#  Dynamic programming principle for delayed stochastic recursive optimal
control problem and HJB equation with non-Lipschitz generator††thanks: This
paper is supported by National Key R&D Program of China (Grant
Nos.2022YFA1006101, 2018YFA0703900), National Natural Science Foundation of
China (Grant Nos.12101291, 12371445), Guangdong Basic and Applied Basic
Research Foundation (grant No. 2022A1515012017), the Taishan Scholars Climbing
Program of Shandong (grant No. TSPD20210302), and Science and Technology
Commission of Shanghai Municipality (Grant No.22ZR1407600).

Jiaqiang Wen , Zhen Wu , Qi Zhang Department of Mathematics, Southern
University of Science and Technology, Shenzhen 518055, China (Email:
wenjq@sustech.edu.cn). School of Mathematics, Shandong University, Jinan
250100, China (Email:wuzhen@sdu.edu.cn). Corresponding author. School of
Mathematical Sciences, Fudan University, Shanghai 200433, China
(qzh@fudan.edu.cn).

Abstract. In this paper, we study the delayed stochastic recursive optimal
control problem with a non-Lipschitz generator, in which both the dynamics of
the control system and the recursive cost functional depend on the past path
segment of the state process in a general form. First, the dynamic programming
principle for this control problem is obtained. Then, by the generalized
comparison theorem of backward stochastic differential equations and the
stability of viscosity solutions, we establish the connection between the
value function and the viscosity solution of the associated Hamilton-Jacobi-
Bellman equation. Finally, an application to the consumption-investment
problem under the delayed continuous-time Epstein-Zin utility with a non-
Lipschitz generator is presented.

Key words: delayed stochastic recursive control problem, non-Lipschitz
generator, dynamic programming principle, Hamilton-Jacobi-Bellman equation,
stochastic differential utility.

AMS subject classifications. 90C39, 60H10, 93E20.

##  1 Introduction

The classical stochastic control theory appears with the birth of stochastic
analysis and developed rapidly in recent decades due to its wide range of
applications (see Yong–Zhou [28]), and one of its important applications is
the optimal consumption-investment problem under stochastic differential
utility (SDU, for short), which was put forward by Duffie–Epstein [6] in a
conditional expectation form. Actually, SDU is equivalent to the nonlinear
backward stochastic differential equations (BSDEs, for short) and its control
problem is the stochastic recursive control problem introduced firstly by Peng
[22] and developed by Wu–Yu [27], Li–Peng [16], Pu–Zhang [23], to name but a
few. Due to the importance of BSDEs in modern mathematical finance (see El
Karoui–Peng–Quenez [7]), both stochastic recursive control theory and the
optimal consumption-investment problem of stochastic differential utilities
achieved great progresses in the past few decades.

On the other hand, the research of many natural and social phenomena shows
that the future development of many processes depends on not only their
present state but also essentially on their historical information. For
example, the immune response of the vaccine of COVID-19 depends on the
complete history of the vaccinator’s exposure to microbial infection antigens;
the quality of a harvest does not only depend on the current temperature of
the environment but also on the whole pattern of the past temperatures; the
price of the option depends not only on its current value but also on the
whole history of the underlying price, etc. In the theory of stochastic
processes, such phenomenons can be described by the stochastic differential
delayed equations (SDDEs, for short). See Mohammed [18, 19], Chang [2], and
the references therein.

Hence, an interesting problem is whether the stochastic differential utilities
can be developed to stochastic control problems with delay. The answer is
positive, although most problems remain practically intractable due to the
complex infinite-dimensional state-space framework. Here we name but a few
existing results. For example, Larssen [14] obtained the dynamic programming
principle for the systems of stochastic control with delay. Later,
Larssen–Risebro [15] further studied the optimal consumption problems based on
the stochastic delay equations and presented the financial application of
dynamic programming principle. Arriojas–Hu–Mohammed–Pap [1] got a delayed
Black–Scholes formula. Fuhrman–Masiero–Tessitore [10] studied a forward-
backward system with delay, its application to optimal stochastic control and
the regularity of associated Hamilton-Jacobi-Bellman (HJB, for short)
equations. Chen–Wu [3] was concerned with one kind of delayed stochastic
recursive optimal control problem, proved the dynamic programming principle,
and demonstrated that the value function is a viscosity solution of the
corresponding HJB equation. Shi–Xu–Zhang [25] studied a general kind of
stochastic differential delayed equations, obtained the stochastic maximal
principle, and show the application to stochastic portfolio optimization model
with delay and bounded memory. Tang–Zhang [26] obtained the dynamic
programming principle in a path-dependent case and studied the associated
path-dependent Bellman equations. Li–Wang–Wu [17] focused on a class of
linear-quadratic optimal control for time-delay stochastic recursive utility.
It should be pointed out that all the above-mentioned works concerning delayed
systems are based on the condition of Lipschitz generator. While in many SDU
models the generator is non-Lipschitz with respect to the utility and
consumption, see (1.5) below for an example.

In this paper, inspired by the financial applications, we are interested in
studying the delayed stochastic recursive control problem with non-Lipschitz
generator. The dynamic programming principle for the delayed systems will be
obtained, and the connection between the value function and the viscosity
solution of the associated Hamilton-Jacobi-Bellman equation will be shown. In
order to present our work more clearly, we present the problem in details.
Consider the following controlled SDDE

$\left\\{\begin{split}\displaystyle
dX(s)&=b(s,X_{s},v(s))dt+\sigma(s,X_{s},v(s))dW(t),\quad s\in[t,T],\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle X(s)&=\gamma(s),\quad
s\in[t-\delta,t],\end{split}\right.$ (1.1)

where $\gamma$ comes from the continuous function space
$C([0,t];\mathbb{R}^{n})$ regarded as the past value of the equation, $X_{s}$
is the segment of the path of $X(\cdot)$ from $s-\delta$ up to time $s$ for a
fixed $\delta>0$, and $v(\cdot)$ is an admissible control process arisen from
the admissible control set ${\cal U}$ (see (2.1) below). For the state
equation (1.1), the cost functional is associated with the solution of the
following BSDE for $s\in[t,T]$,

$Y^{t,\gamma;v}(s)=\Phi(X_{T}^{t,\gamma;v})+\int_{s}^{T}g(r,X_{r}^{t,\gamma;v},Y^{t,\gamma;v}(r),Z^{t,\gamma;v}(r),v(r))dr-\int_{s}^{T}Z^{t,\gamma;v}(r)dW(r),$
(1.2)

and the cost functional is defined as below

$J(t,\gamma;v)\triangleq Y^{t,\gamma;v}(t),\quad t\in[0,T].$ (1.3)

Since the recursive utility can be regarded as the conditional expectation
form of the solution of BSDE (1.2), the stochastic recursive control system in
form contains the following SDU

$U^{t,\gamma;c}(t)=\mathbb{E}^{\mathscr{F}_{t}}\left[\Phi(X_{T}^{t,\gamma;c})+\int_{t}^{T}g\left(r,X_{r}^{t,\gamma;c},U^{t,\gamma;c}(r),c(r)\right)dr\right],$
(1.4)

where $c(\cdot)$ is a consumption process serving as the control. Indeed,
(1.4) is a general form of the SDU in Duffie–Epstein [6], involving the delay.
The goal of the optimal control problem is to find an optimal control process
$\tilde{v}\in{\cal U}$ to maximize the cost functional (1.3) for given
$(t,\gamma)$.

In this paper, we are committed to relaxing the Lipschitz restriction to the
generator, i.e., the generator $g(\gamma,u,c)$ in (1.4) is continuous but not
necessarily Lipschitz with respect to both the utility variable $u$ and the
consumption variable $c$. For the variable $u$, the generator could be of
polynomial growth in our assumptions. These settings will make a lot of
difference in the deduction of the dynamic programming principle and bring
troubles in the verification of conditions for the stability of viscosity
solutions, which plays the key role in the connection between the value
function and the viscosity solution of the associated HJB equation. Let’s
present an significant example for the non-Lipschitz generator with respect to
utility and consumption, i.e., the well-known continuous-time Epstein-Zin
utility with a form

$g(u,c)=\frac{\vartheta}{1-\frac{1}{\psi}}(1-r)u\bigg{[}\Big{(}\frac{c}{\left((1-r)u\right)^{\frac{1}{1-r}}}\Big{)}^{1-\frac{1}{\psi}}-1\bigg{]},$
(1.5)

where $\vartheta>0$ represents the rate of time preference, $0<\psi\neq 1$ is
the elasticity of intertemporal substitution, and $0<r\neq 1$ is the
coefficient of relative risk aversion. Note that the generator $g(u,c)$ in
(1.5) is non-Lipschitz in $u$ and $c$ but monotonic with respect to $u$ in a
wide range of parameters.

