J_{2}(t,x;\bar{u}_{1}(\cdot\,;t,x,u_{2}(\cdot)),u_{2}(\cdot))$ is minimized.
For any given initial pair $(t,x)$ and control $u_{2}(\cdot)$ of the leader,
by the standard results of LQ control problems (see [79, Chapter 6]), the
follower admits a unique optimal strategy
$\bar{u}_{1}(\cdot\,;t,x,u_{2}(\cdot))$. Then by some straightforward
calculations, the leader’s problem can be stated as follows: Find a control
$u_{2}(\cdot)$ to minimize

$J_{2}(t,x;\bar{u}_{1}(\cdot\,;t,x,u_{2}(\cdot)),u_{2}(\cdot))=\int_{t}^{1}\big{[}Y(s)+u_{2}(s)+|u_{2}(s)|^{2}\big{]}ds,$
(5.18)

with the backward evolution equation

$\left\\{\begin{array}[]{ll}\displaystyle\dot{Y}(s)={1\over 2-s}Y(s)+{1\over
2-s}u_{2}(s),\quad s\in[t,1],\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle Y(1)=0.\end{array}\right.$ (5.19)

Note that

$Y(s)={1\over s-2}\int_{s}^{1}u_{2}(r)dr.$

Then

$\displaystyle J_{2}(t,x;\bar{u}_{1}(\cdot),u_{2}(\cdot))$
$\displaystyle=\int_{t}^{1}\Big{[}{1\over
s-2}\int_{s}^{1}u_{2}(r)dr+u_{2}(s)+|u_{2}(s)|^{2}\Big{]}ds$
$\displaystyle=\int_{t}^{1}\Big{\\{}\big{[}\ln(2-s)-\ln(2-t)+1\big{]}u_{2}(s)+|u_{2}(s)|^{2}\Big{\\}}ds.$

It follows that the unique optimal control of the leader is given by

$\bar{u}_{2}(s;t,x)={\ln(2-t)-\ln(2-s)-1\over 2},\quad s\in[t,1].$

In particular, at the initial pair $(0,x)$, the unique optimal control of the
leader is

$\bar{u}_{2}(s;0,x)={\ln 2-\ln(2-s)-1\over 2},\quad s\in[0,1].$

Let $\bar{X}(\cdot)\equiv\bar{X}(\cdot;0,x)$ be the state process with initial
pair $(0,x)$ and optimal controls
$(\bar{u}_{1}(s;t,x,\bar{u}_{2}),\,\bar{u}_{2}(\cdot))$. For any given
$t\in(0,1)$, at the initial pair $(t,\bar{X}(t;0,x))$, the unique optimal
control of the leader is

$\bar{u}_{2}(s;t,\bar{X}(t;0,x))={\ln(2-t)-\ln(2-s)-1\over 2},\quad
s\in[t,1].$

Thus, on the time interval $[t,1]$,

$\bar{u}_{2}(\cdot;0,x)\neq\bar{u}_{2}(\cdot;t,\bar{X}(t;0,x)),$

which implies that the leader’s problem is time-inconsistent. By Theorem 2.7,
we can easily obtain the time-consistent equilibrium strategy of the leader,
which is given by

$\bar{\Psi}(s,x)=-{1\over 2},\quad(s,x)\in[0,1]\times\mathbb{R}^{n}.$

######  Remark 5.4.

We refer the reader to [56, 72, 14, 4, 59] for some theoretical results and
financial applications of Stackelberg games. It is worthy of pointing out that
the well-known principal-agent problem (see [14]) can be regarded as a special
case.

##  6 Verification Theorem

In this section, we shall show that the function $\bar{\Psi}(\cdot\,,\cdot)$,
determined by (2.19), is an equilibrium strategy of Problem (N). In other
words, we would like to rigorously prove the verification theorem (i.e.,
Theorem 2.7). To do this, we assume that the equilibrium HJB equation (2.18)
admits a classical solution and the function $\bar{\Psi}(\cdot\,,\cdot)$
defined by (2.19) is a feedback strategy. We also assume that all the involved
functions are bounded and differentiable with bounded derivatives.

Let $(\bar{X}(\cdot),\bar{Y}(\cdot),\bar{Z}(\cdot))$ and
$(\bar{Y}^{0}(\cdot),\bar{Z}^{0}(\cdot,\cdot))$ be the solutions to FBSDE
(1.1) and BSVIE (1.3), respectively, corresponding to the strategy
$\bar{\Psi}(\cdot\,,\cdot)$ and the initial pair $(0,\xi)$. For any
$t\in[0,T)$, $u\in L_{{\cal F}_{t}}^{2}(\Omega;U)$ and
$\varepsilon\in[0,T-t)$, define the strategy
$\Psi^{\varepsilon}(\cdot\,,\cdot)$ by (2.15). With the initial pair
$(t,\bar{X}(t))\in\mathscr{D}$, take the strategy
$\Psi^{\varepsilon}(\cdot\,,\cdot)$, then the corresponding state equation
(1.1) and cost functional (1.2)–(1.3) become

$\left\\{\begin{aligned}
dX^{\varepsilon}(s)&=b(s,X^{\varepsilon}(s),\bar{\Psi}(s,X^{\varepsilon}(s)))ds+\sigma(s,X^{\varepsilon}(s),\bar{\Psi}(s,X^{\varepsilon}(s)))dW(s),\quad
s\in[t+\varepsilon,T];\\\
dX^{\varepsilon}(s)&=b(s,X^{\varepsilon}(s),u)ds+\sigma(s,X^{\varepsilon}(s),u)dW(s),\quad
s\in[t,t+\varepsilon),\\\
dY^{\varepsilon}(s)&=-g(s,X^{\varepsilon}(s),\bar{\Psi}(s,X^{\varepsilon}(s)),Y^{\varepsilon}(s),Z^{\varepsilon}(s))ds+Z^{\varepsilon}(s)dW(s),\quad
s\in[t+\varepsilon,T];\\\
dY^{\varepsilon}(s)&=-g(s,X^{\varepsilon}(s),u,Y^{\varepsilon}(s),Z^{\varepsilon}(s))ds+Z^{\varepsilon}(s)dW(s),\quad
s\in[t,t+\varepsilon),\\\ X^{\varepsilon}(t)&=\bar{X}(t),\quad
Y^{\varepsilon}(T)=h(X^{\varepsilon}(T)),\end{aligned}\right.$ (6.1)

and

$J(t,\bar{X}(t);\Psi^{\varepsilon}(\cdot))=Y^{0,\varepsilon}(t),$ (6.2)

respectively, with

$\left\\{\begin{aligned}
&Y^{0,\varepsilon}(r)=h^{0}(r,X^{\varepsilon}(r),X^{\varepsilon}(T),Y^{\varepsilon}(r))-\int_{r}^{T}Z^{0,\varepsilon}(r,s)dW(s)\\\
&\quad+\int_{r}^{T}g^{0}(r,s,X^{\varepsilon}(r),X^{\varepsilon}(s),\bar{\Psi}(s,X^{\varepsilon}(s)),Y^{\varepsilon}(s),Z^{\varepsilon}(s),Y^{0,\varepsilon}(s),Z^{0,\varepsilon}(r,s))dr,\quad
r\in[t+\varepsilon,T];\\\
&Y^{0,\varepsilon}(r)=h^{0}(r,X^{\varepsilon}(r),X^{\varepsilon}(T),Y^{\varepsilon}(r))-\int_{r}^{T}Z^{0,\varepsilon}(r,s)dW(s)\\\
&\quad+\int_{t+\varepsilon}^{T}g^{0}(r,s,X^{\varepsilon}(r),X^{\varepsilon}(s),\bar{\Psi}(s,X^{\varepsilon}(s)),Y^{\varepsilon}(s),Z^{\varepsilon}(s),Y^{0,\varepsilon}(s),Z^{0,\varepsilon}(r,s))ds\\\
&\quad+\int_{r}^{t+\varepsilon}g^{0}(r,s,X^{\varepsilon}(r),X^{\varepsilon}(s),u,Y^{\varepsilon}(s),Z^{\varepsilon}(s),Y^{0,\varepsilon}(s),Z^{0,\varepsilon}(r,s))ds,\quad
r\in[t,t+\varepsilon).\end{aligned}\right.$ (6.3)

Next, let us deduce the PDEs associated with the FBSDE (6.1) and BSVIE (6.3).

By the Feynman–Kac formula for BSDEs (see Pardoux–Peng [49], for example), we
get

$\displaystyle Y^{\varepsilon}(s)=\Theta(s,X^{\varepsilon}(s)),\quad
Z^{\varepsilon}(s)=\Theta_{x}(s,X^{\varepsilon}(s))\sigma(s,X^{\varepsilon}(s),\bar{\Psi}(s,X^{\varepsilon}(s))),\quad
s\in[t+\varepsilon,T],$

where $\Theta(\cdot,\cdot)$ is the unique solution to the first PDE in (2.18).
Then on the time interval $[t,t+\varepsilon]$, we can rewrite (6.1) as
follows:

$\left\\{\begin{aligned}
dX^{\varepsilon}(s)&=b(s,X^{\varepsilon}(s),u)ds+\sigma(s,X^{\varepsilon}(s),u)dW(s),\quad
s\in[t,t+\varepsilon],\\\
dY^{\varepsilon}(s)&=-g(s,X^{\varepsilon}(s),u,Y^{\varepsilon}(s),Z^{\varepsilon}(s))ds+Z^{\varepsilon}(s)dW(s),\quad
s\in[t,t+\varepsilon],\\\ X^{\varepsilon}(t)&=\bar{X}(t),\quad
Y^{\varepsilon}(t+\varepsilon)=\Theta(t+\varepsilon,X^{\varepsilon}(t+\varepsilon)).\end{aligned}\right.$

Note that the control $u\in L^{2}_{{\cal F}_{t}}(\Omega;U)$ is ${\cal
F}_{t}$-measurable. Then by the Feynman–Kac formula for BSDEs again, we get

$Y^{\varepsilon}(s)=\Theta^{\varepsilon}(s,X^{\varepsilon}(s)),\quad
Z^{\varepsilon}(s)=\Theta^{\varepsilon}_{x}(s,X^{\varepsilon}(s))\sigma(s,X^{\varepsilon}(s),u),\quad
s\in[t,t+\varepsilon],$ (6.4)

where $\Theta^{\varepsilon}(\cdot,\cdot)$ is the unique classical solution to
the following perturbation PDE:

$\left\\{\begin{aligned}
&\Theta^{\varepsilon}_{s}(s,x)+\Theta^{\varepsilon}_{x}(s,x)b(s,x,u)+\hbox{tr$\,$}[\Theta^{\varepsilon}_{xx}(s,x)a(s,x,u)]\\\
&\quad+g(s,x,u,\Theta^{\varepsilon}(s,x),\Theta^{\varepsilon}_{x}(s,x)\sigma(s,x,u))=0,\quad
s\in[t,t+\varepsilon],\\\
&\Theta^{\varepsilon}(t+\varepsilon,x)=\Theta(t+\varepsilon,x).\end{aligned}\right.$
(6.5)

######  Remark 6.1.

Indeed, (6.5) is a PDE with random parameters, because $u\in L^{2}_{{\cal
F}_{t}}(\Omega;U)$ is a random variable. However, note that $u$ is ${\cal
F}_{t}$-measurable and (6.5) is considered only on $[t,t+\varepsilon]$. The
random PDE (6.5) can be treated as a deterministic one.

By 2.2, on the time interval $[t+\varepsilon,T]$, we get

$Y^{0,\varepsilon}(s)=\Theta^{0}(s,s,x,x,\Theta(s,X^{\varepsilon}(s))),$ (6.6)

where $\Theta^{0}(\cdot)$ is the solution of the second PDE in (2.18).
Motivated by Wang–Yong–Zhang [67], we introduce the following auxiliary
processes with two time variables:

$\left\\{\begin{aligned}
dY^{0,\varepsilon}(r;s)&=-g^{0}\big{(}r,s,X^{\varepsilon}(r),X^{\varepsilon}(s),\bar{\Psi}(s,X^{\varepsilon}(s)),Y^{\varepsilon}(s),Z^{\varepsilon}(s),Y^{0,\varepsilon}(s),Z^{0,\varepsilon}(r;s)\big{)}ds\\\
&\quad+Z^{0,\varepsilon}(r;s)dW(s),\quad s\in[(t+\varepsilon)\vee
r,T],~{}r\in[t,T];\\\
dY^{0,\varepsilon}(r;s)&=-g^{0}\big{(}r,s,X^{\varepsilon}(r),X^{\varepsilon}(s),u,Y^{\varepsilon}(s),Z^{\varepsilon}(s),Y^{0,\varepsilon}(s),Z^{0,\varepsilon}(r;s)\big{)}ds\\\
&\quad+Z^{0,\varepsilon}(r;s)dW(s),\quad(r,s)\in\Delta^{*}[t,t+\varepsilon];\\\
Y^{0,\varepsilon}(r;T)&=h^{0}(r,X^{\varepsilon}(r),X^{\varepsilon}(T),Y^{\varepsilon}(r)),\quad
r\in[t,T],\end{aligned}\right.$ (6.7)

which can give the unique solution of BSVIE (6.3) by

$Y^{0,\varepsilon}(s)=Y^{0,\varepsilon}(s;s),\quad
Z^{0,\varepsilon}(r,s)=Z^{0,\varepsilon}(r;s),\quad(r,s)\in\Delta^{*}[t,T].$
(6.8)

Notice that for any fixed $r\in[t,T]$, (6.7) is a BSDE. Recall the
representations (6.4) and (6.6). Then by the Feynman–Kac formula for BSDEs
again, we get that for any $r\in[t,t+\varepsilon]$,

$\displaystyle
Y^{0,\varepsilon}(r;s)=\Theta^{0}(r,s,X^{\varepsilon}(r),X^{\varepsilon}(s),\Theta^{\varepsilon}(r,X^{\varepsilon}(r))),\quad
s\in[t+\varepsilon,T],$ $\displaystyle
Z^{0,\varepsilon}(r;s)=\Theta^{0}_{x}(r,s,X^{\varepsilon}(r),X^{\varepsilon}(s),\Theta^{\varepsilon}(r,X^{\varepsilon}(r)))\sigma(s,X^{\varepsilon}(s),\bar{\Psi}(s,X^{\varepsilon}(s))),\quad
s\in[t+\varepsilon,T].$ (6.9)

On the other hand, by the flow property of the auxiliary process
$Y^{0,\varepsilon}(\cdot;\cdot)$, we have

$\displaystyle Y^{0,\varepsilon}(r;r)$
$\displaystyle=Y^{0,\varepsilon}(r;t+\varepsilon)+\int_{r}^{t+\varepsilon}g^{0}(r,s,X^{\varepsilon}(r),X^{\varepsilon}(s),u,Y^{\varepsilon}(s),Z^{\varepsilon}(s),Y^{0,\varepsilon}(s),Z^{0,\varepsilon}(r;s))ds$
$\displaystyle\quad-\int_{r}^{t+\varepsilon}Z^{0,\varepsilon}(r;s)dW(s),\quad
r\in[t,t+\varepsilon].$

Substituting (6.9) into the above and noting (6.8), we get

$\displaystyle Y^{0,\varepsilon}(r)$
$\displaystyle=\Theta^{0}(r,t+\varepsilon,X^{\varepsilon}(r),X^{\varepsilon}(t+\varepsilon),\Theta^{\varepsilon}(r,X^{\varepsilon}(r)))-\int_{r}^{t+\varepsilon}Z^{0,\varepsilon}(r,s)dW(s)$
$\displaystyle\quad+\int_{r}^{t+\varepsilon}g^{0}(r,s,X^{\varepsilon}(r),X^{\varepsilon}(s),u,Y^{\varepsilon}(s),Z^{\varepsilon}(s),Y^{0,\varepsilon}(s),Z^{0,\varepsilon}(r,s))ds,\quad
r\in[t,t+\varepsilon].$ (6.10)

Then by 2.2 (recalling (6.4)–(6.5)), we have the following representation:

$\displaystyle
Y^{0,\varepsilon}(r)=\Theta^{0,\varepsilon}(r,r,X^{\varepsilon}(r),X^{\varepsilon}(r),\Theta^{\varepsilon}(r,X^{\varepsilon}(r))),$
$\displaystyle
Z^{0,\varepsilon}(r,s)=\Theta_{x}^{0,\varepsilon}(r,s,X^{\varepsilon}(r),X^{\varepsilon}(s),\Theta^{\varepsilon}(r,X^{\varepsilon}(r)))\sigma(s,X^{\varepsilon}(s),u),\quad(r,s)\in\Delta^{*}[t,t+\varepsilon],$
(6.11)

where $\Theta^{0,\varepsilon}(\cdot)$ is the unique solution to the following
PDE:

$\left\\{\begin{aligned}
&\Theta_{s}^{0,\varepsilon}(r,s,\tilde{x},x,y)+\Theta_{x}^{0,\varepsilon}(r,s,\tilde{x},x,y)b(s,x,u)+\hbox{tr$\,$}[\Theta_{xx}^{0,\varepsilon}(r,s,\tilde{x},x,y)a(s,x,u)]\\\
&\quad+g^{0}\big{(}r,s,\tilde{x},x,u,\Theta^{\varepsilon}(s,x),\Theta^{\varepsilon}_{x}(s,x)\sigma(s,x,u),\Theta^{0,\varepsilon}(s,s,x,x,\Theta^{\varepsilon}(s,x)),\\\
&\qquad\quad\Theta_{x}^{0,\varepsilon}(r,s,\tilde{x},x,y)\sigma(s,x,u)\big{)}=0,\quad(r,s)\in\Delta^{*}[t,t+\varepsilon],\\\
&\Theta^{0,\varepsilon}(r,t+\varepsilon,\tilde{x},x,y)=\Theta^{0}(r,t+\varepsilon,\tilde{x},x,y),\quad
r\in[t,t+\varepsilon].\end{aligned}\right.$ (6.12)

######  Remark 6.2.

Note that both (6.5) and (6.12) are semilinear parabolic equations. To
guarantee the well-posedness of PDEs (6.5) and (6.12), we assume that the
following non-degenerate condition holds: There exist two constants
$\lambda_{0},\lambda_{1}>0$ such that

$\lambda_{0}I\leqslant
a(t,x,u)\leqslant\lambda_{1}I,\quad\forall(t,x,u)\in[0,T]\times\mathbb{R}^{n}\times
U.$ (6.13)

Under the assumption (6.13), we have the following convergence result of the
families $\\{\Theta^{\varepsilon}(\cdot)\\}_{\varepsilon>0}$ and
$\\{\Theta^{0,\varepsilon}(\cdot)\\}_{\varepsilon>0}$.

######  Proposition 6.3.

Let (6.13) hold. Then the PDEs (6.5) and (6.12) admit unique classical
solutions $\Theta^{\varepsilon}(\cdot)$ and $\Theta^{0,\varepsilon}(\cdot)$,
respectively. Moreover, there exists a constant $K>0$, only depending on
$\|\Theta(\cdot)\|_{C^{{\alpha\over 2},2+\alpha}}$ and
$\|\Theta^{0}(\cdot)\|_{C^{{\alpha\over 2},{\alpha\over
2},\alpha,\alpha,1+\alpha}}$, such that

$\|\Theta^{\varepsilon}(\cdot)-\Theta(\cdot)\|_{C^{0,2}[t,t+\varepsilon]}+\|\Theta^{0,\varepsilon}(\cdot)-\Theta^{0}(\cdot)\|_{C^{0,0,0,0,1}[t,t+\varepsilon]}\leqslant
K\varepsilon^{\alpha\over 2},$ (6.14)

where $\alpha\in(0,1)$ is a constant.

######  Remark 6.4.

The proof of 6.3 is sketched in Section 7 as a byproduct of Theorem 2.10. We
emphasize that for 6.3, the assumption (6.13) should not be necessary, because
one could replace the analytic approach by a probabilistic argument (see [49,
67]).

######  Remark 6.5.

The estimate (6.14) plays the same role as the convergence assumption (H3) in
Wei–Yong–Yu [70], which was proved only for some special cases (see [70,
Theorem 6.2]). In 6.3, we can show that (6.14) holds in general. The key point
is that (6.14) is only a byproduct of the stability of semilinear parabolic
equations, while the assumption (H3) in [70] is concerned with the fully
nonlinear PDEs. The deeper reason is that in our paper the main technique is
the Feynman–Kac formula for BSVIEs/BSDEs, while in [70] they heavily rely on
the HJB equation approach.

