# Backward doubly stochastic differential equations and SPDEs with quadratic
growth††thanks: Ying Hu was partially supported by Lebesgue Center of
Mathematics “Investissements d’avenir” program-ANR-11-LABX-0020-01, by
CAESARS-ANR-15-CE05-0024 and by MFG-ANR16-CE40-0015-01. Jiaqiang Wen was
supported by National Natural Science Foundation of China (Grant No.
12101291), Natural Science Foundation of Guangdong Province of China (Grant
No. 2214050003543), and SUSTech start-up fund (Grant No. Y01286233). Jie Xiong
was supported by National Natural Science Foundation of China (Grants No.
61873325 and 11831010), and SUSTech start-up fund (Grant No. Y01286120).

Ying Hu , Jiaqiang Wen , Jie Xiong Univ. Rennes, CNRS, IRMAR - UMR 6625,
F-35000 Rennes, France ((Email: ying.hu@univ-rennes1.fr).Department of
Mathematics, Southern University of Science and Technology, Shenzhen,
Guangdong, 518055, China ((Email: wenjq@sustech.edu.cn).Department of
Mathematics and SUSTech International center for Mathematics, Southern
University of Science and Technology, Shenzhen, Guangdong, 518055, China
(Email: xiongj@sustech.edu.cn).

Abstract. In this paper, we initiate the study of backward doubly stochastic
differential equations (BDSDEs, for short) with quadratic growth. The
existence, comparison, and stability results for one-dimensional BDSDEs are
proved when the generator $f(t,Y,Z)$ grows in $Z$ quadratically and the
terminal value is bounded, by introducing some new ideas. Moreover, in this
framework, we use BDSDEs to give a probabilistic representation for the
solutions of semilinear stochastic partial differential equations (SPDEs, for
short) in Sobolev spaces, and use it to prove the existence and uniqueness of
such SPDEs, thus extending the nonlinear Feynman-Kac formula.

Key words: backward doubly stochastic differential equation, stochastic
partial differential equation, quadratic growth, Feynman-Kac formula, Sobolev
solution.

AMS subject classifications. 60H10, 60H15, 60H30.

##  1 Introduction

The nonlinear Feynman-Kac formula was introduced by Pardoux–Peng [25] after
they proved the existence and uniqueness of nonlinear backward stochastic
differential equations (BSDEs, for short) in [24]. It provides a probabilistic
representation for a large class of semilinear partial differential equations
(PDEs, for short). A few years later, in order to give a probabilistic
representation for semilinear stochastic PDEs (SPDEs, for short), a new class
of BSDEs called the backward doubly stochastic differential equations (BDSDEs,
for short) were introduced by them too in [26]. In order to present the work
more clearly, we describe the problem in detail.

Let $(\Omega,\mathscr{F},\mathbb{P})$ be a complete probability space on which
two standard independent Brownian motions $\\{W_{t};0\leqslant t<\infty\\}$
and $\\{B_{t};0\leqslant t<\infty\\}$, with values in $\mathbb{R}^{d}$ and
$\mathbb{R}^{l}$, respectively, are defined. Let $T>0$ be a fixed terminal
time and denote by $\mathcal{N}$ the class of $\mathbb{P}$-null sets of
$\mathscr{F}$, where

$\mathscr{F}_{t}\triangleq\mathscr{F}_{t}^{W}\vee\mathscr{F}_{t,T}^{B},\quad\forall
t\in[0,T].$

In the above, for any process $\\{\eta_{t};0\leqslant t\leqslant T\\},$
$\mathscr{F}_{s,t}^{\eta}=\sigma\\{\eta_{r}-\eta_{s};s\leqslant r\leqslant
t\\}\vee\mathcal{N}$ and $\mathscr{F}_{t}^{\eta}=\mathscr{F}_{0,t}^{\eta}$.
For each $(t,x)\in[0,T]\times\mathbb{R}^{n}$, let $\\{X^{t,x}_{s};t\leqslant
s\leqslant T\\}$ be the solution of the following stochastic differential
equations (SDEs, for short):

$X^{t,x}_{s}=x+\int_{t}^{s}b(X^{t,x}_{r})dr+\int_{t}^{s}\sigma(X^{t,x}_{r})dW_{r},\quad~{}t\leqslant
s\leqslant T,$

and consider the following backward doubly stochastic differential equations:
for $t\leqslant s\leqslant T$,

$\displaystyle
Y^{t,x}_{s}=h(X^{t,x}_{T})+\int_{s}^{T}f(r,X^{t,x}_{r},Y^{t,x}_{r},Z^{t,x}_{r})dr+\int_{s}^{T}g(r,X^{t,x}_{r},Y^{t,x}_{r},Z^{t,x}_{r})d\overleftarrow{B}_{r}-\int_{s}^{T}Z^{t,x}_{r}dW_{r},$
(1.1)

where the $dW$ integral is a forward Itô’s integral and the
$d\overleftarrow{B}$ integral is a backward one. The coefficient $h$ is called
the terminal value and the coefficient $f$ is called the generator. The
solution of (1.1) is the pair $(Y,Z)$ of $\mathscr{F}$-measurable processes.
For convenience, hereafter, by a quadratic BDSDE, or BDSDE with quadratic
growth, we mean that in (1.1), the generator $f$ grows in $Z$ quadratically.
Meanwhile, we call $h$ the bounded terminal value if it is bounded.

Pardoux–Peng [26] introduced BDSDEs (1.1), gave the well-posedness of
solutions under globally Lipschitz coefficients, and used it to prove the
existence and uniqueness of the following semilinear SPDEs:

$\displaystyle u(s,x)=$ $\displaystyle
h(x)+\int_{s}^{T}\left\\{\mathcal{L}u(r,x)+f\big{(}r,x,u(r,x),(\sigma^{\top}\nabla
u)(r,x)\big{)}\right\\}dr$ (1.2)
$\displaystyle+\int_{s}^{T}g\big{(}r,x,u(r,x),(\sigma^{\top}\nabla
u)(r,x)\big{)}d\overleftarrow{B}_{r},\quad t\leqslant s\leqslant T,$

where $\sigma^{\top}$ denotes the transposed of $\sigma$, and

$\displaystyle\mathcal{L}=\sum_{i=1}^{n}b_{i}\frac{\partial}{\partial
x_{i}}+\frac{1}{2}\sum_{i,j=1}^{n}a_{ij}\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}},\quad~{}\left(a_{ij}\right)=\sigma\sigma^{\top}.$

This result is summarized as the nonlinear stochastic Feynman-Kac formula,
which permits us to solve SPDE by BDSDE and, conversely, one can use SPDE to
solve BDSDE too. Since then, this important theory has attracted a lot of
attention. For example, Buckdahn–Ma [10, 11] introduced the stochastic
viscosity solution to the nonlinear SPDEs, which connect BDSDEs to extend the
nonlinear stochastic Feynman-Kac formula. Bally–Matoussi [3] obtained the
connection between the solution of BDSDEs and the Sobolev solution of related
SPDEs. Zhang–Zhao [37] studied infinite horizon BDSDEs and got the stationary
solutions of related SPDEs. Xiong [33] solved a long standing open problem
about the SPDE characterization of the super-Brownian motion by making use of
the BDSDE. Buckdahn–Li–Xing [9] studied mean-field BDSDEs and the associated
nonlocal semilinear SPDEs. Besides, Shi–Gu–Liu [28] obtained the comparison
theorem of solutions of BDSDEs. Han–Peng–Wu [16] got the maximum principle for
backward doubly stochastic control systems. Hu–Matoussi–Zhang [18] studied the
Wong-Zakai approximations of BDSDEs. For more recent developments of BDSDEs,
we refer the readers to Boufoussi–Van Casteren–Mrhardy [6], Gomez et al. [15],
Shi–Wen–Xiong [29], Wu–Zhang [31], Zhang–Zhao [38], etc.

Moreover, when the coefficient $g$ is absent, BDSDEs (1.1) are essentially
reduced to the following backward stochastic differential equations:

$\displaystyle
Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds-\int_{t}^{T}Z_{s}dW_{s},\quad~{}0\leqslant
t\leqslant T,$ (1.3)

which were firstly introduced by Pardoux–Peng [24], where they obtained the
existence and uniqueness under Lipschitz coefficients. Since then, BSDE
stimulates some significant developments in many fields, such as partial
differential equation (see Pardoux–Peng [25] and Yong [34]), mathematical
finance (see El Karoui–Peng–Quenez [14]), and stochastic optimal control (see
Yong–Zhou [35]), to mention a few. Meanwhile, based on the wide applications
and motivated by the open problem proposed by Peng [23], many efforts have
been made to relax the conditions on the generator $f$ of BSDE (1.3) for the
existence and/or uniqueness of adapted solutions. For example, Lepeltier–San
Martin [21] proved the existence of adapted solutions for BSDE when the
generator $f$ is continuous and of linear growth in $(Y,Z)$. In 2000,
Kobylanski [20] established the well-posedness for one-dimensional BSDE (1.3)
with $f$ growing in $Z$ quadratically and with bounded terminal value. When
the terminal value is unbounded, Briand–Hu [7, 8] proved the existence and
uniqueness of one-dimensional BSDEs with quadratic growth. Delbaen–Hu–Bao [13]
studied BSDEs with the generator $f$ growing in $Z$ super-quadratically. For
multi-dimensional situation, Hu–Tang [19] proved the existence and uniqueness
of BDSDEs with diagonally quadratic generators, and Xing–Zitkovic [32]
established the existence and uniqueness for a large class of Markovian BSDEs
with quadratic growth. Some other recent developments of quadratic BSDEs can
be found in Bahlali–Eddahbi–Ouknine [1], Barrieu–El Karoui [5], Cheridito–Nam
[12], Hu–Li–Wen [17], Richou [27], Tevzadze [30], and references cited
therein.

Note that in quadratic BSDEs, in order to overcome the difficulty of quadratic
growth, one always would like to distinguish whether the terminal value is
bounded or not. The reason is that when the terminal value is bounded, by
involving bounded mean oscillation martingales (BMO martingale, for short),
some nice estimates and regularities for the solution $(Y,Z)$ could be
obtained. In other words, in this case, one could prove that $Y$ is bounded
and $Z$ belongs to the BMO martingale space, which are useful when proving the
existence and uniqueness. In addition, when proving the existence, the main
idea is to use the exponential transformation, which implies that there is an
essential difference between the study of the one-dimensional case and the
multi-dimensional case. In summary, when studying the quadratic BSDEs, it is
better to distinguish whether the terminal value is bounded or not, and the
framework is a one-dimensional case or a multi-dimensional case.

On the other hand, during the past two decades, stimulated by the broad
applications and the open problem of Peng [23], tremendous efforts have been
made to relax the conditions on the generator $f$ of BDSDEs and extend the
nonlinear stochastic Feynman-Kac formula. However, to the best of our
knowledge, there are few works concerning BDSDEs (1.1) when the generator $f$
grows in $Z$ quadratically. Some related studies are the works of Zhang–Zhao
[39] and Bahlali et al. [2], where they obtained the existence and uniqueness
of BDSDEs when the generator $f$ is of polynomial growth in $Y$ and is of
super-linear (or sub-quadratic) in $Z$, respectively. For quadratic BDSDEs,
the main difficulty is that due to the collection
$\mathbb{F}=\\{\mathscr{F}_{t};0\leqslant t\leqslant T\\}$ is neither
increasing nor decreasing, and it does not constitute a filtration. So the
main technique of BMO martingale used in quadratic BSDEs is useless here, and
one cannot expect to study the regularity of $Z$ in the BMO martingale space.
Besides, the backward Itô’s integral appeared in BDSDE (1.1) will bring extra
troubles when proving the existence and uniqueness of the solutions.

In this paper, we initiate the theory of BDSDEs with quadratic growth. In
particular, we consider the one-dimensional framework of quadratic BDSDEs with
bounded terminal value. First, we study the existence under a general
assumption on the generator $f$. Borrowed some ideas from Kobylanski [20], we
construct an artful exponential transformation (see 3.1), which transforms the
quadratic BDSDE into a classical one. It should be pointed out that the
solution $(Y,Z)$ we considered is in the space
$L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ (see Section 2 for detailed
definitions), since $\mathbb{F}=\\{\mathscr{F}_{t};0\leqslant t\leqslant T\\}$
is not a filtration. Moreover, in order to overcome the troubles that come
from the backward Itô’s integral, we introduce the idea of the comparison
theorem of classical BDSDEs, and some restricted condition (see (3.5)) on the
coefficient $g$ is made, otherwise, $Y$ cannot be bounded (see 3.2).
Furthermore, based on a priori estimate (see 3.5) and the monotone stability
obtained (see 3.7), we prove that BDSDE with quadratic growth admits a
solution $(Y,Z)\in L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ (see Theorem 3.10). Second, in order
to prove the uniqueness, we study a comparison theorem, due to the fact that
the one-dimensional framework allows us to provide a comparison theorem which
directly implies the uniqueness as a by-product (see Theorem 4.4). However, a
stronger assumption on the coefficients $f$ and $g$ than that for the
existence result is used to prove the comparison theorem. In other words, we
assume that the partial derivatives of the coefficients $f$ and $g$ with
respect to $Z$ are linear growth and bounded, respectively. Moreover, based on
the comparison theorem, we obtain a stability result, i.e., the solutions
$(Y^{n},Z^{n})$ of BDSDE with coefficients $(f^{n},g,\xi^{n})$ converge to the
unique solution $(Y,Z)$ of BDSDE with coefficients $(f,g,\xi)$ when the
sequences $\\{f^{n};n\in\mathbb{N}\\}$ and $\\{\xi^{n};n\in\mathbb{N}\\}$
converge to $f$ and $\xi$, respectively (see Theorem 4.6). Finally, we use
BDSDE (1.1) with quadratic growth to give a probabilistic representation for
the solutions of SPDE (1.2) in Sobolev spaces, and use it to prove the
existence and uniqueness of SPDE (1.2) when $f$ grows in $\sigma^{\top}\nabla
u$ quadratically (see Theorem 5.6), thus extending the nonlinear stochastic
Feynman-Kac formula of Pardoux–Peng [26] to quadratic growth situation.

The rest of the paper is organized as follows. In Section 2, some
preliminaries are presented. In Section 3, a priori estimate, the monotonicity
stability, and the existence of the solutions are proved. In Section 4, we
focus on the comparison theorem and thus derive the uniqueness of the
solutions. The stability result is obtained in this section too. In Section 5,
we study the relationship between the solution of BDSDEs with quadratic growth
and the Sobolev solution of related SPDEs. Section 6 concludes the results.

##  2 Preliminaries

Repeat that the triple $(\Omega,\mathscr{F},\mathbb{P})$ is a complete
probability space, and on which two standard independent Brownian motions
$\\{W_{t};0\leqslant t<\infty\\}$ and $\\{B_{t};0\leqslant t<\infty\\}$ are
defined with values in $\mathbb{R}^{d}$ and $\mathbb{R}^{l}$, respectively.
Let $T>0$ be a fixed terminal time and denote by $\mathcal{N}$ the class of
$\mathbb{P}$-null sets of $\mathscr{F}$, where

$\mathscr{F}_{t}\triangleq\mathscr{F}_{t}^{W}\vee\mathscr{F}_{t,T}^{B},\quad~{}\forall
t\in[0,T].$

In the above, for any process $\\{\eta_{t};0\leqslant t\leqslant T\\},$
$\mathscr{F}_{s,t}^{\eta}=\sigma\\{\eta_{r}-\eta_{s};s\leqslant r\leqslant
t\\}\vee\mathcal{N}$ and $\mathscr{F}_{t}^{\eta}=\mathscr{F}_{0,t}^{\eta}$.
Note that the collection $\mathbb{F}=\\{\mathscr{F}_{t};0\leqslant t\leqslant
T\\}$ is not a filtration since it is neither increasing nor decreasing.

Let us introduce some notations that will be used below. Denote by $|\cdot|$
and $\langle\cdot,\cdot\rangle$ the Euclidean norm and dot product,
respectively, throughout the paper. For any $p\geqslant 1$ and Euclidean space
$\mathbb{H}$, denote by $C^{p}(\mathbb{H})$ the set of functions valued in
$\mathbb{H}$ and of class $C^{p}$ with the partial derivations of order less
than or equal to $p$ are bounded. In addition, we introduce the following
spaces:

$\begin{array}[]{ll}\displaystyle
L^{p}_{\mathscr{F}_{T}}(\Omega;\mathbb{H})=\Big{\\{}\xi:\Omega\to\mathbb{H}\bigm{|}\xi\hbox{
is $\mathscr{F}_{T}$-measurable,
}\|\xi\|_{L^{p}}\triangleq\big{(}\mathbb{E}|\xi|^{p}\big{)}^{1\over
p}<\infty\Big{\\}},\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
L_{\mathscr{F}_{T}}^{\infty}(\Omega;\mathbb{H})=\Big{\\{}\xi:\Omega\to\mathbb{H}\bigm{|}\xi\hbox{
is $\mathscr{F}_{T}$-measurable, }\|\xi\|_{\infty}\triangleq\mathop{\rm
esssup}_{\omega\in\Omega}|\xi(\omega)|<\infty\Big{\\}},\end{array}$

and for any $t\in[0,T)$ and $s\in[t,T]$, define

$\displaystyle\displaystyle L_{\mathbb{F}}^{p}(t,T;\mathbb{H})\negthinspace=$
$\displaystyle\
\negthinspace\Big{\\{}\varphi:[t,T]\negthinspace\times\negthinspace\Omega\to\mathbb{H}\bigm{|}\varphi_{s}\hbox{
is $\mathscr{F}_{s}$-measurable},\
\|\varphi\|_{L_{\mathbb{F}}^{p}(t,T)}\triangleq\negthinspace\Big{(}\mathbb{E}\int^{T}_{t}\negthinspace|\varphi_{s}|^{p}ds\Big{)}^{1\over
p}\negthinspace<\negthinspace\negthinspace\infty\Big{\\}},$
$\displaystyle\displaystyle L_{\mathbb{F}}^{\infty}(t,T;\mathbb{H})=$
$\displaystyle~{}\Big{\\{}\varphi:[t,T]\times\Omega\to\mathbb{H}\bigm{|}\varphi_{s}\hbox{
is $\mathscr{F}_{s}$-measurable,
}\|\varphi\|_{L^{\infty}_{\mathbb{F}}(t,T)}\triangleq\mathop{\rm
esssup}_{(s,\omega)\in[t,T]\times\Omega}|\varphi_{s}(\omega)|<\infty\Big{\\}},$
$\displaystyle\displaystyle S_{\mathbb{F}}^{p}(t,T;\mathbb{H})=$
$\displaystyle~{}\Big{\\{}\varphi:[t,T]\times\Omega\to\mathbb{H}\bigm{|}\varphi_{s}\hbox{
is $\mathscr{F}_{s}$-measurable, continuous, and }$
$\displaystyle\qquad\qquad\qquad\qquad\quad\
\|\varphi\|_{S_{\mathbb{F}}^{p}(t,T)}\triangleq\Big{\\{}\mathbb{E}\Big{[}\sup_{s\in[t,T]}|\varphi_{s}|^{p}\Big{]}\Big{\\}}^{\frac{1}{p}}<\infty\Big{\\}}.$

Consider the following backward doubly stochastic differential equations:

$Y_{t}=\xi+\int_{t}^{T}f\left(s,Y_{s},Z_{s}\right)ds+\int_{t}^{T}g\left(s,Y_{s},Z_{s}\right)d\overleftarrow{B}_{s}-\int_{t}^{T}Z_{s}dW_{s},\quad~{}0\leqslant
t\leqslant T,$ (2.1)

where $\xi:\Omega\rightarrow\mathbb{R}^{k}$ is a random variable,
$f:\Omega\times\mathbb{R}^{k}\times\mathbb{R}^{k\times
d}\rightarrow\mathbb{R}^{k}$ and
$g:\Omega\times\mathbb{R}^{k}\times\mathbb{R}^{k\times
d}\rightarrow\mathbb{R}^{k\times l}$ are two coefficients. Note that the $dW$
integral is a forward Itô’s integral and the $d\overleftarrow{B}$ integral is
a backward one, and both integrals are particular cases of Itô-Skorohod
integral (see Nualart [22]).

######  Definition 2.1.

A pair of measurable processes $(Y,Z)\in
S^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{k})\times
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{k\times d})$ is called a solution of BDSDE
(2.1), if $\mathbb{P}$-almost surely, it satisfies (2.1). In addition, if $Y$
is bounded, then we call the pair $(Y,Z)$ a bounded solution.

Now, we recall the following propositions concerning BDSDEs, which come from
Pardoux–Peng [26] and Shi et al. [28]. The first one is the Itô’s formula for
backward doubly stochastic differential equations, the second one is the
existence and uniqueness for the classical BDSDEs, and the third one is the
comparison theorem for one-dimensional classical BDSDEs.