The main difficulties of our problem come from the delayed term and the non-
Lipschitz generator. For the former one, we need to overcome the trouble from
the infinite-dimensional of the space $C([0,t];\mathbb{R}^{n})$ in the
verification that the optimal value function satisfies the HJB equation. We
manage to do it with the help of the semigroup property of generator in the
infinite-dimensional setting, by borrowing ideas from
Fuhrman–Masiero–Tessitore [10] and Chen–Wu [3]. The non-Lipschitz generator
brings troubles in the establishment of the relationship between the value
function and its associated HJB equation. We conquer it by applying the
techniques for the solvability of BSDE with monotonic coefficients such as in
Pardoux [20] and the stability of viscosity solutions in Fleming–Soner [9].

The paper is organized as follows. In Section 2, some preliminaries are
presented. In Section 3, we give the main results, i.e., the dynamic
programming principle for delayed stochastic recursive optimal control problem
and the relationship between the value function of the delayed stochastic
recursive optimal control problem and the viscosity solution of associated HJB
equation, which develop the results as in [10, 3, 23]. In Section 4, an
example concerning continuous-time Epstein-Zin utility with delay and non-
Lipschitz generator is presented to demonstrate the applications of our
theoretical results to mathematical finance. Some auxiliaries are presented in
Section 5.

##  2 Preliminaries

Let $(\Omega,\mathscr{F},\mathbb{F},\mathbb{P})$ be a complete filtered
probability space and a $d$-dimensional standard Brownian motion
$\\{W(t)\;;0\leqslant t<\infty\\}$ is defined in it, where
$\mathbb{F}=\\{\mathscr{F}_{t};0\leqslant t<\infty\\}$ is the natural
filtration of the Brownian motion $W(\cdot)$ with $\mathscr{F}_{0}$ containing
$\mathcal{N}$, the class of all $\mathbb{P}$-null sets of $\mathscr{F}$. Let
$T>0$ be a fixed terminal time and $\delta\geqslant 0$ be a given finite time
delay. For each $t\in[0,T]$, we denote by $\mathbb{C}_{t}\triangleq
C([t-\delta,t];\mathbb{R}^{n})$ the set of bounded continuous
$\mathbb{R}^{n}$-valued paths on $[t-\delta,t]$ and

$\mathbb{C}=\bigcup_{t\in[0,T]}\mathbb{C}_{t}.$

For each $\gamma_{t}\in\mathbb{C}_{t}$, it denotes the path from $t-\delta$ up
to time $t$, i.e.,

$\displaystyle\gamma_{t}=\gamma_{t}(\theta)_{t-\delta\leqslant\theta\leqslant
t},\quad~{}t\in[0,T],$

and its value at time $r$ is denoted by $\gamma_{t}(r)$. For each
$t,s\in[0,T]$ and $\gamma_{t},\gamma_{s}\in\mathbb{C}$, we denote

$\displaystyle\|\gamma_{t}\|_{\mathbb{C}}\triangleq$
$\displaystyle\sup_{t-\delta\leqslant r\leqslant t}|\gamma_{t}(r)|,$
$\displaystyle\|\gamma_{t}-\gamma_{s}\|_{\mathbb{C}}\triangleq$
$\displaystyle\sup_{(t-\delta)\wedge(s-\delta)\leqslant r\leqslant t\vee
s}|\gamma_{t}((t-\delta)\vee r\wedge t)-\gamma_{s}((s-\delta)\vee r\wedge
s)|.$

Then, it is obvious that $\mathbb{C}_{t}$ is a Banach space with respect to
$\|\cdot\|$. Next, for $t\in[0,T]$, $q\geqslant 1$ and Euclidean space
$\mathbb{H}$ (which could be $\mathbb{R}^{n}$, $\mathbb{R}^{n\times m}$,
$\mathbb{S}^{n}$, etc.), we introduce the following spaces of functions and
processes:

$\displaystyle\displaystyle
L^{2q}(\Omega,\mathscr{F}_{t};\mathbb{H})\triangleq$
$\displaystyle\Big{\\{}\xi:\Omega\to\mathbb{H}\bigm{|}\xi\hbox{ is
$\mathscr{F}_{t}$-measurable, }\mathbb{E}|\xi|^{2q}<\infty\Big{\\}},$
$\displaystyle\displaystyle M_{\mathbb{F}}^{2q}(t,T;\mathbb{H})\triangleq$
$\displaystyle\bigg{\\{}\varphi:[t,T]\times\Omega\to\mathbb{H}\bigm{|}\varphi(\cdot)\hbox{
is $\mathbb{F}$-adapted and}\
\mathbb{E}\int^{T}_{t}|\varphi(s)|^{2q}ds<\infty\bigg{\\}},$ $\displaystyle
S^{2q}_{\mathbb{F}}(t,T;\mathbb{H})\triangleq$
$\displaystyle\bigg{\\{}\varphi:[t,T]\times\Omega\to\mathbb{H}\bigm{|}\varphi(\cdot)\hbox{
is $\mathbb{F}$-adapted, continuous and}\
\mathbb{E}\sup_{s\in[t,T]}|\varphi(s)|^{2q}<\infty\bigg{\\}},$
$\displaystyle\displaystyle
L^{2q}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})\triangleq$
$\displaystyle~{}\bigg{\\{}\gamma:\Omega\to\mathbb{C}_{t}\bigm{|}\gamma(\cdot)\hbox{
is $\mathbb{F}$-adapted and}\
\mathbb{E}\|\gamma_{t}\|_{\mathbb{C}}^{2q}<\infty\bigg{\\}}$

and

${\cal U}\triangleq\Big{\\{}v\in
M_{\mathbb{F}}^{2}(0,T;\mathbb{R}^{m})\big{|}\hbox{ the process $v$ takes
value in a compact set $U\in\mathbb{R}^{m}$}\Big{\\}}.$ (2.1)

Here ${\cal U}$ is called the admissible control set.

For an element of the space $L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$, we
have the following proposition.

######  Proposition 2.1 (Lemma 3.4 in [3]).

For any bounded $\Upsilon\in L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$, there
is a sequence

$\Upsilon^{n}=\sum^{n}_{i=1}\gamma^{i}I_{A_{i}}$

converging to $\Upsilon$ in the space
$L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$, where
$\gamma^{i}\in\mathbb{C}_{t}$ with $1\leqslant i\leqslant n$, $n\in\mathbb{N}$
and $\\{A_{i}\\}^{n}_{i=1}$ is a partition of $(\Omega,\mathscr{F}_{t})$.

For the coefficients of the state equation (1.1), we assume the conditions as
follows.

  * (A1)

$b(t,\gamma,v):[0,T]\times\mathbb{C}\times U\rightarrow\mathbb{R}^{n}$ and
$\sigma(t,\gamma,v):[0,T]\times\mathbb{C}\times
U\rightarrow\mathbb{R}^{n\times d}$ are joint measurable and continuous with
respect to $t$.

  * (A2)

There is a constant $L\geqslant 0$ such that for any $t\in[0,T]$,
$\gamma,\bar{\gamma}\in\mathbb{C}$ and $v,\bar{v}\in U$, we have

$|b(t,\gamma,v)-b(t,\bar{\gamma},\bar{v})|+|\sigma(t,\gamma,v)-\sigma(t,\bar{\gamma},\bar{v})|\leqslant
L\left(\|\gamma-\bar{\gamma}\|_{\mathbb{C}}+|v-\bar{v}|\right).$

For the sake of the proof of dynamic programming principle, we would like to
consider a general stochastic differential equation (SDE, for short) with a
random initial variable. By using a standard argument, we have the following
2.2, which is a non-essential extension of Theorem 2.3 in
Fuhrman–Masiero–Tessitore [10].