###  6.1 Proof of Theorem 2.7

For any fixed $t\in[0,T)$, $\varepsilon\in[0,T-t]$ and $u\in L_{{\cal
F}_{t}}^{2}(\Omega;U)$, let $\Theta^{\varepsilon}(\cdot)$ and
$\Theta^{0,\varepsilon}(\cdot)$ be the unique classical solution to PDEs (6.5)
and (6.12), respectively. With the representations (6.4) and (6.11), by (6.10)
we can represent $Y^{0,\varepsilon}(t)$ as follows:

$\displaystyle
Y^{0,\varepsilon}(t)=\mathbb{E}_{t}\Big{[}\Theta^{0}(t,t+\varepsilon,X^{\varepsilon}(t),X^{\varepsilon}(t+\varepsilon),\Theta^{\varepsilon}(t,X^{\varepsilon}(t)))+\int_{t}^{t+\varepsilon}g^{0,\varepsilon}(t,s,u)ds\Big{]},$
(6.15)

where

$\displaystyle
g^{0,\varepsilon}(t,s,u):=g^{0}\Big{(}t,\,s,\,X^{\varepsilon}(t),\,X^{\varepsilon}(s),\,u,\,\Theta^{\varepsilon}(X^{\varepsilon}(s),s),\Theta^{\varepsilon}_{x}(s,X^{\varepsilon}(s))\sigma(s,X^{\varepsilon}(s),u),$
$\displaystyle\qquad\qquad\quad\Theta^{0,\varepsilon}(s,s,X^{\varepsilon}(s),X^{\varepsilon}(s),\Theta^{\varepsilon}(s,X^{\varepsilon}(s))),\,\Theta_{x}^{0,\varepsilon}(t,s,X^{\varepsilon}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,X^{\varepsilon}(t)))\sigma(s,X^{\varepsilon}(s),u)\Big{)}.$
(6.16)

Note that $X^{\varepsilon}(t)=\bar{X}(t)$. Applying Itô’s formula to the
mapping
$s\mapsto\Theta^{0}(t,s,X^{\varepsilon}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,X^{\varepsilon}(t)))$
yields that

$\displaystyle Y^{0,\varepsilon}(t)$
$\displaystyle=\mathbb{E}_{t}\Big{\\{}\Theta^{0}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(t,\bar{X}(t)))+\int_{t}^{t+\varepsilon}\Big{[}\Theta_{s}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))$
$\displaystyle\qquad+\Theta_{x}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))b(s,X^{\varepsilon}(s),u)+g^{0,\varepsilon}(t,s,u)$
$\displaystyle\qquad+\hbox{tr$\,$}\big{[}\Theta_{xx}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))a(s,X^{\varepsilon}(s),u)\big{]}\Big{]}ds\Big{\\}}.$

Using the fact
$\Theta^{\varepsilon}(t+\varepsilon,\cdot)=\Theta(t+\varepsilon,\cdot)$, we
get

$\displaystyle Y^{0,\varepsilon}(t)$
$\displaystyle=\mathbb{E}_{t}\Big{\\{}\Theta^{0}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(t,\bar{X}(t)))-\Theta^{0}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(t+\varepsilon,\bar{X}(t+\varepsilon)))$
$\displaystyle\qquad+\Theta^{0}(t,t,\bar{X}(t),\bar{X}(t),\Theta(t+\varepsilon,\bar{X}(t+\varepsilon)))+\int_{t}^{t+\varepsilon}\Big{[}\Theta_{s}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))$
$\displaystyle\qquad+\Theta_{x}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))b(s,X^{\varepsilon}(s),u)+g^{0,\varepsilon}(t,s,u)$
$\displaystyle\qquad+\hbox{tr$\,$}\big{[}\Theta_{xx}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))a(s,X^{\varepsilon}(s),u)\big{]}\Big{]}ds\Big{\\}}.$

Then by applying the Itô’s formula to the mapping
$s\mapsto\Theta^{0}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(s,\bar{X}(s)))$,
we have

$\displaystyle Y^{0,\varepsilon}(t)$
$\displaystyle=\mathbb{E}_{t}\Big{\\{}\Theta^{0}(t,t,\bar{X}(t),\bar{X}(t),\Theta(t+\varepsilon,\bar{X}(t+\varepsilon)))$
$\displaystyle\qquad-{1\over
2}\int_{t}^{t+\varepsilon}\hbox{tr$\,$}\Big{[}\Theta^{0}_{yy}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(s,\bar{X}(s)))\Theta_{x}^{\varepsilon}(s,\bar{X}(s))\bar{\sigma}(s)[\Theta_{x}^{\varepsilon}(s,\bar{X}(s))\bar{\sigma}(s)]^{\top}\Big{]}ds$
$\displaystyle\qquad-\int_{t}^{t+\varepsilon}\Theta^{0}_{y}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(s,\bar{X}(s)))\Big{[}\Theta_{s}^{\varepsilon}(s,\bar{X}(s))+\Theta_{x}^{\varepsilon}(s,\bar{X}(s))\bar{b}(s)$
$\displaystyle\qquad+\hbox{tr$\,$}[\Theta_{xx}^{\varepsilon}(s,\bar{X}(s))\bar{a}(s)]\Big{]}ds+\int_{t}^{t+\varepsilon}\Big{[}\Theta_{s}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))$
$\displaystyle\qquad+\Theta_{x}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))b(s,X^{\varepsilon}(s),u)+g^{0,\varepsilon}(t,s,u)$
$\displaystyle\qquad+\hbox{tr$\,$}[\Theta_{xx}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))a(s,X^{\varepsilon}(s),u)]\Big{]}ds\Big{\\}},$
(6.17)

where

$\displaystyle\bar{\varphi}(s):=\varphi(s,\bar{X}(s),\bar{\Psi}(s,\bar{X}(s))),\,\,s\in[t,t+\varepsilon],\quad\hbox{for}\quad\varphi(\cdot)=b(\cdot),\sigma(\cdot),a(\cdot).$
(6.18)

Recalling (6.5) and (2.18), we get that on $[t,t+\varepsilon]$,

$\displaystyle\Theta_{s}^{\varepsilon}(s,\bar{X}(s))+\Theta_{x}^{\varepsilon}(s,\bar{X}(s))\bar{b}(s)+\hbox{tr$\,$}[\Theta_{xx}^{\varepsilon}(s,\bar{X}(s))\bar{a}(s)]$
$\displaystyle\quad=\Theta_{x}^{\varepsilon}(s,\bar{X}(s))[\bar{b}(s)-b(s,\bar{X}(s),u)]+\hbox{tr$\,$}\big{\\{}\Theta_{xx}^{\varepsilon}(s,\bar{X}(s))[\bar{a}(s)-a(s,\bar{X}(s),u)]\big{\\}}-g^{\varepsilon}(s,u),$
(6.19)

and

$\displaystyle\Theta_{s}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))+\Theta_{x}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))b(s,X^{\varepsilon}(s),u)$
$\displaystyle+\hbox{tr$\,$}[\Theta_{xx}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))a(s,X^{\varepsilon}(s),u)]$
$\displaystyle\quad=\Theta_{x}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))[b(s,X^{\varepsilon}(s),u)-\bar{b}^{\varepsilon}(s)]$
$\displaystyle\qquad+\hbox{tr$\,$}\big{\\{}\Theta_{xx}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))[a(s,X^{\varepsilon}(s),u)-\bar{a}^{\varepsilon}(s)]\big{\\}}-\bar{g}^{0,\varepsilon}(t,s),$
(6.20)

where

$\displaystyle\bar{\varphi}^{\varepsilon}(s):=\varphi(s,X^{\varepsilon}(s),\bar{\Psi}(s,X^{\varepsilon}(s))),\quad\hbox{for}\quad\varphi(\cdot)=b(\cdot),\,\sigma(\cdot),\,a(\cdot);$
$\displaystyle
g^{\varepsilon}(s,u):=g\big{(}s,\bar{X}(s),u,\Theta^{\varepsilon}(s,\bar{X}(s)),\Theta^{\varepsilon}_{x}(s,\bar{X}(s))\sigma(s,\bar{X}(s),u)\big{)};$
$\displaystyle\bar{g}^{0,\varepsilon}(t,s):=g^{0}\big{(}t,s,\bar{X}(t),X^{\varepsilon}(s),\bar{\Psi}(s,X^{\varepsilon}(s)),\Theta(s,X^{\varepsilon}(s)),\Theta_{x}(s,X^{\varepsilon}(s))\bar{\sigma}^{\varepsilon}(s),$
$\displaystyle\quad\Theta^{0}(s,s,X^{\varepsilon}(s),X^{\varepsilon}(s),\Theta(s,X^{\varepsilon}(s))),\Theta_{x}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta(t,\bar{X}(t)))\bar{\sigma}^{\varepsilon}(s)\big{)}.$
(6.21)

Substituting (6.19) and (6.20) into (6.17) yields that

$\displaystyle Y^{0,\varepsilon}(t)$
$\displaystyle=\mathbb{E}_{t}\Big{\\{}\Theta^{0}(t,t,\bar{X}(t),\bar{X}(t),\Theta(t+\varepsilon,\bar{X}(t+\varepsilon)))$
$\displaystyle\qquad+\int_{t}^{t+\varepsilon}\Theta^{0}_{y}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(s,\bar{X}(s)))\Big{[}g^{\varepsilon}(s,u)+\Theta_{x}^{\varepsilon}(s,\bar{X}(s))$
$\displaystyle\qquad\quad\times[b(s,\bar{X}(s),u)-\bar{b}(s)]+\hbox{tr$\,$}\big{\\{}\Theta_{xx}^{\varepsilon}(s,\bar{X}(s))[a(s,\bar{X}(s),u)-\bar{a}(s)]\big{\\}}\Big{]}ds$
$\displaystyle\qquad-{1\over
2}\int_{t}^{t+\varepsilon}\hbox{tr$\,$}\Big{[}\Theta^{0}_{yy}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(s,\bar{X}(s)))\Theta_{x}^{\varepsilon}(s,\bar{X}(s))\bar{\sigma}(s)[\Theta_{x}^{\varepsilon}(s,\bar{X}(s))\bar{\sigma}(s)]^{\top}\Big{]}ds$
$\displaystyle\qquad+\int_{t}^{t+\varepsilon}\Big{[}g^{0,\varepsilon}(t,s,u)-\bar{g}^{0,\varepsilon}(t,s)+\Theta_{x}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))[b(s,X^{\varepsilon}(s),u)-\bar{b}^{\varepsilon}(s)]$
$\displaystyle\qquad\quad+\hbox{tr$\,$}\big{\\{}\Theta_{xx}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))[a(s,X^{\varepsilon}(s),u)-\bar{a}^{\varepsilon}(s)]\big{\\}}\Big{]}ds\Big{\\}}.$
(6.22)

Applying the above arguments to $\bar{Y}^{0}(t)$, we have

$\displaystyle J(t,\bar{X}(t);\bar{\Psi}(\cdot))$
$\displaystyle\equiv\bar{Y}^{0}(t)=\mathbb{E}_{t}\Big{\\{}\Theta^{0}(t,t,\bar{X}(t),\bar{X}(t),\Theta(t+\varepsilon,\bar{X}(t+\varepsilon)))$
$\displaystyle\quad-{1\over
2}\int_{t}^{t+\varepsilon}\hbox{tr$\,$}\big{\\{}\Theta^{0}_{yy}(t,t,\bar{X}(t),\bar{X}(t),\Theta(s,\bar{X}(s)))\Theta_{x}(s,\bar{X}(s))\bar{\sigma}(s)[\Theta_{x}(s,\bar{X}(s))\bar{\sigma}(s)]^{\top}\big{\\}}ds$
$\displaystyle\quad+\int_{t}^{t+\varepsilon}\Theta^{0}_{y}(t,t,\bar{X}(t),\bar{X}(t),\Theta(s,\bar{X}(s)))\bar{g}(s,\bar{\Psi}(s,\bar{X}(s)))ds\Big{\\}},$
(6.23)

where

$\displaystyle\bar{g}(s,u)$
$\displaystyle:=g\big{(}s,\bar{X}(s),u,\Theta(s,\bar{X}(s)),\Theta_{x}(s,\bar{X}(s))\sigma(s,\bar{X}(s),u)\big{)},\quad(s,u)\in[t,t+\varepsilon]\times
U.$ (6.24)

Combining (6.22) with (6.23) together, we get

$\displaystyle Y^{0,\varepsilon}(t)-\bar{Y}(t)$
$\displaystyle=\mathbb{E}_{t}\Big{\\{}\int_{t}^{t+\varepsilon}\Theta^{0}_{y}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(s,\bar{X}(s)))\Big{[}g^{\varepsilon}(s,u)+\Theta_{x}^{\varepsilon}(s,\bar{X}(s))$
$\displaystyle\quad\times[b(s,\bar{X}(s),u)-\bar{b}(s)]+\hbox{tr$\,$}\big{[}\Theta_{xx}^{\varepsilon}(s,\bar{X}(s))[a(s,\bar{X}(s),u)-\bar{a}(s)]\big{]}\Big{]}ds$
$\displaystyle\quad-{1\over
2}\int_{t}^{t+\varepsilon}\hbox{tr$\,$}\Big{[}\Theta^{0}_{yy}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(s,\bar{X}(s)))\Theta_{x}^{\varepsilon}(s,\bar{X}(s))\bar{\sigma}(s)[\Theta_{x}^{\varepsilon}(s,\bar{X}(s))\bar{\sigma}(s)]^{\top}\Big{]}ds$
$\displaystyle\quad+\int_{t}^{t+\varepsilon}\Big{[}g^{0,\varepsilon}(t,s,u)-\bar{g}^{0,\varepsilon}(t,s)+\Theta_{x}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))[b(s,X^{\varepsilon}(s),u)-\bar{b}^{\varepsilon}(s)]$
$\displaystyle\quad+\hbox{tr$\,$}\big{\\{}\Theta_{xx}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))[a(s,X^{\varepsilon}(s),u)-\bar{a}^{\varepsilon}(s)]\big{\\}}\Big{]}ds$
$\displaystyle\quad+{1\over
2}\int_{t}^{t+\varepsilon}\hbox{tr$\,$}\big{[}\Theta^{0}_{yy}(t,t,\bar{X}(t),\bar{X}(t),\Theta(s,\bar{X}(s)))\Theta_{x}(s,\bar{X}(s))\bar{\sigma}(s)[\Theta_{x}(s,\bar{X}(s))\bar{\sigma}(s)]^{\top}\big{]}ds$
$\displaystyle\quad-\int_{t}^{t+\varepsilon}\Theta^{0}_{y}(t,t,\bar{X}(t),\bar{X}(t),\Theta(s,\bar{X}(s)))\bar{g}(s,\bar{\Psi}(s,\bar{X}(s)))ds\Big{\\}}.$
(6.25)

By the standard results of SDEs, we get

$\displaystyle\mathbb{E}_{t}\Big{[}\sup_{s\in[t,t+\varepsilon]}\big{(}|\bar{X}(s)|^{2}+|X^{\varepsilon}(s)|^{2}\big{)}\Big{]}\leqslant
K(1+|\bar{X}(t)|^{2}),$
$\displaystyle\mathbb{E}_{t}\Big{[}\sup_{s\in[t,t+\varepsilon]}|\bar{X}(s)-X^{\varepsilon}(s)|^{2}\Big{]}\leqslant
K(1+|\bar{X}(t)|^{2})\varepsilon.$ (6.26)

By 6.3, we have

$\displaystyle\mathbb{E}_{t}\big{[}|\Theta^{\varepsilon}(s,\bar{X}(s))-\Theta(s,\bar{X}(s))|+|\Theta_{x}^{\varepsilon}(s,\bar{X}(s))-\Theta_{x}(s,\bar{X}(s))|$
$\displaystyle\quad+|\Theta_{xx}^{\varepsilon}(s,\bar{X}(s))-\Theta_{xx}(s,\bar{X}(s))|\big{]}\leqslant
K\varepsilon^{\alpha\over 2}.$ (6.27)

It follows that

$\displaystyle\mathbb{E}_{t}\Big{[}\big{|}\Theta^{0}_{y}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(s,\bar{X}(s)))-\Theta^{0}_{y}(t,t,\bar{X}(t),\bar{X}(t),\Theta(s,\bar{X}(s)))\big{|}$
$\displaystyle\quad+\big{|}\Theta^{0}_{yy}(t,t,\bar{X}(t),\bar{X}(t),\Theta^{\varepsilon}(s,\bar{X}(s)))-\Theta^{0}_{yy}(t,t,\bar{X}(t),\bar{X}(t),\Theta(s,\bar{X}(s)))\big{|}$
$\displaystyle\quad+\big{|}\Theta_{x}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))-\Theta_{x}^{0}(t,s,\bar{X}(t),\bar{X}(s),\Theta(t,\bar{X}(t)))\big{|}$
$\displaystyle\quad+\big{|}\Theta_{xx}^{0}(t,s,\bar{X}(t),X^{\varepsilon}(s),\Theta^{\varepsilon}(t,\bar{X}(t)))-\Theta_{xx}^{0}(t,s,\bar{X}(t),\bar{X}(s),\Theta(t,\bar{X}(t)))\big{|}\Big{]}$
$\displaystyle\qquad\leqslant K\varepsilon^{\alpha\over
2}+K\varepsilon^{1\over 2}(1+|\bar{X}(t)|)\leqslant K\varepsilon^{\alpha\over
2}(1+|\bar{X}(t)|),$ (6.28)

and

$\displaystyle\mathbb{E}_{t}\Big{[}|g^{\varepsilon}(s,u)-\bar{g}(s,u)|+|\bar{g}^{0,\varepsilon}(t,s)-\bar{g}^{0}(t,s,\bar{\Psi}(s,\bar{X}(s)))|$
$\displaystyle\quad+|\bar{b}^{\varepsilon}(s)-\bar{b}(s)|+|\bar{\sigma}^{\varepsilon}(s)-\bar{\sigma}(s)|\Big{]}\leqslant
K\varepsilon^{\alpha\over 2}(1+|\bar{X}(t)|),$ (6.29)

where

$\displaystyle\bar{g}^{0}(t,s,u)$
$\displaystyle:=g^{0}\Big{(}t,\,s,\,\bar{X}(t),\,\bar{X}(s),\,u,\,\Theta(s,\bar{X}(s)),\,\Theta_{x}(s,\bar{X}(s))\sigma(s,\bar{X}(s),u),$
$\displaystyle\qquad\quad\Theta^{0}(s,s,\bar{X}(s),\bar{X}(s),\Theta(s,\bar{X}(s))),\,\Theta_{x}^{0}(t,s,\bar{X}(t),\bar{X}(s),\Theta(t,\bar{X}(t)))\sigma(s,\bar{X}(s),u)\Big{)},$
(6.30)

and the form of other functions is given in (6.18), (6.21), and (6.24).
Moreover, by 6.3 and (6.26), we have

$\displaystyle\mathbb{E}_{t}\Big{[}|g^{0,\varepsilon}(t,s,u)-\bar{g}^{0}(t,s,u)|\Big{]}\leqslant
K\varepsilon^{\alpha\over 2}(1+|\bar{X}(t)|),$ (6.31)

where $g^{0,\varepsilon}(t,s,u)$ is given by (6.16). With the above estimates
(6.26)–(6.31), from (6.25) we have

$\displaystyle Y^{0,\varepsilon}(t)-\bar{Y}(t)$
$\displaystyle=\mathbb{E}_{t}\Big{\\{}\int_{t}^{t+\varepsilon}\Theta^{0}_{y}(t,t,\bar{X}(t),\bar{X}(t),\Theta(s,\bar{X}(s)))\Big{[}\bar{g}(s,u)-\bar{g}(s,\bar{\Psi}(s,\bar{X}(s)))$
$\displaystyle\qquad+\Theta_{x}(s,\bar{X}(s))[b(s,\bar{X}(s),u)-\bar{b}(s)]+\hbox{tr$\,$}\big{\\{}\Theta_{xx}(s,\bar{X}(s))[a(s,\bar{X}(s),u)-\bar{a}(s)]\big{\\}}\Big{]}ds$
$\displaystyle\qquad+\int_{t}^{t+\varepsilon}\Big{[}\bar{g}^{0}(t,s,u)-\bar{g}(t,s,\bar{\Psi}(s,\bar{X}(s)))+\Theta^{0}_{x}(t,s,\bar{X}(t),\bar{X}(s),\Theta(s,\bar{X}(s)))$
$\displaystyle\qquad\times[b(s,\bar{X}(s),u)-\bar{b}(s)]+\hbox{tr$\,$}\big{\\{}\Theta^{0}_{xx}(t,s,\bar{X}(t),\bar{X}(s),\Theta(s,\bar{X}(s)))[a(s,\bar{X}(s),u)-\bar{a}(s)]\big{\\}}\Big{]}ds\Big{\\}}$
$\displaystyle\quad+o(\varepsilon)(1+|\bar{X}(t)|).$

Thus,

$\displaystyle\mathop{\underline{\rm lim}}_{\varepsilon\to
0^{+}}{J(t,\bar{X}(t);\Psi^{\varepsilon}(\cdot))-J(t,\bar{X}(t);\bar{\Psi}(\cdot))\over\varepsilon}=\mathop{\underline{\rm
lim}}_{\varepsilon\to
0^{+}}{Y^{0,\varepsilon}(t)-\bar{Y}^{0}(t)\over\varepsilon}$
$\displaystyle\quad=\bar{\Theta}^{0}_{y}(t)\Big{\\{}\bar{\Theta}_{x}(t)[b(t,\bar{X}(t),u)-\bar{b}(t)]+\hbox{tr$\,$}\big{\\{}\bar{\Theta}_{xx}(t)[a(t,\bar{X}(t),u)-\bar{a}(t)]\big{\\}}$
$\displaystyle\qquad+\bar{g}(t,t,u)-\bar{g}(t,t,\bar{\Psi}(t,\bar{X}(t)))\Big{\\}}+\bar{\Theta}^{0}_{x}(t)[b(t,\bar{X}(t),u)-\bar{b}(t)]$
$\displaystyle\qquad+\hbox{tr$\,$}\big{\\{}\bar{\Theta}^{0}_{xx}(t)[a(t,\bar{X}(t),u)-\bar{a}(t)]\big{\\}}+\bar{g}^{0}(t,t,u)-\bar{g}(t,t,\bar{\Psi}(t,\bar{X}(t))),$

where

$\displaystyle\bar{\Theta}(t):=\Theta(t,\bar{X}(t)),\quad\bar{\Theta}^{0}(t):=\Theta^{0}(t,t,\bar{X}(t),\bar{X}(t),\Theta(t,\bar{X}(t))),\quad
t\in[0,T].$

Then by the local optimality condition (2.17) of $\bar{\Psi}(\cdot,\cdot)$, we
have

$\mathop{\underline{\rm lim}}_{\varepsilon\to
0^{+}}{J(t,\bar{X}(t);\Psi^{\varepsilon}(\cdot))-J(t,\bar{X}(t);\bar{\Psi}(\cdot))\over\varepsilon}\geqslant
0,$

which completes the proof.