######  Proposition 2.2.

Let $\Phi\in C^{2}(\mathbb{R}^{k})$, $\alpha\in
S^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{k})$, $\beta\in
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{k})$, $\gamma\in
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{k\times l})$, $\delta\in
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{k\times d})$ be such that

$\displaystyle\alpha_{t}=\alpha_{0}+\int_{0}^{t}\beta_{s}ds+\int_{0}^{t}\gamma_{s}d\overleftarrow{B}_{s}+\int_{0}^{t}\delta_{s}dW_{s},\quad~{}0\leqslant
t\leqslant T.$

Then

$\displaystyle\Phi(\alpha_{t})=$ $\displaystyle\
\Phi(\alpha_{0})+\int_{0}^{t}\langle\Phi^{\prime}(\alpha_{s}),\beta_{s}\rangle
ds+\int_{0}^{t}\langle\Phi^{\prime}(\alpha_{s}),\gamma_{s}d\overleftarrow{B}_{s}\rangle+\int_{0}^{t}\langle\Phi^{\prime}(\alpha_{s}),\delta_{s}dW_{s}\rangle$
$\displaystyle-\frac{1}{2}\int_{0}^{t}\mathrm{Tr}[\Phi^{\prime\prime}(\alpha_{s})\gamma_{s}\gamma_{s}^{\top}]ds+\frac{1}{2}\int_{0}^{t}\mathrm{Tr}[\Phi^{\prime\prime}(\alpha_{s}))\delta_{s}\delta_{s}^{\top}]ds.$

######  Assumption 1.

The terminal value $\xi$ comes from
$L^{2}_{\mathscr{F}_{T}}(\Omega;\mathbb{R}^{k})$, and for any
$y\in\mathbb{R}^{k}$ and $z\in\mathbb{R}^{k\times d}$,

$f(\cdot,y,z)\in
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{k})\quad~{}\hbox{and}\quad~{}g(\cdot,y,z)\in
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{k\times l}).$

Moreover, there exist two positive constants $C$ and $\alpha$ with
$\alpha\in(0,1)$ such that for any $t\in[0,T]$,
$y_{1},y_{2}\in\mathbb{R}^{k}$, $z_{1},z_{2}\in\mathbb{R}^{k\times d}$,

$\displaystyle\displaystyle|f(t,y_{1},z_{1})-f(t,y_{2},z_{2})|^{2}\leqslant$
$\displaystyle\ C\big{[}|y_{1}-y_{2}|^{2}+|z_{1}-z_{2}|^{2}\big{]},$
$\displaystyle\displaystyle|g(t,y_{1},z_{1})-g(t,y_{2},z_{2})|^{2}\leqslant$
$\displaystyle\ C|y_{1}-y_{2}|^{2}+\alpha|z_{1}-z_{2}|^{2}.$ (2.2)

######  Proposition 2.3.

Under 1, BDSDEs (2.1) admit a unique solution $(Y,Z)\in
S^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{k})\times
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{k\times d})$.

######  Proposition 2.4.

Suppose that the coefficients $(f^{1},g,\xi^{1})$ and $(f^{2},g,\xi^{2})$
satisfy 1, and let $(Y^{1},Z^{1})$ and $(Y^{2},Z^{2})$ be the solutions of
one-dimensional BDSDEs (2.1) with parameters $(f^{1},g,\xi^{1})$ and
$(f^{2},g,\xi^{2})$, respectively. If for any
$(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$,

$\xi^{1}\leqslant\xi^{2},\quad~{}f^{1}(t,y,z)\leqslant
f^{2}(t,y,z),\quad~{}\text{a.s.,}$

then we have that for any $t\in[0,T]$,

$Y^{1}_{t}\leqslant Y^{2}_{t},\quad~{}\text{a.s.}$

##  3 Quadratic BDSDE

In this section, we study the existence of one-dimensional BDSDEs with
quadratic growth, i.e., $k=1$. Before proving the existence, we study a priori
estimate and the monotone stability, which are important for us to prove
existence later. In order to justify the assumptions that were used to prove
the existence, we give two examples. It should be pointed out that the first
example also implies the main idea to prove existence, which is the central
technique throughout this section.

######  Example 3.1.

Consider the following backward doubly stochastic differential equation,

$Y_{t}=\xi+C\int_{t}^{T}|Z_{s}|^{2}ds+\alpha\int_{t}^{T}Z_{s}d\overleftarrow{B}_{s}-\int_{t}^{T}Z_{s}dW_{s},\quad~{}0\leqslant
t\leqslant T,$ (3.1)

where $C>0$ and $\alpha\in(-1,1)$ are two constants. Let $\beta$ be a positive
constant which will be given later. Applying the exponential change of
variable to $y=\exp(\beta Y)$ implies that

$\displaystyle\displaystyle y_{T}=$ $\displaystyle~{}y_{t}-C\int_{t}^{T}\beta
y_{s}|Z_{s}|^{2}ds-\alpha\int_{t}^{T}\beta
y_{s}Z_{s}d\overleftarrow{B}_{s}+\int_{t}^{T}\beta y_{s}Z_{s}dW_{s}$
$\displaystyle-\frac{\alpha^{2}}{2}\int_{t}^{T}\beta^{2}y_{s}|Z_{s}|^{2}ds+\frac{1}{2}\int_{t}^{T}\beta^{2}y_{s}|Z_{s}|^{2}ds$
$\displaystyle\displaystyle=$
$\displaystyle~{}y_{t}-\Big{[}C-\frac{1-\alpha^{2}}{2}\beta\Big{]}\int_{t}^{T}\beta
y_{s}|Z_{s}|^{2}ds-\alpha\int_{t}^{T}\beta
y_{s}Z_{s}d\overleftarrow{B}_{s}+\int_{t}^{T}\beta y_{s}Z_{s}dW_{s}.$

Let $\beta=\frac{2C}{1-\alpha^{2}}$ and denote by $z=\beta yZ$, then the above
equation can be rewritten as

$\displaystyle y_{t}=\exp(\beta\xi)+\int_{t}^{T}\alpha
z_{s}d\overleftarrow{B}_{s}-\int_{t}^{T}z_{s}dW_{s},\quad~{}0\leqslant
t\leqslant T,$ (3.2)

which essential is a linear backward doubly stochastic differential equation.
Then the classical existence and uniqueness (see 2.3) implies that when

$\exp(\beta\xi)\in L^{2}_{\mathscr{F}_{T}}(\Omega;\mathbb{R}),$ (3.3)

there exists a unique solution $(y,z)\in
S^{2}_{\mathbb{F}}(0,T;\mathbb{R})\times
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$ for BDSDE (3.2). Note that, $\beta$ is
positive constant, so taking $\xi\in
L^{\infty}_{\mathscr{F}_{T}}(\Omega;\mathbb{R})$ is a sufficient condition to
insure that $\xi$ satisfies (3.3). In fact, if $\xi\in
L_{\mathscr{F}_{T}}^{\infty}(\Omega;\mathbb{R})$, then $\exp(\xi)\in
L_{\mathscr{F}_{T}}^{\infty}(\Omega;\mathbb{R})$ and (3.3) holds, thus Eq.
(3.2) admits a unique solution $(y,z)\in
S^{2}_{\mathbb{F}}(0,T;\mathbb{R})\times
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$. Moreover, for BDSDE (3.2), by the
comparison theorem (see 2.4), one has

$\exp(-\beta\|\xi\|_{\infty})\leqslant
y_{t}\leqslant\exp(\beta\|\xi\|_{\infty}),\quad~{}\forall t\in[0,T],$

which implies that $y_{t}$ is positive and bounded. Now, we can define $(Y,Z)$
by

$Y_{t}=\frac{\ln(y_{t})}{\beta},\quad~{}Z_{t}=\frac{z_{t}}{\beta
y_{t}},\quad~{}\forall t\in[0,T].$

Then, it is easy to check that the pair $(Y,Z)\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ defined above is a solution of BDSDE
(3.1). Finally, the uniqueness of (3.1) in
$L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ follows from the uniqueness of (3.2)
and the fact that the exponential change of variable is no longer formal.

######  Example 3.2.

Consider the following linear backward doubly stochastic differential
equation,

$Y_{t}=\xi+\int_{t}^{T}(H+d_{s}Y_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}Z_{s}dW_{s},\quad~{}0\leqslant
t\leqslant T,$ (3.4)

where $H$ is a constant and $d$ is a bounded deterministic function. From 2.3,
BDSDE (3.4) admits a unique solution $(Y,Z)$. Moreover, Proposition 2.1 of
Bally–Matoussi [3] implies that the solution $Y$ is given explicitly by

$Y_{t}=\mathbb{E}\left[\Psi(t,T)\xi+H\int_{t}^{T}d_{s}\Psi(t,s)ds\bigg{|}\mathscr{F}_{t}\right],\quad\
0\leqslant t\leqslant T,$

where

$\Psi(t,s)=\exp\left(\int_{t}^{s}d_{r}d\overleftarrow{B}_{r}+\frac{1}{2}\int_{t}^{s}|d_{r}|^{2}dr\right),\quad\
t\leqslant s\leqslant T,$

which is not bounded even if the terminal value $\xi$ is bounded.

######  Remark 3.3.

On the one hand, 3.1 tells us that why we require the boundedness of the
terminal value $\xi$. On the other hand, 3.2 implies that even in the
classical situation with bounded terminal value $\xi$, the normal Lipschitz
condition of the coefficient $g$ with respect to $y$ (see (2.2)) could not
guarantee that $Y$ is bounded.

###  3.1 A priori estimate

In this subsection, we prove a priori estimate for the solution $(Y,Z)$ of
quadratic BDSDEs. First, we introduce the assumption.

######  Assumption 2.

Suppose that the terminal value $\xi\in
L^{\infty}_{\mathscr{F}_{T}}(\Omega;\mathbb{R})$. Let
$a:[0,T]\rightarrow\mathbb{R}$ and $b:[0,T]\rightarrow\mathbb{R}^{+}$ be two
functions, and $C$ and $\alpha$ be two positive constants with
$\alpha\in(0,1)$. For all
$(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$,

$\displaystyle f(t,y,z)=a_{0}(t,y,z)y+f_{0}(t,y,z)$

with

$\displaystyle a_{0}(t,y,z)\leqslant a_{t},\quad~{}|f_{0}(t,y,z)|\leqslant
b_{t}+C|z|^{2},\quad~{}\hbox{a.s.}$

and

$\displaystyle|g(t,y,z)|^{2}\leqslant\alpha|z|^{2},\quad~{}\hbox{a.s.}$ (3.5)

######  Remark 3.4.

We point out that the boundedness of the solution $Y$ requires us to make the
condition (3.5) concerning the coefficient $g$; if not, the solution $Y$ may
not be bounded even if the terminal value $\xi$ is bounded (see 3.2 and 3.3).
Here we give an example that satisfying (3.5):

$g(t,y,z)=\sin(y)l(z)\quad~{}\hbox{with}\quad~{}|l(z)|^{2}\leqslant\alpha|z|^{2},\quad~{}\forall(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}.$

######  Proposition 3.5 (A priori estimate).

Let $(Y,Z)\in L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ be a solution of BDSDE (2.1) with
parameters $(f,g,\xi)$, which satisfy 2 with $a,b,C,\alpha$, such that

$a^{+}=\max(a,0),\quad~{}b\in L^{1}(0,T;\mathbb{R}).$

Then for every $t\in[0,T]$,

$\displaystyle\displaystyle Y_{t}\leqslant$ $\displaystyle\
\Big{[}\sup_{\Omega}(Y_{T})\Big{]}^{+}\cdot\exp\Big{(}\int_{t}^{T}a_{s}ds\Big{)}+\int_{t}^{T}b_{s}\exp\Big{(}\int_{t}^{s}a_{r}dr\Big{)}ds,\quad~{}\hbox{a.s.}$
(3.6) $\displaystyle\displaystyle\bigg{(}\hbox{resp. }Y_{t}\geqslant$
$\displaystyle\
\Big{[}\inf_{\Omega}(Y_{T})\Big{]}^{-}\cdot\exp\Big{(}\int_{t}^{T}a_{s}ds\Big{)}-\int_{t}^{T}b_{s}\exp\Big{(}\int_{t}^{s}a_{r}dr\Big{)}ds,\quad~{}\hbox{a.s.}\bigg{)}.$
(3.7)

Moreover, there exists a positive constant $K$ depending only on
$\|Y\|_{\infty},\ \|a^{+}\|_{L^{1}}$, $C$ and $\alpha$ such that

$\mathbb{E}\int_{0}^{T}|Z_{s}|^{2}ds\leqslant K.$ (3.8)

###### Proof.

Consider the following linear ordinary differential equation,

$\varphi_{t}=\Big{[}\sup_{\Omega}(Y_{T})\Big{]}^{+}+\int_{t}^{T}(a_{s}\varphi_{s}+b_{s})ds,\quad~{}0\leqslant
t\leqslant T.$

Then someone could compute that for $0\leqslant t\leqslant T$,

$\varphi_{t}=\Big{[}\sup_{\Omega}(Y_{T})\Big{]}^{+}\cdot\exp\Big{(}\int_{t}^{T}a_{s}ds\Big{)}+\int_{t}^{T}b_{s}\exp\Big{(}\int_{t}^{s}a_{r}dr\Big{)}ds.$

So the result (3.6) holds if we can prove that $Y_{t}\leqslant\varphi_{t}$ for
all $t\in[0,T]$. Applying Itô’s formula (see 2.2) to the process
$Y_{t}-\varphi_{t}$ and to an increasing $C^{2}(\mathbb{R})$ function $\Phi$,
which will be determined later, deduce that

$\displaystyle\displaystyle\Phi(Y_{t}-\varphi_{t})=$ $\displaystyle\
\Phi(Y_{T}-\varphi_{T})+\int_{t}^{T}\Phi^{\prime}(Y_{s}-\varphi_{s})[f(s,Y_{s},Z_{s})-(a_{s}\varphi_{s}+b_{s})]ds$
$\displaystyle\
+\int_{t}^{T}\Phi^{\prime}(Y_{s}-\varphi_{s})g(s,Y_{s},Z_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}\Phi^{\prime}(Y_{s}-\varphi_{s})Z_{s}dW_{s}$
$\displaystyle\
+\frac{1}{2}\int_{t}^{T}\Phi^{\prime\prime}(Y_{s}-\varphi_{s})|g(s,Y_{s},Z_{s})|^{2}ds-\frac{1}{2}\int_{t}^{T}\Phi^{\prime\prime}(Y_{s}-\varphi_{s})|Z_{s}|^{2}ds.$

We denote that for every $s\in[t,T]$,

$\tilde{a}_{s}=a_{0}(s,Y_{s},Z_{s}).$

From 2, note that the function $\Phi$ is increasing, we have that for every
$s\in[t,T]$,

$\displaystyle\displaystyle\Phi^{\prime}$
$\displaystyle(Y_{s}-\varphi_{s})\big{[}f(s,Y_{s},Z_{s})-(a_{s}\varphi_{s}+b_{s})\big{]}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\
\Phi^{\prime}(Y_{s}-\varphi_{s})\big{[}\tilde{a}_{s}Y_{s}+b_{s}+C|Z_{s}|^{2}-(a_{s}\varphi_{s}+b_{s})\big{]}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\
\Phi^{\prime}(Y_{s}-\varphi_{s})\big{[}\tilde{a}_{s}(Y_{s}-\varphi_{s})+(\tilde{a}_{s}-a_{s})\varphi_{s}+C|Z_{s}|^{2}\big{]},$

and

$\Phi^{\prime\prime}(Y_{s}-\varphi_{s})|g(s,Y_{s},Z_{s})|^{2}\leqslant\alpha\Phi^{\prime\prime}(Y_{s}-\varphi_{s})|Z_{s}|^{2}.$

Due to that $(\tilde{a}_{s}-a_{s})\varphi_{s}\leqslant 0$, we obtain

$\displaystyle\displaystyle\Phi(Y_{t}-\varphi_{t})\leqslant$ $\displaystyle\
\Phi(Y_{T}-\varphi_{T})+\int_{t}^{T}\tilde{a}_{s}\Phi^{\prime}(Y_{s}-\varphi_{s})(Y_{s}-\varphi_{s})ds$
$\displaystyle\
+\int_{t}^{T}\Big{(}C\Phi^{\prime}-\frac{1-\alpha}{2}\Phi^{\prime\prime}\Big{)}(Y_{s}-\varphi_{s})|Z_{s}|^{2}ds$
(3.9) $\displaystyle\
+\int_{t}^{T}\Phi^{\prime}(Y_{s}-\varphi_{s})g(s,Y_{s},Z_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}\Phi^{\prime}(Y_{s}-\varphi_{s})Z_{s}dW_{s}.$

Set $M=\|Y\|_{\infty}+\|\varphi\|_{\infty}$ and define the function $\Phi$ on
the interval $[-M,M]$ by

$\Phi(u)=\left\\{\begin{array}[]{ll}\displaystyle
e^{\frac{2C}{1-\alpha}u}-1-\frac{2C}{1-\alpha}u-\frac{2C^{2}}{(1-\alpha)^{2}}u^{2},&u\in[0,M],\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
0,&u\in[-M,0].\end{array}\right.$

Then it is easy to check that for all $u\in[-M,M]$,

$\displaystyle\displaystyle\Phi(u)\geqslant 0,\ \hbox{ and }\ \Phi(u)=$
$\displaystyle\ 0\ \hbox{ if and only if }\ u\leqslant 0,$
$\displaystyle\displaystyle\Phi^{\prime}(u)\geqslant$ $\displaystyle\ 0,$
$\displaystyle\displaystyle 0\leqslant u\Phi^{\prime}(u)\leqslant$
$\displaystyle\ (M+1)\frac{2C}{1-\alpha}\Phi(u),$ $\displaystyle\displaystyle
C\Phi^{\prime}-\frac{1-\alpha}{2}\Phi^{\prime\prime}\leqslant$ $\displaystyle\
0.$

Hence, if we set

$k_{t}=a^{+}_{t}(M+1)\frac{2C}{1-\alpha},$

then the function $k$ is positive and deterministic. Note that
$Y_{T}-\varphi_{T}\leqslant 0$ implies that $\Phi(Y_{T}-\varphi_{T})=0$, and
thus for all $0\leqslant t\leqslant T$, one has

$\displaystyle 0\leqslant$ $\displaystyle\
\Phi(Y_{t}-\varphi_{t})\leqslant\int_{t}^{T}k_{s}\Phi(Y_{s}-\varphi_{s})ds$
(3.10)
$\displaystyle+\int_{t}^{T}\Phi^{\prime}(Y_{s}-\varphi_{s})g(s,Y_{s},Z_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}\Phi^{\prime}(Y_{s}-\varphi_{s})Z_{s}dW_{s}.$

Note that the term $\Phi^{\prime}(Y_{s}-\varphi_{s})$ is bounded on the
interval $t\leqslant s\leqslant T$, so the terms
$\Phi^{\prime}(Y_{s}-\varphi_{s})Z_{s}$ and
$\Phi^{\prime}(Y_{s}-\varphi_{s})g(s,Y_{s},Z_{s})$ belong to the spaces
$L_{\mathbb{F}}^{2}(t,T;\mathbb{R}^{d})$ and
$L_{\mathbb{F}}^{2}(t,T;\mathbb{R}^{l})$, respectively. Taking expectations on
both side of the inequality (3.10) implies that

$0\leqslant\mathbb{E}\Phi(Y_{t}-\varphi_{t})\leqslant\mathbb{E}\int_{t}^{T}k_{s}\Phi(Y_{s}-\varphi_{s})ds.$

Then Gronwall’s inequality yields that

$\mathbb{E}\Phi(Y_{t}-\varphi_{t})=0,\quad~{}\forall t\in[0,T].$

Therefore, note that $\Phi(u)\geqslant 0$, for all $t\in[0,T],$

$\Phi(Y_{t}-\varphi_{t})=0,\quad~{}\hbox{ a.s.}$

Finally, again, note that $\Phi(u)=0$ if and only if $u\leqslant 0$, we obtain
that for every $t\in[0,T],$

$Y_{t}-\varphi_{t}\leqslant 0,\quad~{}\hbox{ a.s.},$

which implies that the inequality (3.6) holds. Similarly, by using the same
computation, one can prove that (3.7) holds too.

Finally, in order to prove the estimate (3.8), we use again (3.1) with $t=0,\
\varphi=0$, $M=\|Y\|_{\infty}$, and define $\Phi$ on $[-M,M]$ by

$\Phi(u)=\frac{1-\alpha}{2C^{2}}\Big{[}\exp\Big{(}\frac{2C}{1-\alpha}(u+M)\Big{)}-1-\frac{2C}{1-\alpha}(u+M)\Big{]}.$

It is easy to check that, for every $u\in[-M,M]$,

$\displaystyle\displaystyle\Phi(u)\geqslant$ $\displaystyle\ 0,$
$\displaystyle\displaystyle\Phi^{\prime}(u)\geqslant$ $\displaystyle\ 0,$
$\displaystyle\displaystyle 0\leqslant u\Phi^{\prime}(u)\leqslant$
$\displaystyle\
\frac{M}{C}\Big{[}\exp\Big{(}\frac{4CM}{1-\alpha}\Big{)}-1\Big{]},$
$\displaystyle\displaystyle\frac{1-\alpha}{2}\Phi^{\prime\prime}-C\Phi^{\prime}=$
$\displaystyle\ 1.$

Then (3.1) implies that

$\displaystyle 0\leqslant\Phi(Y_{0})\leqslant$ $\displaystyle\
\Phi(Y_{T})+\int_{0}^{T}a_{s}^{+}\frac{M}{C}\Big{[}\exp\Big{(}\frac{4CM}{1-\alpha}\Big{)}-1\Big{]}ds-\int_{0}^{T}|Z_{s}|^{2}ds$
$\displaystyle\
+\int_{0}^{T}\Phi^{\prime}(Y_{s})g(s,Y_{s},Z_{s})d\overleftarrow{B}_{s}-\int_{0}^{T}\Phi^{\prime}(Y_{s})Z_{s}dW_{s},$

which leads that

$\displaystyle\mathbb{E}\int_{0}^{T}|Z_{s}|^{2}ds\leqslant\Phi(M)+\frac{M}{C}\Big{[}\exp\Big{(}\frac{4CM}{1-\alpha}\Big{)}-1\Big{]}\|a^{+}\|_{L^{1}([0,T];\mathbb{R})}.$

This completes the proof. ∎

An immediate consequence of this proposition is the following corollary.