######  Proposition 2.2.

Assume (A1)-(A2) and $q\geqslant 1$. Then for any $t\in[0,T]$,
$v\in\mathcal{U}$, $\Upsilon\in L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$,
the following SDE

$\left\\{\begin{split}\displaystyle&dX^{t,\Upsilon;v}(s)=b(s,X^{t,\Upsilon;v}_{s},v(s))dt+\sigma(s,X^{t,\Upsilon;v}_{s},v(s))dW(t),\quad
s\in[t,T],\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle&X^{t,\Upsilon;v}(s)=\Upsilon_{t}(s),\quad
s\in[t-\delta,t],\end{split}\right.$ (2.2)

has a unique strong solution $X^{t,\Upsilon;v}(\cdot)\in
S_{\mathbb{F}}^{2q}(t,T;\mathbb{R}^{n})$. Moreover, there is a constant
$C\geqslant 0$ depending only on $K,\ L$ and $T$ such that for any
$s\in[t,T],v,v^{\prime}\in\mathcal{U},\Upsilon,\Upsilon^{\prime}\in
L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$, we have

$\mathbb{E}^{\mathscr{F}_{t}}\Big{[}\|X^{t,\Upsilon;v}_{s}\|^{2q}_{\mathbb{C}}\Big{]}\leqslant
C\Big{(}1+\|\Upsilon_{t}\|_{\mathbb{C}}^{2q}+\mathbb{E}^{\mathscr{F}_{t}}\int_{t}^{T}|v(s)|^{2q}ds\Big{)}$

and

$\mathbb{E}^{\mathscr{F}_{t}}\Big{[}\|X^{t,\Upsilon;v}_{s}-X^{t,\Upsilon^{\prime};v^{\prime}}_{s}\|^{2q}_{\mathbb{C}}\Big{]}\leqslant
C\Big{(}\|\Upsilon_{t}-\Upsilon^{\prime}_{t}\|_{\mathbb{C}}^{2q}+\mathbb{E}^{\mathscr{F}_{t}}\int_{t}^{T}|v(s)-v^{\prime}(s)|^{2q}ds\Big{)}.$

With $X^{t,\Upsilon;v}(\cdot)$ being the solution of the state equation (2.2),
we consider the following type of BSDE, for $s\in[t,T]$,

$Y^{t,\Upsilon;v}(s)=\Phi(X_{T}^{t,\Upsilon;v})+\int_{s}^{T}g(r,X_{r}^{t,\Upsilon;v},Y^{t,\Upsilon;v}(r),Z^{t,\Upsilon;v}(r),v(r))dr-\int_{s}^{T}Z^{t,\Upsilon;v}(r)dW(r).$
(2.3)

For the generator $(\Phi,g)$ of BSDE (2.3), we give the following assumptions.

  * (A3)

The generator $\Phi(\gamma):\mathbb{C}\rightarrow\mathbb{R}$ and
$g(t,\gamma,y,z,v):[0,T]\times\mathbb{C}\times\mathbb{R}\times\mathbb{R}^{d}\times
U\rightarrow\mathbb{R}$ are joint measurable, and $g(t,\gamma,y,z,v)$ is
continuous with respect to $(t,y,v)$.

  * (A4)

There is a positive constant $\tilde{L}$ such that for any
$t\in[0,T],\gamma,\bar{\gamma}\in\mathbb{C},y\in\mathbb{R},z,\bar{z}\in\mathbb{R}^{d},v\in
U$,

$\left|\Phi(\gamma)-\Phi\left(\bar{\gamma}\right)\right|+\left|g(t,\gamma,y,z,v)-g\left(t,\bar{\gamma},y,\bar{z},v\right)\right|\leqslant\tilde{L}\left(\left\|\gamma-\bar{\gamma}\right\|_{\mathbb{C}}+\left|z-\bar{z}\right|\right).$

  * (A5)

There is a constant $\mu\in\mathbb{R}$ such that for any
$t\in[0,T],\gamma\in\mathbb{C},y,\bar{y}\in\mathbb{R},z\in\mathbb{R}^{d},v\in
U$,

$(y-\bar{y})[g(t,\gamma,y,z,v)-g(t,\gamma,\bar{y},z,v)]\leqslant\mu|y-\bar{y}|^{2}.$

  * (A6)

For a given $p\geqslant 1$, there is a constant $M\geqslant 0$ such that for
any $t\in[0,T],\gamma\in\mathbb{C},y\in\mathbb{R},z\in\mathbb{R}^{d},v\in U$,

$\left|g(t,\gamma,y,z,v)-g\left(t,\gamma,0,z,v\right)\right|\leqslant
M(1+|y|^{p}).$

Note that (A3) implies that $g(t,0,0,0,v)$ is bounded for all $t\in[0,T]$ and
$v\in U$. Moreover, under the condition (A1)-(A6), by a similar method to
Pardoux [20], we can prove that BSDE (2.3) admits a unique adapted solution.
Here, we conclude the existence, uniqueness, and some useful estimates to the
solution of BSDE (2.3).

######  Proposition 2.3.

Assume (A1)-(A6) and $q\geqslant 1$. Then for any $t\in[0,T],$
$v\in\mathcal{U}$, $\Upsilon\in L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$,
BSDE (2.3) admits a unique strong solution
$(Y^{t,\Upsilon;v}(\cdot),Z^{t,\Upsilon;v}(\cdot))\in
S_{\mathbb{F}}^{2q}(t,T;\mathbb{R})\times
M_{\mathbb{F}}^{2}(t,T;\mathbb{R}^{d})$. Moreover, there is a constant
$C\geqslant 0$ depending only on $K,L,\tilde{L},\mu,M$ and $T$ such that for
any $s\in[t,T],v\in\mathcal{U},\Upsilon,\Upsilon^{\prime}\in
L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$, we have

$\displaystyle\mathbb{E}^{\mathscr{F}_{t}}\Big{\\{}\sup_{s\in[t,T]}|Y^{t,\Upsilon;v}(s)|^{2q}+\int_{t}^{T}|Y^{t,\Upsilon;v}(s)|^{2q-2}|Z^{t,\Upsilon;v}(s)|^{2}\Big{\\}}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle
C\Big{\\{}1+\|\Upsilon_{t}\|^{2q}_{\mathbb{C}}+\mathbb{E}^{\mathscr{F}_{t}}\int_{t}^{T}|g(r,0,0,0,v(r))|^{2q}dr\Big{\\}}$

and

$|Y^{t,\Upsilon;v}(t)-Y^{t,\Upsilon^{\prime};v}(t)|\leqslant
C\|\Upsilon_{t}-\Upsilon^{\prime}_{t}\|_{\mathbb{C}}.$

In the theory of the dynamic programming principle, the comparison theorem of
BSDE plays a key role, without the exception of our circumstances that the
generator of BSDE is continuous and monotonic rather than the Lipschitz
continuous with respect to its items. The following comparison theorem is an
immediate consequence from Fan–Jiang [8] which is applicable under our
circumstances.

######  Proposition 2.4.

Assume that $(\Phi,g)$ and $\left(\Phi^{\prime},g^{\prime}\right)$ are two
pair of generators of BSDE (2.3) with respect to the solutions $(Y,Z)$ and
$\left(Y^{\prime},Z^{\prime}\right)$, respectively. Assume that for all
$t\in[0,T],\gamma\in\mathbb{C},v\in U$,
$\Phi(\gamma)\leqslant\Phi^{\prime}(\gamma)$ and

$g(t,\gamma,Y^{\prime}(t),Z^{\prime}(t),v)\leqslant
g^{\prime}(t,\gamma,Y^{\prime}(t),Z^{\prime}(t),v)\ \ {\rm(or}\
g(t,\gamma,Y(t),Z(t),v)\leqslant g^{\prime}(t,\gamma,Y(t),Z(t),v){\rm)}\ \ \
\hbox{a.s.}$

Then $Y(t)\leqslant Y^{\prime}(t)$ a.s., for any $t\in[0,T]$.

##  3 Main results

In this section, we present the main results, i.e., the dynamic programming
principle and the connection between the value function and the viscosity
solution of the associated Hamilton-Jacobi-Bellman equation.

###  3.1 Dynamic programming principle

In this subsection, we show the dynamic programming principle of the
stochastic recursive control problem, where the generator $g$ with delay is
not necessarily Lipschitz but monotonic and continuous. First, we introduce a
family of backward semigroups which was put forward by Peng [21].