##  7 Some Proofs

###  7.1 Proof of Theorem 2.10

For the ease of presentation, in the rest of the paper we restrict to the case
with $m=1$ only. However, all our results hold true in the multiple
dimensional situation. To begin with, let us first adopt some notations.

Some Notations: For any functions $\varsigma:[S,T]\to\mathbb{R}$ and
$\nu:\mathbb{R}^{n}\to\mathbb{R}$, with $\alpha\in(0,1)$ and $S\in[0,T)$, let

$\|\varsigma(\cdot)\|_{\alpha\over 2}=\sup_{s_{1},s_{2}\in[S,T],\,s_{1}\neq
s_{2}}{|\varsigma(s_{1})-\varsigma(s_{2})|\over|s_{1}-s_{2}|^{\alpha\over
2}},\quad\|\nu(\cdot)\|_{\alpha}=\sup_{x_{1},x_{2}\in\mathbb{R}^{n},\,x_{1}\neq
x_{2}}{|\nu(x_{1})-\nu(x_{2})|\over|x_{1}-x_{2}|^{\alpha}}.$

For any $\varphi:[S,T]\times\mathbb{R}^{n}\to\mathbb{R}$, let

$\displaystyle\|\varphi(\cdot\,,\cdot)\|_{C^{{\alpha\over
2},\alpha}([S,T]\times\mathbb{R}^{n};\mathbb{R})}$
$\displaystyle=\|\varphi(\cdot\,,\cdot)\|_{L^{\infty}([S,T]\times\mathbb{R}^{n};\mathbb{R})}+\sup_{x\in\mathbb{R}^{n}}\|\varphi(\cdot\,,x)\|_{\alpha\over
2}+\sup_{s\in[S,T]}\|\varphi(s,\cdot)\|_{\alpha},$
$\displaystyle\|\varphi(\cdot\,,\cdot)\|_{C^{{\alpha\over
2},1+\alpha}([S,T]\times\mathbb{R}^{n};\mathbb{R})}$
$\displaystyle=\|\varphi(\cdot\,,\cdot)\|_{C^{0,1}([S,T]\times\mathbb{R}^{n};\mathbb{R})}+\|\varphi(\cdot\,,\cdot)\|_{C^{{\alpha\over
2},\alpha}([S,T]\times\mathbb{R}^{n};\mathbb{R})}$
$\displaystyle\quad+\|\varphi_{x}(\cdot\,,\cdot)\|_{C^{{\alpha\over
2},\alpha}([S,T]\times\mathbb{R}^{n};\mathbb{R})}.$

We will often simply write $C^{{\alpha\over
2},1+\alpha}([S,T]\times\mathbb{R}^{n};\mathbb{R})$ as $C^{{\alpha\over
2},1+\alpha}$ when there is no confusion. Similarly, we can define
$C^{{\alpha\over 2},{\alpha\over
2},\alpha,1+\alpha,2}([S,T]\times[S,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R};\mathbb{R})$,
etc.

For any $\theta(\cdot)\in C^{{\alpha\over 2},1+\alpha}$ and
$\theta^{0}(\cdot)\in C^{{\alpha\over 2},{\alpha\over 2},\alpha,1+\alpha,2}$,
let us consider the following PDE:

$\left\\{\begin{aligned} &\Theta_{s}(s,x)+{\cal
L}[s;\theta(\cdot),\theta^{0}(\cdot)]\Theta(s,x)+\tilde{g}\big{(}s,x,\theta(s,x),\theta_{x}(s,x),\theta^{0}(s,x,s,x,\theta(s,x)),\\\
&\quad\theta_{x}^{0}(s,x,s,x,\theta(s,x)),\theta_{y}^{0}(s,x,s,x,\theta(s,x))\big{)}=0,\\\
&\Theta^{0}_{s}(t,s,\tilde{x},x,y)+{\cal
L}[s;\theta(\cdot),\theta^{0}(\cdot)]\Theta^{0}(t,s,\tilde{x},x,y)+\tilde{g}^{0}\big{(}t,s,\tilde{x},x,\theta(s,x),\theta_{x}(s,x),\\\
&\quad\theta^{0}(s,x,s,x,\theta(s,x)),\theta_{x}^{0}(s,x,s,x,\theta(s,x)),\theta_{y}^{0}(s,x,s,x,\theta(s,x)),\Theta^{0}_{x}(t,s,\tilde{x},x,y)\big{)}=0,\\\
&\Theta(T,x)=h(x),\quad\Theta^{0}_{s}(t,T,\tilde{x},x,y)=h^{0}(t,\tilde{x},x,y),\end{aligned}\right.$
(7.1)

where the differential operator ${\cal L}[s;\theta(\cdot),\theta^{0}(\cdot)]$
is defined by the following:

$\displaystyle{\cal L}[s;\theta(\cdot),\theta^{0}(\cdot)]\varphi(x)$
$\displaystyle=\hbox{tr$\,$}[\varphi_{xx}(x)a(s,x)]+\varphi_{x}(x)\tilde{b}\big{(}s,x,\theta(s,x),\theta_{x}(s,x),\theta^{0}(s,x,s,x,\theta(s,x)),$
$\displaystyle\quad\,\theta^{0}_{x}(s,x,s,x,\theta(s,x)),\theta^{0}_{y}(s,x,s,x,\theta(s,x))\big{)},\quad\forall\varphi(\cdot)\in
C^{2}(\mathbb{R}^{n};\mathbb{R}).$ (7.2)

We first present a result for the well-posedness of PDE (7.1).

######  Lemma 7.1.

Fix a $(t,\tilde{x},y)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}$. Then for
any $\theta(\cdot)\in C^{{\alpha\over 2},1+\alpha}$ and $\theta^{0}(\cdot)\in
C^{{\alpha\over 2},{\alpha\over 2},\alpha,1+\alpha,2}$, the PDE (7.1) admits a
unique classical solution
$(\Theta(\cdot,\cdot),\Theta^{0}(t,\cdot,\tilde{x},\cdot,y))\in
C^{1+{\alpha\over 2},2+\alpha}\times C^{1+{\alpha\over 2},2+\alpha}$.
Moreover, the following relationship holds:

$\displaystyle\Theta(s,x)=\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)h(\mu)d\mu+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)\Big{[}\Theta_{x}(r,\mu)$
$\displaystyle\quad\times\tilde{b}\big{(}r,\mu,\theta(r,\mu),\theta_{x}(r,\mu),\theta^{0}(r,\mu,r,\mu,\theta(r,\mu)),\theta_{x}^{0}(r,\mu,r,\mu,\theta(r,\mu)),\theta_{y}^{0}(r,\mu,r,\mu,\theta(r,\mu))\big{)}$
$\displaystyle\quad+\tilde{g}\big{(}r,\mu,\theta(r,\mu),\theta_{x}(r,\mu),\theta^{0}(r,\mu,r,\mu,\theta(r,\mu)),\theta_{x}^{0}(r,\mu,r,\mu,\theta(r,\mu)),\theta_{y}^{0}(r,\mu,r,\mu,\theta(r,\mu))\big{)}\Big{]}d\mu
dr;$ (7.3)
$\displaystyle\Theta^{0}(t,s,\tilde{x},x,y)=\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)L(t,\tilde{x},\mu,y)d\mu+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)$
$\displaystyle\quad\times\Big{[}\Theta^{0}_{x}(t,r,\tilde{x},\mu,y)\tilde{b}\big{(}r,\mu,\theta(r,\mu),\theta_{x}(r,\mu),\theta^{0}(r,\mu,r,\mu,\theta(r,\mu)),\theta_{x}^{0}(r,\mu,r,\mu,\theta(r,\mu)),$
$\displaystyle\qquad\theta_{y}^{0}(r,\mu,r,\mu,\theta(r,\mu))\big{)}+\tilde{g}^{0}\big{(}t,r,\tilde{x},\mu,\theta(r,\mu),\theta_{x}(r,\mu),\theta^{0}(r,\mu,r,\mu,\theta(r,\mu)),$
$\displaystyle\qquad\theta_{x}^{0}(r,\mu,r,\mu,\theta(r,\mu)),\theta_{y}^{0}(r,\mu,r,\mu,\theta(r,\mu)),\Theta^{0}_{x}(t,r,\tilde{x},\mu,y)\big{)}\Big{]}d\mu
dr,$ (7.4)

where $\Xi(\cdot\,,\cdot\,,\cdot\,,\cdot)$ is given by the following
explicitly:

$\displaystyle\Xi(s,x,r,\mu)={1\over(4\pi(r-s))^{{n\over
2}}(\hbox{det\,}[a(r,\mu)])^{{1\over 2}}}e^{-{\langle
a(r,\mu)^{-1}(x-\mu),(x-\mu)\rangle\over
4(r-s)}},\quad(s,x),(r,\mu)\in[0,T]\times\mathbb{R}^{n}.$ (7.5)

###### Proof.

For any fixed $\theta(\cdot)\in C^{{\alpha\over 2},1+\alpha}$ and
$\theta^{0}(\cdot)\in C^{{\alpha\over 2},{\alpha\over 2},\alpha,1+\alpha,2}$,
denote

$\displaystyle\nu_{1}(s,x)=\theta(s,x),\quad\nu_{2}(s,x)=\theta_{x}(s,x),\quad\nu_{3}(s,x)=\theta^{0}(s,s,x,x,\theta(s,x)),$
$\displaystyle\nu_{4}(s,x)=\theta_{x}^{0}(s,s,x,x,\theta(s,x)),\quad\nu_{5}(s,x)=\theta_{y}^{0}(s,s,x,x,\theta(s,x)),\quad(s,x)\in[0,T]\times\mathbb{R}^{n}.$

Then we have $\nu_{i}(\cdot)\in C^{{\alpha\over 2},\alpha}$, for $i=1,...,5$.
Taking $(t,\tilde{x},y)$ as parameters, by the standard results of parabolic
equations (see [23, Theorem 12, Chapter 1] or [36, Chapter IV, Sections
13–14], for example), we get that PDE (7.1) admits a unique classical solution
and (7.3)–(7.4) hold. ∎

Direct computations show that (see [23, 36], for example)

$\Xi_{\mu}(s,x,r,\mu)=-\Xi_{x}(s,x,r,\mu)-\Xi(s,x,r,\mu)\rho(s,x,r,\mu),$
(7.6)

where

$\left\\{\begin{aligned} &\rho(s,x,r,\mu)={(\hbox{det\,}[a(r,\mu)])_{\mu}\over
2\hbox{det\,}[a(r,\mu)]}+{\langle[a(r,\mu)^{-1}]_{\mu}(x-\mu),(x-\mu)\rangle\over
4(r-s)},\\\
&\langle[a(r,\mu)^{-1}]_{\mu}(x-\mu),(x-\mu)\rangle=\begin{pmatrix}\langle[a(r,\mu)^{-1}]_{\mu_{1}}(x-\mu),(x-\mu)\rangle\\\
\langle[a(r,z)^{-1}]_{\mu_{2}}(x-\mu),(x-\mu)\rangle\\\ \vdots\\\
\langle[a(r,z)^{-1}]_{\mu_{n}}(x-\mu),(x-\mu)\rangle\end{pmatrix}.\end{aligned}\right.$
(7.7)

Moreover, under assumption (H4), it is easy to check that

$\left\\{\begin{aligned} &|\Xi(s,x,r,\mu)|\leqslant K{1\over(r-s)^{{n\over
2}}}e^{{-\lambda|x-\mu|^{2}\over 4(r-s)}},\\\ &|\Xi_{x}(s,x,r,\mu)|\leqslant
K{1\over(r-s)^{{n+1\over 2}}}e^{{-\lambda|x-\mu|^{2}\over 4(r-s)}},\\\
&|\rho(s,x,r,\mu)|\leqslant
K\left(1+{|x-\mu|^{2}\over(r-s)}\right),\end{aligned}\right.$ (7.8)

for some $0<\lambda<\lambda_{0}$. In what follows, we denote

$\tilde{\varphi}(s,x;\theta,\theta^{0}):=\tilde{\varphi}\big{(}s,\,x,\,\theta(s,x),\,\theta_{x}(s,x),\,\theta^{0}(s,x,s,x,\theta(s,x)),\,\theta_{x}^{0}(s,x,s,x,\theta(s,x)),\,\theta_{y}^{0}(s,x,s,x,\theta(s,x))\big{)},$

for $\varphi(\cdot)=\tilde{b}(\cdot),\tilde{g}(\cdot)$, and

$\displaystyle\tilde{g}^{0}(t,s,\tilde{x},x,y;\theta,\theta^{0},\Theta^{0})$
$\displaystyle:=\tilde{g}^{0}\big{(}t,\,s,\,\tilde{x},\,x,\,\theta(s,x),\,\theta_{x}(s,x),\,\theta^{0}(s,x,s,x,\theta(s,x)),\,\theta_{x}^{0}(s,x,s,x,\theta(s,x)),$
$\displaystyle\qquad\quad\theta_{y}^{0}(s,x,s,x,\theta(s,x)),\,\Theta_{x}^{0}(t,s,\tilde{x},x,y)\big{)}.$
(7.9)

First, we establish a $C^{0,1}$-norm estimate for
$(\Theta(\cdot,\cdot),\Theta^{0}(t,\cdot,\tilde{x},\cdot,y))$.

######  Lemma 7.2.

There exists a constant $\kappa>0$, independent of $\theta(\cdot)$ and
$\theta^{0}(\cdot)$, such that

$\displaystyle\|\Theta(\cdot,\cdot)\|_{C^{0,1}}+\sup_{t,\tilde{x},y\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}}\|\Theta^{0}(t,\cdot,\tilde{x},\cdot,y))\|_{C^{0,1}}$
$\displaystyle\quad\leqslant\kappa\Big{[}1+\|h(\cdot)\|_{C^{1}}+\sup_{t,\tilde{x},y\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}}\|h^{0}(t,\tilde{x},\cdot,y)\|_{C^{1}}\Big{]}.$
(7.10)

###### Proof.

By (7.3) and (7.6), using the method of integration by parts, we have

$\displaystyle\Theta_{x}(s,x)$
$\displaystyle=\int_{\mathbb{R}^{n}}\big{[}\Xi(s,x,T,\mu)h_{x}(\mu)-\Xi(s,x,T,\mu)\rho(s,x,T,\mu)h(\mu)\big{]}d\mu$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi_{x}(s,x,r,\mu)\Big{[}\Theta_{x}(r,\mu)\tilde{b}(r,\mu;\theta,\theta^{0})+\tilde{g}(r,\mu;\theta,\theta^{0})\Big{]}d\mu
dr.$ (7.11)

Then from the estimate (7.8), we get

$\displaystyle|\Theta_{x}(s,x)|$
$\displaystyle\leqslant\int_{\mathbb{R}^{n}}K{1\over(r-s)^{{n\over
2}}}e^{{-\lambda|x-\mu|^{2}\over
4(r-s)}}\Big{[}|h_{x}(\mu)|+\Big{(}1+{|x-\mu|^{2}\over(r-s)}\Big{)}|h(\mu)|\Big{]}d\mu$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}K{1\over(r-s)^{{n+1\over
2}}}e^{{-\lambda|x-\mu|^{2}\over 4(r-s)}}(1+|\Theta_{x}(r,\mu)|)d\mu dr$
$\displaystyle\leqslant
K(1+\|h(\cdot)\|_{C^{1}})+\int_{s}^{T}\int_{\mathbb{R}^{n}}K{1\over(r-s)^{{n+1\over
2}}}e^{{-\lambda|x-\mu|^{2}\over 4(r-s)}}|\Theta_{x}(r,\mu)|d\mu dr.$

By Grönwall’s inequality, we obtain

$|\Theta_{x}(s,x)|\leqslant
K(1+\|h(\cdot)\|_{C^{1}}),\quad\forall(s,x)\in[0,T]\times\mathbb{R}^{n}.$

Substituting the above into (7.3) and then by (7.8) again, we have

$|\Theta(s,x)|\leqslant
K(1+\|h(\cdot)\|_{C^{1}}),\quad\forall(s,x)\in[0,T]\times\mathbb{R}^{n}.$

It follows that

$\|\Theta(\cdot\,,\cdot)\|_{C^{0,1}}\leqslant K(1+\|h(\cdot)\|_{C^{1}}).$
(7.12)

Similar to (7.11), we get

$\displaystyle\Theta_{x}^{0}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\big{[}\Xi(s,x,T,\mu)h^{0}_{x}(t,\tilde{x},\mu,y)-\Xi(s,x,T,\mu)\rho(s,x,T,\mu)h^{0}(t,\tilde{x},\mu,y)\big{]}d\mu$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi_{x}(s,x,r,\mu)\Big{[}\Theta_{x}^{0}(t,r,\tilde{x},\mu,y)\tilde{b}(r,\mu;\theta,\theta^{0})+\tilde{g}^{0}(t,r,\tilde{x},\mu,y;\theta,\theta^{0},\Theta^{0})\Big{]}d\mu
dr.$ (7.13)

Note that $(t,\tilde{x},y)$ serve only as parameters in (7.4). By the same
argument as the above, we get

$\sup_{t,\tilde{x},y\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}}\|\Theta^{0}(t,\cdot\,,\tilde{x},\cdot\,,y))\|_{C^{0,1}}\leqslant
K\sup_{t,\tilde{x},y\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}}\big{[}1+\|h^{0}(t,\tilde{x},\cdot\,,y)\|_{C^{1}}\big{]}.$

Combining the above with (7.12), the desired estimate (7.10) is obtained. ∎

The following gives the regularity estimate of $\Theta^{0}(t,s,\tilde{x},x,y)$
with respect to the parameters $t$, $\tilde{x}$ and $y$.

######  Lemma 7.3.

There exists a constant $\kappa>0$, independent of $\theta(\cdot)$ and
$\theta^{0}(\cdot)$, such that

$\|\Theta^{0}(\cdot)\|_{C^{{\alpha\over
2},0,\alpha,1,2}}\leqslant\kappa\big{[}1+\|h^{0}(\cdot)\|^{2}_{C^{{\alpha\over
2},\alpha,1,2}}\big{]}.$ (7.14)

###### Proof.

From (7.4) and (7.13), it is easily seen that both
$\Theta^{0}(t,s,\tilde{x},x,y)$ and $\Theta_{x}^{0}(t,s,\tilde{x},x,y)$ are
differentiable with respect to the parameter $y$. Moreover, the derivatives
are given by

$\displaystyle\Theta_{y}^{0}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)h^{0}_{y}(t,\tilde{x},\mu,y)d\mu$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y)\big{[}\tilde{b}(r,\mu;\theta,\theta^{0})+\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x},\mu,y;\theta,\theta^{0},\Theta^{0})\big{]}d\mu
dr,$ $\displaystyle\Theta_{xy}^{0}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\big{[}\Xi(s,x,T,\mu)h^{0}_{xy}(t,\tilde{x},\mu,y)-\Xi(s,x,T,\mu)\rho(s,x,T,\mu)h^{0}_{y}(t,\tilde{x},\mu,y)\big{]}d\mu$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi_{x}(s,x,r,\mu)\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y)\big{[}\tilde{b}(r,\mu;\theta,\theta^{0})+\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x},\mu,y;\theta,\theta^{0},\Theta^{0})\big{]}d\mu
dr.$

Applying the arguments employed in the proof of 7.2, we have

$\|\Theta^{0}_{y}(\cdot)\|_{L^{\infty}}+\|\Theta^{0}_{xy}(\cdot)\|_{L^{\infty}}\leqslant
K\big{(}1+\|h^{0}(\cdot)\|_{C^{0,0,1,1}}\big{)}.$ (7.15)

By continuing the above argument, we get

$\displaystyle\Theta_{yy}^{0}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)h^{0}_{yy}(t,\tilde{x},\mu,y)d\mu+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)$
$\displaystyle\quad\times\Big{\\{}\Theta_{xyy}^{0}(t,r,\tilde{x},\mu,y)\big{[}\tilde{b}(r,\mu;\theta,\theta^{0})+\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x},\mu,y;\theta,\theta^{0},\Theta^{0})\big{]}$
$\displaystyle\quad+\big{\langle}\tilde{g}_{p^{0}p^{0}}^{0}(t,r,\tilde{x},\mu,y;\theta,\theta^{0},\Theta^{0})\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y),\,\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y)\big{\rangle}\Big{\\}}d\mu
dr.$ $\displaystyle\Theta_{xyy}^{0}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\big{[}\Xi(s,x,T,\mu)h^{0}_{xyy}(t,\tilde{x},\mu,y)-\Xi(s,x,T,\mu)\rho(s,x,T,\mu)h^{0}_{yy}(t,\tilde{x},\mu,y)\big{]}d\mu$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi_{x}(s,x,r,\mu)\Big{\\{}\Theta_{xyy}^{0}(t,r,\tilde{x},\mu,y)\big{[}\tilde{b}(r,\mu;\theta,\theta^{0})+\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x},\mu,y;\theta,\theta^{0},\Theta^{0})\big{]}$
$\displaystyle\quad+\big{\langle}\tilde{g}_{p^{0}p^{0}}^{0}(t,r,\tilde{x},\mu,y;\theta,\theta^{0},\Theta^{0})\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y),\,\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y)\big{\rangle}\Big{\\}}d\mu
dr.$

Note from (7.15) that

$\big{|}\big{\langle}\tilde{g}_{p^{0}p^{0}}^{0}(t,r,\tilde{x},\mu,y;\theta,\theta^{0},\Theta^{0})\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y),\,\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y)\big{\rangle}\big{|}\leqslant
K\big{(}1+\|h^{0}(\cdot)\|^{2}_{C^{0,0,1,1}}\big{)}.$