######  Corollary 3.6.

Let $(Y,Z)\in L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ be a solution of BDSDE with parameters
$(f,g,\xi)$, where $\xi$ belongs to
$L^{\infty}_{\mathscr{F}_{T}}(\Omega;\mathbb{R})$, and $f$ and $g$ satisfy the
condition 2 with $a,b$, $C>0$ and $\alpha\in(0,1)$ such that $a^{+},b\in
L^{1}(0,T;\mathbb{R})$, then

$\|Y\|_{\infty}\leqslant\big{(}\|\xi\|_{\infty}+\|b\|_{L^{1}([0,T];\mathbb{R})}\big{)}\exp\big{(}\|a^{+}\|_{L^{1}([0,T];\mathbb{R})}\big{)}.$

###  3.2 Monotone stability

In this subsection, we would like to study the monotone stability of BDSDEs
(2.1) with quadratic growth, which is also important for us to prove the
existence later.

######  Proposition 3.7 (Monotone stability).

Let $(f,g,\xi)$ be a set of parameters and let $(f^{n},g,\xi^{n})$ be a
sequence of parameters such that:

  * $\rm(i)$

There exist two positive constants $C$ and $\alpha$ with $0<\alpha<1$ such
that for each $n\in\mathbb{N}$,

$\displaystyle|f^{n}(t,y,z)|\leqslant
C(1+|y|+|z|^{2}),\quad~{}|g(t,y,z)|^{2}\leqslant\alpha|z|^{2},\quad~{}\forall
t\in[0,T],y\in\mathbb{R},z\in\mathbb{R}^{d},$ (3.11)
$\displaystyle|g(t,y_{1},z_{1})-g(t,y_{2},z_{2})|^{2}\leqslant
C|y_{1}-y_{2}|^{2}+\alpha|z_{1}-z_{2}|^{2},\quad~{}\forall
t\in[0,T],y_{1},y_{2}\in\mathbb{R},z_{1},z_{2}\in\mathbb{R}^{d}.$

  * $\rm(ii)$

The sequence $\\{f^{n};n\in\mathbb{N}\\}$ converges to $f$ locally uniformly
on $[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$, and for each $n\in\mathbb{N},$
the sequence $\\{\xi^{n};n\in\mathbb{N}\\}$ converges to $\xi$ in
$L^{\infty}_{\mathscr{F}_{T}}(\Omega;\mathbb{R})$.

  * $\rm(iii)$

For each $n\in\mathbb{N}$, the BDSDE with parameters $(f^{n},g,\xi^{n})$ has a
solution

$(Y^{n},Z^{n})\in L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d}),$

and the sequence $\\{Y^{n};n\in\mathbb{N}\\}$ is monotonic such that for every
$n\in\mathbb{N}$, $\|Y^{n}\|_{\infty}\leqslant M$, where $M$ is a positive
constant.

Then there exists a pair of processes $(Y,Z)\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ such that

$\left\\{\begin{aligned} \displaystyle&\lim_{n\rightarrow\infty}Y^{n}=Y\hbox{
uniformy on }[0,T],\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle&\\{Z^{n};n\in\mathbb{N}\\}\hbox{ converges to $Z$ in
}L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d}).\end{aligned}\right.$

Moreover, $(Y,Z)$ is a solution of BDSDE with parameters $(f,g,\xi)$. In
particular, if the sequence $\\{Y^{n};n\in\mathbb{N}\\}$ has continuous paths
for every $n\in\mathbb{N}$, then $Y$ has continuous paths too.

###### Proof.

From the assumptions we see that for all $t\in[0,T]$, the sequence
$\\{Y^{n}_{t};n\in\mathbb{N}\\}$ is monotonic and bounded, and therefore it
has a limit, which denoted by $Y$. Without lose of generality, we consider
that the sequence $\\{Y^{n};n\in\mathbb{N}\\}$ is monotonic increasing to $Y$.
Moreover, from 3.5, there is a positive constant $K$ such that,

$\mathbb{E}\int_{0}^{T}|Z^{n}|^{2}ds\leqslant K,\quad~{}\forall
n\in\mathbb{N}.$

Then there exists a process $Z\in L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ such
that a subsequence of $\\{Z^{n};n\in\mathbb{N}\\}$ converges to $Z$ weakly in
$L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$. For simplicity presentation, by
otherwise choosing a subsequence, we may assume that the whole sequence
$\\{Z^{n};n\in\mathbb{N}\\}$ converges to $Z$ weakly in
$L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$.

Next, we would like to divide the proof into two steps, and denote the
following notations:

$\displaystyle\displaystyle\Delta Y^{n}\triangleq Y-Y^{n},\quad\ \Delta
Y^{m,n}\triangleq Y^{m}-Y^{n},$ $\displaystyle\displaystyle\Delta
Z^{n}\triangleq Z-Z^{n},\quad\ \Delta Z^{m,n}\triangleq Z^{m}-Z^{n}.$

Step 1. The sequence $\\{Z^{n};n\in\mathbb{N}\\}$ strongly converges to $Z$ in
$L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$, i.e.,

$\lim_{n\rightarrow\infty}\mathbb{E}\int_{0}^{T}|\Delta Z^{n}_{t}|^{2}dt=0.$

The main arguments of this step come from Zhang [36] and Kobylanski [20]. It
is easy to see that the pair $(\Delta Y^{m,n},\Delta Z^{m,n})$ satisfies the
following equation,

$\left\\{\begin{array}[]{ll}\displaystyle-d\Delta
Y^{m,n}_{t}=[f^{m}(t,Y^{m}_{t},Z^{m}_{t})-f^{n}(t,Y^{n}_{t},Z^{n}_{t})]dt\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle\qquad\qquad\quad\ \
+[g(t,Y^{m}_{t},Z^{m}_{t})-g(t,Y^{n}_{t},Z^{n}_{t})]d\overleftarrow{B}_{t}-\Delta
Z^{m,n}_{t}dW_{t},\quad~{}t\in[0,T],\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle\quad\ \Delta
Y^{m,n}_{T}=\ \xi^{m}-\xi^{n}.\end{array}\right.$

By assumptions, there exists a constant $C_{0}$ depending on $C$ and $M$, such
that for all $m,n\in\mathbb{N}$, $t\in[0,T]$,

$\displaystyle|f^{m}(t,Y^{m}_{t},Z^{m}_{t})-f^{n}(t,Y^{n}_{t},Z^{n}_{t})|\leqslant$
$\displaystyle\ C_{0}[1+|\Delta Z^{m,n}_{t}|^{2}+|\Delta
Z^{n}_{t}|^{2}+|Z_{t}|^{2}],$ (3.12)
$\displaystyle|g(t,Y^{m}_{t},Z^{m}_{t})-g(t,Y^{n}_{t},Z^{n}_{t})|^{2}\leqslant$
$\displaystyle\ C|\Delta Y^{m,n}_{t}|^{2}+\alpha|\Delta Z^{m,n}_{t}|^{2}.$

Let $\Phi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a smooth increasing
function which will be specified later. For $m\geqslant n$, using Itô’s
formula (see 2.2) to $\Phi(\Delta Y^{m,n})$ we have that

$\displaystyle\displaystyle d\Phi(\Delta Y^{m,n}_{t})=$
$\displaystyle-\Phi^{\prime}(\Delta
Y^{m,n}_{t})[f^{m}(t,Y^{m}_{t},Z^{m}_{t})-f^{n}(t,Y^{n}_{t},Z^{n}_{t})]dt$
$\displaystyle-\Phi^{\prime}(\Delta
Y^{m,n}_{t})[g(t,Y^{m}_{t},Z^{m}_{t})-g(t,Y^{n}_{t},Z^{n}_{t})]d\overleftarrow{B}_{t}+\Phi^{\prime}(\Delta
Y^{m,n}_{t})\Delta Z^{m,n}_{t}dW_{t}$
$\displaystyle-\frac{1}{2}\Phi^{\prime\prime}(\Delta
Y^{m,n}_{t})|g(t,Y^{m}_{t},Z^{m}_{t})-g(t,Y^{n}_{t},Z^{n}_{t})|^{2}dt+\frac{1}{2}\Phi^{\prime\prime}(\Delta
Y^{m,n}_{t})|\Delta Z^{m,n}_{t}|^{2}dt$ $\displaystyle\displaystyle\geqslant$
$\displaystyle-C_{0}\Phi^{\prime}(\Delta Y^{m,n}_{t})[1+|\Delta
Z^{m,n}_{t}|^{2}+|\Delta Z^{n}_{t}|^{2}+|Z_{t}|^{2}]dt$
$\displaystyle-\Phi^{\prime}(\Delta
Y^{m,n}_{t})[g(t,Y^{m}_{t},Z^{m}_{t})-g(t,Y^{n}_{t},Z^{n}_{t})]d\overleftarrow{B}_{t}+\Phi^{\prime}(\Delta
Y^{m,n}_{t})\Delta Z^{m,n}_{t}dW_{t}$
$\displaystyle-\frac{C}{2}\Phi^{\prime\prime}(\Delta Y^{m,n}_{t})|\Delta
Y^{m,n}_{t}|^{2}dt-\frac{\alpha}{2}\Phi^{\prime\prime}(\Delta
Y^{m,n}_{t})|\Delta Z^{m,n}_{t}|^{2}dt+\frac{1}{2}\Phi^{\prime\prime}(\Delta
Y^{m,n}_{t})|\Delta Z^{m,n}_{t}|^{2}dt,$

where we have use the inequality (3.12).

In the following, we expect $\Delta Z^{m,n}$ and $\Delta Z^{n}$ to be closed.
For this purpose, define

$\Phi(u)=\frac{1-\alpha}{8C_{0}^{2}}\Big{[}\exp(\frac{4C_{0}}{1-\alpha}u)-\frac{4C_{0}}{1-\alpha}u-1\Big{]}.$

It is straightforward to check that for $u\geqslant 0$, we have

$\displaystyle\Phi(u)\geqslant 0\quad\hbox{with}\quad\Phi(0)=0,$
$\displaystyle\Phi^{\prime}(u)\geqslant
0\quad\hbox{with}\quad\Phi^{\prime}(0)=0,$
$\displaystyle\frac{1-\alpha}{2}\Phi^{\prime\prime}(u)=4C_{0}\Phi^{\prime}(u)+2.$

Note that $Y^{m}\geqslant Y^{n}$ when $m\geqslant n$. Then

$\displaystyle\displaystyle d\Phi(\Delta Y^{m,n}_{t})\geqslant$
$\displaystyle-C_{0}\Phi^{\prime}(\Delta Y^{m,n}_{t})[1+|\Delta
Z^{m,n}_{t}|^{2}+|\Delta Z^{n}_{t}|^{2}+|Z_{t}|^{2}]dt$
$\displaystyle-\Phi^{\prime}(\Delta
Y^{m,n}_{t})[g(t,Y^{m}_{t},Z^{m}_{t})-g(t,Y^{n}_{t},Z^{n}_{t})]d\overleftarrow{B}_{t}+\Phi^{\prime}(\Delta
Y^{m,n}_{t})\Delta Z^{m,n}_{t}dW_{t}$
$\displaystyle+[4C_{0}\Phi^{\prime}(\Delta Y^{m,n}_{t})+2]|\Delta
Z^{m,n}_{t}|^{2}dt-\frac{C}{2}\Phi^{\prime\prime}(\Delta Y^{m,n}_{t})|\Delta
Y^{m,n}_{t}|^{2}dt$ $\displaystyle\displaystyle=$ $\displaystyle-
C_{0}\Phi^{\prime}(\Delta Y^{m,n}_{t})[1+|\Delta
Z^{n}_{t}|^{2}+|Z_{t}|^{2}]dt-\frac{C}{2}\Phi^{\prime\prime}(\Delta
Y^{m,n}_{t})|\Delta Y^{m,n}_{t}|^{2}dt$ $\displaystyle-\Phi^{\prime}(\Delta
Y^{m,n}_{t})[g(t,Y^{m}_{t},Z^{m}_{t})-g(t,Y^{n}_{t},Z^{n}_{t})]d\overleftarrow{B}_{t}+\Phi^{\prime}(\Delta
Y^{m,n}_{t})\Delta Z^{m,n}_{t}dW_{t}$
$\displaystyle+[3C_{0}\Phi^{\prime}(\Delta Y^{m,n}_{t})+2]|\Delta
Z^{m,n}_{t}|^{2}dt,$

and thus

$\displaystyle\mathbb{E}\Phi(\Delta
Y^{m,n}_{0})+\mathbb{E}\int_{0}^{T}[3C_{0}\Phi^{\prime}(\Delta
Y^{m,n}_{t})+2]|\Delta Z^{m,n}_{t}|^{2}dt-
C_{0}\mathbb{E}\int_{0}^{T}\Phi^{\prime}(\Delta Y^{m,n}_{t})|\Delta
Z^{n}_{t}|^{2}dt$ $\displaystyle\displaystyle\leqslant$
$\displaystyle\mathbb{E}\Big{[}\Phi(\Delta
Y^{m,n}_{T})+C_{0}\int_{0}^{T}\Phi^{\prime}(\Delta
Y^{m,n}_{t})[1+|Z_{t}|^{2}]dt+\frac{C}{2}\int_{t}^{T}\Phi^{\prime\prime}(\Delta
Y^{m,n}_{t})|\Delta Y^{m,n}_{t}|^{2}dt\Big{]}.$

Due to that $\\{Z^{m};m\geqslant 1\\}$ converges to $Z$ weakly in
$L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$, so next we would like to fix $n$ and
pass to the limit as $m\rightarrow\infty$. The convergence of
$Y^{m}\rightarrow Y$ being pointwise, and $Y^{m}$ being bounded by $M$, by
Lebesgue’s dominated convergence theorem, we have that

$\displaystyle\liminf_{m\rightarrow\infty}\mathbb{E}\int_{0}^{T}[3C_{0}\Phi^{\prime}(\Delta
Y^{n}_{t})+2]|\Delta Z^{m,n}_{t}|^{2}dt-
C_{0}\mathbb{E}\int_{0}^{T}\Phi^{\prime}(\Delta Y^{n}_{t})|\Delta
Z^{n}_{t}|^{2}dt$ $\displaystyle\leqslant$
$\displaystyle\mathbb{E}\Big{[}\Phi(\Delta
Y^{n}_{T})+C_{0}\int_{0}^{T}\Phi^{\prime}(\Delta
Y^{n}_{t})[1+|Z_{t}|^{2}]dt+\frac{C}{2}\int_{0}^{T}\Phi^{\prime\prime}(\Delta
Y^{n}_{t})|\Delta Y^{n}_{t}|^{2}dt\Big{]},$

and as

$\displaystyle\liminf_{m\rightarrow\infty}\bigg{[}-\mathbb{E}\int_{0}^{T}\Phi^{\prime}(\Delta
Y^{n}_{t})|\Delta
Z^{m,n}_{t}|^{2}dt\bigg{]}\leqslant-\mathbb{E}\int_{0}^{T}\Phi^{\prime}(\Delta
Y^{n}_{t})|\Delta Z^{n}_{t}|^{2}dt,$

we obtain that

$\displaystyle\liminf_{m\rightarrow\infty}\mathbb{E}\int_{0}^{T}[2C_{0}\Phi^{\prime}(\Delta
Y^{n}_{t})+2]|\Delta Z^{m,n}_{t}|^{2}dt$ (3.13) $\displaystyle\leqslant$
$\displaystyle\mathbb{E}\Big{[}\Phi(\Delta
Y^{n}_{T})+C_{0}\int_{0}^{T}\Phi^{\prime}(\Delta
Y^{n}_{t})[1+|Z_{t}|^{2}]dt+\frac{C}{2}\int_{0}^{T}\Phi^{\prime\prime}(\Delta
Y^{n}_{t})|\Delta Y^{n}_{t}|^{2}dt\Big{]}.$

Noting that by the convexity of the lower semi-continuous functional

$J(Z)=\mathbb{E}\int_{0}^{T}|\Delta Z^{n}_{t}|^{2}dt,$

we have

$\displaystyle\mathbb{E}\int_{0}^{T}|\Delta
Z^{n}_{t}|^{2}dt\leqslant\liminf_{m\rightarrow\infty}\mathbb{E}\int_{0}^{T}|\Delta
Z^{m,n}_{t}|^{2}dt.$ (3.14)

Hence, combining (3.13) and (3.14) we have that

$\displaystyle\mathbb{E}\int_{0}^{T}|\Delta
Z^{n}_{t}|^{2}dt\leqslant\mathbb{E}\Big{[}\Phi(\Delta
Y^{n}_{T})+C_{0}\int_{0}^{T}\Phi^{\prime}(\Delta
Y^{n}_{t})[1+|Z_{t}|^{2}]dt+\frac{C}{2}\int_{0}^{T}\Phi^{\prime\prime}(\Delta
Y^{n}_{t})|\Delta Y^{n}_{t}|^{2}dt\Big{]}.$

By Lebesgue’s dominated convergence theorem, and note that
$\Phi(0)=\Phi^{\prime}(0)=0$, we have that as $n\rightarrow\infty$, the right-
hand side of the above inequality converges to 0, as well as the left-hand
side. Now, passing to the limit as $n\rightarrow\infty$ on the above
inequality, we obtain that

$\limsup_{n\rightarrow\infty}\mathbb{E}\int_{0}^{T}|\Delta
Z^{n}_{t}|^{2}dt=0,$

which implies that the sequence $\\{Z^{n};n\in\mathbb{N}\\}$ strongly
converges to $Z$ in $L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$.

Step 2. The uniform convergence of a subsequence of
$\\{Y^{n};n\in\mathbb{N}\\}$ to $Y$.

At this stage we know that for all $t\in[0,T]$, ${\rm
lim}_{n\rightarrow\infty}Y^{n}_{t}=Y_{t}$, and the sequence
$\\{Z^{n};n\in\mathbb{N}\\}$ converges to $Z$ in
$L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$. Then from Lemma 2.5 of Kobylanski
[20], there exists a subsequence $\\{Z^{n_{j}};n_{j}\in\mathbb{N}\\}$ of
$\\{Z^{n};n\in\mathbb{N}\\}$ such that $Z^{n_{j}}$ converges almost surely to
$Z$ and such that

$\widetilde{Z}\triangleq\sup_{j}|Z^{n_{j}}|\in
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d}).$

For simplicity presentation, by otherwise choosing a subsequence, we would
like to still assume that the whole sequence $\\{Z^{n};n\in\mathbb{N}\\}$
converges almost surely to $Z$ in $L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$,
and thus we have

$Z^{n}\rightarrow Z\ \hbox{ a.s. }dt\otimes
d\mathbb{P}\quad~{}\hbox{and}\quad~{}\widetilde{Z}=\sup_{n}|Z^{n}|\in
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d}).$

Recall that the sequence $\\{f^{n};n\in\mathbb{N}\\}$ converges locally
uniformly to $f$, we obtain that

$\lim_{n\rightarrow\infty}f^{n}\left(t,Y_{t}^{n},Z_{t}^{n}\right)=f\left(t,Y_{t},Z_{t}\right),\quad~{}t\in[0,T],\
a.s.$

In addition, due to that $f^{n}$ satisfies the condition (3.11), one has

$\left|f^{n}\left(t,Y_{t}^{n},Z_{t}^{n}\right)\right|\leqslant
C(1+M+\sup_{n}\left|Z_{t}^{n}\right|^{2})\leqslant
C_{0}(1+\widetilde{Z}_{t}^{2}).$

Thus, Lebesgue’s dominated convergence theorem gives

$\lim_{n\rightarrow\infty}\int_{0}^{T}f^{n}(t,Y^{n}_{t},Z^{n}_{t})dt=\int_{0}^{T}f(t,Y_{t},Z_{t})dt,\quad~{}a.s.$

Similarly, we have

$\lim_{n\rightarrow\infty}\mathbb{E}\int_{0}^{T}|g(t,Y^{n}_{t},Z_{t}^{n})-g(t,Y_{t},Z_{t})|^{2}dt\leqslant\lim_{n\rightarrow\infty}\mathbb{E}\int_{0}^{T}[C|\Delta
Y^{n}_{t}|^{2}+\alpha|\Delta Z^{n}_{t}|^{2}]dt=0,$

and therefore

$g(t,Y^{n},Z^{n})\rightarrow g(t,Y,Z)\ \hbox{ a.s. }dt\otimes
d\mathbb{P}\quad~{}\hbox{and}\quad~{}\sup_{n}|g(t,Y^{n},Z^{n})|\in
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d}).$

On the other hand, from the continuity properties of stochastic integral, we
get that

$\displaystyle\lim_{n\rightarrow\infty}\sup_{0\leqslant t\leqslant
T}\Big{|}\int_{t}^{T}Z^{n}_{s}dW_{s}-\int_{t}^{T}Z_{s}dW_{s}\Big{|}=0\quad\hbox{in
probability,}$ $\displaystyle\lim_{n\rightarrow\infty}\sup_{0\leqslant
t\leqslant
T}\Big{|}\int_{t}^{T}g(s,Y^{n}_{s},Z^{n}_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}g(s,Y_{s},Z_{s})d\overleftarrow{B}_{s}\Big{|}=0\quad\hbox{in
probability.}$ (3.15)

Extracting a subsequence again if necessary, we may assume that the above
convergence is $\mathbb{P}$-a.s. Finally, we have

$\displaystyle\displaystyle|\Delta Y^{m,n}_{t}|\leqslant$
$\displaystyle~{}|\Delta
Y^{m,n}_{T}|+\int_{t}^{T}|f^{m}(s,Y^{m}_{s},Z^{m}_{s})-f^{n}(s,Y^{n}_{s},Z^{n}_{s})|ds$
$\displaystyle+\Big{|}\int_{t}^{T}[g(s,Y^{m}_{s},Z^{m}_{s})-g(s,Y^{n}_{s},Z^{n}_{s})]d\overleftarrow{B}_{s}\Big{|}+\Big{|}\int_{t}^{T}\Delta
Z^{m,n}_{s}dW_{s}\Big{|}.$

Now fix $n$, and taking limits on $m$ and supremum over $t\in[0,T]$, we obtain
that for almost all $\omega\in\Omega$,

$\displaystyle\displaystyle\sup_{0\leqslant t\leqslant T}|\Delta
Y^{n}_{t}|\leqslant$ $\displaystyle\ |\Delta
Y^{n}_{T}|+\int_{t}^{T}|f(s,Y_{s},Z_{s})-f^{n}(s,Y^{n}_{s},Z^{n}_{s})|ds$
$\displaystyle+\sup_{0\leqslant t\leqslant
T}\Big{|}\int_{t}^{T}[g(s,Y_{s},Z_{s})-g(s,Y^{n}_{s},Z^{n}_{s})]d\overleftarrow{B}_{s}\Big{|}+\sup_{0\leqslant
t\leqslant T}\Big{|}\int_{t}^{T}\Delta Z^{n}_{s}dW_{s}\Big{|},$

from which we deduce that the sequence $\\{Y^{n};n\in\mathbb{N}\\}$ converges
to $Y$ uniformly for $t\in[0,T]$. In particular, $Y$ is continuous if the
sequence $\\{Y^{n};n\in\mathbb{N}\\}$ is.