For any $t\in[0,T],s\in(t,T],\gamma\in\mathbb{C},v\in{\cal U}$, and $\xi\in
L^{2}(\Omega,\mathscr{F}_{s};\mathbb{R})$, we define

$G_{\theta,s}^{t,\gamma;v}[\xi]\triangleq\widetilde{Y}^{t,\gamma;v}(\theta),\quad\theta\in[t,s],$

where
$(\widetilde{Y}^{t,\gamma;v}(\cdot),\widetilde{Z}^{t,\gamma;v}(\cdot))\in
S_{\mathbb{F}}^{2}(t,s;\mathbb{R})\times
M_{\mathbb{F}}^{2}(t,s;\mathbb{R}^{d})$ is the solution of the following BSDE
over the time interval $[t,s]$:

$\widetilde{Y}^{t,\gamma;v}(\theta)=\xi+\int_{\theta}^{s}f\big{(}r,X_{r}^{t,\gamma;v},\widetilde{Y}^{t,\gamma;v}(r),\widetilde{Z}^{t,\gamma;v}(r),v(r)\big{)}dr-\int_{\theta}^{s}\widetilde{Z}^{t,\gamma;v}(r)dW(r),$

and $X_{r}^{t,\gamma;v}$ is the solution of the state equation (1.1)
representing the segment of the path of the process $X^{t,x;v}(\cdot)$ from
$r-\delta$ up to time $r$. By the uniqueness of BSDE in 2.3, for
$\xi=Y^{t,\gamma;v}(s)$, where $Y^{t,\gamma;v}(\cdot)$ is the solution of BSDE
(1.2), we have

$G_{t,T}^{t,\gamma;v}[\Phi(X_{T}^{t,\gamma;v})]=G_{t,s}^{t,\gamma;v}[Y^{t,\gamma;v}(s)].$

On the other hand, coming back to the control system (1.1)-(1.3), we have that
the related value function, which represents the control problem to maximize
the cost functional, is defined by

$u(t,\gamma)\triangleq\mathop{\rm
esssup}_{v\in\mathcal{U}}J(t,\gamma;v),\quad(t,\gamma)\in[0,T]\times\mathbb{C}.$
(3.1)

Similar to the classical situation, the value function $u(t,\gamma)$ defined
in (3.1) is a deterministic function in the setting of non-Lipschitz
generator.

######  Lemma 3.1.

Assume (A1)-(A6). Then the value function $u(t,\gamma)$ defined in (3.1) is a
deterministic function.

For clear presentation, the proof of the above lemma together with the proofs
of the following lemmas are presented in the appendix (see Section 5).

######  Lemma 3.2.

For any $t\in[0,T]$ and $\gamma,\bar{\gamma}\in\mathbb{C}$,

$|u(t,\gamma)|\leqslant C\big{(}1+\|\gamma\|_{\mathbb{C}}\big{)}$ (3.2)

and

$|u(t,\gamma)-u(t,\bar{\gamma})|\leqslant\tilde{C}\|\gamma-\bar{\gamma}\|_{\mathbb{C}},$
(3.3)

where $C\geqslant 0$ is the constant in 2.3.

Next, we would like to study some properties for the cost functional $J$ and
the value function $u$ when the initial state is a $\mathbb{C}$-valued random
variable, which will be used in the proof of the dynamic programming principle
later.

######  Lemma 3.3.

For any $t\in[0,T]$, $v\in{\cal U}$ and $\Upsilon\in
L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$, we have

$J(t,\Upsilon;v)=Y^{t,\Upsilon;v}(t).$

######  Lemma 3.4.

For any $t\in[0,T]$, $v\in{\cal U}$ and $\Upsilon\in
L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$, we have

$u(t,\Upsilon)\geqslant Y^{t,\Upsilon;v}(t)\quad\hbox{a.s.},$ (3.4)

and for arbitrary $\varepsilon>0$, there is an admissible control $v\in{\cal
U}$ such that

$u(t,\Upsilon)\leqslant Y^{t,\Upsilon;v}(t)+\varepsilon\quad\hbox{a.s.}$ (3.5)

Based on the above preparations, we can prove the first main result, i.e., the
dynamic programming principle.

######  Theorem 3.5 (Generalized dynamic programming principle).

Assume (A1)-(A6). Then the value function $u(t,\gamma)$ defined in (3.1) for
our optimal control problem system obeys dynamic programming principle, i.e.,
for each $0\leqslant\tau\leqslant T-t$,

$u(t,\gamma)=\mathop{\rm
esssup}_{v\in\mathcal{U}}G_{t,t+\tau}^{t,\gamma;v}[u(t+\tau,X_{t+\tau}^{t,\gamma;v})].$
(3.6)

###### Proof.

Noticing the existence and uniqueness of BSDE (1.2), we have

$\displaystyle u(t,\gamma)$ $\displaystyle=\mathop{\rm
esssup}_{v\in\mathcal{U}}G_{t,T}^{t,\gamma;v}[\Phi(X_{T}^{t,\gamma;v})]=\mathop{\rm
esssup}_{v\in\mathcal{U}}G_{t,t+\tau}^{t,\gamma;v}[Y^{t,\gamma;v}(t+\tau)]=\mathop{\rm
esssup}_{v\in\mathcal{U}}G_{t,t+\tau}^{t,\gamma;v}[Y^{t+\tau,X_{t+\tau}^{t,\gamma;v};v}(t+\tau)].$

Keep in mind that the generator of BSDE (1.2) is not Lipschitz with respect to
$Y$, which leads to the failure of the classical comparison theorem.
Fortunately, a generalized comparison theorem in 2.4 given by Fan–Jiang [8]
works, in which the generator of BSDE is non-Lipschitz and “weakly” monotonic
with respect to the first solution in the solution pair. However, before
applying the generalized comparison theorem, we need verify the necessary
terminal condition, i.e., for $s\in[t,T]$ and $v\in{\cal U}$,
$u(s,X^{t,\gamma,v}_{s})$ is square-integrable and can act as the terminal
value of BSDE. For this, note that for arbitrary $\varepsilon>0$, 3.4 implies
that there is an admissible control $\bar{v}\in{\cal U}$ such that

$Y^{s,X_{s}^{t,\gamma;v};\bar{v}}(s)\leqslant u(s,X_{s}^{t,\gamma;v})\leqslant
Y^{s,X_{s}^{t,\gamma;v};\bar{v}}(s)+\varepsilon.$

Hence we only need to show
$\mathbb{E}|Y^{s,X_{s}^{t,\gamma;v};\bar{v}}(s)|^{2}<\infty$. Actually, it
follows from (A1)-(A6), 2.2 and 2.3, since

$\displaystyle\mathbb{E}|Y^{s,X_{s}^{t,\gamma;v};\bar{v}}(s)|^{2}\leqslant
C_{p}\left(1+\mathbb{E}\|X_{s}^{t,\gamma;v}\|^{2}_{\mathbb{C}}\right)\leqslant
C_{p}\bigg{(}1+\|\gamma\|_{\mathbb{C}}^{2}+\mathbb{E}\int_{t}^{T}\left|v(r)\right|^{2}dr\bigg{)}<\infty.$

Now, by 2.4 we have

$u(t,\gamma)\leqslant\mathop{\rm
esssup}_{v\in\mathcal{U}}G_{t,t+\tau}^{t,\gamma;v}[u(t+\tau,X_{t+\tau}^{t,\gamma;v})].$
(3.7)

On the other hand, 3.4 also implies that, for arbitrary $\varepsilon>0$, there
is an admissible control $\tilde{v}\in\mathcal{U}$ such that

$u(t+\tau,X_{t+\tau}^{t,\gamma;v})-\varepsilon\leqslant
Y^{t+\tau,X_{t+\tau}^{t,\gamma;v};\tilde{v}}(t+\tau).$

Hence by 2.4 again, we have

$\displaystyle\displaystyle u(t,\gamma)$ $\displaystyle\geqslant\mathop{\rm
esssup}_{v\in\mathcal{U}}G_{t,t+\tau}^{t,\gamma;v}[Y^{t+\tau,X_{t+\tau}^{t,\gamma;v};\tilde{v}}(t+\tau)]$
$\displaystyle\geqslant\mathop{\rm
esssup}_{v\in\mathcal{U}}G_{t,t+\tau}^{t,\gamma;v}[u(t+\tau,X_{t+\tau}^{t,\gamma;v})-\varepsilon]$
$\displaystyle\geqslant\mathop{\rm
esssup}_{v\in\mathcal{U}}G_{t,t+\tau}^{t,\gamma;v}[u(t+\tau,X_{t+\tau}^{t,\gamma;v})]-\sqrt{C_{p}}\varepsilon,$
(3.8)

where the last inequality is based on the regular estimation of BSDE (see Peng
[22] or Zhang [29]).

