Multidimensional diffusion processes, volume 233.  Springer Science & Business Media, 1997.
  * [15] Maria Weber.  The fundamental solution of a degenerate partial differential equation of parabolic type.  Transactions of the American Mathematical Society, 71(1):24–37, 1951.
  * [16] Jianjun Zhou.  Viscosity solutions to second order path-dependent hamilton-jacobi-bellman equations and applications.  arXiv preprint arXiv:2005.05309, 2020.