Then by the arguments employed in the proof of 7.2 again, we get

$\|\Theta^{0}_{yy}(\cdot)\|_{L^{\infty}}+\|\Theta^{0}_{xyy}(\cdot)\|_{L^{\infty}}\leqslant
K\big{(}1+\|h^{0}(\cdot)\|^{2}_{C^{0,0,1,2}}\big{)}.$ (7.16)

For any $\tilde{x}_{1},\tilde{x}_{2}\in\mathbb{R}^{n}$, denote

$\delta\Theta^{0}(t,s,x,y)=\Theta^{0,1}(t,s,x,y)-\Theta^{0,2}(t,s,x,y)\quad\hbox{with}\quad\Theta^{0,i}(t,s,x,y)=\Theta^{0}(t,s,\tilde{x}_{i},x,y),\quad
i=1,2.$

Then, we have

$\displaystyle\delta\Theta^{0}(t,s,x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)[h^{0}(t,\tilde{x}_{1},\mu,y)-h^{0}(t,\tilde{x}_{2},\mu,y)]d\mu$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)\Big{[}\delta\Theta^{0}_{x}(t,r,\mu,y)\tilde{b}(r,\mu;\theta,\theta^{0})$
$\displaystyle\quad+\tilde{g}^{0}(t,r,\tilde{x}_{1},\mu,y;\theta,\theta^{0},\Theta^{0,1})-\tilde{g}^{0}(t,r,\tilde{x}_{2},\mu,y;\theta,\theta^{0},\Theta^{0,2})\Big{]}d\mu
dr,$ $\displaystyle\delta\Theta_{y}^{0}(t,s,x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)[h_{y}^{0}(t,\tilde{x}_{1},\mu,y)-h_{y}^{0}(t,\tilde{x}_{2},\mu,y)]d\mu$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)\Big{\\{}\delta\Theta^{0}_{xy}(t,r,\mu,y)\big{[}\tilde{b}(r,\mu;\theta,\theta^{0})+\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x}_{1},\mu,y;\theta,\theta^{0},\Theta^{0,1})\big{]}$
$\displaystyle\quad+\Theta^{0,2}_{xy}(t,r,\mu,y)\big{[}\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x}_{1},\mu,y;\theta,\theta^{0},\Theta^{0,1})-\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x}_{2},\mu,y;\theta,\theta^{0},\Theta^{0,2})\big{]}\Big{\\}}d\mu
dr,$

and

$\displaystyle\delta\Theta_{x}^{0}(t,s,x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)[h_{x}^{0}(t,\tilde{x}_{1},\mu,y)-h_{x}^{0}(t,\tilde{x}_{2},\mu,y)]d\mu$
$\displaystyle\quad+\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)\rho(s,x,T,\mu)[h^{0}(t,\tilde{x}_{1},\mu,y)-h^{0}(t,\tilde{x}_{2},\mu,y)]d\mu$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi_{x}(s,x,r,\mu)\Big{[}\delta\Theta^{0}_{x}(t,r,\mu,y)\tilde{b}(r,\mu;\theta,\theta^{0})$
$\displaystyle\quad+\tilde{g}^{0}(t,r,\tilde{x}_{1},\mu,y;\theta,\theta^{0},\Theta^{0,1})-\tilde{g}^{0}(t,r,\tilde{x}_{2},\mu,y;\theta,\theta^{0},\Theta^{0,2})\Big{]}d\mu
dr,$ $\displaystyle\delta\Theta_{xy}^{0}(t,s,x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)[h_{xy}^{0}(t,\tilde{x}_{1},\mu,y)-h_{xy}^{0}(t,\tilde{x}_{2},\mu,y)]d\mu$
$\displaystyle\quad+\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)\rho(s,x,T,\mu)[h_{y}^{0}(t,\tilde{x}_{1},\mu,y)-h_{y}^{0}(t,\tilde{x}_{2},\mu,y)]d\mu$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi_{x}(s,x,r,\mu)\Big{\\{}\delta\Theta^{0}_{xy}(t,r,\mu,y)\big{[}\tilde{b}(r,\mu;\theta,\theta^{0})+\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x}_{1},\mu,y;\theta,\theta^{0},\Theta^{0,1})\big{]}$
$\displaystyle\quad+\Theta^{0,2}_{xy}(t,r,\mu,y)\big{[}\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x}_{1},\mu,y;\theta,\theta^{0},\Theta^{0,1})-\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x}_{2},\mu,y;\theta,\theta^{0},\Theta^{0,2})\big{]}\Big{\\}}d\mu
dr.$

Using the estimates (7.8) and (7.15), we get

$\displaystyle|\delta\Theta^{0}(t,s,x,y)|$ $\displaystyle\leqslant
K\int_{\mathbb{R}^{n}}{1\over(r-s)^{{n\over 2}}}e^{{-\lambda|x-\mu|^{2}\over
4(r-s)}}\|h^{0}(\cdot)\|_{C^{0,\alpha,0,0}}d\mu|\tilde{x}_{1}-\tilde{x}_{2}|^{\alpha}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{1\over(r-s)^{n\over
2}}e^{{-\lambda|x-\mu|^{2}\over
4(r-s)}}\Big{[}|\tilde{x}_{1}-\tilde{x}_{2}|^{\alpha}+|\delta\Theta^{0}_{x}(t,r,\mu,y)|\Big{]}d\mu
dr,$ $\displaystyle|\delta\Theta_{y}^{0}(t,s,x,y)|$ $\displaystyle\leqslant
K\int_{\mathbb{R}^{n}}{1\over(r-s)^{{n\over 2}}}e^{{-\lambda|x-\mu|^{2}\over
4(r-s)}}\|h^{0}(\cdot)\|_{C^{0,\alpha,0,1}}d\mu|\tilde{x}_{1}-\tilde{x}_{2}|^{\alpha}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{1\over(r-s)^{n\over
2}}e^{{-\lambda|x-\mu|^{2}\over
4(r-s)}}\Big{\\{}|\delta\Theta^{0}_{xy}(t,r,\mu,y)|$
$\displaystyle\quad+\big{[}|\tilde{x}_{1}-\tilde{x}_{2}|^{\alpha}+|\delta\Theta^{0}_{x}(t,r,\mu,y)|\big{]}|\Theta^{0,2}_{xy}(t,r,\mu,y)|\Big{\\}}d\mu
dr,$

and

$\displaystyle|\delta\Theta_{x}^{0}(t,s,x,y)|$ $\displaystyle\leqslant
K\int_{\mathbb{R}^{n}}{1\over(r-s)^{{n\over 2}}}e^{{-\lambda|x-\mu|^{2}\over
4(r-s)}}\Big{[}\|h^{0}(\cdot)\|_{C^{0,\alpha,1,0}}+\Big{(}1+{|x-\mu|^{2}\over(r-s)}\Big{)}\|h^{0}(\cdot)\|_{C^{0,\alpha,0,0}}\Big{]}d\mu|\tilde{x}_{1}-\tilde{x}_{2}|^{\alpha}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{1\over(r-s)^{n+1\over
2}}e^{{-\lambda|x-\mu|^{2}\over
4(r-s)}}\Big{[}|\tilde{x}_{1}-\tilde{x}_{2}|^{\alpha}+|\delta\Theta^{0}_{x}(t,r,\mu,y)|\Big{]}d\mu
dr,$ $\displaystyle|\delta\Theta_{xy}^{0}(t,s,x,y)|$ $\displaystyle\leqslant
K\int_{\mathbb{R}^{n}}{1\over(r-s)^{{n\over 2}}}e^{{-\lambda|x-\mu|^{2}\over
4(r-s)}}\Big{[}\|h^{0}(\cdot)\|_{C^{0,\alpha,1,1}}+\Big{(}1+{|x-\mu|^{2}\over(r-s)}\Big{)}\|h^{0}(\cdot)\|_{C^{0,\alpha,0,1}}\Big{]}d\mu|\tilde{x}_{1}-\tilde{x}_{2}|^{\alpha}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{1\over(r-s)^{n+1\over
2}}e^{{-\lambda|x-\mu|^{2}\over
4(r-s)}}\Big{\\{}|\delta\Theta^{0}_{xy}(t,r,\mu,y)|$
$\displaystyle\quad+\big{[}|\tilde{x}_{1}-\tilde{x}_{2}|^{\alpha}+|\delta\Theta^{0}_{x}(t,r,\mu,y)|\big{]}|\Theta^{0,2}_{xy}(t,r,\mu,y)|\Big{\\}}d\mu
dr.$

Note from (7.15) that $\Theta^{0,2}_{xy}(\cdot)$ is globally bounded, and the
estimate of $\|\Theta^{0,2}_{xy}(\cdot)\|_{L^{\infty}}$ is independent of
$(\theta(\cdot),\theta^{0}(\cdot))$. Thus, by the definition of the seminorm
$\|\cdot\|_{\alpha}$ and the Grönwall’s inequality, we obtain

$\displaystyle\sup_{(t,s,x,y)}\left[\|\Theta^{0}(t,s,\cdot\,,x,y)\|_{\alpha}+\|\Theta_{x}^{0}(t,s,\cdot\,,x,y)\|_{\alpha}+\|\Theta_{y}^{0}(t,s,\cdot\,,x,y)\|_{\alpha}+\|\Theta_{xy}^{0}(t,s,\cdot\,,x,y)\|_{\alpha}\right]$
$\displaystyle\qquad\leqslant
K\big{(}1+\|h^{0}(\cdot)\|_{C^{0,\alpha,1,1}}\big{)}.$

By continuing the above arguments, we can also have

$\displaystyle\sup_{(t,s,x,y)}\big{[}\|\Theta_{yy}^{0}(t,s,\cdot\,,x,y)\|_{\alpha}+\|\Theta_{xyy}^{0}(t,s,\cdot\,,x,y)\|_{\alpha}\big{]}\leqslant
K\big{(}1+\|h^{0}(\cdot)\|^{2}_{C^{0,\alpha,1,2}}\big{)}.$

Similarly, for the parameter $t$, we have

$\displaystyle\sup_{(s,\tilde{x},x,y)}\Big{[}\|\Theta^{0}(\cdot\,,s,\tilde{x},x,y)\|_{\alpha\over
2}+\|\Theta_{x}^{0}(\cdot,s,\tilde{x},x,y)\|_{\alpha\over
2}+\|\Theta_{y}^{0}(\cdot,s,\tilde{x},x,y)\|_{\alpha\over
2}+\|\Theta_{xy}^{0}(\cdot\,,s,\tilde{x},x,y)\|_{\alpha\over 2}$
$\displaystyle\qquad\quad+\|\Theta_{yy}^{0}(\cdot\,,s,\tilde{x},x,y)\|_{\alpha\over
2}+\|\Theta_{xyy}^{0}(\cdot\,,s,\tilde{x},x,y)\|_{\alpha\over
2}\Big{]}\leqslant K\big{(}1+\|h^{0}(\cdot)\|_{C^{{\alpha\over
2},0,1,2}}\big{)}.$

Combining the above together, we get the estimate (7.14) immediately. ∎

######  Remark 7.4.

Note that (7.14) is a global prior estimate and the constant $\kappa>0$ is
independent of the choice of $(\theta(\cdot),\theta^{0}(\cdot))$. Thus, we
could always assume that

$\|\theta(\cdot)\|_{C^{0,1}}+\|\theta^{0}(\cdot)\|_{C^{{\alpha\over
2},0,\alpha,1,2}}\leqslant\kappa\big{[}1+\|h^{0}(\cdot)\|^{2}_{C^{{\alpha\over
2},\alpha,1,2}}\big{]},$ (7.17)

where $\kappa>0$ is same as that given in 7.3.

Next, we are going to establish the $C^{1+\alpha}$-norm estimate for
$\Theta^{0}(t,s,\tilde{x},\cdot,y)$ and $\Theta(s,\cdot)$. To achieve this, we
need to make some preparations. By making the transforms
$x-\mu=(\sqrt{T-s})\tilde{\mu}$ and $x-\mu=(\sqrt{r-s})\tilde{\mu}$ in the
first integral term and the second integral term of (7.11)–(7.13),
respectively, we have

$\displaystyle\Theta_{x}(s,x)$
$\displaystyle=\int_{\mathbb{R}^{n}}\tilde{\Xi}(s,x,T,x-\sqrt{T-s}\tilde{\mu})\big{[}h_{x}(x-\sqrt{T-s}\tilde{\mu})-\rho(s,x,T,x-\sqrt{T-s}\tilde{\mu})h(x-\sqrt{T-s}\tilde{\mu})\big{]}d\tilde{\mu}$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\tilde{\Xi}_{x}(s,x,r,x-\sqrt{r-s}\tilde{\mu})\Big{[}\Theta_{x}(r,x-\sqrt{r-s}\tilde{\mu})\tilde{b}(r,x-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})$
$\displaystyle\quad+\tilde{g}(r,x-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})\Big{]}d\tilde{\mu}dr,$
(7.18)

and

$\displaystyle\Theta^{0}_{x}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\tilde{\Xi}(s,x,T,x-\sqrt{T-s}\tilde{\mu})\Big{[}h^{0}_{x}(t,\tilde{x},x-\sqrt{T-s}\tilde{\mu},y)$
$\displaystyle\quad-\rho(s,x,T,x-\sqrt{T-s}\tilde{\mu})h^{0}(t,\tilde{x},x-\sqrt{T-s}\tilde{\mu},y)\Big{]}d\tilde{\mu}$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\tilde{\Xi}_{x}(s,x,r,x-\sqrt{r-s}\tilde{\mu})\Big{[}\Theta^{0}_{x}(t,r,\tilde{x},x-\sqrt{r-s}\tilde{\mu},y)\tilde{b}(r,x-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})$
$\displaystyle\quad+\tilde{g}^{0}(t,r,\tilde{x},x-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0},\Theta^{0})\Big{]}d\tilde{\mu}dr,$
(7.19)

where

$\displaystyle\widetilde{\Xi}(s,x,r,x-\sqrt{r-s}\tilde{\mu})$
$\displaystyle:=-\Xi(s,x,r,x-\sqrt{r-s}\tilde{\mu})(r-s)^{n\over 2}$
$\displaystyle=-{1\over(4\pi)^{n\over
2}(\hbox{det\,}[a(r,x-\sqrt{r-s}\tilde{\mu})])^{1\over 2}}e^{-{\langle
a(r,x-\sqrt{r-s}\tilde{\mu})^{-1}\tilde{\mu},\tilde{\mu}\rangle\over 4}},$
$\displaystyle\rho(s,x,r,x-\sqrt{r-s}\tilde{\mu})$
$\displaystyle={(\hbox{det\,}[a(r,x-\sqrt{r-s}\tilde{\mu})])_{\mu}\over
2\hbox{det\,}[a(r,x-\sqrt{r-s}\tilde{\mu})]}+{\langle[a(r,x-\sqrt{r-s}\tilde{\mu})^{-1}]_{\mu}\tilde{\mu},\,\tilde{\mu}\rangle\over
4},$ $\displaystyle\widetilde{\Xi}_{x}(s,x,r,x-\sqrt{r-s}\tilde{\mu})$
$\displaystyle:=-\Xi_{x}(s,x,r,x-\sqrt{r-s}\tilde{\mu})(r-s)^{n\over 2}$
$\displaystyle={1\over(4\pi)^{n\over
2}(\hbox{det\,}[a(r,x-\sqrt{r-s}\tilde{\mu})])^{1\over 2}}e^{-{\langle
a(r,x-\sqrt{r-s}\tilde{\mu})^{-1}\tilde{\mu},\tilde{\mu}\rangle\over
4}}{a(r,x-\sqrt{r-s}\tilde{\mu})^{-1}\over 2\sqrt{r-s}}\tilde{\mu}.$ (7.20)

By some straightforward calculations, it is clearly seen that

$\displaystyle|\widetilde{\Xi}(s,x,r,x-\sqrt{r-s}\tilde{\mu})|\leqslant
Ke^{-\lambda|\tilde{\mu}|^{2}},\quad$
$\displaystyle|\rho(s,x,r,x-\sqrt{r-s}\tilde{\mu})|\leqslant
K(1+|\tilde{\mu}|^{2}),$
$\displaystyle|\widetilde{\Xi}_{x}(s,x,r,x-\sqrt{r-s}\tilde{\mu})|\leqslant{K\over\sqrt{r-s}}e^{-\lambda|\tilde{\mu}|^{2}},\quad$
$\displaystyle\|\widetilde{\Xi}(s,\cdot,r,\cdot-\sqrt{r-s}\tilde{\mu})\|_{\alpha}\leqslant
Ke^{-\lambda|\tilde{\mu}|^{2}},$
$\displaystyle\|\rho(s,\cdot,r,\cdot-\sqrt{r-s}\tilde{\mu})\|_{\alpha}\leqslant
K(1+|\tilde{\mu}|^{2}),\quad$
$\displaystyle\|\widetilde{\Xi}_{x}(s,\cdot,r,\cdot-\sqrt{r-s}\tilde{\mu})\|_{\alpha}\leqslant{K\over\sqrt{r-s}}e^{-\lambda|\tilde{\mu}|^{2}},$
(7.21)

for some $0<\lambda<\lambda_{0}$.

######  Proposition 7.5.

There exist two constants $0<\varepsilon\leqslant T$ and $\bar{\kappa}>0$ such
for any $\theta(\cdot)\in C^{{\alpha\over 2},1+\alpha}$ and
$\theta^{0}(\cdot)\in C^{{\alpha\over 2},{\alpha\over 2},\alpha,1+\alpha,2}$
with

$\|\theta_{x}(\cdot)\|_{C^{0,\alpha}([T-\varepsilon,T])}+\|\theta^{0}_{x}(\cdot)\|_{C^{0,0,0,\alpha,0}([T-\varepsilon,T])}\leqslant
2\bar{\kappa}\big{[}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,0}}\big{]},$
(7.22)

the unique solution $(\Theta(\cdot),\Theta^{0}(\cdot))$ of PDE (7.1) satisfies

$\displaystyle\|\Theta_{x}(\cdot)\|_{C^{0,\alpha}([T-\varepsilon,T])}+\|\Theta^{0}_{x}(\cdot)\|_{C^{0,0,0,\alpha,0}([T-\varepsilon,T])}\leqslant
2\bar{\kappa}\big{[}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,0}}\big{]}.$
(7.23)

Moreover, there exists a constant $\widehat{\kappa}>0$, which depends on
$\bar{\kappa}$, such that

$\displaystyle\|\Theta^{0}_{xy}(\cdot)\|_{C^{0,0,0,\alpha,0}([T-\varepsilon,T])}\leqslant\widehat{\kappa}\big{[}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,1}}\big{]}.$
(7.24)

###### Proof.

For any $x_{1},\,x_{2}\in\mathbb{R}^{n}$, from (7.18) and then by the estimate
(7.21), we have

$\displaystyle|\Theta_{x}(s,x_{1})-\Theta_{x}(s,x_{2})|$
$\displaystyle\leqslant\int_{\mathbb{R}^{n}}Ke^{-\lambda|\tilde{\mu}|^{2}}(1+|\tilde{\mu}|^{2})d\tilde{\mu}\|h(\cdot)\|_{C^{1}}|x_{1}-x_{2}|^{\alpha}$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}{K\over\sqrt{r-s}}e^{-\lambda|\tilde{\mu}|^{2}}d\tilde{\mu}dr\big{[}1+\|\Theta_{x}(\cdot)\|_{L^{\infty}}\big{]}|x_{1}-x_{2}|^{\alpha}$
$\displaystyle\quad+\int_{\mathbb{R}^{n}}Ke^{-\lambda|\tilde{\mu}|^{2}}\big{|}h_{x}(x_{1}-\sqrt{T-s}\tilde{\mu})-h_{x}(x_{2}-\sqrt{T-s}\tilde{\mu})\big{|}d\tilde{\mu}$
$\displaystyle\quad+\int_{\mathbb{R}^{n}}Ke^{-\lambda|\tilde{\mu}|^{2}}(1+|\tilde{\mu}|^{2})\big{|}h(x_{1}-\sqrt{T-s}\tilde{\mu})-h(x_{2}-\sqrt{T-s}\tilde{\mu})\big{|}d\tilde{\mu}$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}{K\over\sqrt{r-s}}e^{-\lambda|\tilde{\mu}|^{2}}\Big{[}\big{|}\Theta_{x}(r,x_{1}-\sqrt{r-s}\tilde{\mu})-\Theta_{x}(r,x_{2}-\sqrt{r-s}\tilde{\mu})\big{|}$
$\displaystyle\qquad+\big{|}\tilde{b}(r,x_{1}-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})-\tilde{b}(r,x_{2}-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})\big{|}\|\Theta_{x}(\cdot)\|_{L^{\infty}}$
$\displaystyle\qquad+\big{|}\tilde{g}(r,x_{1}-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})-\tilde{g}(r,x_{2}-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})\big{|}\Big{]}d\tilde{\mu}dr.$
(7.25)