Finally, due to the arbitrariness of $\varepsilon$, (3.6) follows from (3.7)
and (3.1). ∎

###  3.2 Hamilton–Jacobi–Bellman equation

In the classical case, Peng [22] established that the value function is as a
viscosity solution to its associated Hamilton-Jacobi-Bellman equation. In this
subsection, our objective is to establish a connection between the value
function defined in equation (3.1) and the viscosity solution of the HJB
equation (3.9) in the setting of a non-Lipschitz generator with delay.

To prove it, we need an additional assumption: namely, that the generator $g$
of BSDE (1.2) is independent of the second unknown variable $Z$, otherwise,
obtaining the desired result would be significantly challenging. The good news
is that this scenario finds important applications in finance (as described in
Section 4). Specifically, in this case, the generator $g(t,\gamma,y,z,v)$ in
BSDE (1.2) is simplified to $g(t,\gamma,y,v)$, and the HJB equation is a
second-order fully nonlinear parabolic partial differential equation (PDE, for
short) with the form

$\left\\{\begin{split}\displaystyle&\frac{\partial}{\partial
t}u(t,\gamma)+H(t,\gamma,u(t,\gamma),\nabla_{\gamma}u(t,\gamma),\nabla_{\gamma}^{2}u(t,\gamma))=0,\quad~{}(t,\gamma)\in[0,T)\times\mathbb{C},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle&u(T,\gamma)=\Phi(\gamma),\end{split}\right.$ (3.9)

where the Hamiltonian
$H:[0,T]\times\mathbb{C}\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{S}^{n}\rightarrow\mathbb{R}$
is defined as below

$H(t,\gamma,r,p,A)\triangleq\sup_{v\in U}\left\\{\langle
p,b(t,\gamma,v)\rangle+\frac{1}{2}Tr\left(\sigma(t,\gamma,v)\sigma^{*}(t,\gamma,v)A\right)+g\left(t,\gamma,r,v\right)\right\\}.$
(3.10)

In the above, $\mathbb{S}^{n}$ denotes the matrix space of $n\times n$
symmetric matrices, and the gradient $\nabla_{\gamma}u$ (resp., Hessian
$\nabla_{\gamma}^{2}u$) represents the first (resp., second) Gâteaux
derivative of $u$ with respect to $\gamma\in\mathbb{C}$, which is an $n$-tuple
of finite Borel measures on $[0,T]$ and the dual space of $\mathbb{C}$. See
e.g. Fuhrman–Masiero–Tessitore [10].

Before recalling the definition of the viscosity solution of HJB equations, we
specify the notation $C^{1,2}_{\text{Lip}}([0,T]\times\mathbb{C};\mathbb{R})$
which is the space of all continuous functions from $[0,T]\times\mathbb{C}$ to
$\mathbb{R}$ such that their partial derivatives up to order one with respect
to time variable and order two with respect to state variable are globally
bounded and Lipschitz continuous.

######  Definition 3.6.

Let $u:[0,T]\times\mathbb{C}\rightarrow\mathbb{R}$ be a continuous function.

  * (a)

We call $u$ a viscosity subsolution of HJB equation (3.9), if for any
$\gamma\in\mathbb{C}$, we have $u(T,\gamma)\leqslant\Phi(\gamma)$, and for any
$\phi\in C^{1,2}_{\text{Lip}}([0,T]\times\mathbb{C};\mathbb{R})$,
$(t,\gamma)\in[0,T]\times\mathbb{C}$, whenever $\phi-u$ attains a global
minimum at $(t,\gamma)$, the following holds:

$\frac{\partial}{\partial
t}\phi(t,\gamma)+H(t,\gamma,\phi,\nabla_{\gamma}\phi,\nabla_{\gamma}^{2}\phi)\geqslant
0.$

  * (b)

We call $u$ a viscosity supersolution of HJB equation (3.9), if for any
$\gamma\in\mathbb{C}$, we have $u(T,\gamma)\geqslant\Phi(\gamma)$, and for any
$\phi\in C^{1,2}_{\text{Lip}}([0,T]\times\mathbb{C};\mathbb{R})$,
$(t,\gamma)\in[0,T]\times\mathbb{C}$, whenever $\phi-u$ attains a global
maximum at $(t,\gamma)$, the following holds:

$\frac{\partial}{\partial
t}\phi(t,\gamma)+H(t,\gamma,\phi,\nabla_{\gamma}\phi,\nabla_{\gamma}^{2}\phi)\leqslant
0.$

  * (c)

We call $u$ a viscosity solution of HJB equation (3.9), if it is both a
viscosity subsolution and viscosity supersolution.

The definition of viscosity solution presented here is a delayed version of
Crandall–Ishii–Lions [5] which first introduces the theory of viscosity
solutions for standard PDEs. Besides, some preliminaries are needed. First of
all, we point out the continuity of the values function defined in (3.1) with
respect to both the time and state variables.

######  Proposition 3.7.

Assume (A1)-(A6). The value function
$u(t,\gamma):[0,T]\times\mathbb{C}\rightarrow\mathbb{R}$ defined in (3.1) is
continuous with respect to $t$ and $\gamma$.

######  Remark 3.8.

A direct conclusion of 3.2 is that $u(t,\gamma)$ is Lipschitz continuous with
respect to $\gamma$, and the continuity of $u(t,\gamma)$ with respect to $t$
is similar to the proof of Proposition 5.5 in Peng [22]. There is no essential
difference between the classical Lipschitz generator and the non-Lipschitz
generator in our setting, so we leave out the proof of 3.7 here.

Based on the generator $g$ of BSDE (1.2) for $n\in\mathbb{N}$, we define a
sequence of smooth functions $g_{n}$ as below:

$g_{n}(t,\gamma,y,v)\triangleq\left(\rho_{n}*g(t,\gamma,\cdot,v)\right)(y),$
(3.11)

where $\rho_{n}:\mathbb{R}\rightarrow\mathbb{R}^{+}$ is a family of
sufficiently smooth functions with the compact supports in
$\left[-\frac{1}{n},\frac{1}{n}\right]$ and satisfies

$\int_{\mathbb{R}}\rho_{n}(a)da=1.$

As a result, concerning the generators $g_{n}$ for $n\in\mathbb{N}$, we can
construct a sequence of BSDEs, for $s\in[t,T]$,

$Y^{t,\gamma,n;v}(s)=\Phi(X_{T}^{t,\gamma;v})+\int_{s}^{T}g_{n}(r,X_{r}^{t,\gamma;v},Y^{t,\gamma,n;v}(r),v(r))dr-\int_{s}^{T}Z^{t,\gamma,n;v}(r)dW(r).$
(3.12)

Then, based on the above BSDE (3.12), the cost functional of a sequence of
stochastic recursive control problems can be defined similarly, i.e., for
$n\in\mathbb{N}$,

$J_{n}(t,\gamma;v)\triangleq Y^{t,\gamma,n;v}(t),\quad\hbox{for $t\in[0,T],\
\gamma\in\mathbb{C},\ v\in{\cal U}$}.$

At the same time, the value function of a sequence of stochastic recursive
control problems has the corresponding forms, for $n\in\mathbb{N}$,

$u_{n}(t,\gamma)\triangleq\mathop{\rm
esssup}_{v\in\mathcal{U}}J_{n}(t,\gamma;v),\quad\hbox{for $t\in[0,T],\
\gamma\in\mathbb{C}$}.$ (3.13)

Also, Hamiltonian appears in the following form, for $n\in\mathbb{N}$,

$H_{n}(t,\gamma,r,p,A)\triangleq\sup_{v\in U}\left\\{\langle
p,b(t,\gamma,v)\rangle+\frac{1}{2}Tr\left(\sigma(t,\gamma,v)\sigma^{*}(t,\gamma,v)A\right)+g_{n}\left(t,\gamma,r,v\right)\right\\}.$
(3.14)