Finally, we could pass to the limit in

$\displaystyle
Y^{n}_{t}=Y^{n}_{T}+\int_{t}^{T}f^{n}(s,Y^{n}_{s},Z^{n}_{s})ds+\int_{t}^{T}g(s,Y^{n}_{s},Z^{n}_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}Z^{n}_{s}dW_{s},\quad~{}0\leqslant
t\leqslant T,$

deducing that the pair $(Y,Z)$ is a solution of BDSDE with parameters
$(f,g,\xi)$. ∎

######  Remark 3.8.

It should be point out that the solution here we obtain might not be unique,
due to that the limit generator $f$ is of quadratic growth in $z$, but not
satisfies a comparison theorem. A stability result of BDSDE will be given (see
Theorem 4.6) after the comparison theorem is proved in Section 4.

###  3.3 Existence

Based on the results of a priori estimate and the monotone stability, now we
can prove the existence of solutions of one-dimensional BDSDEs (2.1) with
quadratic growth.

######  Assumption 3.

Suppose that the terminal value $\xi$ belongs to
$L^{\infty}_{\mathscr{F}_{T}}(\Omega;\mathbb{R})$, and
$c:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is a continuous increasing
function. Moreover, there exist some constants $a$, $b$, $C$, $d\in\mathbb{R}$
and $\alpha\in(0,1)$ such that for all
$t\in[0,T],y,\bar{y}\in\mathbb{R},z,\bar{z}\in\mathbb{R}^{d}$,

$\displaystyle f(t,y,z)=a_{0}(t,y,z)y+f_{0}(t,y,z)$

with

$\displaystyle d\leqslant a_{0}(t,y,z)\leqslant
a,\quad~{}|f_{0}(t,y,z)|\leqslant
b+c(|y|)|z|^{2},\quad~{}|g(t,y,z)|^{2}\leqslant\alpha|z|^{2},\quad~{}\hbox{a.s.}$
$\displaystyle|g(t,y_{1},z_{1})-g(t,y_{2},z_{2})|^{2}\leqslant
C|y_{1}-y_{2}|^{2}+\alpha|z_{1}-z_{2}|^{2},\quad~{}\hbox{a.s.}$

######  Remark 3.9.

Under 3, the coefficient $g$ is bounded with respect to $y$ and $z$. However,
if the coefficient $g$ is independent of $y$, then it is of linear growth and
Lipschitz continuous with respect to $z$. In fact, 3 implies that

$g(t,y,0)=0,\quad~{}\forall(t,y)\in[0,T]\times\mathbb{R}.$

Here we present two examples that satisfying 3:

$\displaystyle\displaystyle\hbox{(i)}\quad~{}g(t,y,z)=$ $\displaystyle\
\alpha\sin(y)\cos(z),\quad~{}\forall(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d};$
$\displaystyle\displaystyle\hbox{(ii)}\quad~{}g(t,y,z)=$ $\displaystyle\
l(z),\quad~{}|l(z)|^{2}\leqslant\alpha|z|^{2},\quad~{}\forall(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}.$

The following theorem is the main result of this section.

######  Theorem 3.10 (Existence).

Suppose that the parameters $(f,g,\xi)$ satisfy 3, then BDSDE (2.1) has at
least one solution $(Y,Z)\in L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$, where $Y$ is continuous. Moreover,
there exists a maximal solution $(Y^{*},Z^{*})$ $($resp. a minimal solution
$(Y_{*},Z_{*}))$ such that for any set of parameters $(\hat{f},g,\hat{\xi})$,
if

$\hat{f}\leqslant f\quad\hbox{and}\quad\hat{\xi}\leqslant\xi\quad(\hbox{resp.
}\hat{f}\geqslant f\quad\hbox{and}\quad\hat{\xi}\geqslant\xi),$

one has

$\hat{Y}\leqslant Y^{*}\quad(\hbox{resp. }\hat{Y}\geqslant Y_{*}),$

where $(\hat{Y},\hat{Z})$ is any solution of BDSDE with parameters
$(\hat{f},g,\hat{\xi})$.

###### Proof.

The main idea here we used is that, first, using an exponential change and a
truncation argument in order to control the growth of the generator $f$ in $z$
and $y$, respectively. Second, finding a good approximation of $f$ and
constructing a sequence of uniformly Lipschitz continuous functions. Third,
combine the previous result (3.5 and 3.7) to get the solution.

First, instead of 3, we assume that the coefficient $f$ satisfies the
following condition: There exist some constants
$\tilde{a},\tilde{d}\in\mathbb{R}$ and $B,C\in\mathbb{R}^{+}$ such that for
all $(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$,

$\displaystyle f(t,y,z)=a_{0}(t,y,z)y+f_{0}(t,y,z)$ (3.16)

with

$\displaystyle\tilde{d}\leqslant
a_{0}(t,y,z)\leqslant\tilde{a},\quad~{}|f_{0}(t,y,z)|\leqslant
B+C|z|^{2},\quad~{}\hbox{a.s.}$ (3.17)

Let $(F,g,\zeta)$ be a set of parameters of BDSDE such that

$F\leqslant f\quad\hbox{and}\quad\zeta\leqslant\xi,$ (3.18)

and suppose that it has a solution $(\tilde{Y},\tilde{Z})\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$. The purpose is to find a solution
$(Y,Z)\in L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ of BDSDE with parameters $(f,g,\xi)$
such that

$\tilde{Y}\leqslant Y.$

From 3.5, if we set

$\tilde{K}=(\|\xi\|_{\infty}+BT)\exp(\tilde{a}^{+}T),$

then any solution $(Y,Z)$ of BDSDE with parameters $(f,g,\xi)$ has the
following estimate,

$\|Y\|_{\infty}\leqslant\tilde{K}.$

Define

$M=\max(\tilde{K},\|\tilde{Y}\|_{\infty}).$ (3.19)

Step 1. The exponential transformation.

In order to make a presentation more clearly, we rewrite BDSDE (2.1) as
follows,

$\displaystyle
Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds+\int_{t}^{T}g(s,Y_{s},Z_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}Z_{s}dW_{s},\quad~{}0\leqslant
t\leqslant T.$ (3.20)

Let $\beta$ be a positive constant which will be determined later. Applying
the exponential change of variable (see 2.2) to $u=\exp(\beta Y)$ transforms
formally BDSDE (3.20) with parameters $(f,g,\xi)$ in a BDSDE with parameters
$(\bar{f},\bar{g},\bar{\xi})$ of the following,

$\displaystyle
u_{t}=\bar{\xi}+\int_{t}^{T}\bar{f}(s,u_{s},v_{s})ds+\int_{t}^{T}\bar{g}(s,u_{s},v_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}v_{s}dW_{s},\quad~{}0\leqslant
t\leqslant T,$ (3.21)

where

$\displaystyle\bar{f}(t,u,v)=\beta uf(t,\frac{\ln u}{\beta},\frac{v}{\beta
u})+\frac{1}{2}\beta^{2}u\Big{[}g(t,\frac{\ln u}{\beta},\frac{v}{\beta
u})^{2}-\Big{(}\frac{v}{\beta u}\Big{)}^{2}\Big{]},$ (3.22)
$\displaystyle\bar{g}(t,u,v)=\beta ug(t,\frac{\ln u}{\beta},\frac{v}{\beta
u}),\qquad\bar{\xi}=\exp(\beta\xi),\qquad v=\beta uZ.$

It is easy to check that $\bar{g}$ and $\bar{\xi}$ satisfy 3. Similarly, the
exponential change $u=\exp(\beta\tilde{Y})$ transforms formally the BDSDE with
parameters $(F,g,\zeta)$ in a BDSDE with parameters
$(\bar{F},\bar{g},\bar{\zeta})$ where

$\displaystyle\bar{F}(t,u,v)=\beta uF(s,\frac{\ln u}{\beta},\frac{v}{\beta
u})+\frac{1}{2}\beta^{2}u\Big{[}g(s,\frac{\ln u}{\beta},\frac{v}{\beta
u})^{2}-\Big{(}\frac{v}{\beta
u}\Big{)}^{2}\Big{]},\qquad\bar{\zeta}=\exp(\beta\zeta).$

We consider a function $\Phi:\mathbb{R}\rightarrow[0,1]$ such that

$\Phi(u)=\left\\{\begin{array}[]{ll}\displaystyle 1,\qquad\hbox{if
}u\in[\exp(-\beta M),\exp(\beta M)],\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle 0,\qquad\hbox{if
}u\notin[\exp(-\beta(M+1)),\exp(\beta(M+1))],\end{array}\right.$

and we set that for $(t,u,v)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$,

$\tilde{f}(t,u,v)=\Phi(u)\bar{f}(t,u,v),\quad~{}\tilde{F}(t,u,v)=\Phi(u)\bar{F}(t,u,v),\quad~{}l(u)=\Phi(u)\big{(}\tilde{a}u\ln
u+B\beta u).$

Now, let

$\beta=\frac{2C}{1-\alpha},$ (3.23)

then, note that (3.17), we have

$\Phi(u)\Big{(}\tilde{d}u\ln u-B\beta
u-(\frac{C}{\beta}+\frac{1}{2})\frac{|v|^{2}}{u}\Big{)}\leqslant\tilde{f}(t,u,v)\leqslant
l(u).$

We remark that $l$ is a Lipschitz continuous function bounded from above by a
constant, denoted by $L$. Moreover, note that if we define

$\tilde{y}=\exp(\beta\tilde{Y})\quad~{}\hbox{and}\quad~{}\tilde{z}=\beta\tilde{Z}\exp(\beta\tilde{Y}),$

then the pair $(\tilde{y},\tilde{z})$ is a solution of BDSDE with parameters
$(\tilde{F},\bar{g},\bar{\zeta})$.

Step 2. The approximation.

We can approximate $\tilde{f}$ by a decreasing sequence of uniformly Lipschitz
continuous functions $\\{\tilde{f}^{n};n\in\mathbb{N}\\}$ such that for all
$(t,u,v)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$,

$\tilde{F}(t,u,v)\leqslant\tilde{f}(t,u,v)\leqslant\tilde{f}^{n}(t,u,v)\leqslant
l(u)+\frac{1}{2^{n}}.$

Indeed, such a sequence exists. For instance, denote by $m$ an integral, and
for each integral $n\geqslant m$, we would like to define the sequence of the
functions $\\{\tilde{f}^{n};n\in\mathbb{N}\\}$ by

$\tilde{f}_{n}(t,y,z)=\sup\left\\{\tilde{f}(t,p,q)-n|p-y|-n|q-z|:(p,q)\in\mathbb{Q}^{1+d}\right\\},$
(3.24)

where $\mathbb{Q}$ denotes the rational number. It is easy to see that
$\tilde{f}_{n}$ is well defined and it is globally Lipschitz continuous in $y$
and $z$ with the constant $n$. In addition, the sequence
$\\{\tilde{f}^{n};n\geqslant m\\}$ is decreasing and converges pointwise to
the function $\tilde{f}$.

Then the classical results of the existence and uniqueness (see 2.3) and the
comparison theorem (see 2.4) for Lipschitz continuous coefficients give us
that, for every $n\in\mathbb{N}$, there exists a unique solution
$(y^{n},z^{n})$ of the BDSDE with parameters
$(\tilde{f}^{n},\bar{g},\bar{\xi})$ such that

$\tilde{y}\leqslant y^{n+1}\leqslant y^{n}\leqslant y^{1}.$

Next, we indicate that the sequence $\\{y^{n};n\in\mathbb{N}\\}$ is bounded
and positive. It should be point out that by the definition of $\bar{g}$ (see
(3.22)) and 3, it is certainly that

$|\bar{g}(t,u,v)|^{2}\leqslant\alpha|v|^{2},\quad~{}\forall(t,u,v)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d},$
(3.25)

which implies that $\bar{g}(t,u,v)=0$ when $v=0$. Hence, again, 2.3 implies
that, on the one hand, the pair of processes $\big{\\{}\big{(}\exp(\beta
M),0\big{)};t\in[0,T]\big{\\}}$ is the unique solution of BDSDE with
parameters $\big{(}0,\bar{g},\exp(\beta M)\big{)}$; on the other hand, the
pair of processes $\big{\\{}\big{(}\exp(-\beta M),0\big{)};t\in[0,T]\big{\\}}$
is the unique solution of BDSDE with parameters $\big{(}0,\bar{g},\exp(-\beta
M)\big{)}$. Moreover, since for $n\in\mathbb{N}$ large enough,

$\exp(\beta\xi)\leqslant\exp(\beta M)\quad\text{ and
}\quad\tilde{f}^{n}\big{(}t,\exp(\beta M),0\big{)}\leqslant
0,\quad~{}t\in[0,T],$

and for all $n$,

$\exp(-\beta M)\leqslant\exp(\beta\xi)\quad\text{ and }\quad
0\leqslant\tilde{f}^{n}\big{(}t,\exp(-\beta M),0\big{)},\quad~{}t\in[0,T],$

then 2.4 gives us that for $n\in\mathbb{N}$ large enough,

$\exp(-\beta M)\leqslant y^{n+1}_{t}\leqslant y^{n}_{t}\leqslant\exp(\beta
M),\quad~{}\hbox{a.s.}\ t\in[0,T].$ (3.26)

Now, we come back to the previous problem by denoting

$\displaystyle\displaystyle\tilde{G}^{n}(t,y,z)$
$\displaystyle=\frac{\tilde{f}^{n}(t,e^{\beta y},\beta e^{\beta y}z)}{\beta
e^{\beta y}}+\frac{\beta}{2}\big{[}z^{2}-g(t,y,z)^{2}\big{]},$
$\displaystyle\displaystyle\tilde{G}(t,y,z)$
$\displaystyle=\frac{\tilde{f}(t,e^{\beta y},\beta e^{\beta y}z)}{\beta
e^{\beta y}}+\frac{\beta}{2}\big{[}z^{2}-g(t,y,z)^{2}\big{]}$
$\displaystyle=\Phi(e^{\beta y})f(t,y,z)+(1-\Phi(e^{\beta
y}))\frac{\beta}{2}\big{[}z^{2}-g(t,y,z)^{2}\big{]}.$

Notice that $\beta$ is defined in (3.23). Then, the pair $(Y^{n},Z^{n})$
defined by

$Y^{n}_{t}=\frac{\ln
y^{n}_{t}}{\beta}\quad~{}\hbox{and}\quad~{}Z^{n}_{t}=\frac{z^{n}_{t}}{\beta
y^{n}_{t}}$

is a solution of the BDSDE with parameters $(\tilde{G}^{n},g,\xi)$. Let us
recapitulate what we have obtained.

  * (i)

There exist two positive constants $K$ and $C$ such that for all
$(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$,

$|\tilde{G}^{n}(t,y,z)|\leqslant K+C|z|^{2},\quad n\in\mathbb{N}.$

  * (ii)

The sequence $\\{\tilde{G}^{n};n\in\mathbb{N}\\}$ converges to $\tilde{G}$
locally uniformly on $[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$.

  * (iii)

For each $n\in\mathbb{N}$, the BDSDE with parameters $(\tilde{G}^{n},g,\xi)$
has a solution

$(Y^{n},Z^{n})\in L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d}),$

such that the sequence $\\{Y^{n};n\in\mathbb{N}\\}$ is decreasing, and there
exists a constant $\bar{K}>0$ such that for every $n\in\mathbb{N}$, we have
that $\|Y^{n}\|_{\infty}\leqslant\bar{K}$.

  * (iv)

For each $n\in\mathbb{N}$, $\tilde{Y}\leqslant Y^{n}$.

Therefore, applying 3.7, we have that there exists a pair of processes
$(Y,Z)\in L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ such that, the sequence
$\\{Y^{n};n\in\mathbb{N}\\}$ converges uniformly to $Y$, the sequence
$\\{Z^{n};n\in\mathbb{N}\\}$ converges to $Z$ in
$L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$, and $(Y,Z)$ is a solution of BDSDE
with parameters $(\tilde{G},g,\xi)$, i.e.,

$\displaystyle
Y_{t}=\xi+\int_{t}^{T}\tilde{G}(s,Y_{s},Z_{s})ds+\int_{t}^{T}g(s,Y_{s},Z_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}Z_{s}dW_{s},\quad~{}0\leqslant
t\leqslant T.$

Finally, we declare that

$\|Y\|_{\infty}\leqslant M.$

In fact, it follows from 3.5 as

$\displaystyle\tilde{G}(t,y,z)=\tilde{a}_{0}(t,y,z)y+\tilde{f}_{0}(t,y,z)$

with

$\displaystyle\tilde{a}_{0}(t,y,z)=\Phi(e^{\beta y})a_{0}(t,y,z)\leqslant
a^{+},$

and

$\displaystyle\tilde{f}_{0}(t,y,z)=\Phi(e^{\beta
y})f_{0}(t,y,z)+(1-\Phi(e^{\beta
y}))\frac{\beta}{2}\big{[}z^{2}-g(t,y,z)^{2}\big{]},$

where the functions $a_{0}$ and $f_{0}$ come from (3.16). Then, notice that
(3.23), we have

$\displaystyle|\tilde{f}_{0}(t,y,z)|\leqslant B+\frac{3C}{1-\alpha}|z|^{2}.$

So, by 3.5, we obtain

$\|Y\|_{\infty}\leqslant(\|\xi\|_{\infty}+BT)\exp(a^{+}T)=\tilde{K}\leqslant
M.$

Hence the pair $(Y,Z)$ is also a solution of the BDSDE with parameters
$(f,g,\xi)$. In addition,

$\tilde{Y}\leqslant Y\quad~{}\hbox{and}\quad~{}\|Y\|_{\infty}\leqslant M.$

Step 3. The truncation.

Finally, we suppose that the generator $f$ satisfies 3. Let $(F,g,\zeta)$ be a
set of parameters of BDSDE such that the condition (3.18) hold true and denote
by $(\tilde{Y},\tilde{Z})\in L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ a solution of BDSDE with the
parameters $(F,g,\zeta)$. For all
$(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$, we set

$\check{f}(t,y,z)=a_{0}(t,y,z)y+f_{0}(t,\phi_{M}(y)y,z)\quad~{}\hbox{and}\quad~{}\check{F}=F(t,\phi_{M}(y)y,z),$

where $M$ is defined in (3.19), and

$\phi_{n}(u)=\left\\{\begin{array}[]{ll}\displaystyle 1,\qquad\hbox{if
}|u|\leqslant n,\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
0,\qquad\hbox{if }|u|\geqslant n+1.\end{array}\right.$

It is easy to check that the generator $\check{f}$ satisfies the condition
(3.17), and $(\tilde{Y},\tilde{Z})$ is also a solution of the BDSDE with
parameters $(\check{F},g,\zeta)$. Then, by Step 1 and Step 2, we deduce that
BDSDE with parameters $(\check{f},g,\xi)$ has a solution $(Y,Z)$ such that

$\tilde{Y}\leqslant Y\quad~{}\hbox{and}\quad~{}\|Y\|_{\infty}\leqslant M,$

which implies that the pair of process $(Y,Z)$ is also a solution of BDSDE
with parameters $(f,g,\xi)$. then the existence of a maximal solution is
proved.

Similarly, by using the same argument but with the change of variable
$u=\exp(-\beta y)$, one can prove the existence of a minimal solution. This
completes the proof. ∎

######  Remark 3.11.

Some ideas for proving Theorem 3.10 are borrowed from Kobylanski [20],
however, due to the emergence of the backward Itô’s integral appeared in BDSDE
(2.1), some new ideas and technics have been introduced and used. For example,
the main idea of Kobylanski [20] is to take the exponential change to
transform a quadratic BDSDE into a classical BDSDE essentially. While, this is
not enough for us, since the presence of the backward Itô’s integral. In order
to overcome the troubles, we introduce the idea of the comparison theorem of
classical BDSDEs, which is evident from 3.1.

##  4 Comparison Theorem

As we all know that the one-dimensional circumstance allows us to establish a
comparison theorem for the solutions of BDSDEs, which implies the uniqueness
of solutions of BDSDEs as a by-product. So in this section, we would like to
prove a comparison theorem for the solutions of BDSDEs with quadratic growth.
For $i=1,2$, we consider the following equation:

$Y_{t}^{i}=\xi^{i}+\int_{t}^{T}f^{i}\left(s,Y_{s}^{i},Z_{s}^{i}\right)ds+\int_{t}^{T}g\left(s,Y_{s}^{i},Z_{s}^{i}\right)d\overleftarrow{B}_{s}-\int_{t}^{T}Z_{s}^{i}dW_{s},\quad~{}0\leqslant
t\leqslant T.$ (4.1)

Then, under 3, there exists a pair of measurable processes $(Y^{i},Z^{i})\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ that satisfies the corresponding BDSDE
(4.1), where $i=1,2$. Moreover, 3.5 implies that there exists a positive
constant $K$ such that

$\|Y^{i}\|_{\infty}+\mathbb{E}\int_{0}^{t}|Z^{i}_{s}|^{2}ds\leqslant
K,\quad~{}\forall t\in[0,T].$

In order to prove the comparison theorem, we assume that the terminal value
$\xi$ is bounded, and the generator $f$ is locally Lipschitz continuous and is
of quadratic growth in $Z$ in a strong sense.