Note that for $\tilde{\varphi}(\cdot)=\tilde{b}(\cdot),\tilde{g}(\cdot)$,

$\displaystyle\tilde{\varphi}(r,x_{1}-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})-\tilde{\varphi}(r,x_{2}-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})$
$\displaystyle\quad=\tilde{\varphi}\Big{(}r,x_{1}-\sqrt{r-s}\tilde{\mu},\theta(r,x_{1}-\sqrt{r-s}\tilde{\mu}),\theta_{x}(r,x_{1}-\sqrt{r-s}\tilde{\mu}),$
$\displaystyle\qquad\quad\,\theta^{0}(r,x_{1}-\sqrt{r-s}\tilde{\mu},r,x_{1}-\sqrt{r-s}\tilde{\mu},\theta(r,x_{1}-\sqrt{r-s}\tilde{\mu})),$
$\displaystyle\qquad\quad\,\theta_{x}^{0}(r,x_{1}-\sqrt{r-s}\tilde{\mu},r,x_{1}-\sqrt{r-s}\tilde{\mu},\theta(r,x_{1}-\sqrt{r-s}\tilde{\mu})),$
$\displaystyle\qquad\quad\,\theta_{y}^{0}(r,x_{1}-\sqrt{r-s}\tilde{\mu},r,x_{1}-\sqrt{r-s}\tilde{\mu},\theta(r,x_{1}-\sqrt{r-s}\tilde{\mu}))\Big{)}$
$\displaystyle\qquad-\tilde{\varphi}\Big{(}r,x_{2}-\sqrt{r-s}\tilde{\mu},\theta(r,x_{2}-\sqrt{r-s}\tilde{\mu}),\theta_{x}(r,x_{2}-\sqrt{r-s}\tilde{\mu}),$
$\displaystyle\qquad\quad\,\theta^{0}(r,x_{2}-\sqrt{r-s}\tilde{\mu},r,x_{2}-\sqrt{r-s}\tilde{\mu},\theta(r,x_{2}-\sqrt{r-s}\tilde{\mu})),$
$\displaystyle\qquad\quad\,\theta_{x}^{0}(r,x_{2}-\sqrt{r-s}\tilde{\mu},r,x_{2}-\sqrt{r-s}\tilde{\mu},\theta(r,x_{2}-\sqrt{r-s}\tilde{\mu})),$
$\displaystyle\qquad\quad\,\theta_{y}^{0}(r,x_{2}-\sqrt{r-s}\tilde{\mu},r,x_{2}-\sqrt{r-s}\tilde{\mu},\theta(r,x_{2}-\sqrt{r-s}\tilde{\mu}))\Big{)}.$

Then by 7.3 (or (7.17)), we get

$\displaystyle|\tilde{\varphi}(r,x_{1}-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})-\tilde{\varphi}(r,x_{2}-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})|$
$\displaystyle\quad\leqslant
K\big{(}1+\|\theta_{x}(r,\cdot)\|_{C^{\alpha}}+\|\theta^{0}_{x}(r,\cdot,r,\cdot,\cdot)\|_{C^{0,\alpha,0}}\big{)}|x_{1}-x_{2}|^{\alpha},$

where $K>0$ depends on $h(\cdot)$ and $h^{0}(\cdot)$. Substituting the above
into (7.25) and then by 7.2, we have

$\displaystyle|\Theta_{x}(s,x_{1})-\Theta_{x}(s,x_{2})|$
$\displaystyle\leqslant
K\big{(}1+\|h(\cdot)\|_{C^{1+\alpha}}\big{)}|x_{1}-x_{2}|^{\alpha}$
$\displaystyle\quad+\int_{s}^{T}{K\over\sqrt{r-s}}\Big{[}\|\Theta_{x}(r,\cdot)\|_{\alpha}+\|\theta_{x}(r,\cdot)\|_{C^{\alpha}}+\|\theta^{0}_{x}(r,\cdot,r,\cdot,\cdot)\|_{C^{0,\alpha,0}}\Big{]}dr|x_{1}-x_{2}|^{\alpha},$

which implies that

$\|\Theta_{x}(s,\cdot)\|_{\alpha}\leqslant
K\big{(}1+\|h(\cdot)\|_{C^{1+\alpha}}\big{)}+\int_{s}^{T}{K\over\sqrt{r-s}}\Big{[}\|\Theta_{x}(r,\cdot)\|_{\alpha}+\|\theta_{x}(r,\cdot)\|_{C^{\alpha}}+\|\theta^{0}_{x}(r,\cdot,r,\cdot,\cdot)\|_{C^{0,\alpha,0}}\Big{]}dr.$
(7.26)

By the same argument as the above (noting (7.19)), we also have

$\displaystyle\|\Theta^{0}_{x}(t,s,\tilde{x},\cdot,y)\|_{\alpha}$
$\displaystyle\leqslant
K\big{(}1+\|h^{0}(t,\tilde{x},\cdot,y)\|_{C^{1+\alpha}}\big{)}$
$\displaystyle\quad+\int_{s}^{T}{K\over\sqrt{r-s}}\Big{[}\|\Theta^{0}_{x}(t,s,\tilde{x},\cdot,y)\|_{\alpha}+\|\theta_{x}(r,\cdot)\|_{\alpha}+\|\theta^{0}_{x}(r,\cdot,r,\cdot,\cdot)\|_{C^{0,\alpha,0}}\Big{]}dr,$
(7.27) $\displaystyle\|\Theta^{0}_{xy}(t,s,\tilde{x},\cdot,y)\|_{\alpha}$
$\displaystyle\leqslant
K\big{(}1+\|h^{0}(t,\tilde{x},\cdot,\cdot)\|_{C^{1+\alpha,1}}\big{)}$
$\displaystyle\quad+\int_{s}^{T}{K\over\sqrt{r-s}}\Big{[}\|\Theta^{0}_{xy}(t,s,\tilde{x},\cdot,y)\|_{\alpha}+\|\theta_{x}(r,\cdot)\|_{\alpha}+\|\theta^{0}_{x}(r,\cdot,r,\cdot,\cdot)\|_{C^{0,\alpha,0}}\Big{]}dr.$
(7.28)

Combining (7.26) and (7.27) yields that

$\displaystyle\|\Theta_{x}(\cdot)\|_{C^{0,\alpha}([T-\varepsilon,T])}+\|\Theta^{0}_{x}(\cdot)\|_{C^{0,0,0,\alpha,0}([T-\varepsilon,T])}$
$\displaystyle\quad\leqslant\bar{\kappa}\big{[}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,0}}\big{]}+\bar{\kappa}\sqrt{\varepsilon}\big{[}\|\Theta_{x}(\cdot)\|_{C^{0,\alpha}([T-\varepsilon,T])}$
$\displaystyle\qquad+\|\Theta^{0}_{x}(\cdot)\|_{C^{0,0,0,\alpha,0}([T-\varepsilon,T])}+\|\theta_{x}(\cdot)\|_{C^{0,\alpha}([T-\varepsilon,T])}+\|\theta^{0}_{x}(\cdot)\|_{C^{0,0,0,\alpha,0}([T-\varepsilon,T])}\big{]},$
(7.29)

where $\bar{\kappa}>\kappa$, only depending on
$(h(\cdot),\tilde{g}(\cdot),h^{0}(\cdot),\tilde{g}^{0}(\cdot))$, is a fixed
constant. Let $\varepsilon$ be small enough such that
$\bar{\kappa}\sqrt{\varepsilon}\leqslant{1\over 4}$ and

$\|\theta_{x}(\cdot)\|_{C^{0,\alpha}([T-\varepsilon,T])}+\|\theta^{0}_{x}(\cdot)\|_{C^{0,0,0,\alpha,0}([T-\varepsilon,T])}\leqslant
2\bar{\kappa}\big{[}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,0}}\big{]}.$
(7.30)

Then from (7.29), we get

$\displaystyle\|\Theta_{x}(\cdot)\|_{C^{0,\alpha}([T-\varepsilon,T])}+\|\Theta^{0}_{x}(\cdot)\|_{C^{0,0,0,\alpha,0}([T-\varepsilon,T])}\leqslant
2\bar{\kappa}\big{[}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,0}}\big{]}.$

Substituting (7.30) into (7.28) also yields (7.24) immediately. The proof is
complete. ∎

The following gives the $C^{\alpha\over 2}$-norm estimate for
$\Theta^{0}(t,\cdot,\tilde{x},x,y)$ and $\Theta(\cdot,x)$.

######  Proposition 7.6.

There exists a constant $\check{\kappa}>0$ such for any $\theta(\cdot)\in
C^{{\alpha\over 2},1+\alpha}$ and $\theta^{0}(\cdot)\in C^{{\alpha\over
2},{\alpha\over 2},\alpha,1+\alpha,2}$ satisfying (7.22), the unique solution
of PDE (7.1) satisfies

$\displaystyle\|\Theta(\cdot,x)\|_{\alpha\over
2}+\|\Theta_{x}(\cdot,x)\|_{\alpha\over
2}+\|\Theta^{0}(t,\cdot,\tilde{x},x,y)\|_{\alpha\over
2}+\|\Theta_{x}^{0}(t,\cdot,\tilde{x},x,y)\|_{\alpha\over
2}+\|\Theta_{y}^{0}(t,\cdot,\tilde{x},x,y)\|_{\alpha\over 2}$
$\displaystyle+\|\Theta_{xy}^{0}(t,\cdot,\tilde{x},x,y)\|_{\alpha\over
2}\,\leqslant\,\check{\kappa}\big{(}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,0}}\big{)},\quad\forall(t,\tilde{x},x,y)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R},$
(7.31)

on the time interval $[T-\varepsilon,T]$.

###### Proof.

For any $T-\varepsilon\leqslant s_{1}\leqslant s_{2}\leqslant T$, by (7.18) we
have

$\displaystyle|\Theta_{x}(s_{1},x)-\Theta_{x}(s_{2},x)|$
$\displaystyle\leqslant\int_{\mathbb{R}^{n}}\Big{[}|\tilde{\Xi}(s_{1},x,T,x-\sqrt{T-s_{1}}\tilde{\mu})|\|h_{x}(\cdot)\|_{\alpha}|\tilde{\mu}|^{\alpha}|s_{1}-s_{2}|^{\alpha\over
2}$
$\displaystyle\quad+\big{|}\tilde{\Xi}(s_{1},x,T,x-\sqrt{T-s_{1}}\tilde{\mu})-\tilde{\Xi}(s_{2},x,T,x-\sqrt{T-s_{2}}\tilde{\mu})\big{|}\|h_{x}(\cdot)\|_{L^{\infty}}$
$\displaystyle\quad+|\tilde{\Xi}(s_{1},x,T,x-\sqrt{T-s_{1}}\tilde{\mu})\rho(s_{1},x,T,x-\sqrt{T-s_{1}}\tilde{\mu})|\|h(\cdot)\|_{\alpha}|\tilde{\mu}|^{\alpha}|s_{1}-s_{2}|^{\alpha\over
2}$
$\displaystyle\quad+\big{|}\tilde{\Xi}(s_{1},x,T,x-\sqrt{T-s_{1}}\tilde{\mu})\rho(s_{1},x,T,x-\sqrt{T-s_{1}}\tilde{\mu})$
$\displaystyle\qquad-\tilde{\Xi}(s_{2},x,T,x-\sqrt{T-s_{2}}\tilde{\mu})\rho(s_{2},x,T,x-\sqrt{T-s_{2}}\tilde{\mu})\big{|}\|h(\cdot)\|_{L^{\infty}}\Big{]}d\tilde{\mu}$
$\displaystyle\quad+\int_{s_{1}}^{s_{2}}\int_{\mathbb{R}^{n}}|\tilde{\Xi}_{x}(s_{1},x,r,x-\sqrt{r-s_{1}}\tilde{\mu})|\|\Theta_{x}(\cdot)\tilde{b}(\cdot;\theta,\theta^{0})+\tilde{g}\big{(}\cdot;\theta,\theta^{0}\big{)}\|_{L^{\infty}}d\tilde{\mu}dr$
$\displaystyle\quad+\int_{s_{2}}^{T}\int_{\mathbb{R}^{n}}\big{|}\tilde{\Xi}_{x}(s_{1},x,r,x-\sqrt{r-s_{1}}\tilde{\mu})-\tilde{\Xi}_{x}(s_{2},x,r,x-\sqrt{r-s_{2}}\tilde{\mu})\big{|}$
$\displaystyle\qquad\times\|\Theta_{x}(\cdot)\tilde{b}(\cdot;\theta,\theta^{0})+\tilde{g}\big{(}\cdot;\theta,\theta^{0}\big{)}\|_{L^{\infty}}d\tilde{\mu}dr$
$\displaystyle\quad+\int_{s_{2}}^{T}\int_{\mathbb{R}^{n}}|\tilde{\Xi}_{x}(s_{1},x,r,x-\sqrt{r-s_{1}}\tilde{\mu})|\Big{[}\big{|}\Theta_{x}(r,x-\sqrt{r-s_{1}}\tilde{\mu})\tilde{b}(r,x-\sqrt{r-s_{1}}\tilde{\mu};\theta,\theta^{0})$
$\displaystyle\qquad-\Theta_{x}(r,x-\sqrt{r-s_{2}}\tilde{\mu})\tilde{b}(r,x-\sqrt{r-s_{2}}\tilde{\mu};\theta,\theta^{0})\big{|}+\big{|}\tilde{g}(r,x-\sqrt{r-s_{1}}\tilde{\mu};\theta,\theta^{0})$
$\displaystyle\qquad-\tilde{g}(r,x-\sqrt{r-s_{2}}\tilde{\mu};\theta,\theta^{0})\big{|}\Big{]}d\tilde{\mu}dr.$
(7.32)

Note that (recalling (7.20))

$\displaystyle\int_{s_{2}}^{T}\int_{\mathbb{R}^{n}}\big{|}\tilde{\Xi}_{x}(s_{1},x,r,x-\sqrt{r-s_{1}}\tilde{\mu})-\tilde{\Xi}_{x}(s_{2},x,r,x-\sqrt{r-s_{2}}\tilde{\mu})\big{|}d\tilde{\mu}dr$
$\displaystyle\quad\leqslant K|s_{1}-s_{2}|^{\alpha\over
2}+K\int_{s_{2}}^{T}\Big{|}{1\over\sqrt{r-s_{2}}}-{1\over\sqrt{r-s_{1}}}\Big{|}dr\leqslant
K|s_{1}-s_{2}|^{\alpha\over 2}.$

Then by (7.17), 7.5 and (7.21), from (7.32) we obtain

$\displaystyle|\Theta_{x}(s_{1},x)-\Theta_{x}(s_{2},x)|\leqslant
K\big{(}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,0}}\big{)}|s_{1}-s_{2}|^{\alpha\over
2},$ $\displaystyle\forall T-\varepsilon\leqslant s_{1}\leqslant
s_{2}\leqslant T,\,x\in\mathbb{R}^{n},$ (7.33)

where $K>0$ depends on $\bar{\kappa}$. Recall (7.19) and note that

$\displaystyle\Theta^{0}_{y}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\tilde{\Xi}(s,x,T,x-\sqrt{T-s}\tilde{\mu})h^{0}_{y}(t,\tilde{x},x-\sqrt{T-s}\tilde{\mu},y)d\tilde{\mu}$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\tilde{\Xi}(s,x,r,x-\sqrt{r-s}\tilde{\mu})\Big{[}\Theta^{0}_{xy}(t,r,\tilde{x},x-\sqrt{r-s}\tilde{\mu},y)\tilde{b}(r,x-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})$
$\displaystyle\quad+\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x},x-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0},\Theta^{0})\Theta^{0}_{xy}(t,r,\tilde{x},x-\sqrt{r-s}\tilde{\mu},y)\Big{]}d\tilde{\mu}dr,$

and

$\displaystyle\Theta^{0}_{xy}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{\mathbb{R}^{n}}\tilde{\Xi}(s,x,T,x-\sqrt{T-s}\tilde{\mu}\Big{[}h^{0}_{xy}(t,\tilde{x},x-\sqrt{T-s}\tilde{\mu},y)$
$\displaystyle\quad-\rho(s,x,T,x-\sqrt{T-s}\tilde{\mu})h^{0}_{y}(t,\tilde{x},x-\sqrt{T-s}\tilde{\mu},y)\Big{]}d\tilde{\mu}$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\tilde{\Xi}_{x}(s,x,r,x-\sqrt{r-s}\tilde{\mu})\Big{[}\Theta^{0}_{xy}(t,r,\tilde{x},x-\sqrt{r-s}\tilde{\mu},y)\tilde{b}(r,x-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0})$
$\displaystyle\quad+\tilde{g}_{p^{0}}^{0}(t,r,\tilde{x},x-\sqrt{r-s}\tilde{\mu};\theta,\theta^{0},\Theta^{0})\Theta^{0}_{xy}(t,r,\tilde{x},x-\sqrt{r-s}\tilde{\mu},y)\Big{]}d\tilde{\mu}dr.$

By the same argument as the above, we also have

$\displaystyle|\Theta^{0}_{x}(t,s_{1},\tilde{x},x,y)-\Theta^{0}_{x}(t,s_{2},\tilde{x},x,y)|\leqslant
K\big{(}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,0}}\big{)}|s_{1}-s_{2}|^{\alpha\over
2},$
$\displaystyle|\Theta^{0}_{y}(t,s_{1},\tilde{x},x,y)-\Theta^{0}_{y}(t,s_{2},\tilde{x},x,y)|+|\Theta^{0}_{xy}(t,s_{1},\tilde{x},x,y)-\Theta^{0}_{xy}(t,s_{2},\tilde{x},x,y)|$
$\displaystyle\quad\leqslant
K\big{(}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,1}}\big{)}|s_{1}-s_{2}|^{\alpha\over
2},$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\forall
T-\varepsilon\leqslant s_{1}\leqslant s_{2}\leqslant
T,\,(t,y,\tilde{x},x)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n}.$
(7.34)

The rest can be obtained easily. ∎

######  Remark 7.7.

Note that the constant $\check{\kappa}>0$ in 7.6 is independent of the
${\alpha\over 2}$-Hölder norm of
$(\theta(\cdot,\cdot),\theta^{0}(\cdot,\cdot,\cdot,\cdot,\cdot))$ with respect
to the second argument $s$. We assume that

$\displaystyle\|\theta(\cdot,x)\|_{\alpha\over
2}+\|\theta_{x}(\cdot,x)\|_{\alpha\over
2}+\|\theta^{0}(t,\cdot,\tilde{x},x,y)\|_{\alpha\over
2}+\|\theta_{x}^{0}(t,\cdot,\tilde{x},x,y)\|_{\alpha\over
2}+\|\theta_{y}^{0}(t,\cdot,\tilde{x},x,y)\|_{\alpha\over 2}$
$\displaystyle+\|\theta_{xy}^{0}(t,\cdot,\tilde{x},x,y)\|_{\alpha\over
2}\leqslant\check{\kappa}\big{[}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,0}}\big{]},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall(t,\tilde{x},x,y)\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R},$
(7.35)

on the time interval $[T-\varepsilon,T]$.

The following is concerned with the local solvability of the equilibrium HJB
equation (2.25).

######  Proposition 7.8.

There exists a constant $\bar{\varepsilon}\in(0,\varepsilon]$ such that the
equilibrium HJB equation (2.25) admits a unique classical solution on the time
interval $[T-\bar{\varepsilon},T]$.

###### Proof.

Denote

${\cal B}_{\varepsilon}=\big{\\{}(\theta(\cdot),\theta^{0}(\cdot))\in
C^{{\alpha\over 2},1+\alpha}\times C^{{\alpha\over 2},{\alpha\over
2},\alpha,1+\alpha,2}~{}\big{|}~{}(\theta(\cdot),\theta^{0}(\cdot))\hbox{
satisfies }\eqref{lem:v-par-estimate},\eqref{Prop:Holder1}\hbox{ and
}\eqref{Prop:Holder-t-v}\big{\\}}.$

For any $(\theta(\cdot),\theta^{0}(\cdot))\in{\cal B}_{\varepsilon}$, by 7.1,
PDE (7.1) admits a unique classical solution
$(\Theta(\cdot),\Theta^{0}(\cdot))$. Moreover, from 7.2, 7.3, 7.5 and 7.6, we
know that $(\Theta(\cdot),\Theta^{0}(\cdot))\in{\cal B}_{\varepsilon}$. Thus,
the mapping $\Gamma(\cdot,\cdot):{\cal B}_{\varepsilon}\to{\cal
B}_{\varepsilon}$, given by

$\Gamma(\theta(\cdot),\theta^{0}(\cdot))=(\Theta(\cdot),\Theta^{0}(\cdot)),$

is well-defined. For any $(\theta^{i}(\cdot),\theta^{0,i}(\cdot))\in{\cal
B}_{\varepsilon}$ ($i=1,2$), let

$(\Theta^{i}(\cdot),\Theta^{0,i}(\cdot))=\Gamma(\theta^{i}(\cdot),\theta^{0,i}(\cdot)),\quad
i=1,2.$

Denote

$\displaystyle\delta\theta(s,x)=\theta^{1}(s,x)-\theta^{2}(s,x),\quad\delta\theta^{0}(t,s,\tilde{x},x,y)=\theta^{0,1}(t,s,\tilde{x},x,y)-\theta^{0,2}(t,s,\tilde{x},x,y),$
$\displaystyle\delta\Theta(s,x)=\Theta^{1}(s,x)-\Theta^{2}(s,x),\quad\delta\Theta^{0}(t,s,\tilde{x},x,y)=\Theta^{0,1}(t,s,\tilde{x},x,y)-\Theta^{0,2}(t,s,\tilde{x},x,y),$
$\displaystyle\delta\tilde{\varphi}(s,x)=\tilde{\varphi}(s,x;\theta^{1},\theta^{0,1})-\tilde{\varphi}(s,x;\theta^{2},\theta^{0,2}),\quad\hbox{for}\quad\varphi(\cdot)=b(\cdot),g(\cdot),$
$\displaystyle\delta\tilde{g}^{0}(t,s,\tilde{x},x,y)=\tilde{g}^{0}(t,s,\tilde{x},x,y;\theta^{1},\theta^{0,1},\Theta^{0,1})-\tilde{g}^{0}(t,s,\tilde{x},x,y;\theta^{1},\theta^{0,2},\Theta^{0,2}).$

We hope to show that

$\|\delta\Theta(\cdot)\|_{C^{{\alpha\over
2},1+\alpha}}+\|\delta\Theta^{0}(\cdot)\|_{C^{{\alpha\over 2},{\alpha\over
2},\alpha,1+\alpha,2}}\leqslant{1\over
2}\big{[}\|\delta\theta(\cdot)\|_{C^{{\alpha\over
2},1+\alpha}}+\|\delta\theta^{0}(\cdot)\|_{C^{{\alpha\over 2},{\alpha\over
2},\alpha,1+\alpha,2}}\big{]},$ (7.36)

on some time interval $[T-\bar{\varepsilon},T]\subseteq[T-\varepsilon,T]$.
Thus, $\Gamma(\cdot)$ is a contraction mapping and then it admits a unique
fixed point $(\Theta(\cdot),\Theta^{0}(\cdot))\in C^{{\alpha\over
2},1+\alpha}\times C^{{\alpha\over 2},{\alpha\over 2},\alpha,1+\alpha,2}$.
Then we get the well-posedness of the following equation:

$\displaystyle\Theta(s,x)=\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)h(\mu)d\mu+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)\Big{[}\Theta_{x}(r,\mu)\tilde{b}\big{(}r,\mu,\Theta(r,\mu),\Theta_{x}(r,\mu),$
$\displaystyle\quad\,\,\Theta^{0}(r,\mu,r,\mu,\Theta(r,\mu)),\Theta_{x}^{0}(r,\mu,r,\mu,\Theta(r,\mu)),\Theta_{y}^{0}(r,\mu,r,\mu,\Theta(r,\mu))\big{)}$
$\displaystyle\quad+\tilde{g}\big{(}r,\mu,\Theta(r,\mu),\Theta_{x}(r,\mu),\Theta^{0}(r,\mu,r,\mu,\Theta(r,\mu)),\Theta_{x}^{0}(r,\mu,r,\mu,\Theta(r,\mu)),\Theta_{y}^{0}(r,\mu,r,\mu,\Theta(r,\mu))\big{)}\Big{]}d\mu
dr;$
$\displaystyle\Theta^{0}(t,s.\tilde{x},x,y)=\int_{\mathbb{R}^{n}}\Xi(s,x,T,\mu)L(t,\tilde{x},\mu,y)d\mu+\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)$
$\displaystyle\quad\times\Big{[}\Theta^{0}_{x}(t,r,\tilde{x},\mu,y)\tilde{b}\big{(}r,\mu,\Theta(r,\mu),\Theta_{x}(r,\mu),\Theta^{0}(r,\mu,r,\mu,\Theta(r,\mu)),\Theta_{x}^{0}(r,\mu,r,\mu,\Theta(r,\mu)),$
$\displaystyle\quad\,\,\Theta_{y}^{0}(r,\mu,r,\mu,\Theta(r,\mu))\big{)}+\tilde{g}^{0}\big{(}t,r,\tilde{x},\mu,\Theta(r,\mu),\Theta_{x}(r,\mu),\Theta^{0}(r,\mu,r,\mu,\Theta(r,\mu)),$
$\displaystyle\quad\,\,\Theta_{x}^{0}(r,\mu,r,\mu,\Theta(r,\mu)),\Theta_{y}^{0}(r,\mu,r,\mu,\Theta(r,\mu)),\Theta^{0}_{x}(t,r,\tilde{x},\mu,y)\big{)}\Big{]}d\mu
dr.$

By 7.1, $(\Theta(\cdot),\Theta^{0}(\cdot))$ is the unique classical solution
of equilibrium HJB equation (2.25) on $[T-\bar{\varepsilon},T]$.