Besides, we need a series of lemmas to prove the main result in this
subsection. The following 3.9 proves that the smoother generators $g_{n}$
defined in (3.11) are uniformly convergent to the generator $g$ in a compact
subset of their domain. 3.10 implies the uniform convergence of the solutions
$Y^{t,\gamma,n;v}$ of BSDEs (3.12) with smoother generators to the solution
$Y^{t,\gamma;v}$ of BSDE (1.2) in the space of
$L^{2q}(\Omega,\mathscr{F}_{t};\mathbb{R})$. 3.11 displays that the value
functions $u_{n}$ defined in (3.13) are uniformly convergent to the value
function $u$ defined in (3.1) in a compact subset of their domain. Based on
the uniform convergence of $g_{n}$, $Y^{t,\gamma,n;v}$, and $u_{n}$ to $g$,
$Y^{t,\gamma;v}$, and $u$, respectively, it comes without a surprise in 3.12
that the Hamiltonians $H_{n}$ defined in (3.14) are convergent to $H$ defined
in (3.10) as well. To end the preparation, we prove the stability property of
viscosity solutions in 3.13 in order to obtain the connection between the
value function $u$ defined in (3.1) and the viscosity solution of HJB equation
(3.9). Indeed, all the proofs of these mentioned lemmas will be presented in
Section 5.

######  Lemma 3.9.

Assume (A1)-(A6). Then the sequence $g_{n}$ defined in (3.11) converges to $g$
uniformly in each compact subset of their domain.

######  Lemma 3.10.

Assume (A1)-(A6). Then for all $v\in{\cal U}$, we have

$\lim_{n\rightarrow\infty}\sup_{(t,\gamma)\in
K}\mathbb{E}\left[\left|Y^{t,\gamma,n;v}(t)-Y^{t,\gamma;v}(t)\right|^{2}\right]=0,$

where $K$ is an arbitrary compact subset of $[0,T]\times\mathbb{C}$, and
$Y^{t,\gamma;v}(\cdot)$ and $Y^{t,\gamma,n;v}(\cdot)$ are the solutions of
BSDE (1.2) and BSDE (3.12), respectively.

######  Lemma 3.11.

Assume (A1)-(A6). Then the sequence $u_{n}$ defined in (3.13) converges to $u$
defined in (3.1) uniformly in each compact subset of their domain.

######  Lemma 3.12.

Assume (A1)-(A6). Then the sequence $H_{n}$ defined in (3.14) converges to $H$
defined in (3.10) uniformly in each compact subset of their domain.

######  Lemma 3.13 (Stability).

Let $u_{n}$ be a viscosity subsolution (resp. supersolution) to the following
$\mathrm{PDE}$

$\frac{\partial}{\partial
t}u_{n}(t,\gamma)+H_{n}\left(t,\gamma,u_{n}(t,\gamma),D_{x}u_{n}(t,\gamma),D_{x}^{2}u_{n}(t,\gamma)\right)=0,\quad(t,\gamma)\in[0,T)\times\mathbb{C},$

where
$H_{n}(t,\gamma,r,p,A):[0,T]\times\mathbb{C}\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{S}^{n}\rightarrow\mathbb{R}$
is continuous and satisfies the ellipticity condition

$H_{n}(t,\gamma,r,p,X)\leqslant H_{n}(t,\gamma,r,p,Y),\quad\text{ whenever
}X\leqslant Y.$ (3.15)

Assume that $H_{n}$ and $u_{n}$ converge to $H$ and $u$, respectively,
uniformly in each compact subset of their own domains. Then $u$ is a viscosity
subsolution (resp. supersolution) of the limit equation

$\frac{\partial}{\partial
t}u(t,\gamma)+H\left(t,\gamma,u(t,\gamma),D_{x}u(t,\gamma),D_{x}^{2}u(t,\gamma)\right)=0.$
(3.16)

Based on above lemmas, we are going to establish the connection between the
value function defined in (3.1) and the viscosity solution of HJB equation
(3.9).

######  Theorem 3.14.

Assume (A1)-(A6) hold. Then $u$ defined in (3.1) is a viscosity solution of
HJB equation (3.9).

###### Proof.

We divide the proof into two steps.

Step 1. Assume that for all $(t,\gamma,v)\in[0,T]\times\mathbb{C}\times U$,
$|g(t,\gamma,0,v)|$ is uniformly bounded.

Then from the assumption, we have the global Lipschitz condition of
$g_{n}(t,\gamma,y,v)$ with respect to $y$. In fact, for any
$(t,\gamma,v)\in[0,T]\times\mathbb{C}\times U$ and $y,\bar{y}\in\mathbb{R}$,
(A6) implies

$\displaystyle\left|g_{n}\left(t,\gamma,y,v\right)-g_{n}\left(t,\gamma,\bar{y},v\right)\right|$
$\displaystyle=$
$\displaystyle\left|\int_{|a|\leqslant\frac{1}{n}}g(t,\gamma,a,v)\left[\rho_{n}\left(y-a\right)-\rho_{n}\left(\bar{y}-a\right)\right]da\right|$
$\displaystyle\leqslant$
$\displaystyle\max_{a\in\left[-\frac{1}{n},\frac{1}{n}\right]}|g(t,\gamma,a,v)|\int_{|a|\leqslant\frac{1}{n}}\left|\rho_{n}\left(y-a\right)-\rho_{n}\left(\bar{y}-a\right)\right|da$
$\displaystyle\leqslant$
$\displaystyle\max_{a\in\left[-\frac{1}{n},\frac{1}{n}\right]}\left[|g(t,\gamma,0,v)|+M\left(1+|a|^{p}\right)\right]\int_{|a|\leqslant\frac{1}{n}}C_{p}(n)\left|y-\bar{y}\right|da$
$\displaystyle\leqslant$ $\displaystyle C_{p}(M,n)\left|y-\bar{y}\right|.$

So $g_{n}$ satisfies the assumption of Theorem 7.3 in Peng [21], by which we
know that $u_{n}(t,\gamma)$ is a viscosity solution of the following equation,
for $n\in\mathbb{N}$,

$\left\\{\begin{array}[]{l}\displaystyle\frac{\partial}{\partial
t}u_{n}(t,\gamma)+H_{n}\left(t,\gamma,u_{n}(t,\gamma),\nabla_{\gamma}u_{n}(t,\gamma),\nabla_{\gamma}^{2}u_{n}(t,\gamma)\right)=0,\quad~{}(t,\gamma)\in[0,T)\times\mathbb{C},\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
u_{n}(T,\gamma)=\Phi(\gamma).\end{array}\right.$

Here $u_{n}$ and $H_{n}$ are defined in (3.13) and (3.14), respectively. Now,
by 3.9-3.12, we have that the uniform convergence of $g_{n}$ to $g$,
$Y^{t,\gamma,n;v}(t)$ to $Y^{t,\gamma;v}(t)$, $u_{n}$ to $u$ and $H_{n}$ to
$H$ holds in each compact subset of their own domains as $n\rightarrow\infty$.
In addition, $H_{n}$ satisfies the ellipticity condition (3.15). Hence, from
the stability of viscosity solution (3.13), we have that $u$ is a viscosity
solution of the limit equation

$\frac{\partial}{\partial
t}u(t,\gamma)+H\left(t,\gamma,u(t,\gamma),\nabla_{\gamma}u(t,\gamma),\nabla_{\gamma}^{2}u(t,\gamma)\right)=0.$

As for the terminal value of the above equation, by Lemma 3.11, we have

$u(T,\gamma)=\lim_{n\to\infty}u_{n}(T,\gamma)=\Phi(\gamma).$

Thereby $u$ is a viscosity solution of HJB equation (3.9).

Step 2. For all $(t,\gamma,v)\in$ $[0,T]\times\mathbb{C}\times U$,
$|g(t,\gamma,0,v)|$ is not necessary to be uniformly bounded.