######  Assumption 4.

There exist three functions $l$, $l_{\varepsilon}$,
$k:[0,T]\rightarrow\mathbb{R}$ and three positive constants $C$,
$\varepsilon$, and $\alpha\in(0,1)$ such that for all $t\in[0,T],\ y\in[-M,M]$
and $z\in\mathbb{R}^{d}$, the coefficients $f$ and $g$ satisfy the following
conditions:

$\displaystyle|f(t,y,z)|$ $\displaystyle\leqslant l(t)+C|z|^{2},\qquad\
|g(t,y,z)|^{2}\leqslant\alpha|z|^{2},\quad~{}\text{ a.s., }$ (4.2)
$\displaystyle\frac{\partial f}{\partial y}(t,y,z)$ $\displaystyle\leqslant
l_{\varepsilon}(t)+\varepsilon|z|^{2},\quad~{}\Big{|}\frac{\partial
g}{\partial y}(t,y,z)\Big{|}^{2}\leqslant C,\quad~{}\text{a.s.},$
$\displaystyle\left|\frac{\partial f}{\partial z}(t,y,z)\right|$
$\displaystyle\leqslant k(t)+C|z|,\quad\ \ \Big{|}\frac{\partial g}{\partial
z}(t,y,z)\Big{|}^{2}\leqslant\alpha.\quad~{}\text{ a.s. }$

Similar to Kobylanski [20], following the method used by Barles–Murat [4] for
PDEs, we would like to divide the proof of the comparison theorem into two
steps. First, we consider the comparison theorem under the condition that the
generator $f$ satisfies a structure condition (STR). Then, we show the proof
by giving a change of variable that transforms a BDSDE with the generator $f$
satisfying 4 into a BDSDE with the generator $\widetilde{f}$ satisfying this
structure condition (STR).

######  Definition 4.1.

We say that a generator $f$ satisfies the structure condition (STR), if there
exists a constant $a>0$ and a function $b:[0,T]\rightarrow\mathbb{R}$ such
that

$\frac{\partial f}{\partial y}(t,y,z)+a\left|\frac{\partial f}{\partial
z}\right|^{2}(t,y,z)\leqslant
b(t),\quad~{}(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}.$ (4.3)

We have the following comparison theorem under the structure condition (STR).

######  Proposition 4.2.

Assume that both $\xi^{1}$ and $\xi^{2}$ are bounded, the coefficient $g$
satisfies 4, and either $f^{1}$ or $f^{2}$ satisfies the structure condition
(STR) with the constant $a>0$ and the function $b\in L^{1}([0,T];\mathbb{R})$.
For $i=1,2$, denote by $(Y^{i},Z^{i})$ a solution of BDSDE (4.1) with
parameters $(f^{i},g,\xi^{i})$, respectively. Moreover, suppose that for all
$(t,y,z)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$,

$\xi^{1}\leqslant\xi^{2},\quad~{}f^{1}(t,y,z)\leqslant
f^{2}(t,y,z),\quad~{}\text{a.s.},$ (4.4)

then for any $t\in[0,T]$, we have

$\displaystyle Y_{t}^{1}\leqslant Y_{t}^{2},\quad\hbox{a.s.}$

######  Remark 4.3.

(i) It should be point out that in the condition (STR), when
$a>1/(p-1)(1-\alpha)$ and the function $b$ is bounded, 4.2 holds true in the
space $L_{\mathbb{F}}^{p}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$.

(ii) 4.2 still holds true if $f^{1}\left(t,Y_{t}^{2},Z_{t}^{2}\right)\leqslant
f^{2}\left(t,Y_{t}^{2},Z_{t}^{2}\right)$ a.s. for all $t\in[0,T]$ and the
generator $f^{1}$ satisfies the condition (STR), or if either
$f^{1}\left(t,Y_{t}^{1},Z_{t}^{1}\right)\leqslant
f^{2}\left(t,Y_{t}^{1},Z_{t}^{1}\right)$ a.s. for all $t\in[0,T]$ and the
generator $f^{2}$ satisfies the condition (STR).

###### Proof of 4.2.

For simplicity of presentation, we denote that for $t\in[0,T]$,

$\widehat{Y}_{t}=Y^{1}_{t}-Y^{2}_{t}\quad~{}\hbox{and}\quad~{}\widehat{Z}_{t}=Z^{1}_{t}-Z^{2}_{t},$

and for any $s\in[t,T]$,

$\widehat{f}(s)=f^{1}(s,Y^{1}_{s},Z^{1}_{s})-f^{2}(s,Y^{2}_{s},Z^{2}_{s})\quad~{}\hbox{and}\quad~{}\widehat{g}(s)=g(s,Y^{1}_{s},Z^{1}_{s})-g(s,Y^{2}_{s},Z^{2}_{s}).$

Then, note that $\widehat{Y}^{+}=\max\\{0,\widehat{Y}\\}$, using Itô’s formula
(see 2.2) to $\big{(}\widehat{Y}^{+}\big{)}^{p}$ for $p\in\mathbb{N}$ with
$p>2$ implies that

$\displaystyle(\widehat{Y}_{t}^{+})^{p}=$ $\displaystyle\
p\int_{t}^{T}(\widehat{Y}^{+}_{s})^{p-1}\widehat{f}(s)ds+p\int_{t}^{T}(\widehat{Y}^{+}_{s})^{p-1}\widehat{g}(s)d\overleftarrow{B}_{s}-p\int_{t}^{T}(\widehat{Y}^{+}_{s})^{p-1}\widehat{Z}_{s}dW_{s}$
(4.5)
$\displaystyle+\frac{p(p-1)}{2}\int_{t}^{T}(\widehat{Y}^{+}_{s})^{p-2}|\widehat{g}(s)|^{2}ds-\frac{p(p-1)}{2}\int_{t}^{T}(\widehat{Y}^{+}_{s})^{p-2}|\widehat{Z}_{s}|^{2}ds.$

For the term $\widehat{f}$ in the above, note (4.4), we have that

$\displaystyle\widehat{f}(s)=$
$\displaystyle~{}f^{1}(s,Y^{1}_{s},Z^{1}_{s})-f^{2}(s,Y^{2}_{s},Z^{2}_{s})$
$\displaystyle=$
$\displaystyle~{}f^{1}(s,Y^{1}_{s},Z^{1}_{s})-f^{2}(s,Y^{1}_{s},Z^{1}_{s})+f^{2}(s,Y^{1}_{s},Z^{1}_{s})-f^{2}(s,Y^{2}_{s},Z^{2}_{s})$
$\displaystyle\leqslant$
$\displaystyle~{}f^{2}(s,Y^{1}_{s},Z^{1}_{s})-f^{2}(s,Y^{2}_{s},Z^{2}_{s})$
$\displaystyle\leqslant$ $\displaystyle~{}\Big{(}\int_{0}^{1}\frac{\partial
f^{2}}{\partial
y}(*)d\lambda\Big{)}\widehat{Y}_{s}+\Big{(}\int_{0}^{1}\frac{\partial
f^{2}}{\partial z}(*)d\lambda\Big{)}\widehat{Z}_{s},$

where

$(*)=(s,\lambda Y^{1}_{s}+(1-\lambda)Y^{2}_{s},\lambda
Z^{1}_{s}+(1-\lambda)Z^{2}_{s}).$

Then, for the first term in the right hand side of (4.5), we have that

$\displaystyle(\widehat{Y}^{+}_{s})^{p-1}\widehat{f}(s)\leqslant(\widehat{Y}^{+}_{s})^{p}\int_{0}^{1}\bigg{(}\frac{\partial
f^{2}}{\partial y}+a\left|\frac{\partial f^{2}}{\partial
z}\right|^{2}\bigg{)}(*)d\lambda+\frac{1}{4a}(\widehat{Y}^{+}_{s})^{p-2}|\widehat{Z}_{s}|^{2}{\bf
1}_{\\{\widehat{Y}^{+}\geqslant 0\\}},$ (4.6)

where we have used the well-known inequality
$2\beta\gamma\leqslant|\beta|^{2}+|\gamma|^{2}$ with

$\displaystyle\beta=\sqrt{2a}\frac{\partial f^{2}}{\partial
z}(*)\left(Y_{s}^{+}\right)^{p/2},$
$\displaystyle\gamma=\frac{1}{\sqrt{2a}}\left(Y_{s}^{+}\right)^{(p-2)/2}Z_{s}\mathbf{1}_{\left\\{Y^{+}\geqslant
0\right\\}}.$

For the term $\widehat{g}$ in (4.5), note 4 satisfied by the coefficient $g$,
we have that

$\displaystyle|\widehat{g}(s)|^{2}=$
$\displaystyle~{}\bigg{\\{}\Big{(}\int_{0}^{1}\frac{\partial g}{\partial
y}(*)d\lambda\Big{)}\widehat{Y}_{s}+\Big{(}\int_{0}^{1}\frac{\partial
g}{\partial z}(*)d\lambda\Big{)}\widehat{Z}_{s}\bigg{\\}}^{2}$
$\displaystyle\leqslant$ $\displaystyle~{}\Big{(}\int_{0}^{1}\frac{\partial
g}{\partial
y}(*)d\lambda\Big{)}^{2}|\widehat{Y}_{s}|^{2}+2\frac{\sqrt{\varepsilon}}{\sqrt{\varepsilon}}\Big{(}\int_{0}^{1}\frac{\partial
g}{\partial y}(*)d\lambda\Big{)}\Big{(}\int_{0}^{1}\frac{\partial g}{\partial
z}(*)d\lambda\Big{)}|\widehat{Y}_{s}\widehat{Z}_{s}|+\Big{(}\int_{0}^{1}\frac{\partial
g}{\partial z}(*)d\lambda\Big{)}^{2}|\widehat{Z}_{s}|^{2}$
$\displaystyle\leqslant$
$\displaystyle~{}\widetilde{C}|\widehat{Y}_{s}|^{2}+(\alpha+\varepsilon)|\widehat{Z}_{s}|^{2},$

where

$\widetilde{C}=C+\frac{C\alpha}{\varepsilon}\quad~{}\hbox{and}\quad~{}\varepsilon=\frac{1-\alpha}{2}.$

So, for the fourth term on the right hand side of (4.5), we have that

$\displaystyle(\widehat{Y}^{+}_{s})^{p-2}|\widehat{g}(s)|^{2}\leqslant\widetilde{C}(\widehat{Y}^{+}_{s})^{p}+\frac{1+\alpha}{2}(\widehat{Y}^{+}_{s})^{p-2}|\widehat{Z}_{s}|^{2}{\bf
1}_{\\{\widehat{Y}^{+}\geqslant 0\\}}.$ (4.7)

Combining (4.5)-(4.7) and (STR), one has

$\displaystyle(\widehat{Y}_{t}^{+})^{p}+\frac{p}{4}\left((p-1)(1-\alpha)-\frac{1}{a}\right)\int_{t}^{T}(\widehat{Y}^{+}_{s})^{p-2}|\widehat{Z}_{s}|^{2}{\bf
1}_{\\{\widehat{Y}^{+}\geqslant 0\\}}ds$ (4.8) $\displaystyle\leqslant$
$\displaystyle\
p\int_{t}^{T}\big{[}b(s)+\frac{1}{2}(p-1)\widetilde{C}\big{]}(\widehat{Y}^{+}_{s})^{p}ds+p\int_{t}^{T}(\widehat{Y}^{+}_{s})^{p-1}\widehat{g}(s)d\overleftarrow{B}_{s}-p\int_{t}^{T}(\widehat{Y}^{+}_{s})^{p-1}\widehat{Z}_{s}dW_{s},$

where the function $b$ comes from (4.3). Note that $\widehat{Y}$ is bounded,
which implies that $(\widehat{Y}^{+}_{s})^{p-1}\widehat{g}(s)$ and
$(\widehat{Y}^{+}_{s})^{p-1}\widehat{Z}_{s}$ belong to the space
$L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{d})$. Take the expectation on both side of
(4.8),

$\displaystyle\mathbb{E}(\widehat{Y}_{t}^{+})^{p}+\frac{p}{4}\left((p-1)(1-\alpha)-\frac{1}{a}\right)\mathbb{E}\int_{t}^{T}(\widehat{Y}^{+}_{s})^{p-2}|\widehat{Z}_{s}|^{2}{\bf
1}_{\\{\widehat{Y}^{+}\geqslant 0\\}}ds$ $\displaystyle\leqslant$
$\displaystyle\
p\int_{t}^{T}\big{[}b(s)+\frac{1}{2}(p-1)\widetilde{C}\big{]}(\widehat{Y}^{+}_{s})^{p}ds.$

It is easy to choose a large enough $p$ such that

$\frac{p}{4}\left((p-1)(1-\alpha)-\frac{1}{a}\right)\geqslant 0.$

Finally, Gronwall’s inequality deduce that

$\mathbb{E}(\widehat{Y}_{t}^{+})^{p}\leqslant 0,\quad~{}\forall t\in[0,T].$

Therefore, for all $t\in[0,T]$ we have that

$Y^{1}_{t}\leqslant Y^{2}_{t},\quad~{}a.s.$

This completes the proof. ∎

Based on the first step concerning the structure condition (STR), next we are
going to prove the comparison theorem under 4.

######  Theorem 4.4 (Comparison theorem).

Let $(\xi^{1},f^{1},g)$ and $(\xi^{2},f^{2},g)$ be two parameters of BDSDE
(4.1) with bounded terminal values, and suppose that

  * (i)

$\xi^{1}\leqslant\xi^{2}$, a.s., and $f^{1}(t,y,z)\leqslant f^{2}(t,y,z)$,
a.s., for all $(t,y,z)\in[0,T]\times[-M,M]\times\mathbb{R}^{d}.$

  * (ii)

Either $(f^{1},g)$ or $(f^{2},g)$ satisfies 4.

Then if $(Y^{1},Z^{1})$ and $(Y^{2},Z^{2})$ are the associated solutions of
BDSDE (4.1) with parameters $(\xi^{1},f^{1},g)$ and $(\xi^{2},f^{2},g)$,
respectively, one has that for any $t\in[0,T]$,

$\displaystyle Y_{t}^{1}\leqslant Y_{t}^{2},\quad~{}a.s.$

######  Remark 4.5.

We point out that Theorem 4.4 still holds true if
$f^{1}\left(t,Y_{t}^{2},Z_{t}^{2}\right)\leqslant
f^{2}\left(t,Y_{t}^{2},Z_{t}^{2}\right)$ a.s. for all $t\in[0,T]$ and the
generator $f^{1}$ satisfies 4, or if either
$f^{1}\left(t,Y_{t}^{1},Z_{t}^{1}\right)\leqslant
f^{2}\left(t,Y_{t}^{1},Z_{t}^{1}\right)$ a.s. for all $t\in[0,T]$ and the
generator $f^{2}$ satisfies 4.

###### Proof of Theorem 4.4.

The main idea of the proof is to look for a change of variable, which could
transform a coefficient $f$ that satisfies 4 into a coefficient
$\widetilde{f}$ that satisfies the structure condition (STR), and then use 4.2
to getting the goal.

Let $(Y,Z)\in L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ be a solution of BDSDE (4.1) with
parameters $(\xi,f,g)$, where $\xi$ is a bounded terminal value. Take
$M\in\mathbb{R}$ such that $\|Y\|_{\infty}<M$, and consider the change of
variable $y=\phi(\widetilde{y})$, where $\phi$ is a regular increasing
function yet to be chosen. Denote

$\displaystyle\displaystyle
Y=\phi(\widetilde{Y}),\quad~{}w(Y)=\phi^{\prime}(\widetilde{Y}),\quad~{}Z=\phi^{\prime}(\widetilde{Y})\widetilde{Z}=w(Y)\widetilde{Z},$

then

$\displaystyle\widetilde{Y}=\phi^{-1}(Y),\quad~{}\widetilde{Z}=\frac{Z}{\phi^{\prime}(\widetilde{Y})}=\frac{Z}{w(Y)},$

where $(\widetilde{Y},\widetilde{Z})$ is a solution of the following BDSDE
with parameters $(\widetilde{\xi},\widetilde{f},\widetilde{g})$:

$\widetilde{Y}_{t}=\widetilde{\xi}+\int_{t}^{T}\widetilde{f}(s,\widetilde{Y}_{s},\widetilde{Z}_{s})ds+\int_{t}^{T}\widetilde{g}(s,\widetilde{Y}_{s},\widetilde{Z}_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}\widetilde{Z}_{s}dW_{s},$
(4.9)

where

$\displaystyle\widetilde{\xi}$ $\displaystyle=\phi^{-1}(\xi),$ (4.10)
$\displaystyle\widetilde{g}(t,\widetilde{y},\widetilde{z})$
$\displaystyle=\frac{g(t,y,z)}{\phi^{\prime}(\widetilde{y})}=\frac{g(t,y,z)}{w(y)}=\frac{g\big{(}t,\phi(\widetilde{y}),\phi^{\prime}(\widetilde{y})\widetilde{z}\big{)}}{\phi^{\prime}(\widetilde{y})},$
$\displaystyle\widetilde{f}(t,\widetilde{y},\widetilde{z})$
$\displaystyle=\frac{1}{\phi^{\prime}(\widetilde{y})}\left(f\left(t,\phi(\widetilde{y}),\phi^{\prime}(\widetilde{y})\widetilde{z}\right)+\frac{1}{2}\phi^{\prime\prime}(\widetilde{y})\left[\widetilde{z}^{2}-\widetilde{g}(t,\widetilde{y},\widetilde{z})^{2}\right]\right).$

In fact, applying Itô’s formula to $\phi(\widetilde{Y})$, we get that

$\displaystyle\phi(\widetilde{Y}_{t})=$ $\displaystyle\
\phi(\widetilde{\xi})+\int_{t}^{T}\left[\phi^{\prime}(\widetilde{Y}_{s})\widetilde{f}(s,\widetilde{Y}_{s},\widetilde{Z}_{s})+\frac{1}{2}\phi^{\prime\prime}(\widetilde{Y}_{s})\widetilde{g}(s,\widetilde{Y}_{s},\widetilde{Z}_{s})^{2}-\frac{1}{2}\phi^{\prime\prime}(\widetilde{Y}_{s})\widetilde{Z}_{s}^{2}\right]ds$
$\displaystyle\
+\int_{t}^{T}\phi^{\prime}(\widetilde{Y}_{s})\widetilde{g}(s,\widetilde{Y}_{s},\widetilde{Z}_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}\phi^{\prime}(\widetilde{Y}_{s})\widetilde{Z}_{s}dW_{s},$

which, note that $Y=\phi(\widetilde{Y})$, comparing with BDSDE (4.1) implies
(4.10). In other words, BDSDEs (4.1) and (4.9) are equivalent. It is easy to
check that the coefficient $\widetilde{g}$ satisfies 4. Next, we are going to
verify that the coefficient $\widetilde{f}$ given by (4.10) satisfies the
structure condition (STR). Note that

$\displaystyle
y=\phi(\widetilde{y}),\quad~{}w(y)=\phi^{\prime}(\widetilde{y})=w(\phi(\widetilde{y})),\quad~{}z=\phi^{\prime}(\widetilde{y})\widetilde{z}=w(y)\widetilde{z}=w(\phi(\widetilde{y}))\widetilde{z},$
$\displaystyle\frac{\partial
y}{\partial\widetilde{y}}=w,\quad~{}\frac{\partial
z}{\partial\widetilde{y}}=w^{\prime}w\widetilde{z},\quad~{}\frac{\partial
z}{\partial\widetilde{z}}=w,$
$\displaystyle\phi^{\prime\prime}(\widetilde{y})=\frac{\partial
w}{\partial\widetilde{y}}=\frac{\partial w}{\partial y}\cdot\frac{\partial
y}{\partial\widetilde{y}}=w^{\prime}w,\quad~{}\frac{\partial
w^{\prime}}{\partial\widetilde{y}}=\frac{\partial w^{\prime}}{\partial
y}\cdot\frac{\partial y}{\partial\widetilde{y}}=w^{\prime\prime}w,$
$\displaystyle\frac{\partial\widetilde{g}}{\partial\widetilde{y}}(t,\widetilde{y},\widetilde{z})=\frac{\partial\widetilde{g}}{\partial
y}(t,\widetilde{y},\widetilde{z})\cdot\frac{\partial
y}{\partial\widetilde{y}}=\frac{\partial\big{(}g(t,y,z)/w\big{)}}{\partial
y}\cdot\frac{\partial y}{\partial\widetilde{y}}=\frac{\partial g}{\partial
y}(t,y,z)-\frac{w^{\prime}}{w}g(t,y,z),$
$\displaystyle\frac{\partial\widetilde{g}}{\partial\widetilde{z}}(t,\widetilde{y},\widetilde{z})=\frac{\partial\widetilde{g}}{\partial
z}(t,\widetilde{y},\widetilde{z})\cdot\frac{\partial
z}{\partial\widetilde{z}}=\frac{\partial\big{(}g(t,y,z)/w\big{)}}{\partial
z}\cdot\frac{\partial z}{\partial\widetilde{z}}=\frac{\partial g}{\partial
z}(t,y,z).$