In the following, let us show that (7.36) really holds for some
$\bar{\varepsilon}>0$.

Step 1. From (7.3)–(7.4), we have

$\displaystyle\delta\Theta(s,x)$
$\displaystyle=\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)\Big{[}\Theta^{1}_{x}(r,\mu)\delta\tilde{b}(r,\mu)+\delta\Theta_{x}(r,\mu)\tilde{b}(r,\mu;\theta^{2},\theta^{0,2})+\delta\tilde{g}(r,\mu)\Big{]}d\mu
dr,$ (7.37) $\displaystyle\delta\Theta^{0}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)\Big{[}\Theta_{x}^{0,1}(t,r,\tilde{x},\mu,y)\delta\tilde{b}(r,\mu)+\Theta_{x}^{0,1}(t,r,\tilde{x},\mu,y)\tilde{b}(r,\mu;\theta^{2},\theta^{0,2})$
$\displaystyle\quad+\delta\tilde{g}^{0}(t,r,\tilde{x},\mu,y)\Big{]}d\mu dr.$
(7.38)

By 7.2 and the estimate (7.8), from (7.37) we get

$|\delta\Theta(s,x)|\leqslant
K\int_{s}^{T}\int_{\mathbb{R}^{n}}{e^{-{\lambda|x-\mu|^{2}\over
4(r-s)}}\over(r-s)^{{n\over
2}}}\Big{[}|\delta\tilde{b}(r,\mu)|+|\delta\Theta_{x}(r,\mu)|+|\delta\tilde{g}(r,\mu)|\Big{]}d\mu
dr.$ (7.39)

For $\varphi(\cdot)=b(\cdot),g(\cdot)$, by 7.3 (or (7.17)) we have

$\displaystyle|\delta\tilde{\varphi}(s,x)|$ $\displaystyle\leqslant
K\Big{[}|\delta\theta(s,x)|+|\delta\theta_{x}(s,x)|+|\theta^{0,1}(s,x,s,x,\theta^{1}(s,x))-\theta^{0,2}(s,x,s,x,\theta^{2}(s,x))|$
$\displaystyle\quad+|\theta_{x}^{0,1}(s,x,s,x,\theta^{1}(s,x))-\theta_{x}^{0,2}(s,x,s,x,\theta^{2}(s,x))|$
$\displaystyle\quad+|\theta_{y}^{0,1}(s,x,s,x,\theta^{1}(s,x))-\theta_{y}^{0,2}(s,x,s,x,\theta^{2}(s,x))|\Big{]}$
$\displaystyle\leqslant
K\Big{[}|\delta\theta(s,x)|+|\delta\theta_{x}(s,x)|+|\delta\theta^{0}(s,x,s,x,\theta^{1}(s,x))|$
$\displaystyle\quad+|\delta\theta_{x}^{0}(s,x,s,x,\theta^{1}(s,x))|+|\delta\theta_{y}^{0}(s,x,s,x,\theta^{1}(s,x))|\Big{]}.$
(7.40)

Substituting the above into (7.39) yields that

$\displaystyle|\delta\Theta(s,x)|$ $\displaystyle\leqslant
K(T-s)\big{[}\|\delta\theta(\cdot)\|_{L^{\infty}}+\|\delta\theta_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{y}(\cdot)\|_{L^{\infty}}\big{]}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{e^{-{\lambda|x-\mu|^{2}\over
4(r-s)}}\over(r-s)^{{n\over 2}}}|\delta\Theta_{x}(r,\mu)|d\mu dr.$ (7.41)

Similar to (7.40), using 7.3 (or (7.17)) again, we have

$\displaystyle|\delta\tilde{g}^{0}(t,s,\tilde{x},x,y)|$
$\displaystyle\leqslant
K\Big{[}|\delta\theta(s,x)|+|\delta\theta_{x}(s,x)|+|\delta\theta^{0}(s,x,s,x,\theta^{1}(s,x))|+|\delta\theta_{x}^{0}(s,x,s,x,\theta^{1}(s,x))|$
$\displaystyle\quad+|\delta\theta_{y}^{0}(s,x,s,x,\theta^{1}(s,x))|+\delta\Theta^{0}(t,s,\tilde{x},x,y)\Big{]}.$
(7.42)

Substituting the above into (7.38), we get

$\displaystyle|\delta\Theta^{0}(t,s,\tilde{x},x,y)|$ $\displaystyle\leqslant
K(T-s)\big{[}\|\delta\theta(\cdot)\|_{L^{\infty}}+\|\delta\theta_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{y}(\cdot)\|_{L^{\infty}}\big{]}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{e^{-{\lambda|x-\mu|^{2}\over
4(r-s)}}\over(r-s)^{{n\over
2}}}|\delta\Theta^{0}_{x}(t,r,\tilde{x},\mu,y)|d\mu dr.$ (7.43)

From (7.11) and (7.13), we have

$\displaystyle\delta\Theta_{x}(s,x)$
$\displaystyle=\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi_{x}(s,x,r,\mu)\Big{[}\Theta^{1}_{x}(r,\mu)\delta\tilde{b}(r,\mu)+\delta\Theta_{x}(r,\mu)\tilde{b}(r,\mu;\theta^{2},\theta^{0,2})+\delta\tilde{g}(r,\mu)\Big{]}d\mu
dr,$ (7.44) $\displaystyle\delta\Theta_{x}^{0}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi_{x}(s,x,r,\mu)\Big{[}\Theta_{x}^{0,1}(t,r,\tilde{x},\mu,y)\delta\tilde{b}(r,\mu)+\Theta_{x}^{0,1}(t,r,\tilde{x},\mu,y)\tilde{b}(r,\mu;\theta^{2},\theta^{0,2})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\delta\tilde{g}^{0}(t,r,\tilde{x},\mu,y)\Big{]}d\mu
dr.$ (7.45)

By 7.2 and the estimates (7.8), (7.40) and (7.42), we get

$\displaystyle|\delta\Theta_{x}(s,x)|$ $\displaystyle\leqslant
K\sqrt{T-s}\big{[}\|\delta\theta(\cdot)\|_{L^{\infty}}+\|\delta\theta_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{y}(\cdot)\|_{L^{\infty}}\big{]}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{e^{-{\lambda|x-\mu|^{2}\over
4(r-s)}}\over(r-s)^{n+1\over 2}}|\delta\Theta_{x}(r,\mu)|d\mu dr,$ (7.46)
$\displaystyle|\delta\Theta^{0}_{x}(t,s,\tilde{x},x,y)|$
$\displaystyle\leqslant
K\sqrt{T-s}\big{[}\|\delta\theta(\cdot)\|_{L^{\infty}}+\|\delta\theta_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{y}(\cdot)\|_{L^{\infty}}\big{]}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{e^{-{\lambda|x-\mu|^{2}\over
4(r-s)}}\over(r-s)^{n+1\over
2}}|\delta\Theta^{0}_{x}(t,r,\tilde{x},\mu,y)|d\mu dr.$ (7.47)

Combining (7.41), (7.43), (7.46) and (7.47) together, and then by the
Grönwall’s inequality, we get

$\displaystyle\|\delta\Theta(\cdot)\|_{C^{0,1}([T-\varepsilon,T])}+\|\delta\Theta^{0}(\cdot)\|_{C^{0,0,0,1,0}([T-\varepsilon,T])}$
$\displaystyle\quad\leqslant
K\sqrt{\varepsilon}\big{[}\|\delta\theta(\cdot)\|_{C^{0,1}([T-\varepsilon,T])}+\|\delta\theta^{0}(\cdot)\|_{C^{0,0,0,1,1}([T-\varepsilon,T])}\big{]}.$
(7.48)

Step 2. From the proof of 7.3, by some direct computations, it is easily seen
that

$\displaystyle\delta\Theta^{0}_{y}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)\Big{[}\Theta^{0,1}_{xy}(t,r,\tilde{x},\mu,y)\delta\tilde{b}(r,\mu)$
$\displaystyle\quad+\delta\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y)\tilde{b}(r,\mu;\theta^{2},\theta^{0,2})+\delta\tilde{g}^{0}_{p^{0}}(t,r,\tilde{x},\mu,y)\Theta^{0,1}_{xy}(t,r,\tilde{x},\mu,y)$
$\displaystyle\quad+\tilde{g}^{0}_{p^{0}}(t,r,\tilde{x},\mu,y;\theta^{1},\theta^{0,1},\Theta^{0,1})\delta\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y)\Big{]}d\mu
dr,$ $\displaystyle\delta\Theta^{0}_{xy}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi_{x}(s,x,r,\mu)\Big{[}\Theta^{0,1}_{xy}(t,r,\tilde{x},\mu,y)\delta\tilde{b}(r,\mu)$
$\displaystyle\quad+\delta\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y)\tilde{b}(r,\mu;\theta^{2},\theta^{0,2})+\delta\tilde{g}^{0}_{p^{0}}(t,r,\tilde{x},\mu,y)\Theta^{0,1}_{xy}(t,r,\tilde{x},\mu,y)$
$\displaystyle\quad+\tilde{g}^{0}_{p^{0}}(t,r,\tilde{x},\mu,y;\theta^{1},\theta^{0,1},\Theta^{0,1})\delta\Theta_{xy}^{0}(t,r,\tilde{x},\mu,y)\Big{]}d\mu
dr,$

and

$\displaystyle\delta\Theta^{0}_{yy}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi(s,x,r,\mu)\Big{[}\Theta^{0,1}_{xyy}(t,r,\tilde{x},\mu,y)\delta\tilde{b}(r,\mu)$
$\displaystyle\quad+\delta\Theta^{0}_{xyy}(t,r,\tilde{x},\mu,y)\tilde{b}(r,\mu;\theta^{2},\theta^{0,2})+\delta\tilde{g}^{0}_{p^{0}}(t,r,\tilde{x},\mu,y)\Theta^{0,1}_{xyy}(t,r,\tilde{x},\mu,y)$
$\displaystyle\quad+\tilde{g}^{0}_{p^{0}}(t,r,\tilde{x},\mu,y;\theta^{1},\theta^{0,1},\Theta^{0,1})\delta\Theta_{xyy}^{0}(t,r,\tilde{x},\mu,y)$
$\displaystyle\quad+\langle\delta\tilde{g}^{0}_{p^{0}p^{0}}(t,r,\tilde{x},\mu,y)\Theta^{0,1}_{xy}(t,r,\tilde{x},\mu,y),\,\Theta^{0,1}_{xy}(t,r,\tilde{x},\mu,y)\rangle$
$\displaystyle\quad+\langle\tilde{g}^{0}_{p^{0}p^{0}}(t,r,\tilde{x},\mu,y;\theta^{2},\theta^{0,2},\Theta^{0,2})\delta\Theta^{0}_{xy}(t,r,\tilde{x},\mu,y),\,\Theta^{0,1}_{xy}(t,r,\tilde{x},\mu,y)\rangle$
$\displaystyle\quad+\langle\tilde{g}^{0}_{p^{0}p^{0}}(t,r,\tilde{x},\mu,y;\theta^{2},\theta^{0,2},\Theta^{0,2})\Theta^{0,2}_{xy}(t,r,\tilde{x},\mu,y),\,\delta\Theta^{0}_{xy}(t,r,\tilde{x},\mu,y)\rangle\Big{]}d\mu
dr,$ $\displaystyle\delta\Theta^{0}_{xyy}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi_{x}(s,x,r,\mu)\Big{[}\Theta^{0,1}_{xyy}(t,r,\tilde{x},\mu,y)\delta\tilde{b}(r,\mu)$
$\displaystyle\quad+\delta\Theta^{0}_{xyy}(t,r,\tilde{x},\mu,y)\tilde{b}(r,\mu;\theta^{2},\theta^{0,2})+\delta\tilde{g}^{0}_{p^{0}}(t,r,\tilde{x},\mu,y)\Theta^{0,1}_{xyy}(t,r,\tilde{x},\mu,y)$
$\displaystyle\quad+\tilde{g}^{0}_{p^{0}}(t,r,\tilde{x},\mu,y;\theta^{1},\theta^{0,1},\Theta^{0,1})\delta\Theta_{xyy}^{0}(t,r,\tilde{x},\mu,y)$
$\displaystyle\quad+\langle\delta\tilde{g}^{0}_{p^{0}p^{0}}(t,r,\tilde{x},\mu,y)\Theta^{0,1}_{xy}(t,r,\tilde{x},\mu,y),\,\Theta^{0,1}_{xy}(t,r,\tilde{x},\mu,y)\rangle$
$\displaystyle\quad+\langle\tilde{g}^{0}_{p^{0}p^{0}}(t,r,\tilde{x},\mu,y;\theta^{2},\theta^{0,2},\Theta^{0,2})\delta\Theta^{0}_{xy}(t,r,\tilde{x},\mu,y),\,\Theta^{0,1}_{xy}(t,r,\tilde{x},\mu,y)\rangle$
$\displaystyle\quad+\langle\tilde{g}^{0}_{p^{0}p^{0}}(t,r,\tilde{x},\mu,y;\theta^{2},\theta^{0,2},\Theta^{0,2})\Theta^{0,2}_{xy}(t,r,\tilde{x},\mu,y),\,\delta\Theta^{0}_{xy}(t,r,\tilde{x},\mu,y)\rangle\Big{]}d\mu
dr.$

Then by 7.3 and the estimate (7.8), we get

$\displaystyle|\delta\Theta^{0}_{y}(t,s,\tilde{x},x,y)|$
$\displaystyle\leqslant
K(T-s)\Big{[}\|\delta\theta(\cdot)\|_{L^{\infty}}+\|\delta\theta_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{y}(\cdot)\|_{L^{\infty}}\Big{]}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{e^{-{\lambda|x-\mu|^{2}\over
4(r-s)}}\over(r-s)^{{n\over
2}}}\Big{[}|\delta\Theta^{0}_{x}(t,r,\tilde{x},\mu,y)|+|\delta\Theta^{0}_{xy}(t,r,\tilde{x},\mu,y)|\Big{]}d\mu
dr,$ $\displaystyle|\delta\Theta^{0}_{xy}(t,s,\tilde{x},x,y)|$
$\displaystyle\leqslant
K\sqrt{T-s}\Big{[}\|\delta\theta(\cdot)\|_{L^{\infty}}+\|\delta\theta_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{y}(\cdot)\|_{L^{\infty}}\Big{]}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{e^{-{\lambda|x-\mu|^{2}\over
4(r-s)}}\over(r-s)^{{n+1\over
2}}}\Big{[}|\delta\Theta^{0}_{x}(t,r,\tilde{x},\mu,y)|+|\delta\Theta^{0}_{xy}(t,r,\tilde{x},\mu,y)|\Big{]}d\mu
dr,$

and

$\displaystyle|\delta\Theta^{0}_{yy}(t,s,\tilde{x},x,y)|$
$\displaystyle\leqslant
K(T-s)\Big{[}\|\delta\theta(\cdot)\|_{L^{\infty}}+\|\delta\theta_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{y}(\cdot)\|_{L^{\infty}}\Big{]}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{e^{-{\lambda|x-\mu|^{2}\over
4(r-s)}}\over(r-s)^{{n\over
2}}}\Big{[}|\delta\Theta^{0}_{x}(t,r,\tilde{x},\mu,y)|+|\delta\Theta^{0}_{xy}(t,r,\tilde{x},\mu,y)|+|\delta\Theta^{0}_{xyy}(t,r,\tilde{x},\mu,y)|\Big{]}d\mu
dr,$ $\displaystyle|\delta V_{xyy}(t,y,z,s,x)|$ $\displaystyle\leqslant
K\sqrt{T-s}\Big{[}\|\delta\theta(\cdot)\|_{L^{\infty}}+\|\delta\theta_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{y}(\cdot)\|_{L^{\infty}}\Big{]}$
$\displaystyle\quad+K\int_{s}^{T}\int_{\mathbb{R}^{n}}{e^{-{\lambda|x-\mu|^{2}\over
4(r-s)}}\over(r-s)^{{n+1\over
2}}}\Big{[}|\delta\Theta^{0}_{x}(t,r,\tilde{x},\mu,y)|+|\delta\Theta^{0}_{xy}(t,r,\tilde{x},\mu,y)|+|\delta\Theta^{0}_{xyy}(t,r,\tilde{x},\mu,y)|\Big{]}d\mu
dr.$

Combining the above with the estimate (7.48) together, by Grönwall’s
inequality again we get

$\displaystyle\|\delta\Theta(\cdot)\|_{C^{0,1}([T-\varepsilon,T])}+\|\delta\Theta^{0}(\cdot)\|_{C^{0,,0,1,2}([T-\varepsilon,T])}$
$\displaystyle\quad\leqslant\sqrt{\varepsilon}K\big{[}\|\delta\theta(\cdot)\|_{C^{0,1}([T-\varepsilon,T])}+\|\delta\theta^{0}(\cdot)\|_{C^{0,0,0,1,1}([T-\varepsilon,T])}\big{]}.$

For any $\tilde{x}_{1},\tilde{x}_{2}\in\mathbb{R}^{n}$, denote

$\delta\widehat{\Theta}^{0}(t,s,x,y)=\delta\Theta^{0}(t,s,\tilde{x}_{1},x,y)-\delta\Theta^{0}(t,s,\tilde{x}_{2},x,y).$

Then from (7.45), we have

$\displaystyle\delta\widehat{\Theta}_{x}^{0}(t,s,x,y)$
$\displaystyle=\int_{s}^{T}\int_{\mathbb{R}^{n}}\Xi_{x}(s,x,r,\mu)\Big{\\{}[\Theta_{x}^{0,1}(t,r,\tilde{x}_{1},\mu,y)-\Theta_{x}^{0,1}(t,r,\tilde{x}_{2},\mu,y)]\delta\tilde{b}(r,\mu)$
$\displaystyle\quad+\delta\widehat{\Theta}^{0}_{x}(t,r,\mu,y)\tilde{b}(r,\mu;\theta^{2},\theta^{0,2})+\delta\tilde{g}^{0}(t,r,\tilde{x}_{1},\mu,y)-\delta\tilde{g}^{0}(t,r,\tilde{x}_{2},\mu,y)\Big{\\}}d\mu
dr.$

Note that

$\displaystyle|\delta\tilde{g}^{0}(t,r,\tilde{x}_{1},\mu,y)-\delta\tilde{g}^{0}(t,r,\tilde{x}_{2},\mu,y)|$
$\displaystyle\quad\leqslant
K|\delta\widehat{\Theta}_{x}^{0}(t,s,x,y)|+K|\delta\Theta_{x}^{0}(t,s,\tilde{x}_{1},x,y)|\big{[}1+\|\Theta^{0,1}_{x}(t,s,\cdot,x,y)\|_{\alpha}+\|\Theta^{0,2}_{x}(t,s,\cdot,x,y)\|_{\alpha}\big{]}|\tilde{x}_{1}-\tilde{x}_{2}|^{\alpha}.$