In order to prove the result, we construct the sequence of functions

$g_{m}(t,\gamma,y,v)\triangleq
g(t,\gamma,y,v)-g(t,\gamma,0,v)+\Pi_{m}(g(t,\gamma,0,v)),\quad\hbox{for}\quad
m\in\mathbb{N},$

where

$\Pi_{m}(x)=\frac{\inf(m,|x|)}{|x|}x.$

For these $g_{m}$, $m\in\mathbb{N}$, we get a sequence of BSDEs on the
interval $[t,T]$

$Y^{t,\gamma,m;v}(t)=\Phi\left(X_{T}^{t,\gamma;v}\right)+\int_{t}^{T}g_{m}\left(s,X_{s}^{t,\gamma;v},Y^{t,\gamma,m;v}(s),v(s)\right)ds-\int_{t}^{T}Z^{t,\gamma,m;v}(s)dW(s).$

Similarly, we define the corresponding cost functional

$J_{m}(t,\gamma;v)\triangleq Y^{t,\gamma,m;v}(t),\quad\text{ for }\
v\in\mathcal{U},\ t\in[0,T],\ \gamma\in\mathbb{C},$

the value function

$u_{m}(t,\gamma)\triangleq\operatorname{esssup}_{v\in\mathcal{U}}J_{m}(t,\gamma;v),\quad\text{
for }\ t\in[0,T],\ \gamma\in\mathbb{C},$

and the Hamiltonian, for
$(t,\gamma,r,p,A)\in[0,T]\times\mathbb{C}\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{S}^{n}$,

$H_{m}(t,\gamma,r,p,A)\triangleq\sup_{v\in U}\left\\{\langle
p,b(t,\gamma,v)\rangle+\frac{1}{2}Tr\left(\sigma(t,\gamma,v)\sigma^{*}(t,\gamma,v)A\right)+g_{m}\left(t,\gamma,r,v\right)\right\\}.$

Note that $g_{m}(t,\gamma,0,v)=\Pi_{m}(g(t,\gamma,0,v))$, which implies that
$g_{m}(t,\gamma,0,v)$ is uniformly bounded. In addition, it is easy to check
that each $g_{m}$ satisfies the assumptions (A3)-(A6). Hence $g_{m}$ satisfies
the assumptions of Step 1. From Step 1, we see that $u_{m}$ is a viscosity
solution of the following equation

$\left\\{\begin{array}[]{l}\displaystyle\frac{\partial}{\partial
t}u_{m}(t,\gamma)+H_{m}\left(t,\gamma,u_{m}(t,\gamma),D_{x}u_{m}(t,\gamma),D_{x}^{2}u_{m}(t,\gamma)\right)=0,\quad~{}(t,\gamma)\in[0,T)\times\mathbb{C},\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
u_{m}(T,\gamma)=\Phi(\gamma).\end{array}\right.$ (3.17)

Next, we prove that as $m\rightarrow\infty$, the uniform convergence of
$g_{m}$ to $g$, $Y^{t,\gamma,m;v}(t)$ to $Y^{t,\gamma;v}(t)$, $u_{m}$ to $u$
and $H_{m}$ to $H$ holds in each compact subset of their own domains too. It
should be pointed out that due to the different definitions of $g_{m}$ here
from $g_{n}$ in (3.11), only the proof for the convergence of $g_{m}$ to $g$
is very different from 3.9, and the other convergence can be proved similarly
as the proof of 3.10-3.12 in turn.

From (A3), we have the continuity of $g$, so for each compact set
$K\subset[0,T]\times\mathbb{C}\times\mathbb{R}\times U$ and any
$(t,\gamma,y,v)\in K$, $g(t,\gamma,y,v)$ is bounded by a positive integer
$M_{K}$. This implies the uniform convergence of $g_{m}$ to $g$ in $K$. Hence,
when $m\geqslant M_{K}$, we have

$\displaystyle\sup_{(t,\gamma,y,v)\in
K}\left|g_{m}(t,\gamma,y,v)-g(t,\gamma,y,v)\right|=\sup_{(t,\gamma,y,v)\in
K}\left|g(t,\gamma,0,v)-\Pi_{m}(g(t,\gamma,0,v))\right|=0.$

Hence as $m\rightarrow\infty$, $g_{m}$ uniform converges to $g$ in each
compact subset of their domain.

Finally, by 3.13 again, even if $|g(t,\gamma,0,v)|$ is not necessary to be
uniformly bounded for any $(t,\gamma,v)\in[0,T]\times\mathbb{C}\times U$, we
have that $u$ satisfies the limit equation of (3.17), which together with the
fact $u(T,\gamma)=\Phi(\gamma)$ shows that $u$ is a viscosity solution of HJB
equation (3.9). The proof is complete. ∎

##  4 Applications

As stated in the Introduction, the stochastic differential formulation of
recursive utility was introduced by Duffie and Epstein [6], which is actually
a stochastic recursive control system. In this section, based on the
connection between SDU and BSDE, we present an example to illustrate the
application of our study. It should be pointed out that the inspiration of the
model comes from Ivanov–Swishchuk [12], and for completeness, we would like to
present the model in details.

Suppose that an investor can get a profit from his/her invested production.
Denote the capital, the labor, and the consumption rate of the investor at
time $t$ by $X(t)$, $\pi(t)$ and $c(t)$, respectively. In 1928, based on the
assumption that the income of production is a general function of the capital
and the labor, the following model was introduced by Ramsey [24]:

$\frac{\mathrm{d}X(t)}{\mathrm{d}t}=f(X(t),\pi(t))-c(t).$

As we all know, when his/her investment contains a risky asset, such as stock,
options, and futures, the investment involves uncertainty. Hence we would like
to extend Ramsey’s model to the following stochastic situation:

$\mathrm{d}X(t)=[f(X(t),\pi(t))-c(t)]\mathrm{d}t+\sigma(X(t))\mathrm{d}W(t).$

However, the fitness of the model has been questioned for the basic assumption
that there is no delay through the investment, which is not reasonable. In
fact, the empirical evidence presents to us that the volatility of investment
depends not only on the current state of the process but through its whole
history. Actually, as some examples shown in the Introduction, it is natural
that the future development of many natural and social phenomena depends on
their present state and historical information. Therefore, it is reasonable to
believe that in a consistent model of his investment, we should consider the
past value of the capital during the investment. Hence, our model is further
modified as

$\left\\{\begin{aligned} \displaystyle\mathrm{d}X(t)=&\
[f(X_{t},\pi(t))-c(t)]\mathrm{d}t+\sigma(X_{t})\mathrm{d}W(t),\quad
t\in[0,T],\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle X(t)=&\
\gamma(t),\quad t\in[-\delta,0],\end{aligned}\right.$

where $X_{t}$ represents the segment of the path of the process $X(\cdot)$
from $t-\delta$ to $t$. A special case is that $X_{t}=X(t-\delta)$ as in
Chen–Wu [3, 4]. The function $f$ can be reasonably specific. For example,
$f(X_{t},\pi(t))=Kf(X_{t})^{\alpha}\pi^{\beta}(t)$, where $K,\alpha,\beta$ are
some constants as in Gandolfo [11]. For simplicity, similar to
Ivanov–Swishchuk [12], we set $\alpha=\beta=1$ to rewrite our model as

$\left\\{\begin{aligned} \displaystyle\mathrm{d}X(t)=&\
[K\pi(t)f(X_{t})-c(t)]\mathrm{d}t+\sigma(X_{t})\mathrm{d}W(t),\quad
t\in[0,T],\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle X(t)=&\
\gamma(t),\quad t\in[-\delta,0],\end{aligned}\right.$ (4.1)

where $f(\gamma):\mathbb{C}\rightarrow\mathbb{R}$ and
$\sigma(\gamma):\mathbb{C}\rightarrow\mathbb{R}$, and we require that the
coefficients of (4.1) satisfy (A1)-(A2) for $n=d=1$. Now, we suppose that the
SDU preference of the investor is a continuous time Epstein-Zin utility, and
the utility is depicted by BSDE

$\left\\{\begin{array}[]{l}\displaystyle
dV(t)=-\frac{\vartheta}{1-\frac{1}{\psi}}(1-r)V(t)\left[\left(\frac{c(t)}{\left((1-r)V(t)\right)^{\frac{1}{1-r}}}\right)^{1-\frac{1}{\psi}}-1\right]dt+Z(t)dW(t),\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
V(T)=h\left(X_{T}\right),\end{array}\right.$ (4.2)

where $h(\gamma):\mathbb{C}\rightarrow\mathbb{R}$ is a given Lipschitz
continuous function. In BSDE (4.2), the coefficient $\vartheta>0$ represents
the rate of time preference, the coefficient $r\in(0,1)\cup(1,\infty)$ is the
relative risk aversion, and $0<\psi\neq 1$ represents the elasticity of
intertemporal substitution. The investor’s optimization goal is to maximize
the utility $\max_{(\pi,c)\in\mathcal{U}}V(0)$, where

$\mathcal{U}\triangleq\left\\{(\pi,c)\mid(\pi,c):[0,T]\times\Omega\rightarrow[a_{1},a_{2}]\times\left[b_{1},b_{2}\right]\
\text{and}\ \mathbb{F}\text{-adapted},\ \text{for given}\
a_{1},a_{2},b_{1},b_{2}\in\mathbb{R}\right\\}$

is the admissible control set. Clearly, the generator of BSDE (4.2) does not
satisfy the Lipschitz condition with respect to the utility and the
consumption at all times, however, we can find the applications of our study
in the non-Lipschitz situation. Note that there are four cases that the
generator is monotonic with respect to the utility as shown by Proposition 3.2
in Kraft–Seifried–Steffensen [13]. In view of the polynomial growth condition
(A6) with respect to the utility, here we choose two situations for further
investigation:

  * (i)

$r>1$ and $\psi>1$,

  * (ii)

$r<1$ and $\psi<1$.