From (4.10), we could compute that

$\displaystyle\displaystyle\frac{\partial\widetilde{f}}{\partial\widetilde{y}}(t,\widetilde{y},\widetilde{z})$
$\displaystyle=-\frac{w^{\prime}}{w}\left(f\left(t,y,z\right)+\frac{1}{2}w^{\prime}w[\widetilde{z}^{2}-\widetilde{g}(t,\widetilde{y},\widetilde{z})^{2}]\right)$
$\displaystyle\quad\ +\frac{1}{w}\bigg{(}\frac{\partial f}{\partial
y}(t,y,z)w+\frac{\partial f}{\partial
z}(t,y,z)w^{\prime}w\widetilde{z}+\frac{1}{2}[w^{\prime\prime}w^{2}+(w^{\prime})^{2}w]\cdot[\widetilde{z}^{2}-\widetilde{g}(t,\widetilde{y},\widetilde{z})^{2}]$
$\displaystyle\quad\
+w^{\prime}w\widetilde{g}(t,\widetilde{y},\widetilde{z})\Big{[}\frac{w^{\prime}}{w}g(t,y,z)-\frac{\partial
g}{\partial y}(t,y,z)\Big{]}\bigg{)}$
$\displaystyle=-\frac{w^{\prime}}{w}\left(f\left(t,y,z\right)+\frac{1}{2}\frac{w^{\prime}}{w}[z^{2}-g(t,y,z)^{2}]\right)$
$\displaystyle\quad\ +\frac{\partial f}{\partial
y}(t,y,z)+\frac{w^{\prime}}{w}\frac{\partial f}{\partial
z}(t,y,z)z+\frac{1}{2}\Big{[}\frac{w^{\prime\prime}}{w}+\Big{(}\frac{w^{\prime}}{w}\Big{)}^{2}\Big{]}\cdot[z^{2}-g(t,y,z)^{2}]$
$\displaystyle\quad\
+\Big{(}\frac{w^{\prime}}{w}\Big{)}^{2}g(t,y,z)^{2}-\frac{w^{\prime}}{w}g(t,y,z)\cdot\frac{\partial
g}{\partial y}(t,y,z)$
$\displaystyle=\frac{1}{2}\frac{w^{\prime\prime}}{w}z^{2}+\frac{w^{\prime}}{w}\left(\frac{\partial
f}{\partial z}(t,y,z)z-f(t,y,z)\right)+\frac{\partial f}{\partial y}(t,y,z)$
$\displaystyle\quad\
+\Big{[}\Big{(}\frac{w^{\prime}}{w}\Big{)}^{2}-\frac{1}{2}\frac{w^{\prime\prime}}{w}\Big{]}g(t,y,z)^{2}-\frac{w^{\prime}}{w}g(t,y,z)\cdot\frac{\partial
g}{\partial y}(t,y,z),$
$\displaystyle\displaystyle\frac{\partial\widetilde{f}}{\partial\widetilde{z}}(t,\widetilde{y},\widetilde{z})$
$\displaystyle=\frac{\partial f}{\partial
z}(t,y,z)+\frac{w^{\prime}}{w}\Big{[}z-g(t,y,z)\cdot\frac{\partial g}{\partial
z}(t,y,z)\Big{]}.$

We now show that a good choice of $\phi$ allows $\widetilde{f}$ to satisfy the
structure condition (STR). Indeed, if $\phi$ is such that $w>0$,
$w^{\prime}>0$, and $w^{\prime\prime}<0$, note that (4.2) with
$0<\alpha^{2}<\alpha<1$, then

$\displaystyle\left(\frac{\partial\widetilde{f}}{\partial\widetilde{y}}+a\left|\frac{\partial\widetilde{f}}{\partial\widetilde{z}}\right|^{2}\right)(t,\widetilde{y},\widetilde{z})$
$\displaystyle=\frac{1}{2}\frac{w^{\prime\prime}}{w}z^{2}+\frac{w^{\prime}}{w}\Big{(}\frac{\partial
f}{\partial
z}z-f\Big{)}+\Big{[}\Big{(}\frac{w^{\prime}}{w}\Big{)}^{2}-\frac{1}{2}\frac{w^{\prime\prime}}{w}\Big{]}g^{2}-\frac{w^{\prime}}{w}g\cdot\frac{\partial
g}{\partial y}$ $\displaystyle\quad+\frac{\partial f}{\partial
y}+a\left|\frac{\partial f}{\partial
z}+\frac{w^{\prime}}{w}\Big{[}z-g\cdot\frac{\partial g}{\partial
z}\Big{]}\right|^{2}$
$\displaystyle\leqslant\Big{[}\alpha\Big{(}\frac{w^{\prime}}{w}\Big{)}^{2}+\frac{1-\alpha}{2}\frac{w^{\prime\prime}}{w}\Big{]}|z|^{2}+\frac{w^{\prime}}{w}\Big{(}k(t)|z|+l(t)+2C|z|^{2}\Big{)}+\Big{[}\Big{(}\frac{w^{\prime}}{w}\Big{)}^{2}-\frac{1}{2}\frac{w^{\prime\prime}}{w}\Big{]}l(t)\quad$
$\displaystyle\quad+\frac{w^{\prime}}{w}C\alpha|z|+l_{\varepsilon}(t)+\varepsilon|z|^{2}+a\left(k(t)+\Big{(}C+\frac{w^{\prime}}{w}(1+\alpha)\Big{)}|z|\right)^{2}$
$\displaystyle\leqslant|z|^{2}\left[\alpha\Big{(}\frac{w^{\prime}}{w}\Big{)}^{2}+\frac{1-\alpha}{2}\frac{w^{\prime\prime}}{w}+\frac{w^{\prime}}{w}2C+\varepsilon+a\Big{(}C+\frac{w^{\prime}}{w}(1+\alpha)\Big{)}^{2}\right]$
$\displaystyle\quad+|z|\left[\frac{w^{\prime}}{w}\Big{[}k(t)+C\alpha\Big{]}+2ak(t)\Big{(}C+\frac{w^{\prime}}{w}(1+\alpha)\Big{)}\right]$
$\displaystyle\quad+\Big{[}\Big{(}\frac{w^{\prime}}{w}\Big{)}^{2}-\frac{1}{2}\frac{w^{\prime\prime}}{w}\Big{]}l(t)+\frac{w^{\prime}}{w}l(t)+l_{\varepsilon}(t)+ak(t)^{2}$
$\displaystyle\leqslant|z|^{2}\left[\frac{1-\alpha}{2}\frac{w^{\prime\prime}}{w}+\frac{w^{\prime}}{w}2C+\Big{(}\frac{w^{\prime}}{w}\Big{)}^{2}+\varepsilon+2a\left(C+\frac{w^{\prime}}{w}(1+\alpha)\right)^{2}\right]$
$\displaystyle\quad+\Big{[}\Big{(}\frac{w^{\prime}}{w}\Big{)}^{2}-\frac{1}{2}\frac{w^{\prime\prime}}{w}\Big{]}l(t)+\frac{w^{\prime}}{w}l(t)+l_{\varepsilon}(t)+ak(t)^{2}+\frac{1}{4(1-\alpha)}\Big{[}k(t)+C\alpha\Big{]}^{2}+ak(t)^{2},$

where in the last inequality we have used the inequality
$2\beta_{i}\gamma_{i}\leqslant|\beta_{i}|^{2}+|\gamma_{i}|^{2}$ for $i=1,2$,
with

$\displaystyle\beta_{1}$
$\displaystyle=\sqrt{1-\alpha}\frac{w^{\prime}}{w}|z|,\quad~{}\gamma_{1}=\frac{1}{2\sqrt{1-\alpha}}\Big{[}k(t)+C\alpha\Big{]};$
$\displaystyle\beta_{2}$
$\displaystyle=|z|\sqrt{a}\left(C+\frac{w^{\prime}}{w}(1+\alpha)\right),\quad~{}\gamma_{2}=\sqrt{a}k(t).$

Thus, if we find $\phi$ satisfying all the required assumptions and such that
on $[-M,M]$,

$\frac{1-\alpha}{2}\frac{w^{\prime\prime}}{w}+\frac{w^{\prime}}{w}2C+\Big{(}\frac{w^{\prime}}{w}\Big{)}^{2}<-\delta<0,$

then choosing $a$ and $\varepsilon$ small enough, the coefficient before
$|z|^{2}$ is non-positive for all $y\in[-M,M]$. Therefore (STR) is satisfied.
In fact, by setting

$\phi(y)=\frac{1}{\lambda}\ln\left(\frac{e^{\lambda
A\widetilde{y}}+1}{A}\right)-M,$

then a straightforward yet tedious computation gives us that

$w=A-\exp\\{-\lambda(y+M)\\},\quad~{}w^{\prime}=\lambda\exp\\{-\lambda(y+M)\\},\quad~{}w^{\prime\prime}=-\lambda^{2}\exp\\{-\lambda(y+M)\\},$

which implies that $w^{\prime\prime}<0$ for all $y\in[-M,M]$ and $\lambda>0$.
Moreover, when $A>1$ and $\lambda>0$, it is easy to see that $w>0$ and
$w^{\prime}>0$ on the interval $[-M,M]$. Furthermore, one could choose a
proper $A$ and $\lambda$ such that

$\displaystyle\frac{1-\alpha}{2}\frac{w^{\prime\prime}}{w}+\frac{w^{\prime}}{w}2C+\left(\frac{w^{\prime}}{w}\right)^{2}$
$\displaystyle=\frac{1}{2}\frac{\exp(-\lambda(u+M))}{(A-\exp(-\lambda(u+M)))^{2}}$
$\displaystyle\quad\times\Big{[}\lambda^{2}\Big{(}-(1-\alpha)A+(3-\alpha)\exp(-\lambda(u+M))\Big{)}+\lambda
4C(A-\exp(-\lambda(u+M)))\Big{]}$ $\displaystyle\leqslant-\delta<0.$

The proof is complete. ∎

As an important application of the comparison theorem, we study the stability
of BDSDEs.

######  Theorem 4.6 (Stability).

Let $(f,g,\xi)$ be a set of parameters that satisfying the assumptions of
Theorem 4.4, and let $(f^{n},g,\xi^{n})$ be a sequence of parameters of BDSDEs
such that:

  * (i)

There exist some constants $a$, $b$, $C$, $d\in\mathbb{R}$, $\alpha\in(0,1)$,
and a continuous increasing function
$c:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that for all
$n\in\mathbb{N}$, the coefficients $(f^{n},g,\xi^{n})$ satisfies 3.

  * (ii)

For all $n\in\mathbb{N}$, there exists a solution $\left(Y^{n},Z^{n}\right)$
to the BDSDE with parameters $(f^{n},g,\xi^{n})$.

  * (iii)

The sequence $\\{f^{n};n\in\mathbb{N}\\}$ converges to $f$ locally uniformly
on $[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$, and the sequence
$\\{\xi^{n};n\in\mathbb{N}\\}$ converges to $\xi$ in $L^{\infty}(\Omega)$.

Then there exists a pair of processes $(Y,Z)\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ such that

$\left\\{\begin{aligned} &\lim_{n\rightarrow\infty}Y^{n}=Y\hbox{ uniformy on
}[0,T],\\\ &\\{Z^{n};n\in\mathbb{N}\\}\hbox{ converges to $Z$ in
}L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d}).\end{aligned}\right.$

Moreover, the pair $(Y,Z)$ is a solution of BDSDE with parameters $(f,g,\xi)$.

###### Proof.

For simplicity presentation, we define

$\displaystyle\displaystyle H_{n}=$ $\displaystyle\ \inf_{m\geqslant
n}f^{m},\quad~{}H^{n}=\sup_{m\geqslant n}f^{m},$
$\displaystyle\displaystyle\xi_{n*}=$ $\displaystyle\ \inf_{m\geqslant
n}\xi^{m},\quad~{}\xi^{n*}=\sup_{m\geqslant n}\xi^{m},$

and we consider the minimal solutions $\left(Y_{*}^{n}Z_{*}^{n}\right)$ of
BDSDE with parameters $\left(H^{n},\xi^{n*}\right)$, and the maximal solutions
$\left(Y^{n*},Z^{n*}\right)$ of BDSDE with parameters
$\left(H_{n},\xi_{n*}\right)$, as both:

  * (i)

The sequence $\\{H_{n};n\in\mathbb{N}\\}$ is increasing and converges locally
uniformly to $f$, and the sequence $\\{\xi_{n*};n\in\mathbb{N}\\}$ is
increasing.

  * (ii)

The sequence $\\{H^{n};n\in\mathbb{N}\\}$ is decreasing and converges locally
uniformly to $f$, and the sequence $\\{\xi^{n\ast};n\in\mathbb{N}\\}$ is
decreasing.

Then we have that

  * (i)

The sequence $\\{Y^{n*};n\in\mathbb{N}\\}$ is bounded and decreasing, and for
every $n\in\mathbb{N}$

$Y^{n*}\geqslant Y^{n}.$

Hence, from 3.7, there exists $\left(Y^{*},Z^{*}\right)$ such that
$\\{Y^{n*};n\in\mathbb{N}\\}$ converges uniformly to $Y^{*}$, and
$(Y^{*},Z^{*})$ is a solution of the BDSDE with parameters $(f,g,\xi)$.

  * (ii)

The sequence $\\{Y_{*}^{n};n\in\mathbb{N}\\}$ is bounded and decreasing, and
for every $n\in\mathbb{N}$,

$Y_{*}^{n}\leqslant Y^{n}.$

Hence, from 3.7, there exists $\left(Y_{*},Z_{*}\right)$ such that
$\\{Y_{*}^{n};n\in\mathbb{N}\\}$ converges uniformly to $Y_{*}$, and
$\left(Y_{*},Z_{*}\right)$ is a solution of the BDSDE with parameters
$(f,g,\xi)$.

Finally, by Theorem 4.4, for every $n\in\mathbb{N}$, one has that

$Y_{*}^{n}\leqslant Y^{n}\leqslant
Y^{n*}\quad~{}\text{and}\quad~{}Y_{*}=Y^{*}=Y,$

therefore the sequence $\\{Y^{n};n\in\mathbb{N}\\}$ converges uniformly to
$Y$. This completes the proof. ∎

##  5 Application to SPDE

When the generator $f$ is of linear growth with respect to $y$ and $z$,
Pardoux–Peng [26] used BDSDEs to give a probabilistic representation for the
classical solution of semilinear SPDEs; and Bally–Matoussi [3] and Zhang–Zhao
[37] obtained the relationship between the solution of BDSDEs and the Sobolev
solution of SPDEs. Then, Wu–Zhang [31] got the Sobolev solution of related
SPDEs when $f$ is continuous and locally monotone in $y$. Zhang–Zhao [39] and
Bahlali et al. [2] used BDSDEs to prove the existence and uniqueness of
related SPDEs when the generator $f$ is of polynomial growth in $Y$ and grows
in $Z$ super-linearly (or sub-quadratically), respectively. In this section,
when the generator $f$ is of quadratic growth in $z$, we use BDSDEs to give a
probabilistic representation for the solutions of related semilinear SPDEs in
Sobolev spaces, and use it to prove the existence and uniqueness of Sobolev
solutions of the SPDEs, thus extending the nonlinear stochastic Feynman-Kac
formula.

First, we recall some notations. For Euclidean spaces $\mathbb{H}$ and
$\mathbb{G}$, denote by $C_{l,b}^{k}(\mathbb{H};\mathbb{G})$ the set of
functions of class $C^{k}$ from $\mathbb{H}$ to $\mathbb{G}$, whose partial
derivatives of order less than or equal to $k$ are bounded. Denote by
$\mathcal{C}_{c}^{1,\infty}\left([0,T]\times\mathbb{H}\right)$ the set of
compactly supported functions $\varphi(t,x)$ which are continuously derivable
in the $t$-variable and infinitely continuously derivable in the $x$-variable.

Consider the following forward-backward doubly stochastic differential
equation:

$\displaystyle\displaystyle X^{t,x}_{s}=$ $\displaystyle\
x+\int_{t}^{s}b(X^{t,x}_{r})dr+\int_{t}^{s}\sigma(X^{t,x}_{r})dW_{r},\quad~{}t\leqslant
s\leqslant T,$ (5.1) $\displaystyle\displaystyle Y^{t,x}_{s}=$ $\displaystyle\
h(X^{t,x}_{T})+\int_{s}^{T}f(r,X^{t,x}_{r},Y^{t,x}_{r},Z^{t,x}_{r})dr+\int_{s}^{T}g(r,X^{t,x}_{r},Y^{t,x}_{r},Z^{t,x}_{r})d\overleftarrow{B}_{r}-\int_{s}^{T}Z^{t,x}_{r}dW_{r},$
(5.2)

where the coefficients $b$ and $\sigma$ come from
$C_{l,b}^{2}(\mathbb{R}^{n};\mathbb{R}^{n})$ and
$C_{l,b}^{3}(\mathbb{R}^{n};\mathbb{R}^{n\times d})$, respectively. Then it is
well known that the forward equation (5.1) admits a unique adapted solution,
denoted by $\\{X^{t,x}_{s};t\leqslant s\leqslant T\\}$, which satisfying

$\mathbb{E}\left[\sup_{t\leqslant s\leqslant
T}|X^{t,x}_{s}|^{p}\right]<\infty,\quad~{}\forall p>1.$

Now, we would like to connect the forward-backward system (5.1) and (5.2) with
quadratic growth to the following semilinear stochastic partial differential
equation:

$\displaystyle u(s,x)=$ $\displaystyle
h(x)+\int_{s}^{T}\left\\{\mathcal{L}u(r,x)+f\big{(}r,x,u(r,x),(\sigma^{\top}\nabla
u)(r,x)\big{)}\right\\}dr$ (5.3)
$\displaystyle+\int_{s}^{T}g\big{(}r,x,u(r,x),(\sigma^{\top}\nabla
u)(r,x)\big{)}d\overleftarrow{B}_{r},\quad t\leqslant s\leqslant T,$

where $\sigma^{\top}$ denotes the transposed of $\sigma$, and

$\displaystyle\mathcal{L}=\sum_{i=1}^{n}b_{i}\frac{\partial}{\partial
x_{i}}+\frac{1}{2}\sum_{i,j=1}^{n}a_{ij}\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}},\quad~{}\left(a_{ij}\right)=\sigma\sigma^{\top}.$ (5.4)

Before introducing the definition of Sobolev solutions of SPDE (5.3), we let
$\rho:\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}$ be an integrable continuous
positive function, and $L^{2}(\mathbb{R}^{n};\rho^{-1}(x)\mathrm{d}x)$ be the
weighted $L^{2}$ space endowed with the norm

$\|u\|_{\rho}^{2}\triangleq\int_{\mathbb{R}^{n}}|u(x)|^{2}\rho^{-1}(x)dx.$

Let us take the weight $\rho(x)=\exp\\{F(x)\\}$, where
$F:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a continuous function and there is
a positive constant $R>0$ such that $F\in
C_{l,b}^{2}\left(\mathbb{R}^{n},\mathbb{R}\right)$ when $|x|>R$. For example,
one can take $\rho(x)=\exp\\{\delta|x|\\}$ with $\delta\in\mathbb{R}^{+}$ or
$\rho(x)=(1+|x|)^{q}$ with $q>n+2$.

Let $\mathscr{H}$ be the set of random fields $\left\\{u(t,x);0\leqslant
t\leqslant T,x\in\mathbb{R}^{n}\right\\}$ such that $u(t,x)$ is
$\mathscr{F}_{t,T}^{B}$-measurable, and both $u$ and $\sigma^{\top}\nabla u$
belong to
$L^{2}\left(\Omega\times(0,T)\times\mathbb{R}^{n};\mathrm{d}\mathbb{P}\otimes\mathrm{d}t\otimes\rho^{-1}(x)\mathrm{d}x\right)$.
Then $\mathscr{H}$ is a Banach space endowed with the following norm:

$\|u\|_{\mathscr{H}}^{2}\triangleq\mathbb{E}\left[\int_{\mathbb{R}^{n}}\int_{0}^{T}\left(|u(t,x)|^{2}+\left|(\sigma^{\top}\nabla
u)(t,x)\right|^{2}\right)dt\rho^{-1}(x)dx\right].$

Now we present the definition of Sobolev solution of SPDE (5.3).

######  Definition 5.1.

We say that $u$ is a Sobolev solution of SPDE (5.3), if $u\in\mathscr{H}$ and
for any $\varphi\in\mathcal{C}_{c}^{1,\infty}([0,T]\times\mathbb{R}^{n})$,

$\displaystyle\int_{\mathbb{R}^{n}}\int_{t}^{T}u(s,x)\partial_{s}\varphi(s,x)dsdx+\int_{\mathbb{R}^{n}}u(t,x)\varphi(t,x)dx-\int_{\mathbb{R}^{n}}h(x)\varphi(T,x)dx$
(5.5)
$\displaystyle\quad-\frac{1}{2}\int_{\mathbb{R}^{n}}\int_{t}^{T}(\sigma^{\top}\nabla
u)(s,x)\cdot(\sigma^{\top}\nabla\varphi)(s,x)dsdx-\int_{\mathbb{R}^{n}}\int_{t}^{T}u\mathrm{div}[(b-\widetilde{A})\varphi](s,x)dsdx$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{t}^{T}f\big{(}s,x,u(s,x),(\sigma^{\top}\nabla
u)(s,x)\big{)}\varphi(s,x)dsdx$
$\displaystyle\quad+\int_{\mathbb{R}^{n}}\int_{t}^{T}g\big{(}s,x,u(s,x),(\sigma^{\top}\nabla
u)(s,x)\big{)}\varphi(s,x)dB_{s}dx,$

where $A$ is a $n$-vector whose coordinates are given by
$\widetilde{A}_{j}=\frac{1}{2}\sum_{i=1}^{n}\frac{\partial a_{ij}}{\partial
x_{i}}$ with $1\leqslant j\leqslant n.$

It should be pointed out that, for the classical solution, indeed, if one
supposes that $u$ is a solution of SPDE (5.3) of class $C^{2}$, then, similar
to the method of Pardoux–Peng [26], applying 2.2 to $u(s,X^{t,x}_{s})$ implies
that the pair of processes $\\{Y^{t,x}_{s},Z^{t,x}_{s};t\leqslant s\leqslant
T\\}$ defined by

$Y^{t,x}_{s}=u(s,X_{s}^{t,x})\quad~{}\hbox{and}\quad~{}Z^{t,x}_{s}=(\sigma^{\top}\nabla
u)(s,X_{s}^{t,x})$

is a solution of the backward doubly stochastic differential equation (5.2).