Then by the fact that
$\|\Theta^{0,1}_{x}(t,s,\cdot,x,y)\|_{\alpha}+\|\Theta^{0,2}_{x}(t,s,\cdot,x,y)\|_{\alpha}$
is uniformly bounded (see 7.3), we get

$\displaystyle\delta\widehat{\Theta}_{x}^{0}(t,s,x,y)$ $\displaystyle\leqslant
K\int_{s}^{T}\int_{\mathbb{R}^{n}}{e^{-{\lambda|x-\mu|^{2}\over
4(r-s)}}\over(r-s)^{{n+1\over
2}}}\Big{\\{}|\delta\widehat{\Theta}_{x}^{0}(t,r,\mu,y)|+|\delta\Theta_{x}^{0}(t,r,\tilde{x}_{1},\mu,y)||\tilde{x}_{1}-\tilde{x}_{2}|^{\alpha}$
$\displaystyle\quad+\big{[}\|\delta\theta(\cdot)\|_{L^{\infty}}+\|\delta\theta_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{x}(\cdot)\|_{L^{\infty}}+\|\delta\theta^{0}_{y}(\cdot)\|_{L^{\infty}}\big{]}|\tilde{x}_{1}-\tilde{x}_{2}|^{\alpha}\Big{\\}}d\mu
dr.$

It follows that

$\displaystyle\|\delta\Theta^{0}(\cdot)\|_{C^{0,0,\alpha,1,0}([T-\varepsilon,T])}\leqslant\sqrt{\varepsilon}K\big{[}\|\delta\theta(\cdot)\|_{C^{0,1}([T-\varepsilon,T])}+\|\delta\theta^{0}(\cdot)\|_{C^{0,0,0,1,1}([T-\varepsilon,T])}\big{]}.$

By continuing the above arguments, we have

$\displaystyle\|\delta\Theta(\cdot)\|_{C^{0,1}([T-\varepsilon,T])}+\|\delta\Theta^{0}(\cdot)\|_{C^{{\alpha\over
2},0,\alpha,1,2}([T-\varepsilon,T])}$
$\displaystyle\quad\leqslant\sqrt{\varepsilon}K\big{[}\|\delta\theta(\cdot)\|_{C^{0,1}([T-\varepsilon,T])}+\|\delta\theta^{0}(\cdot)\|_{C^{0,0,0,1,1}([T-\varepsilon,T])}\big{]}.$

Step 3. Recalling (7.18)–(7.19), similar to (7.44)–(7.45), we have

$\displaystyle\delta\Theta_{x}(s,x)$
$\displaystyle=\int_{s}^{T}\int_{\mathbb{R}^{n}}\tilde{\Xi}_{x}(s,x,r,x-\sqrt{r-s}\mu)\Big{[}\delta\tilde{g}(r,x-\sqrt{r-s}\mu)$
$\displaystyle\quad+\Theta^{1}_{x}(r,x-\sqrt{r-s}\mu)\delta\tilde{b}(r,x-\sqrt{r-s}\mu)$
$\displaystyle\quad+\delta\Theta_{x}(r,x-\sqrt{r-s}\mu)\tilde{b}(r,x-\sqrt{r-s}\mu;\theta^{2},\theta^{0,2})\Big{]}d\mu
dr,$ (7.49) $\displaystyle\delta\Theta^{0}_{x}(t,s,\tilde{x},x,y)$
$\displaystyle=\int_{s}^{T}\int_{\mathbb{R}^{n}}\tilde{\Xi}_{x}(s,x,r,x-\sqrt{r-s}\mu)\Big{[}\delta\tilde{g}^{0}(t,r,\tilde{x},x-\sqrt{r-s}\mu,y)$
$\displaystyle\quad+\Theta^{0,1}_{x}(t,r,\tilde{x},x-\sqrt{r-s}\mu,y)\delta\tilde{b}(r,x-\sqrt{r-s}\mu)$
$\displaystyle\quad+\delta\Theta^{0}_{x}(t,r,\tilde{x},x-\sqrt{r-s}\mu,y)\tilde{b}(r,x-\sqrt{r-s}\mu;\theta^{2},\theta^{0,2})\Big{]}d\mu
dr,$ (7.50)

where $\tilde{\Xi}(\cdot)$ is defined by (7.20). For any
$x_{1},x_{2}\in\mathbb{R}^{n}$, we have

$\displaystyle\delta\Theta^{0}_{x}(t,s,\tilde{x},x_{1},y)-\delta\Theta^{0}_{x}(t,s,\tilde{x},x_{2},y)$
$\displaystyle\quad=\int_{s}^{T}\int_{\mathbb{R}^{n}}\big{[}\tilde{\Xi}_{x}(s,x_{1},r,x_{1}-\sqrt{r-s}\mu)-\tilde{\Xi}_{x}(s,x_{2},r,x_{2}-\sqrt{r-s}\mu)\big{]}$
$\displaystyle\qquad\times\Big{[}\delta\tilde{g}^{0}(t,r,\tilde{x},x_{1}-\sqrt{r-s}\mu,y)+\Theta^{0,1}_{x}(t,r,\tilde{x},x_{1}-\sqrt{r-s}\mu,y)\delta\tilde{b}(r,x_{1}-\sqrt{r-s}\mu)$
$\displaystyle\qquad+\delta\Theta^{0}_{x}(t,r,\tilde{x},x_{1}-\sqrt{r-s}\mu,y)\tilde{b}(r,x_{1}-\sqrt{r-s}\mu;\theta^{2},\theta^{0,2})\Big{]}d\mu
dr$
$\displaystyle\qquad+\int_{s}^{T}\int_{\mathbb{R}^{n}}\tilde{\Xi}_{x}(s,x_{2},r,x_{2}-\sqrt{r-s}\mu)\Big{\\{}\big{[}\delta\tilde{g}^{0}(t,r,\tilde{x},x_{1}-\sqrt{r-s}\mu,y)$
$\displaystyle\qquad-\delta\tilde{g}^{0}(t,r,\tilde{x},x_{2}-\sqrt{r-s}\mu,y)\big{]}+\big{[}\Theta^{0,1}_{x}(t,r,\tilde{x},x_{1}-\sqrt{r-s}\mu,y)\delta\tilde{b}(r,x_{1}-\sqrt{r-s}\mu)$
$\displaystyle\qquad-\Theta^{0,1}_{x}(t,r,\tilde{x},x_{2}-\sqrt{r-s}\mu,y)\delta\tilde{b}(r,x_{2}-\sqrt{r-s}\mu)\big{]}$
$\displaystyle\qquad+\big{[}\delta\Theta^{0}_{x}(t,r,\tilde{x},x_{1}-\sqrt{r-s}\mu,y)\tilde{b}(r,x_{1}-\sqrt{r-s}\mu;\theta^{2},\theta^{0,2})$
$\displaystyle\qquad-\delta\Theta^{0}_{x}(t,r,\tilde{x},x_{2}-\sqrt{r-s}\mu,y)\tilde{b}(r,x_{2}-\sqrt{r-s}\mu;\theta^{2},\theta^{0,2})\big{]}\Big{\\}}d\mu
dr.$ (7.51)

Note that on $[T-\varepsilon,T]$, by 7.3 (or (7.17)) and 7.5 we have

$\displaystyle\big{|}[\theta^{0,1}_{x}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1}))-\theta^{0,2}_{x}(r,x_{1},r,x_{1},\theta^{2}(r,x_{1}))]$
$\displaystyle-[\theta^{0,1}_{x}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2}))-\theta^{0,2}_{x}(r,x_{2},r,x_{2},\theta^{2}(r,x_{2}))]\big{|}$
$\displaystyle\quad\leqslant\big{|}[\theta^{0,1}_{x}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1}))-\theta^{0,2}_{x}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1}))]$
$\displaystyle\qquad-[\theta^{0,1}_{x}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2}))-\theta^{0,2}_{x}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2}))]\big{|}$
$\displaystyle\qquad+\big{|}[\theta^{0,2}_{x}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1}))-\theta^{0,2}_{x}(r,x_{1},r,x_{1},\theta^{2}(r,x_{1}))]$
$\displaystyle\qquad-[\theta^{0,2}_{x}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2}))-\theta^{0,2}_{x}(r,x_{2},r,x_{2},\theta^{2}(r,x_{2}))]\big{|}$
$\displaystyle\quad\leqslant\|\delta\theta^{0}_{xy}(\cdot)\|_{L^{\infty}}\|\theta^{1}(\cdot)\|_{C^{0,\alpha}}|x_{1}-x_{2}|^{\alpha}+\|\delta\theta^{0}_{x}(\cdot)\|_{C^{0,0,\alpha,\alpha,0}}|x_{1}-x_{2}|^{\alpha}$
$\displaystyle\qquad+\Big{|}\int_{0}^{1}\theta^{0,2}_{xy}\big{(}r,x_{1},r,x_{1},l\theta^{1}(r,x_{1})+(1-l)\theta^{2}(r,x_{1})\big{)}dl[\theta^{1}(r,x_{1})-\theta^{2}(r,x_{1})]$
$\displaystyle\qquad-\int_{0}^{1}\theta^{0,2}_{xy}\big{(}r,x_{2},r,x_{2},l\theta^{1}(r,x_{2})+(1-l)\theta^{2}(r,x_{2})\big{)}dl[\theta^{1}(r,x_{2})-\theta^{2}(r,x_{2})]\Big{|}$
$\displaystyle\quad\leqslant\Big{\\{}\|\delta\theta^{0}_{xy}(\cdot)\|_{L^{\infty}}\|\theta^{1}(\cdot)\|_{C^{0,\alpha}}+\|\delta\theta^{0}_{x}(\cdot)\|_{C^{0,0,\alpha,\alpha,0}}+\|\theta^{0,2}_{xy}(\cdot)\|_{L^{\infty}}\|\delta\theta(\cdot)\|_{C^{0,\alpha}}$
$\displaystyle\qquad+\big{[}\|\theta^{0,2}_{xyy}(\cdot)\|_{L^{\infty}}(\|\theta^{1}(\cdot)\|_{C^{0,\alpha}}+\|\theta^{2}(\cdot)\|_{C^{0,\alpha}})+\|\theta^{0,2}_{xy}(\cdot)\|_{C^{0,0,\alpha,\alpha,0}}\big{]}\|\delta\theta(\cdot)\|_{L^{\infty}}\Big{\\}}|x_{1}-x_{2}|^{\alpha}$
$\displaystyle\quad\leqslant
K\big{[}\|\delta\theta^{0}_{x}(\cdot)\|_{C^{0,0,\alpha,\alpha,1}}+\|\delta\theta(\cdot)\|_{C^{0,\alpha}}\big{]}|x_{1}-x_{2}|^{\alpha}.$
(7.52)

By the same arguments as the above, we have

$\displaystyle\big{|}[\theta^{0,1}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1}))-\theta^{0,2}(r,x_{1},r,x_{1},\theta^{2}(r,x_{1}))]$
$\displaystyle-[\theta^{0,1}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2}))-\theta^{0,2}(r,x_{2},r,x_{2},\theta^{2}(r,x_{2}))]\big{|}$
$\displaystyle\quad\leqslant
K\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,\alpha,1}}+\|\delta\theta(\cdot)\|_{C^{0,\alpha}}\big{]}|x_{1}-x_{2}|^{\alpha},$
(7.53)

and

$\displaystyle\big{|}[\theta_{y}^{0,1}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1}))-\theta_{y}^{0,2}(r,x_{1},r,x_{1},\theta^{2}(r,x_{1}))]$
$\displaystyle-[\theta_{y}^{0,1}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2}))-\theta_{y}^{0,2}(r,x_{2},r,x_{2},\theta^{2}(r,x_{2}))]\big{|}$
$\displaystyle\quad\leqslant
K\big{[}\|\delta\theta_{y}^{0}(\cdot)\|_{C^{0,0,\alpha,\alpha,1}}+\|\delta\theta(\cdot)\|_{C^{0,\alpha}}\big{]}|x_{1}-x_{2}|^{\alpha}.$
(7.54)

For any $\varphi(\cdot)\in C^{1,2}$, from (7.52) and the fact

$\displaystyle\delta\varphi(x_{1})-\delta\varphi(x_{2})$
$\displaystyle:=\big{[}\varphi(x_{1},\theta^{0,1}_{x}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1})))-\varphi(x_{1},\theta^{0,2}_{x}(r,x_{1},r,x_{1},\theta^{2}(r,x_{1})))\big{]}$
$\displaystyle\quad-\big{[}\varphi(x_{2},\theta^{0,1}_{x}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2})))-\varphi(x_{2},\theta^{0,2}_{x}(r,x_{2},r,x_{2},\theta^{2}(r,x_{2})))\big{]}$
$\displaystyle=\int_{0}^{1}\varphi_{\theta^{0}}\big{(}x_{1},\ell\theta^{0,1}_{x}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1}))+(1-\ell)\theta^{0,2}_{x}(r,x_{1},r,x_{1},\theta^{2}(r,x_{1}))\big{)}d\ell$
$\displaystyle\quad\times\Big{\\{}[\theta^{0,1}_{x}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1}))-\theta^{0,2}_{x}(r,x_{1},r,x_{1},\theta^{2}(r,x_{1}))]$
$\displaystyle\qquad-[\theta^{0,1}_{x}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2}))-\theta^{0,2}_{x}(r,x_{2},r,x_{2},\theta^{2}(r,x_{2}))]\Big{\\}}$
$\displaystyle\quad+\int_{0}^{1}\Big{[}\varphi_{\theta^{0}}\big{(}x_{1},\ell\theta^{0,1}_{x}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1}))+(1-\ell)\theta^{0,2}_{x}(r,x_{1},r,x_{1},\theta^{2}(r,x_{1}))\big{)}$
$\displaystyle\quad-\varphi_{\theta^{0}}\big{(}x_{2},\ell\theta^{0,1}_{x}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2}))+(1-\ell)\theta^{0,2}_{x}(r,x_{2},r,x_{2},\theta^{2}(r,x_{2}))\big{)}\Big{]}d\ell$
$\displaystyle\quad\times\big{[}\theta^{0,1}_{x}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2}))-\theta^{0,2}_{x}(r,x_{2},r,x_{2},\theta^{2}(r,x_{2}))\big{]},$

we have the following estimate:

$\displaystyle|\delta\varphi(x_{1})-\delta\varphi(x_{2})|\leqslant
K\big{[}\|\delta\theta^{0}_{x}(\cdot)\|_{C^{0,0,\alpha,\alpha,1}}+\|\delta\theta(\cdot)\|_{C^{0,\alpha}}\big{]}|x_{1}-x_{2}|^{\alpha}\quad\hbox{on}\quad[T-\varepsilon,T],$
(7.55)

where $K$ depends on $\|\varphi(\cdot)\|_{C^{0,1}}$. Recall that

$\displaystyle\delta\tilde{g}^{0}(t,r,\tilde{x},x_{1},y)-\delta\tilde{g}^{0}(t,r,\tilde{x},x_{2},y)$
$\displaystyle\quad=\Big{[}\tilde{g}^{0}\big{(}t,r,\tilde{x},x_{1},\theta^{1}(r,x_{1}),\theta^{1}_{x}(r,x_{1}),\theta^{0,1}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1})),\theta_{x}^{0,1}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1})),$
$\displaystyle\qquad\,\theta_{y}^{0,1}(r,x_{1},r,x_{1},\theta^{1}(r,x_{1})),\Theta^{0,1}_{x}(t,r,\tilde{x},x_{1},y)\big{)}-\tilde{g}^{0}\big{(}t,r,\tilde{x},x_{1},\theta^{2}(r,x_{1}),\theta^{2}_{x}(r,x_{1}),$
$\displaystyle\qquad\,\theta^{0,2}(r,x_{1},r,x_{1},\theta^{2}(r,x_{1})),\theta_{x}^{0,2}(r,x_{1},r,x_{1},\theta^{2}(r,x_{1})),\theta_{y}^{0,2}(r,x_{1},r,x_{1},\theta^{2}(r,x_{1})),\Theta^{0,2}_{x}(t,r,\tilde{x},x_{1},y)\big{)}\Big{]}$
$\displaystyle\qquad\,-\Big{[}\tilde{g}^{0}\big{(}t,r,\tilde{x},x_{2},\theta^{1}(r,x_{2}),\theta^{1}_{x}(r,x_{2}),\theta^{0,1}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2})),\theta_{x}^{0,1}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2})),$
$\displaystyle\qquad\qquad\theta_{y}^{0,1}(r,x_{2},r,x_{2},\theta^{1}(r,x_{2})),\Theta^{0,1}_{x}(t,r,\tilde{x},x_{2},y)\big{)}-\tilde{g}^{0}\big{(}t,r,\tilde{x},x_{2},\theta^{2}(r,x_{2}),\theta^{2}_{x}(r,x_{2}),$
$\displaystyle\qquad\qquad\theta^{0,2}(r,x_{2},r,x_{2},\theta^{2}(r,x_{2})),\theta_{x}^{0,2}(r,x_{2},r,x_{2},\theta^{2}(r,x_{2})),\theta_{y}^{0,2}(r,x_{2},r,x_{2},\theta^{2}(r,x_{2})),\Theta^{0,2}_{x}(t,r,\tilde{x},x_{2},y)\big{)}\Big{]}.$

Using (7.55), by the estimates (7.52)–(7.54), we get

$\displaystyle|\delta\tilde{g}^{0}(t,r,\tilde{x},x_{1},y)-\delta\tilde{g}^{0}(t,r,\tilde{x},x_{2},y)|$
$\displaystyle\quad\leqslant
K\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}+\|\delta\Theta^{0}_{x}(t,r,\tilde{x},\cdot,y)\|_{\alpha}\big{]}|x_{1}-x_{2}|^{\alpha}.$
(7.56)

Similarly,

$\displaystyle|\delta\tilde{b}(r,x_{1})-\delta\tilde{b}(r,x_{2})|\leqslant
K\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}\big{]}|x_{1}-x_{2}|^{\alpha}.$
(7.57)

By (7.21), 7.2 and 7.5, from (7.51) we obtain

$\displaystyle\|\delta\Theta^{0}_{x}(t,s,\tilde{x},\cdot,y)\|_{\alpha}$
$\displaystyle\leqslant\int_{s}^{T}\int_{\mathbb{R}^{n}}{K\over\sqrt{r-s}}e^{-\lambda|\mu|^{2}}\Big{[}|\delta\tilde{b}(r,x-\sqrt{r-s}\mu)|$
$\displaystyle\quad+|\delta\tilde{g}^{0}(t,r,\tilde{x},x-\sqrt{r-s}\mu,y)|+|\delta\Theta^{0}_{x}(t,r,\tilde{x},x-\sqrt{r-s}\mu,y)|\Big{]}d\mu
dr$
$\displaystyle\quad+\int_{s}^{T}\int_{\mathbb{R}^{n}}{K\over\sqrt{r-s}}e^{-\lambda|\mu|^{2}}\Big{[}|\delta\tilde{b}(r,x_{1}-\sqrt{r-s}\mu)|+\|\delta\tilde{b}(r,\cdot)\|_{\alpha}$
$\displaystyle\quad+|\delta\Theta^{0}_{x}(t,r,\tilde{x},x_{1}-\sqrt{r-s}\mu,y)|+\|\delta\Theta^{0}_{x}(t,r,\tilde{x},\cdot,y)\|_{\alpha}+\|\delta\tilde{g}^{0}(t,r,\tilde{x},\cdot,y)\|_{\alpha}\Big{]}d\mu
dr.$

Substituting the estimates (7.40), (7.42), (7.56) and (7.57) into the above,
and then by Grönwall’s inequality, we get

$\displaystyle\|\delta\Theta^{0}_{x}(t,s,\tilde{x},\cdot,y)\|_{\alpha}\leqslant
K\sqrt{\varepsilon}\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}\big{]}.$
(7.58)

By the same arguments as the above, we have

$\|\delta\Theta_{x}(s,\cdot)\|_{\alpha}\leqslant
K\sqrt{\varepsilon}\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}\big{]}.$

By continuing the above arguments, we get

$\|\delta\Theta(\cdot)\|_{C^{0,1+\alpha}}+\|\delta\Theta^{0}(\cdot)\|_{C^{0,0,0,1+\alpha,2}}\leqslant
K\sqrt{\varepsilon}\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}\big{]}.$

Step 4. For any $T-\varepsilon\leqslant s_{1}\leqslant s_{2}\leqslant T$, by
(7.50) we have