In the following, we discuss this relationship between the Sobolev solution of
SPDEs and the solution of BDSDEs with quadratic growth. In particular, we
first focus on a simple situation to better reflect the idea of solving the
problem.

###  5.1 Simple situation

Consider the following type of backward doubly stochastic differential
equations: for $t\leqslant s\leqslant T$,

$Y^{t,x}_{s}=h(X^{t,x}_{T})+\int_{s}^{T}\big{[}f(r,X^{t,x}_{r},Y^{t,x}_{r},Z^{t,x}_{r})+C(Z^{t,x}_{r})^{2}\big{]}dr+\int_{s}^{T}\alpha
Z^{t,x}_{r}d\overleftarrow{B}_{r}-\int_{s}^{T}Z^{t,x}_{r}dW_{r},$ (5.6)

where $X^{t,x}$ is the solution of (5.1), $C>0$ and $\alpha\in(-1,1)$ are two
constants, and the functions
$f:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
and $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ satisfy the following assumption.

######  Assumption 5.

The function $h$ is bounded, and there exist bounded functions
$l:[0,T]\rightarrow\mathbb{R}^{+}$ and $k:[0,T]\rightarrow\mathbb{R}$ such
that for all $t\in[0,T],$ $x\in\mathbb{R}^{n}$, $y\in\mathbb{R}$ and
$z\in\mathbb{R}^{d}$, the generator $f$ satisfies the following condition:

$\displaystyle|f(t,x,y,z)|\leqslant l(t),\quad~{}\frac{\partial f}{\partial
y}(t,x,y,z)+\frac{\partial f}{\partial z}(t,x,y,z)$ $\displaystyle\leqslant
k(t),\quad~{}\text{a.s. }$

######  Proposition 5.2.

Under 5, the following stochastic partial differential equation

$\displaystyle u(s,x)=$ $\displaystyle
h(x)+\int_{s}^{T}\left\\{\mathcal{L}u(r,x)+f\big{(}r,x,u(r,x),(\sigma^{\top}\nabla
u)(r,x)\big{)}+C\big{(}(\sigma^{\top}\nabla u)(r,x)\big{)}^{2}\right\\}dr$
(5.7) $\displaystyle+\alpha\int_{s}^{T}(\sigma^{\top}\nabla
u)(r,x)d\overleftarrow{B}_{r},\quad~{}t\leqslant s\leqslant T,$

admits a unique Sobolev solution $u\in\mathscr{H}$, and for every $t\in[0,T]$,

$u(s,X_{s}^{t,x})=Y^{t,x}_{s}\quad~{}\hbox{and}\quad~{}(\sigma^{\top}\nabla
u)(s,X_{s}^{t,x})=Z^{t,x}_{s},\quad~{}a.s.,a.e.\ s\in[t,T],\
x\in\mathbb{R}^{n},$

where $\\{(Y_{s}^{t,x},Z_{s}^{t,x});t\leqslant s\leqslant T\\}$ is the unique
solution of quadratic BDSDE (5.6).

###### Proof.

Applying the exponential transformation of variable to $\bar{Y}=\exp\\{\beta
Y\\}$ transforms formally BDSDE (5.6) into the following BDSDE:

$\displaystyle\displaystyle\bar{Y}^{t,x}_{s}=$ $\displaystyle\
\bar{Y}^{t,x}_{T}+\int_{s}^{T}\beta\bar{Y}^{t,x}_{r}f(r,X^{t,x}_{r},Y^{t,x}_{r},Z^{t,x}_{r})dr+\Big{[}C-\frac{1-\alpha^{2}}{2}\beta\Big{]}\int_{s}^{T}\beta\bar{Y}^{t,x}_{r}|Z^{t,x}_{r}|^{2}dr$
$\displaystyle\
+\alpha\int_{s}^{T}\beta\bar{Y}^{t,x}_{r}Z^{t,x}_{r}d\overleftarrow{B}_{r}-\int_{s}^{T}\beta\bar{Y}^{t,x}_{r}Z^{t,x}_{r}dW_{r},\quad~{}t\leqslant
s\leqslant T.$

By taking $\beta=\frac{2C}{1-\alpha^{2}}$, the above equation becomes

$\displaystyle\displaystyle\bar{Y}^{t,x}_{s}=\bar{h}(X^{t,x}_{T})+\int_{s}^{T}\bar{f}(r,X^{t,x}_{r},\bar{Y}^{t,x}_{r},\bar{Z}^{t,x}_{r})dr+\alpha\int_{s}^{T}\bar{Z}^{t,x}_{r}d\overleftarrow{B}_{r}-\int_{s}^{T}\bar{Z}^{t,x}_{r}dW_{r},\quad~{}t\leqslant
s\leqslant T,$ (5.8)

where

$\displaystyle\displaystyle\bar{Z}^{t,x}_{r}=\beta\bar{Y}^{t,x}Z^{t,x}_{r},\quad~{}\bar{h}(X^{t,x}_{T})=\exp\\{\beta
h(X^{t,x}_{T})\\},$
$\displaystyle\displaystyle\bar{f}(r,X^{t,x}_{r},\bar{Y}^{t,x}_{r},\bar{Z}^{t,x}_{r})=\beta\bar{Y}^{t,x}_{r}f(r,X^{t,x}_{r},Y^{t,x}_{r},Z^{t,x}_{r}).$
(5.9)

Under 5, Eq. (5.8) is a classical BDSDE with a globally Lipschitz generator.
Then Bally–Matoussi [3] (see also Zhang–Zhao [38] and Wu–Zhang [31]) implies
that the following SPDE

$\displaystyle\bar{u}(s,x)=$
$\displaystyle\bar{h}(x)+\int_{s}^{T}\left\\{\mathcal{L}\bar{u}(r,x)+\bar{f}\big{(}r,x,\bar{u}(r,x),(\sigma^{\top}\nabla\bar{u})(r,x)\big{)}\right\\}dr$
(5.10)
$\displaystyle+\alpha\int_{s}^{T}(\sigma^{\top}\nabla\bar{u})(r,x)d\overleftarrow{B}_{r},\quad~{}t\leqslant
s\leqslant T,$

admits a unique Sobolev solution $\bar{u}\in\mathscr{H}$, and for every
$t\in[0,T]$,

$\bar{u}(s,X_{s}^{t,x})=\bar{Y}_{s}^{t,x}\quad~{}\text{and}\quad~{}\sigma^{\top}\nabla\bar{u}(s,X_{s}^{t,x})=\bar{Z}_{s}^{t,x},\quad~{}a.s.,a.e.\
s\in[t,T],\ x\in\mathbb{R}^{n}.$

On the other hand, we see that

$\bar{Y}^{t,x}_{s}=\exp(\beta
Y^{t,x}_{s})\quad~{}\hbox{and}\quad~{}\bar{Z}^{t,x}_{s}=\beta\bar{Y}^{t,x}_{s}Z^{t,x}_{s},\quad~{}t\leqslant
s\leqslant T,$

where the pair $(Y^{t,x},Z^{t,x})\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ is the solution of BDSDE (5.6). So we
would like to define

$\displaystyle
u(s,X^{t,x}_{s})\triangleq\frac{\ln\bar{u}(s,X^{t,x}_{s})}{\beta}=\frac{\ln\bar{Y}^{t,x}_{s}}{\beta}=Y^{t,x}_{s},\quad~{}t\leqslant
s\leqslant T,$ (5.11)

and then one can compute that

$\displaystyle(\sigma^{\top}\nabla
u)(s,X^{t,x}_{s})=\frac{(\sigma^{\top}\nabla\bar{u})(s,X^{t,x}_{s})}{\beta\bar{u}(s,X^{t,x}_{s})}=\frac{\bar{Z}^{t,x}_{s}}{\beta\bar{Y}^{t,x}_{s}}=Z^{t,x}_{s},\quad~{}t\leqslant
s\leqslant T.$ (5.12)

Hence, applying Itô’s formula (see 2.2) to $u=\ln\bar{u}/\beta$ could
transforms formally SPDE (5.10) into SPDE (5.7). In fact, by using Itô’s
formula to $u=\ln\bar{u}/\beta$, one has that

$\displaystyle du(s,x)=$ $\displaystyle\
\frac{-1}{\beta\bar{u}(s,x)}\left\\{\mathcal{L}\bar{u}(s,x)+\bar{f}\big{(}s,x,\bar{u}(s,x),(\sigma^{\top}\nabla\bar{u})(s,x)\big{)}\right\\}ds$
(5.13)
$\displaystyle-\frac{\alpha(\sigma^{\top}\nabla\bar{u})(s,x)}{\beta\bar{u}(s,x)}d\overleftarrow{B}_{s}+\frac{\alpha^{2}}{2}\frac{\big{(}(\sigma^{\top}\nabla\bar{u})(s,x)\big{)}^{2}}{\beta\bar{u}(s,x)^{2}}ds.$

From the definitions (5.4) and (5.11), it is easy to compute that

$\displaystyle\mathcal{L}\bar{u}(s,x)=\beta\bar{u}(s,x)\Big{[}\mathcal{L}u(s,x)+\frac{\beta}{2}\big{(}(\sigma^{\top}\nabla
u)(s,x)\big{)}^{2}\Big{]}.$ (5.14)

Then, combining (5.9), (5.11), (5.12), (5.13), (5.14), and note that
$\beta=\frac{2C}{1-\alpha^{2}}$, we deduce that

$\left\\{\begin{aligned} \displaystyle
du(s,x)=&-\left\\{\mathcal{L}u(s,x)+f\big{(}s,x,u(s,x),(\sigma^{\top}\nabla
u)(s,x)\big{)}+C\big{(}(\sigma^{\top}\nabla u)(s,x)\big{)}^{2}\right\\}ds\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle&-\alpha(\sigma^{\top}\nabla
u)(s,x)d\overleftarrow{B}_{s},\quad~{}t\leqslant s\leqslant T,\\\ \vskip 3.0pt
plus 1.0pt minus 1.0pt\cr\displaystyle u(T,x)=&h(x).\end{aligned}\right.$
(5.15)

In other words, $u$ is a Sobolev solution of SPDE (5.7), and for every
$t\in[0,T]$,

$u(s,X_{s}^{t,x})=Y^{t,x}_{s}\quad~{}\hbox{and}\quad~{}(\sigma^{\top}\nabla
u)(s,X_{s}^{t,x})=Z^{t,x}_{s},\quad~{}a.s.,a.e.\ s\in[t,T],\
x\in\mathbb{R}^{n},$

where $(Y^{t,x},Z^{t,x})$ is the unique solution of BDSDE (5.6).

Finally, the uniqueness of solutions of SPDE (5.7) comes from the uniqueness
of BDSDE (5.6). This completes the proof. ∎

######  Remark 5.3.

The idea of the above proof is firstly to transform a simple quadratic BDSDE
into a classical BDSDE, and then by using the relationship between the
classical BDSDEs and SPDEs, one can conversely obtain the relationship between
the BDSDE with quadratic growth and its related SPDE.

###  5.2 General situation

In this subsection, we use BDSDE (5.2) to give a probabilistic representation
for the solutions of SPDE (5.3) in Sobolev space, and use it to prove the
existence and uniqueness of Sobolev solutions of SPDE (5.3), thus obtaining
the nonlinear stochastic Feynman-Kac formula in this framework. The following
lemma is an extension of equivalence of norm principle given in Bally–Matoussi
[3] and plays an important role later.

######  Lemma 5.4.

There exist two positive constants $k_{1}$ and $K_{1}$ which depend on
$T,\rho,b$ and $\sigma$, such that for any $t\leqslant s\leqslant T$ and
$\phi\in
L^{1}\left(\Omega\times\mathbb{R}^{n};\mathrm{~{}d}\mathbb{P}\otimes\rho^{-1}(x)\mathrm{d}x\right)$
which is independent of $\mathscr{F}_{t,s}^{W}$,

$k_{1}\mathbb{E}\int_{\mathbb{R}^{d}}|\phi(x)|\rho^{-1}(x)\mathrm{d}x\leqslant\mathbb{E}\int_{\mathbb{R}^{d}}\left|\phi\left(X_{s}^{t,x}\right)\right|\rho^{-1}(x)\mathrm{d}x\leqslant
K_{1}\mathbb{E}\int_{\mathbb{R}^{d}}|\phi(x)|\rho^{-1}(x)\mathrm{d}x.$

Moreover, for any $\Phi\in
L^{1}\left(\Omega\times[0,T]\times\mathbb{R}^{d};\mathrm{~{}d}\mathbb{P}\otimes\mathrm{d}t\otimes\rho^{-1}(x)\mathrm{d}x\right)$
such that $\Phi(s,\cdot)$ is independent of $\mathscr{F}_{t,s}^{W}$, one has

$\displaystyle
k_{1}\mathbb{E}\int_{\mathbb{R}^{d}}\int_{t}^{T}|\Phi(s,x)|\mathrm{d}s\rho^{-1}(x)\mathrm{d}x$
$\displaystyle\leqslant\mathbb{E}\int_{\mathbb{R}^{d}}\int_{t}^{T}\left|\Phi\left(s,X_{s}^{t,x}\right)\right|\mathrm{d}s\rho^{-1}(x)\mathrm{d}x$
$\displaystyle\leqslant
K_{1}\mathbb{E}\int_{\mathbb{R}^{d}}\int_{t}^{T}|\Phi(s,x)|\mathrm{d}s\rho^{-1}(x)\mathrm{d}x.$

For the coefficients of BDSDE (5.2), we present the following assumptions.

######  Assumption 6.

For all $t\in[0,T],$ $x\in\mathbb{R}^{n}$, $y\in\mathbb{R}$ and
$z\in\mathbb{R}^{d}$, the function $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is
bounded, and the functions
$f:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
and
$g:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{l}$
satisfy 4.

######  Remark 5.5.

Note that under 6, BDSDE (5.2) has a unique solution $(Y^{t,x},Z^{t,x})\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$. Moreover, the coefficients $f$, $g$
and $h$ satisfy that

$\int_{\mathbb{R}^{n}}\int_{0}^{T}\Big{(}|h(x)|^{2}+|f(t,x,0,0)|^{2}+|g(t,x,0,0)|^{2}\Big{)}dt\rho^{-1}(x)dx<\infty.$

Now we are in a position to present the main result of this section.

######  Theorem 5.6 (Feynman-Kac formula).

Under 6, SPDE (5.3) admits a unique Sobolev solution $u\in\mathscr{H}$, and
for every $t\in[0,T]$,

$u(s,X_{s}^{t,x})=Y^{t,x}_{s}\quad~{}\hbox{and}\quad~{}(\sigma^{\top}\nabla
u)(s,X_{s}^{t,x})=Z^{t,x}_{s},\quad~{}a.s.,a.e.\ s\in[t,T],\
x\in\mathbb{R}^{n},$

where $\\{(Y_{s}^{t,x},Z_{s}^{t,x});t\leqslant s\leqslant T\\}$ is the unique
solution of BDSDE (5.2) with quadratic growth.

###### Proof.

Uniqueness. The uniqueness of SPDE (5.3) follows from the uniqueness of BDSDE
(5.2). In fact, if $u^{1}$ and $u^{2}$ are two Sobolev solutions of SPDE
(5.3), then the following two pairs

$\big{(}u^{1}(s,X_{s}^{t,x}),(\sigma^{\top}\nabla
u^{1})(s,X_{s}^{t,x})\big{)}\quad~{}\hbox{and}\quad~{}\big{(}u^{2}(s,X_{s}^{t,x}),(\sigma^{\top}\nabla
u^{2})(s,X_{s}^{t,x})\big{)}$

solve BDSDE (5.2). So the uniqueness of BDSDE (5.2) gives us that

$u^{1}(s,X_{s}^{t,x})=u^{2}(s,X_{s}^{t,x}),\quad a.s.,a.e.\ s\in[t,T],\
x\in\mathbb{R}^{n},$

and in particular

$u^{1}(t,x)=u^{2}(t,x),\quad a.s.,a.e.,\ x\in\mathbb{R}^{n}.$

Next, we prove the existence. Compared with the proof of the simple situation
(see 5.2), the main idea here we used is similar, however, the techniques are
more complex and include an approximation. So the proof will be divided into
six steps with some techniques borrowed from the proof of Theorem 3.10.

Step 1. The exponential transformation.

Let $\beta=\frac{2C}{1-\alpha}$ be a given positive constant. Applying the
exponential change of variable (see 2.2) to $\bar{Y}^{t,x}=\exp(\beta
Y^{t,x})$ could transforms formally BDSDE (5.2) with parameters $(f,g,h)$ into
the following BDSDE with parameters $(\bar{f},\bar{g},\bar{h})$,

$\displaystyle\bar{Y}^{t,x}_{t}=\bar{h}(X^{t,x}_{T})+\int_{t}^{T}\bar{f}(s,X^{t,x}_{s},\bar{Y}^{t,x}_{s},\bar{Z}^{t,x}_{s})ds+\int_{t}^{T}\bar{g}(s,X^{t,x}_{s},\bar{Y}^{t,x}_{s},\bar{Z}^{t,x}_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}\bar{Z}^{t,x}_{s}dW_{s},$
(5.16)

where $\bar{Z}^{t,x}_{s}=\beta\bar{Y}^{t,x}_{s}Z^{t,x}_{s}$ and

$\displaystyle\bar{f}(s,X^{t,x}_{s},\bar{Y}^{t,x}_{s},\bar{Z}^{t,x}_{s})=$
$\displaystyle\
\beta\bar{Y}^{t,x}_{s}f\Big{(}s,X^{t,x}_{s},\frac{\ln\bar{Y}^{t,x}_{s}}{\beta},\frac{\bar{Z}^{t,x}_{s}}{\beta\bar{Y}^{t,x}_{s}}\Big{)}$
$\displaystyle\
+\frac{1}{2}\beta^{2}\bar{Y}^{t,x}_{s}\Big{[}g\Big{(}s,X^{t,x}_{s},\frac{\ln\bar{Y}^{t,x}_{s}}{\beta},\frac{\bar{Z}^{t,x}_{s}}{\beta\bar{Y}^{t,x}_{s}}\Big{)}^{2}-\Big{(}\frac{\bar{Z}^{t,x}_{s}}{\beta\bar{Y}^{t,x}_{s}}\Big{)}^{2}\Big{]},$
$\displaystyle\bar{g}(s,X^{t,x}_{s},\bar{Y}^{t,x}_{s},\bar{Z}^{t,x}_{s})=$
$\displaystyle\
\beta\bar{Y}^{t,x}_{s}g\Big{(}s,X^{t,x}_{s},\frac{\ln\bar{Y}^{t,x}_{s}}{\beta},\frac{\bar{Z}^{t,x}_{s}}{\beta\bar{Y}^{t,x}_{s}}\Big{)},$
$\displaystyle\bar{h}(X^{t,x}_{T})=$ $\displaystyle\ \exp(\beta
h(X^{t,x}_{T})),\quad~{}t\leqslant s\leqslant T.$

Then the pair of processes $(\bar{Y}^{t,x},\bar{Z}^{t,x})$ is the solution of
BDSDE (5.16) with parameters $(\bar{f},\bar{g},\bar{h})$, where $\bar{h}$ is
bounded and $\bar{g}$ satisfies 3.

Step 2. Approximation of SPDE.

Let the sequence $\\{\bar{f}^{n};n\in\mathbb{N}\\}$ be defined as in (3.24).
Then, the same arguments we developed in the proof of Theorem 3.10 deduce that
$\bar{f}^{n}$ is globally Lipschitz continuous with the constant $n$ and
decreasing converges pointwise to the function $\bar{f}$. Now, for each
$(t,x)\in[0,T]\times\mathbb{R}^{n}$ and $n\in\mathbb{N}$, we define the
processes $\bar{u}^{n}$ and $\bar{v}^{n}$ by

$\bar{u}^{n}(t,x)=\bar{Y}^{t,x,n}_{t}\quad~{}\hbox{and}\quad~{}\bar{v}^{n}(t,x)=\bar{Z}^{t,x,n}_{t},$
(5.17)

where $(\bar{Y}^{t,x,n},\bar{Z}^{t,x,n})$ is the unique solution of the
following BDSDE with parameters $(\bar{f}^{n},\bar{g},\bar{h})$: for
$t\in[0,T]$ and $x\in\mathbb{R}^{n}$,

$\displaystyle\bar{Y}^{t,x,n}_{t}=$ $\displaystyle\
\bar{h}(X^{t,x}_{T})+\int_{t}^{T}\bar{f}^{n}(s,X^{t,x}_{s},\bar{Y}^{t,x,n}_{s},\bar{Z}^{t,x,n}_{s})ds$
(5.18)
$\displaystyle+\int_{t}^{T}\bar{g}(s,X^{t,x}_{s},\bar{Y}^{t,x,n}_{s},\bar{Z}^{t,x,n}_{s})d\overleftarrow{B}_{s}-\int_{t}^{T}\bar{Z}^{t,x,n}_{s}dW_{s}.$

Note that the parameters $(\bar{f}^{n},\bar{g},\bar{h})$ of the above BDSDE
(5.18) satisfy that

$\int_{\mathbb{R}^{n}}\int_{0}^{T}\Big{(}|\bar{h}(x)|^{2}+|\bar{f}^{n}(t,x,\bar{y},0)|^{2}+|\bar{g}(t,x,\bar{y},0)|^{2}\Big{)}dt\rho^{-1}(x)dx<\infty,\quad\forall\bar{y}\in\big{[}\exp\\{-\beta
M\\},\exp\\{\beta M\\}\big{]}.$

Then Bally–Matoussi [3] (see also Zhang–Zhao [38] and Wu–Zhang [31]) deduce
that

$\bar{v}^{n}(s,x)=(\sigma^{\top}\nabla\bar{u}^{n})(s,x),$ (5.19)

and $\bar{u}^{n}$ is the unique Sobolev solution to the following SPDE:

$\displaystyle\bar{u}^{n}(s,x)=$
$\displaystyle\bar{h}(x)+\int_{s}^{T}\left\\{\mathcal{L}\bar{u}^{n}(r,x)+\bar{f}^{n}\big{(}r,x,\bar{u}^{n}(r,x),(\sigma^{\top}\nabla\bar{u}^{n})(r,x)\big{)}\right\\}dr$
(5.20)
$\displaystyle+\int_{s}^{T}\bar{g}\big{(}r,x,\bar{u}^{n}(r,x),(\sigma^{\top}\nabla\bar{u}^{n})(r,x)\big{)}d\overleftarrow{B}_{r},\quad
s\in[t,T],$

with

$\bar{u}^{n}\left(s,X_{s}^{t,x}\right)=\bar{Y}_{s}^{t,x,n}\quad~{}\hbox{and}\quad~{}\big{(}\sigma^{\top}\nabla\bar{u}^{n}\big{)}\left(s,X_{s}^{t,x}\right)=\bar{Z}_{s}^{t,x,n},\quad~{}a.s.,a.e.\
s\in[t,T],\ x\in\mathbb{R}^{n},$ (5.21)

where $(\bar{Y}^{t,x,n},\bar{Z}^{t,x,n})$ is the unique solution of BDSDE
(5.18) with parameters $(\bar{f}^{n},\bar{g},\bar{h})$.

On the other hand, similar to (3.26), the comparison theorem for Lipschitz
continuous coefficients (see 2.4) implies that the sequence
$\\{\bar{Y}^{t,x,n};n\in\mathbb{N}\\}$ is bounded and decreasing in $n$ such
that

$\exp(-\beta\|h\|_{\infty})\leqslant\bar{Y}_{s}^{t,x,n+1}\leqslant\bar{Y}_{s}^{t,x,n}\leqslant\exp(\beta\|h\|_{\infty}),\quad~{}\hbox{a.s.}\
s\in[t,T].$ (5.22)

In addition, by using the same arguments we developed in the proofs of 3.5 and
3.7, there exists a positive constant $K$ such that

$\sup_{n\in\mathbb{N}}\mathbb{E}\int_{t}^{T}|\bar{Z}^{t,x,n}_{s}|^{2}ds\leqslant
K,$ (5.23)

and

$\left\\{\begin{aligned}
\displaystyle&\lim_{n\rightarrow\infty}\bar{Y}^{t,x,n}_{s}=\bar{Y}^{t,x}_{s},\quad~{}\hbox{a.s.}\
s\in[t,T],\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle&\lim_{n\rightarrow\infty}\mathbb{E}\int_{t}^{T}|\bar{Z}^{t,x,n}_{s}-\bar{Z}^{t,x}_{s}|^{2}ds=0,\end{aligned}\right.$
(5.24)

where $(\bar{Y}^{t,x},\bar{Z}^{t,x})\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ is the solution of BDSDE (5.16) with
parameters $(\bar{f},\bar{g},\bar{h})$.

Step 3. Convergence of SPDE (5.20)

We point out that the limits which we consider below hold along with a
subsequence, however for simplicity, the subsequence will still be indexed by
$n$. From 5.4, note (5.19), (5.21), (5.22) and (5.23), we have that

$\displaystyle\mathbb{E}\int_{\mathbb{R}^{n}}\int_{t}^{T}\left(\left|\bar{u}^{n}(s,x)\right|^{2}+\left|\bar{v}^{n}(s,x)\right|^{2}\right)ds\rho^{-1}(x)dx$
(5.25) $\displaystyle\leqslant
K_{1}\mathbb{E}\int_{\mathbb{R}^{n}}\Big{(}T\|\bar{Y}^{t,x,n}\|_{L^{\infty}_{\mathbb{F}}(t,T)}^{2}+\int_{t}^{T}|\bar{Z}^{t,x,n}_{s}|^{2}ds\Big{)}\rho^{-1}(x)dx$
$\displaystyle\leqslant
K_{1}\int_{\mathbb{R}^{n}}\big{[}T\exp(2\beta\|h\|_{\infty})+K\big{]}\rho^{-1}(x)dx<\infty.$

Moreover, similar to (3.25), we have

$|\bar{g}\big{(}s,x,\bar{u}^{n}(s,x),\bar{v}^{n}(s,x)\big{)}|^{2}\leqslant\alpha|\bar{v}^{n}(s,x)|^{2},\quad~{}a.s.,a.e.\
s\in[t,T],\ x\in\mathbb{R}^{n}.$

Then, note that $\bar{f}^{n}$ is globally Lipschitz continuous, we deduce from
(5.25) that

$\mathbb{E}\int_{\mathbb{R}^{n}}\int_{t}^{T}\Big{[}|\bar{f}^{n}\big{(}s,x,\bar{u}^{n}(s,x),\bar{v}^{n}(s,x)\big{)}|^{2}+|\bar{g}\big{(}s,x,\bar{u}^{n}(s,x),\bar{v}^{n}(s,x)\big{)}|^{2}\Big{]}ds\rho^{-1}(x)dx\leqslant\widetilde{K},$

where $\widetilde{K}$ is some constant which depends on $\|h\|_{\infty}$, $K$,
$K_{1}$ and $T$. Now, from 5.4, (5.17), (5.24) and Lebesgue dominated
convergence theorem, we have that

$\displaystyle\lim_{n,m}\int_{\mathbb{R}^{n}}\int_{t}^{T}|\bar{u}^{n}(s,x)-\bar{u}^{m}(s,x)|^{2}ds\rho^{-1}(x)dx=0,$
$\displaystyle\lim_{n,m}\int_{\mathbb{R}^{n}}\int_{t}^{T}|(\sigma^{\top}\nabla\bar{u}^{n})(s,x)-(\sigma^{\top}\nabla\bar{u}^{m})(s,x)|^{2}ds\rho^{-1}(x)dx=0.$

Then, using 5.4 again, and note that the space $\mathscr{H}$ is complete, it
follows that there exists $\bar{u}\in\mathscr{H}$ such that

$\displaystyle\lim_{n}\int_{\mathbb{R}^{n}}\int_{t}^{T}|\bar{u}^{n}(s,x)-\bar{u}(s,x)|^{2}ds\rho^{-1}(x)dx=0,$
(5.26)
$\displaystyle\lim_{n}\int_{\mathbb{R}^{n}}\int_{t}^{T}|(\sigma^{\top}\nabla\bar{u}^{n})(s,x)-(\sigma^{\top}\nabla\bar{u})(s,x)|^{2}ds\rho^{-1}(x)dx=0,$
(5.27)
$\displaystyle\int_{\mathbb{R}^{n}}\int_{t}^{T}\Big{(}|\bar{u}(s,x)|^{2}+(\sigma^{\top}\nabla\bar{u})(s,x)|^{2}\Big{)}ds\rho^{-1}(x)dx<\infty.$

_Step 4. We show that for any $t\in[0,T]$,_

$\bar{u}(s,X^{t,x}_{s})=\bar{Y}^{t,x}_{s}\quad\hbox{and}\quad~{}(\sigma^{\top}\nabla\bar{u})(s,X^{t,x}_{s})=\bar{Z}^{t,x}_{s},\quad~{}a.s.,a.e.\
s\in[t,T],\ x\in\mathbb{R}^{n}.$ (5.28)

By triangular inequality, it is easy to see that

$\displaystyle\mathbb{E}\int_{\mathbb{R}^{n}}\int_{t}^{T}|\bar{u}(s,X^{t,x}_{s})-\bar{Y}^{t,x}_{s}|^{2}ds\rho^{-1}(x)dx$
$\displaystyle\leqslant\mathbb{E}\int_{\mathbb{R}^{n}}\int_{t}^{T}|\bar{u}(s,X^{t,x}_{s})-\bar{u}^{n}(s,X^{t,x}_{s})|^{2}ds\rho^{-1}(x)dx$
(5.29)
$\displaystyle\quad+\mathbb{E}\int_{\mathbb{R}^{n}}\int_{t}^{T}|\bar{u}^{n}(s,X^{t,x}_{s})-\bar{Y}^{t,x}_{s}|^{2}ds\rho^{-1}(x)dx.$
(5.30)

From 5.4 and (5.26), one has that the term (5.29) tends to $0$ as $n$ tends to
$\infty$. In addition, 5.4, (5.21) and (5.24) deduce that the term (5.30)
tends to $0$ as $n$ tends to $\infty$. So the first result in (5.28) holds. We
next prove the second result in (5.28), i.e.,

$(\sigma^{\top}\nabla\bar{u})(s,X^{t,x}_{s})=\bar{Z}^{t,x}_{s},\quad~{}a.s.,a.e.\
s\in[t,T],\ x\in\mathbb{R}^{n}.$ (5.31)

It is sufficient to show that the following term

$\displaystyle\mathbb{E}\int_{\mathbb{R}^{n}}\int_{t}^{T}|(\sigma^{\top}\nabla\bar{u})(s,X^{t,x}_{s})-\bar{Z}^{t,x}_{s}|^{2}ds\rho^{-1}(x)dx$
(5.32)
$\displaystyle\leqslant\mathbb{E}\int_{\mathbb{R}^{n}}\int_{t}^{T}|(\sigma^{\top}\nabla\bar{u})(s,X^{t,x}_{s})-(\sigma^{\top}\nabla\bar{u}^{n})(s,X^{t,x}_{s})|^{2}ds\rho^{-1}(x)dx$
$\displaystyle\quad~{}+\mathbb{E}\int_{\mathbb{R}^{n}}\int_{t}^{T}|(\sigma^{\top}\nabla\bar{u}^{n})(s,X^{t,x}_{s})-\bar{Z}^{t,x}_{s}|^{2}ds\rho^{-1}(x)dx$
$\displaystyle\longrightarrow 0\quad\hbox{as}\quad n\rightarrow\infty.$

In fact, similarly, combining 5.4, (5.21), (5.24), and (5.27), it is easy to
see that (5.32) holds, which implies that (5.31) holds too.

_Step 5. We show that $\bar{u}$ is a Sobolev solution to the following SPDE:_

$\displaystyle\bar{u}(s,x)=$
$\displaystyle\bar{h}(x)+\int_{s}^{T}\left\\{\mathcal{L}\bar{u}(r,x)+\bar{f}\big{(}r,x,\bar{u}(r,x),(\sigma^{\top}\nabla\bar{u})(r,x)\big{)}\right\\}dr$
(5.33)
$\displaystyle+\int_{s}^{T}\bar{g}\big{(}r,x,\bar{u}(r,x),(\sigma^{\top}\nabla\bar{u})(r,x)\big{)}d\overleftarrow{B}_{r},\quad
s\in[t,T].$

First, it was shown in Step 3 that $\bar{u}$ belongs to the space
$\mathscr{H}$, so by 5.1, we only need to verify that $\bar{u}$ satisfies Eq.
(5.5) with parameters $(\bar{f},\bar{g},\bar{h})$. Note that for every
$n\in\mathbb{N}$, since $\bar{u}^{n}$ is the Sobolev solution of SPDE (5.20),
so for any $\varphi\in\mathcal{C}_{c}^{1,\infty}([0,T]\times\mathbb{R}^{n})$,
$\bar{u}^{n}$ satisfies

$\displaystyle\int_{\mathbb{R}^{n}}\int_{t}^{T}\bar{u}^{n}(s,x)\partial_{s}\varphi(s,x)dsdx+\int_{\mathbb{R}^{n}}\bar{u}^{n}(t,x)\varphi(t,x)dx-\int_{\mathbb{R}^{n}}\bar{h}(x)\varphi(T,x)dx$
(5.34)
$\displaystyle\quad-\frac{1}{2}\int_{\mathbb{R}^{n}}\int_{t}^{T}(\sigma^{\top}\nabla\bar{u}^{n})(s,x)\cdot(\sigma^{\top}\nabla\varphi)(s,x)dsdx-\int_{\mathbb{R}^{n}}\int_{t}^{T}\bar{u}^{n}\mathrm{div}[(b-\widetilde{A})\varphi](s,x)dsdx$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{t}^{T}\bar{f}^{n}\big{(}s,x,\bar{u}^{n}(s,x),(\sigma^{\top}\nabla\bar{u}^{n})(s,x)\big{)}\varphi(s,x)dsdx$
$\displaystyle\quad+\int_{\mathbb{R}^{n}}\int_{t}^{T}\bar{g}\big{(}s,x,\bar{u}^{n}(s,x),(\sigma^{\top}\nabla\bar{u}^{n})(s,x)\big{)}\varphi(s,x)dB_{s}dx.$

Hence, if we can prove that along a subsequence (5.34) converges to (5.5) with
parameters $(\bar{f},\bar{g},\bar{h})$, then $\bar{u}$ satisfies (5.5) with
parameters $(\bar{f},\bar{g},\bar{h})$.

Due to that $\varphi\in\mathcal{C}_{c}^{1,\infty}([0,T]\times\mathbb{R}^{n})$,
$b\in C_{l,b}^{2}(\mathbb{R}^{n};\mathbb{R}^{n})$ and $\sigma\in
C_{l,b}^{3}(\mathbb{R}^{n};\mathbb{R}^{n\times n})$, then as
$n\rightarrow\infty$, clearly the left hand side of (5.34) tends to the
following term:

$\displaystyle\int_{\mathbb{R}^{n}}\int_{t}^{T}\bar{u}(s,x)\partial_{s}\varphi(s,x)dsdx+\int_{\mathbb{R}^{n}}\bar{u}(t,x)\varphi(t,x)dx-\int_{\mathbb{R}^{n}}\bar{h}(x)\varphi(T,x)dx$
$\displaystyle\quad-\frac{1}{2}\int_{\mathbb{R}^{n}}\int_{t}^{T}(\sigma^{\top}\nabla\bar{u})(s,x)\cdot(\sigma^{\top}\nabla\varphi)(s,x)dsdx-\int_{\mathbb{R}^{n}}\int_{t}^{T}\bar{u}\mathrm{div}[(b-\widetilde{A})\varphi](s,x)dsdx.$

Next, we compute the limit on the right hand side of (5.34). We show that
along a sequence

$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{n}}\int_{t}^{T}\bar{f}^{n}\big{(}s,x,\bar{u}^{n}(s,x),(\sigma^{\top}\nabla\bar{u}^{n})(s,x)\big{)}\varphi(s,x)dsdx$
(5.35)
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{t}^{T}\bar{f}\big{(}s,x,\bar{u}(s,x),(\sigma^{\top}\nabla\bar{u})(s,x)\big{)}\varphi(s,x)dsdx.$

In fact, note that the sequence $\\{\bar{f}^{n};n\in\mathbb{N}\\}$ is globally
Lipschitz continuous and decreasing converges pointwise to the function
$\bar{f}$, and for every $t\in[0,T]$,

$\displaystyle\big{(}\bar{u}^{n}(s,X_{s}^{t,x}),(\sigma^{\top}\nabla\bar{u}^{n})(s,X_{s}^{t,x})\big{)}=\big{(}\bar{Y}_{s}^{t,x,n},\bar{Z}_{s}^{t,x,n}\big{)},\quad~{}a.s.,a.e.\
s\in[t,T],\ x\in\mathbb{R}^{n},$
$\displaystyle\big{(}\bar{u}(s,X_{s}^{t,x}),(\sigma^{\top}\nabla\bar{u})(s,X_{s}^{t,x})\big{)}=\big{(}\bar{Y}_{s}^{t,x},\bar{Z}_{s}^{t,x}\big{)},\quad~{}a.s.,a.e.\
s\in[t,T],\ x\in\mathbb{R}^{n},$

with

$\left\\{\begin{aligned}
\displaystyle&\lim_{n\rightarrow\infty}\bar{Y}^{t,x,n}_{s}=\bar{Y}^{t,x}_{s},\quad~{}\hbox{a.s.}\
s\in[t,T],\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle&\lim_{n\rightarrow\infty}\mathbb{E}\int_{t}^{T}|\bar{Z}^{t,x,n}_{s}-\bar{Z}^{t,x}_{s}|^{2}ds=0.\end{aligned}\right.$

Then, similar to the argument as in the proof of 3.7, we get that

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\int_{t}^{T}\Big{|}$
$\displaystyle\bar{f}^{n}\big{(}s,X_{s}^{t,x},\bar{u}^{n}(s,X_{s}^{t,x}),(\sigma^{\top}\nabla\bar{u}^{n})(s,X_{s}^{t,x})\big{)}$
$\displaystyle-\bar{f}\big{(}s,X_{s}^{t,x},\bar{u}(s,X_{s}^{t,x}),(\sigma^{\top}\nabla\bar{u})(s,X_{s}^{t,x})\big{)}\Big{|}ds=0.$

So combining 5.4 and the Lebesgue dominated convergence theorem we have that

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\int_{\mathbb{R}^{n}}\int_{t}^{T}\Big{|}$
$\displaystyle\bar{f}^{n}\big{(}s,x,\bar{u}^{n}(s,x),(\sigma^{\top}\nabla\bar{u}^{n})(s,x)\big{)}$
$\displaystyle-\bar{f}\big{(}s,x,\bar{u}(s,x),(\sigma^{\top}\nabla\bar{u})(s,x)\big{)}\Big{|}ds\rho^{-1}(x)dx=0,$

which implies that (5.35) holds. Finally, for the second term on the right
hand side of (5.34), similar to (3.15) and the previous discussion, one has
that

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\bigg{|}$
$\displaystyle\int_{\mathbb{R}^{n}}\int_{t}^{T}\bar{g}\big{(}s,x,\bar{u}^{n}(s,x),(\sigma^{\top}\nabla\bar{u}^{n})(s,x)\big{)}\varphi(s,x)dB_{s}dx$
$\displaystyle-\int_{\mathbb{R}^{n}}\int_{t}^{T}\bar{g}\big{(}s,x,\bar{u}(s,x),(\sigma^{\top}\nabla\bar{u})(s,x)\big{)}\varphi(s,x)dB_{s}dx\bigg{|}=0\quad\hbox{in
probability.}$

Therefore $\bar{u}$ satisfies Eq. (5.5) with parameters
$(\bar{f},\bar{g},\bar{h})$, and thus $\bar{u}$ is a Sobolev solution to SPDE
(5.33).

_Step 6. The logarithmic transformation._

On the one hand, we see that SPDE (5.33) with parameters
$(\bar{f},\bar{g},\bar{h})$ has a Sobolev solution $\bar{u}$, and for every
$t\in[0,T]$,

$\bar{u}(s,X^{t,x}_{s})=\bar{Y}^{t,x}_{s}\quad\hbox{and}\quad(\sigma^{\top}\nabla\bar{u})(s,X^{t,x}_{s})=\bar{Z}^{t,x}_{s},\quad~{}a.s.,a.e.\
s\in[t,T],\ x\in\mathbb{R}^{n},$

where the pair $(\bar{Y}^{t,x},\bar{Z}^{t,x})$ is the solution of BDSDE (5.16)
with parameters $(\bar{f},\bar{g},\bar{h})$. On the other hand,

$\bar{Y}^{t,x}_{s}=\exp(\beta
Y^{t,x}_{s})\quad~{}\hbox{and}\quad~{}\bar{Z}^{t,x}_{s}=\beta\bar{Y}^{t,x}_{s}Z^{t,x}_{s},\quad~{}t\leqslant
s\leqslant T,$

where the pair $(Y^{t,x},Z^{t,x})\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{R})\times
L_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{d})$ is the solution of BDSDE (5.2) with
parameters $(f,g,h)$. Hence, similar to the computation from (5.11) to (5.15),
if we define

$\displaystyle
u(s,X^{t,x}_{s})\triangleq\frac{\ln\bar{u}(s,X^{t,x}_{s})}{\beta}=\frac{\ln\bar{Y}^{t,x}_{s}}{\beta}=Y^{t,x}_{s},\quad~{}t\leqslant
s\leqslant T,$

with

$\displaystyle(\sigma^{\top}\nabla
u)(s,X^{t,x}_{s})=\frac{(\sigma^{\top}\nabla\bar{u})(s,X^{t,x}_{s})}{\beta\bar{u}(s,X^{t,x}_{s})}=\frac{\bar{Z}^{t,x}_{s}}{\beta\bar{Y}^{t,x}_{s}}=Z^{t,x}_{s},\quad~{}t\leqslant
s\leqslant T,$

then $u$ is a Sobolev solution of SPDE (5.3) with parameters $(f,g,h)$, and
for every $t\in[0,T]$,

$u(s,X_{s}^{t,x})=Y^{t,x}_{s}\quad~{}\hbox{and}\quad~{}(\sigma^{\top}\nabla
u)(s,X_{s}^{t,x})=Z^{t,x}_{s},\quad~{}a.s.,a.e.\ s\in[t,T],\
x\in\mathbb{R}^{n},$

where $(Y^{t,x},Z^{t,x})$ is the unique solution of BDSDE (5.2) with
parameters $(f,g,h)$. This completes the proof. ∎

##  6 Conclusion Remarks

We initiate the study of quadratic BDSDEs, where the existence, comparison,
and stability results for one-dimensional BDSDEs with quadratic growth and
bounded terminal value are obtained. We remark that when proving a priori
estimate, in order to ensure $Y$ is bounded, we introduce the condition (3.5)
concerning the coefficient $g$, which we think is necessary for the case of
bounded terminal value. However, if someone discusses the solution of
quadratic BDSDEs with unbounded terminal value, the condition (3.5) could be
relaxed because the boundedness of $Y$ is needn’t at that time. We hope to
publish some follow-up works when the terminal value $\xi$ is unbounded or/and
$Y$ is multi-dimensional. Finally, in this framework, we used BDSDEs to prove
the existence and uniqueness of the Sobolev solutions of semilinear SPDEs,
which extend the nonlinear Feynman-Kac formula of Pardoux–Peng [26]. On the
other hand, the existence and uniqueness of viscosity solutions of related
SPDEs are interesting and merit further studies in the near future.

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