$\displaystyle\delta\Theta^{0}_{x}(t,s_{1},\tilde{x},x,y)-\delta\Theta^{0}_{x}(t,s_{2},\tilde{x},x,y)$
$\displaystyle\quad=\int_{s_{1}}^{s_{2}}\int_{\mathbb{R}^{n}}\tilde{\Xi}_{x}(s_{1},x,r,x-\sqrt{r-s_{1}}\mu)\Big{[}\delta\tilde{g}^{0}(t,r,\tilde{x},x-\sqrt{r-s_{1}}\mu,y)+\Theta^{0,1}_{x}(t,r,\tilde{x},x-\sqrt{r-s_{1}}\mu,y)$
$\displaystyle\qquad\quad\times\delta\tilde{b}(r,x-\sqrt{r-s_{1}}\mu)+\delta\Theta^{0}_{x}(t,r,\tilde{x},x-\sqrt{r-s_{1}}\mu,y)\tilde{b}(r,x-\sqrt{r-s_{1}}\mu;\theta^{2},\theta^{0,2})\Big{]}d\mu
dr$
$\displaystyle\qquad+\int_{s_{2}}^{T}\int_{\mathbb{R}^{n}}\big{[}\tilde{\Xi}_{x}(s_{1},x,r,x-\sqrt{r-s_{1}}\mu)-\tilde{\Xi}_{x}(s_{2},x,r,x-\sqrt{r-s_{2}}\mu)\big{]}$
$\displaystyle\qquad\quad\times\Big{[}\delta\tilde{g}^{0}(t,r,\tilde{x},x-\sqrt{r-s_{1}}\mu,y)+\Theta^{0,1}_{x}(t,r,\tilde{x},x-\sqrt{r-s_{1}}\mu,y)\delta\tilde{b}(r,x-\sqrt{r-s_{1}}\mu)$
$\displaystyle\qquad\quad+\delta\Theta^{0}_{x}(t,r,\tilde{x},x-\sqrt{r-s_{1}}\mu,y)\tilde{b}(r,x-\sqrt{r-s_{1}}\mu;\theta^{2},\theta^{0,2})\Big{]}d\mu
dr$
$\displaystyle\qquad+\int_{s_{2}}^{T}\int_{\mathbb{R}^{n}}\tilde{\Xi}_{x}(s_{2},x,r,x-\sqrt{r-s_{2}}\mu)\Big{\\{}\big{[}\delta\tilde{g}^{0}(t,r,\tilde{x},x-\sqrt{r-s_{1}}\mu,y)-\delta\tilde{g}^{0}(t,r,\tilde{x},x-\sqrt{r-s_{2}}\mu,y)\big{]}$
$\displaystyle\qquad\quad+\big{[}\Theta^{0,1}_{x}(t,r,\tilde{x},x-\sqrt{r-s_{1}}\mu,y)\delta\tilde{b}(r,x-\sqrt{r-s_{1}}\mu)-\Theta^{0,1}_{x}(t,r,\tilde{x},x-\sqrt{r-s_{2}}\mu,y)$
$\displaystyle\qquad\quad\times\delta\tilde{b}(r,x-\sqrt{r-s_{2}}\mu)\big{]}+\big{[}\delta\Theta^{0}_{x}(t,r,\tilde{x},x-\sqrt{r-s_{1}}\mu,y)\tilde{b}(r,x-\sqrt{r-s_{1}}\mu;\theta^{2},\theta^{0,2})$
$\displaystyle\qquad\quad-\delta\Theta^{0}_{x}(t,r,\tilde{x},x-\sqrt{r-s_{2}}\mu,y)\tilde{b}(r,x-\sqrt{r-s_{2}}\mu;\theta^{2},\theta^{0,2})\big{]}\Big{\\}}d\mu
dr$ $\displaystyle\quad=:(I)+(II)+(III).$ (7.59)

By 7.2 and (7.21), we get

$|(I)|\leqslant\sqrt{s_{2}-s_{1}}K\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,0,1,1}}+\|\delta\theta(\cdot)\|_{C^{0,1}}+\|\delta\Theta^{0}(\cdot)\|_{C^{0,0,0,0,1}}\big{]}.$
(7.60)

By 7.2 and (7.20)–(7.21), we get

$\displaystyle|(II)|$ $\displaystyle\leqslant
K\Big{[}\Big{|}\int_{s_{2}}^{T}\int_{\mathbb{R}^{n}}\big{[}\tilde{\Xi}_{x}(s_{1},x,r,x-\sqrt{r-s_{1}}\mu)-\tilde{\Xi}_{x}(s_{2},x,r,x-\sqrt{r-s_{1}}\mu)\big{]}d\mu
dr\Big{|}+|s_{2}-s_{1}|^{\alpha\over 2}\sqrt{T-s_{1}}\Big{]}$
$\displaystyle\quad\times\Big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,0,1,1}}+\|\delta\theta(\cdot)\|_{C^{0,1}}+\|\delta\Theta^{0}(\cdot)\|_{C^{0,0,0,1,0}}\Big{]}$
$\displaystyle\leqslant
K\Big{[}\int_{s_{2}}^{T}\int_{\mathbb{R}^{n}}e^{-\lambda|\mu|^{2}\over
4}\Big{|}{1\over\sqrt{r-s_{1}}}-{1\over\sqrt{r-s_{2}}}\Big{|}d\mu
dr+|s_{2}-s_{1}|^{\alpha\over 2}\sqrt{T-s_{1}}\Big{]}$
$\displaystyle\quad\times\Big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,0,1,1}}+\|\delta\theta(\cdot)\|_{C^{0,1}}+\|\delta\Theta^{0}(\cdot)\|_{C^{0,0,0,1,0}}\Big{]}$
$\displaystyle\leqslant K\big{[}|s_{1}-s_{2}|^{1\over
2}+|s_{2}-s_{1}|^{\alpha\over
2}\sqrt{\varepsilon}\big{]}\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,0,1,1}}+\|\delta\theta(\cdot)\|_{C^{0,1}}+\|\delta\Theta^{0}(\cdot)\|_{C^{0,0,0,1,0}}\big{]}.$
(7.61)

By the estimates (7.21), (7.56) and (7.57), we get

$\displaystyle|(III)|\leqslant
K\sqrt{\varepsilon}\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}+\|\delta\Theta^{0}(\cdot)\|_{C^{0,0,0,1+\alpha,0}}\big{]}|s_{1}-s_{2}|^{\alpha\over
2}.$ (7.62)

Combining the estimates (7.60)–(7.62) together, we have

$\displaystyle\|\delta\Theta^{0}_{x}(t,\cdot,\tilde{x},x,y)\|_{\alpha\over
2}\leqslant K\varepsilon^{1-\alpha\over
2}\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}+\|\delta\Theta^{0}(\cdot)\|_{C^{0,0,0,1+\alpha,0}}\big{]}.$

Then it follows from (7.58) that

$\displaystyle\|\delta\Theta^{0}_{x}(t,\cdot,\tilde{x},x,y)\|_{\alpha\over
2}\leqslant K\varepsilon^{1-\alpha\over
2}\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}\big{]}.$

With (7.49), by the same arguments as the above, we have

$\displaystyle\|\delta\Theta_{x}(\cdot,x)\|_{\alpha\over 2}\leqslant
K\varepsilon^{1-\alpha\over
2}\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}\big{]}.$

By continuing the above arguments, we get

$\displaystyle\|\delta\Theta(\cdot)\|_{C^{{\alpha\over
2},1}}+\|\delta\Theta^{0}(\cdot)\|_{C^{0,{\alpha\over 2},0,1,2}}\leqslant
K\varepsilon^{1-\alpha\over
2}\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}\big{]}.$

Step 5. Combining the estimates in Steps 1–4 together, we get

$\displaystyle\|\delta\Theta(\cdot)\|_{C^{{\alpha\over
2},1}}+\|\delta\Theta^{0}(\cdot)\|_{C^{{\alpha\over 2},{\alpha\over
2},\alpha,1+\alpha,2}}\leqslant K\varepsilon^{1-\alpha\over
2}\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}\big{]}.$

Then by choosing an $0<\bar{\varepsilon}\leqslant\varepsilon$ small enough, we
get that on $[T-\bar{\varepsilon},T]$,

$\displaystyle\|\delta\Theta(\cdot)\|_{C^{{\alpha\over
2},1}}+\|\delta\Theta^{0}(\cdot)\|_{C^{{\alpha\over 2},{\alpha\over
2},\alpha,1+\alpha,2}}\leqslant{1\over
2}\big{[}\|\delta\theta^{0}(\cdot)\|_{C^{0,0,\alpha,1+\alpha,2}}+\|\delta\theta(\cdot)\|_{C^{0,1+\alpha}}\big{]}.$

Thus, (7.36) holds and this completes the proof. ∎

Complete the proof of Theorem 2.10. By a routine argument, we can prove that
equilibrium HJB equation (2.25) admits a unique classical solution
$(\Theta(\cdot),\Theta^{0}(\cdot))$ on $[T-\varepsilon,T]$, where
$\varepsilon$ is given by 7.5. Thus, to extend the solution to the whole time
interval $[0,T]$, it suffices to prove a global prior estimate for
$\|\Theta(\cdot)\|_{C^{0,1+\alpha}}$ and
$\|\Theta^{0}(\cdot)\|_{C^{0,0,0,1+\alpha,0}}$.

By (7.26) and (7.27), we get

$\displaystyle\|\Theta_{x}(t,s,\tilde{x},\cdot,y)\|_{\alpha}+\|\Theta_{x}(s,\cdot)\|_{\alpha}$
$\displaystyle\leqslant
K\big{[}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,0}}\big{]}$
$\displaystyle\quad+\int_{s}^{T}{2K\over\sqrt{r-s}}\big{[}\|\Theta^{0}_{x}(t,r,\tilde{x},\cdot,y)\|_{\alpha}+\|\Theta_{x}(r,\cdot)\|_{\alpha}\big{]}dr.$

Let ${\cal M}(\cdot)$ be the unique solution of the following integral
equation

$\displaystyle{\cal
M}(s)=K\big{[}1+\|h(\cdot)\|_{C^{1+\alpha}}+\|h^{0}(\cdot)\|_{C^{0,0,1+\alpha,0}}\big{]}+\int_{s}^{T}{2K\over\sqrt{r-s}}{\cal
M}(r)dr,$

and define

$\widehat{\cal M}\triangleq\sup_{s\in[0,T]}{\cal M}(s).$

Then by the comparison theorem for Volterra integral equations (see [68], for
example), we get

$\|\Theta_{x}(s,\cdot)\|_{C^{\alpha}}+\|\Theta^{0}_{x}(\cdot,s,\cdot,\cdot,\cdot)\|_{C^{0,0,0,\alpha,0}}\leqslant{\cal
M}(s)\leqslant\widehat{\cal M}\quad\hbox{on}\quad[0,T].$

This completes the proof.

###  7.2 Proof of 6.3

We first establish a priori estimate for the second-order derivative terms
$\Theta^{\varepsilon}_{xx}(\cdot)$. Recalling from (6.5), similar to (7.18),
we have

$\displaystyle\Theta^{\varepsilon}_{x}(s,x)$
$\displaystyle=-\int_{\mathbb{R}^{n}}\tilde{\Xi}(s,x,t+\varepsilon,x-\sqrt{t+\varepsilon-s}\mu;u)\Big{[}\Theta_{x}(t+\varepsilon,x-\sqrt{t+\varepsilon-s}\mu)$
$\displaystyle\quad-\tilde{\rho}(s,x,t+\varepsilon,x-\sqrt{t+\varepsilon-s}\mu;u)\Theta(t+\varepsilon,x-\sqrt{t+\varepsilon-s}\mu)\Big{]}d\mu$
$\displaystyle\quad+\int_{s}^{t+\varepsilon}\int_{\mathbb{R}^{n}}\tilde{\Xi}_{x}(s,x,r,x-\sqrt{r-s}\mu;u)\Big{[}\Theta^{\varepsilon}_{x}(r,x-\sqrt{r-s}\mu)b(r,x-\sqrt{r-s}\mu,u)$
$\displaystyle\quad+g\big{(}r,x-\sqrt{r-s}\mu,u,\Theta^{\varepsilon}(r,x-\sqrt{r-s}\mu),\Theta^{\varepsilon}_{x}(r,x-\sqrt{r-s}\mu)\sigma(r,x-\sqrt{r-s}\mu,u)\big{)}\Big{]}d\mu
dr,$

where

$\displaystyle\widetilde{\Xi}(s,x,r,x-\sqrt{r-s}\mu;u):={1\over(4\pi)^{n\over
2}(\hbox{det\,}[a(r,x-\sqrt{r-s}\mu,u)])^{1\over 2}}e^{-{\langle
a(r,x-\sqrt{r-s}\mu,u)^{-1}\mu,\mu\rangle\over 4}},$
$\displaystyle\tilde{\rho}(s,x,r,x-\sqrt{r-s}\mu;u)$
$\displaystyle\quad:={(\hbox{det\,}[a(r,x-\sqrt{r-s}\mu,u)])_{x}\over
2\hbox{det\,}[a(r,x-\sqrt{r-s}\mu,u)]}+{\langle[a(r,x-\sqrt{r-s}\mu,u)^{-1}]_{x}\mu,\mu\rangle\over
4},$ $\displaystyle\widetilde{\Xi}_{x}(s,x,r,x-\sqrt{r-s}\mu;u)$
$\displaystyle\quad:={-1\over(4\pi)^{n\over
2}(\hbox{det\,}[a(r,x-\sqrt{r-s}\mu,u)])^{1\over 2}}e^{-{\langle
a(r,x-\sqrt{r-s}\mu,u)^{-1}\mu,\mu\rangle\over
4}}{a(r,x-\sqrt{r-s}\mu,u)^{-1}\over 2\sqrt{r-s}}\mu.$

Note that the functions $\tilde{\Xi}(\cdot)$ and $\tilde{\rho}(\cdot)$ depend
on the variable $x$ only through the function $a(\cdot)$. Recall that
$a(\cdot)$ is a smooth function with bounded derivatives and it satisfies the
non-degenerate condition (6.13). Thus, it will not cause any difficulties in
the proof. For the ease of presentation, we assume that

$a(s,x,u)=a(s,u),\quad(s,x,u)\in[0,T]\times\mathbb{R}^{n}\times U.$

Then by some straightforward calculations, it is easy to obtain that

$\displaystyle\Theta^{\varepsilon}_{xx}(s,x)$
$\displaystyle=-\int_{\mathbb{R}^{n}}\tilde{\Xi}(s,x,t+\varepsilon,x-\sqrt{t+\varepsilon-s}\mu;u)\Big{[}\Theta_{xx}(t+\varepsilon,x-\sqrt{t+\varepsilon-s}\mu)$
$\displaystyle\qquad-\tilde{\rho}(s,x,t+\varepsilon,x-\sqrt{t+\varepsilon-s}\mu;u)\Theta_{x}(t+\varepsilon,x-\sqrt{t+\varepsilon-s}\mu)\Big{]}d\mu$
$\displaystyle\quad+\int_{s}^{t+\varepsilon}\int_{\mathbb{R}^{n}}\tilde{\Xi}_{x}(s,x,r,x-\sqrt{r-s}\mu;u)\Big{[}\Theta^{\varepsilon}_{x}(r,x-\sqrt{r-s}\mu)b_{x}(r,x-\sqrt{r-s}\mu,u)$
$\displaystyle\qquad+\Theta^{\varepsilon}_{xx}(r,x-\sqrt{r-s}\mu)b(r,x-\sqrt{r-s}\mu,u)+g_{\theta}(r,x-\sqrt{r-s}\mu,u)\Theta^{\varepsilon}_{x}(r,x-\sqrt{r-s}\mu)$
$\displaystyle\qquad+g_{p}(r,x-\sqrt{r-s}\mu,u)\Theta^{\varepsilon}_{xx}(r,x-\sqrt{r-s}\mu)+g_{x}(r,x-\sqrt{r-s}\mu,u)\Big{]}d\mu
dr,$ (7.63)

where $g(r,x-\sqrt{r-s}\mu,u)\equiv
g(r,x-\sqrt{r-s}\mu,u,\Theta^{\varepsilon}(r,x-\sqrt{r-s}\mu),\Theta^{\varepsilon}_{x}(r,x-\sqrt{r-s}\mu)\sigma(r,x-\sqrt{r-s}\mu,u))$.
It follows that

$\displaystyle|\Theta^{\varepsilon}_{xx}(s,x)|$ $\displaystyle\leqslant
K\int_{\mathbb{R}^{n}}e^{-\lambda|\mu|^{2}\over
4}\|\Theta(t+\varepsilon,\cdot)\|_{C^{2}}d\mu$
$\displaystyle\quad+K\int_{s}^{t+\varepsilon}\int_{\mathbb{R}^{n}}{e^{-\lambda|\mu|^{2}\over
4}\over\sqrt{r-\mu}}\Big{[}|\Theta^{\varepsilon}_{x}(r,x-\sqrt{r-s}\mu)|+|\Theta^{\varepsilon}_{xx}(r,x-\sqrt{r-s}\mu)|+1\Big{]}d\mu
dr.$

Then by Grönwall’s inequality, we get

$\|\Theta^{\varepsilon}_{xx}(\cdot,\cdot)\|_{L^{\infty}}\leqslant
K\big{[}1+\|\Theta(t+\varepsilon,\cdot)\|_{C^{2}}\big{]}.$ (7.64)

By 7.2, we also have

$\|\Theta^{\varepsilon}(\cdot,\cdot)\|_{C^{0,1}}\leqslant
K\big{[}1+\|\Theta(t+\varepsilon,\cdot)\|_{C^{1}}\big{]}.$ (7.65)

Thus, similar to (7.26), using (7.63) and the estimates (7.64)–(7.65), we have

$\displaystyle\|\Theta^{\varepsilon}_{xx}(s,\cdot)\|_{\alpha}$
$\displaystyle\leqslant K\int_{\mathbb{R}^{n}}e^{-\lambda|\mu|^{2}\over
4}\|\Theta(t+\varepsilon,\cdot)\|_{C^{2+\alpha}}d\mu$
$\displaystyle\quad+K\int_{s}^{t+\varepsilon}\int_{\mathbb{R}^{n}}{e^{-\lambda|\mu|^{2}\over
4}\over\sqrt{r-s}}\Big{[}\|\Theta^{\varepsilon}(r,\cdot)\|_{\alpha}+\|\Theta^{\varepsilon}_{x}(r,\cdot)\|_{\alpha}+\|\Theta^{\varepsilon}_{xx}(r,\cdot)\|_{\alpha}+1\Big{]}d\mu
dr.$

Then by Grönwall’s inequality again, we have

$\|\Theta^{\varepsilon}_{xx}(\cdot,\cdot)\|_{C^{\alpha}}\leqslant
K\big{[}1+\|\Theta(t+\varepsilon,\cdot)\|_{C^{2+\alpha}}\big{]}.$ (7.66)

For any $t\leqslant s_{1}\leqslant s_{2}\leqslant t+\varepsilon$, by (7.63)
and (7.66) we have

$\displaystyle|\Theta^{\varepsilon}_{xx}(s_{1},x)-\Theta^{\varepsilon}_{xx}(s_{2},x)|$
$\displaystyle\quad\leqslant K\int_{\mathbb{R}^{n}}e^{-\lambda|\mu|^{2}\over
4}\big{[}\|\Theta_{xx}(t+\varepsilon,\cdot)\|_{\alpha}+\|\Theta_{x}(t+\varepsilon,\cdot)\|_{\alpha}\big{]}|s_{1}-s_{2}|^{\alpha\over
2}|\mu|^{\alpha\over 2}d\mu$
$\displaystyle\qquad+K\int_{s_{1}}^{t+\varepsilon}\int_{\mathbb{R}^{n}}{e^{-\lambda|\mu|^{2}\over
4}\over\sqrt{r-s_{1}}}\big{[}\|\Theta^{\varepsilon}(r,\cdot)\|_{\alpha}+\|\Theta^{\varepsilon}_{x}(r,\cdot)\|_{\alpha}+\|\Theta^{\varepsilon}_{xx}(r,\cdot)\|_{\alpha}+1\big{]}|s_{1}-s_{2}|^{\alpha\over
2}|\mu|^{\alpha\over 2}d\mu dr$
$\displaystyle\qquad+K\sqrt{s_{2}-s_{1}}+K\int_{s_{1}}^{s_{2}}\int_{\mathbb{R}^{n}}{e^{-\lambda|\mu|^{2}\over
4}\over\sqrt{r-s_{1}}}d\mu dr$ $\displaystyle\quad\leqslant
K\big{[}1+\|\Theta(t+\varepsilon,\cdot)\|_{C^{2+\alpha}}\big{]}|s_{1}-s_{2}|^{\alpha\over
2}.$ (7.67)

Similar to (7.33), we have that for any $t\leqslant s_{1}\leqslant
s_{2}\leqslant t+\varepsilon$,

$\displaystyle|\Theta^{\varepsilon}(s_{1},x)-\Theta^{\varepsilon}(s_{2},x)|+|\Theta^{\varepsilon}_{x}(s_{1},x)-\Theta^{\varepsilon}_{x}(s_{2},x)|\leqslant
K\big{[}1+\|\Theta(t+\varepsilon,\cdot)\|_{C^{1+\alpha}}\big{]}|s_{1}-s_{2}|^{\alpha\over
2}.$ (7.68)

Moreover, similar to (7.34), we have that for any $t\leqslant r\leqslant
s_{1}\leqslant s_{2}\leqslant t+\varepsilon$,

$\displaystyle|\Theta^{0,\varepsilon}(r,s_{1},\tilde{x},x,y)-\Theta^{0,\varepsilon}(r,s_{2},\tilde{x},x,y)|+|\Theta_{x}^{0,\varepsilon}(r,s_{1},\tilde{x},x,y)-\Theta_{x}^{0,\varepsilon}(r,s_{2},\tilde{x},x,y)|$
$\displaystyle\quad\leqslant
K\big{[}1+\|\Theta(t+\varepsilon,\cdot)\|_{C^{1+\alpha}}+\|\Theta^{0}(r,t+\varepsilon,\cdot\,,\cdot\,,\cdot)\|_{C^{\alpha,1+\alpha,2}}\big{]}|s_{1}-s_{2}|^{\alpha\over
2}.$ (7.69)

Note that $\Theta^{\varepsilon}(t+\varepsilon,x)=\Theta(t+\varepsilon,x)$,
$\Theta^{0,\varepsilon}(r,t+\varepsilon,\tilde{x},x,y)=\Theta^{0}(r,t+\varepsilon,\tilde{x},x,y)$,
and that the functions $\Theta(\cdot)$ and $\Theta^{0}(\cdot)$ are smooth with
bounded derivatives. Then the desired results are obtained from (7.67)–(7.69).

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