In both situations (i) and (ii), we can find suitable powers of utility such
that the generator of BSDE (4.2) is continuous and monotonic but non-Lipschitz
with respect to the utility in its domain. As for the continuity with respect
to the consumption, it is obvious that both situations are Lipschitz
continuous if $b_{1}>0$. In particular, in the case of $b_{1}=0$, only the
situation (i) satisfies the continuity but not Lipschitz continuity with
respect to the consumption.

Therefore, under the assumptions (A3)-(A6), for all suitable non-Lipschitz
situations, applying 3.14 we know that the value function of the investor is a
viscosity solution of HJB equation

$\left\\{\begin{split}\displaystyle&\sup_{(\pi,c)\in[a_{1},a_{2}]\times[b_{1},b_{2}]}\Bigg{\\{}\frac{\partial}{\partial
t}u(t,\gamma)+\frac{1}{2}\sigma^{2}(\gamma)\nabla_{\gamma}^{2}u(t,\gamma)+[K\pi
f(\gamma)-c]\nabla_{\gamma}u(t,\gamma)\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle&\qquad\qquad\quad+\frac{\vartheta}{1-\frac{1}{\psi}}(1-r)u(t,\gamma)\bigg{[}\bigg{(}\frac{c}{\left((1-r)u(t,\gamma)\right)^{\frac{1}{1-r}}}\bigg{)}^{1-\frac{1}{\psi}}-1\bigg{]}\Bigg{\\}}=0,\quad(t,\gamma)\in[0,T)\times\mathbb{C},\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\displaystyle&u(T,\gamma)=h(\gamma).\end{split}\right.$

##  5 Proof of the auxiliary results

We first give the detailed proofs to the lemmas in Subsection 3.1.

###### Proof of 3.1.

The absence of a Lipschitz condition for the generator $g(t,\gamma,y,z,v)$
with respect to $y$ and $v$ brings about significant differences and
challenges compared to the work of Peng [22].

To begin with, we would like to denote by
$\left\\{\mathscr{F}_{t}^{s};t\leqslant s\leqslant T\right\\}$ the natural
filtration of the Brownian motion $\\{W(s)-W(t);t\leqslant s\leqslant T\\}$
augmented by the $\mathbb{P}$-null sets of $\mathscr{F}$, and define two
subspaces of $\mathcal{U}$:

$\displaystyle\mathcal{U}^{t}\triangleq\Big{\\{}v\in\mathcal{U}\ \big{|}\
v(s)\text{ is }\mathscr{F}_{t}^{s}\text{-measurable for }t\leqslant s\leqslant
T\Big{\\}},$
$\displaystyle\overline{\mathcal{U}}^{t}\triangleq\Big{\\{}v\in\mathcal{U}\
\big{|}\ v(s)=\sum_{j=1}^{N}v^{j}(s)1_{A_{j}},\hbox{ where
}v^{j}\in\mathcal{U}^{t}\hbox{ and }\\{A_{j}\\}_{j=1}^{N}\text{ is a partition
of }(\Omega,\mathscr{F}_{t})\Big{\\}}.$

Step 1. We prove

$\mathop{\rm esssup}_{v\in{\cal U}}J(t,\gamma;v)=\mathop{\rm
esssup}_{v\in\overline{{\cal U}}^{t}}J(t,\gamma;v).$ (5.1)

From the fact that $\overline{{\cal U}}^{t}$ is a subset of $\mathcal{U}$, we
have

$\mathop{\rm esssup}_{v\in{\cal U}}J(t,\gamma;v)\geqslant\mathop{\rm
esssup}_{v\in\overline{{\cal U}}^{t}}J(t,\gamma;v).$ (5.2)

So we only need to prove the inverse inequality of (5.2), i.e.,

$\mathop{\rm esssup}_{v\in{\cal U}}J(t,\gamma;v)\leqslant\mathop{\rm
esssup}_{v\in\overline{{\cal U}}^{t}}J(t,\gamma;v).$ (5.3)

Note that $\overline{{\cal U}}^{t}$ is dense in the space $\mathcal{U}$. Hence
for each $v\in{\cal U}$, there is a sequence
$\\{v^{n}\\}_{n=1}^{\infty}\in\overline{{\cal U}}^{t}$ such that

$\lim_{n\rightarrow\infty}\mathbb{E}\int_{t}^{T}|v^{n}(s)-v(s)|^{2}ds=0.$

Moreover, there is a subsequence of $\\{v^{n}\\}_{n=1}^{\infty}$, still
denoted by $\\{v^{n}\\}_{n=1}^{\infty}$ without loss of generality, which
satisfies

$\lim_{n\rightarrow\infty}v^{n}(s)=v(s)\quad\hbox{a.s.}$ (5.4)

Denote by $(Y^{t,\gamma;v^{n}}(\cdot),Z^{t,\gamma;v^{n}}(\cdot))$ the adapted
solution of BSDE (1.2) with $v$ replaced by $v^{n}$. Applying Itô’s lemma to
$|Y^{t,\gamma;v^{n}}(s)-Y^{t,\gamma;v}(s)|^{2}e^{-\beta s}$ on the interval
$[t,T]$, together with the monotonic condition (A5), we have

$\displaystyle\mathbb{E}|Y^{t,\gamma;v^{n}}(t)-Y^{t,\gamma;v}(t)|^{2}+\mathbb{E}\int_{t}^{T}\big{[}\beta|Y^{t,\gamma;v^{n}}(s)-Y^{t,\gamma;v}(s)|^{2}+|Z^{t,\gamma;v^{n}}(s)-Z^{t,\gamma;v}(s)|^{2}\big{]}e^{\beta(s-t)}ds$
$\displaystyle=$ $\displaystyle e^{\beta
T}\mathbb{E}|\Phi(X^{t,\gamma;v^{n}}_{T})-\Phi(X^{t,\gamma;v}_{T})|^{2}+\mathbb{E}\int_{t}^{T}2\big{[}Y^{t,\gamma;v^{n}}(s)-Y^{t,\gamma;v}(s)\big{]}$
$\displaystyle\cdot\big{[}g(s,X^{t,\gamma;v^{n}}_{s},Y^{t,\gamma;v^{n}}(s),Z^{t,\gamma;v^{n}}(s),v^{n}(s))-g(s,X^{t,\gamma;v}_{s},Y^{t,\gamma;v}(s),Z^{t,\gamma;v}(s),v(s))\big{]}e^{\beta(s-t)}ds$
$\displaystyle\leqslant$ $\displaystyle
C_{p}\mathbb{E}\|X^{t,\gamma;v^{n}}_{T}-X^{t,\gamma;v}_{T}\|_{\mathbb{C}}^{2}+\mathbb{E}\int_{t}^{T}\big{[}\frac{6\tilde{L}^{2}}{\beta}\|X^{t,\gamma;v^{n}}_{s}-X^{t,\gamma;v}_{s}\|_{\mathbb{C}}^{2}+(\frac{\beta}{2}+\mu)|Y^{t,\gamma;v^{n}}(s)-Y^{t,\gamma;v}(s)|^{2}$
$\displaystyle\quad+\frac{6\tilde{L}^{2}}{\beta}|Z^{t,\gamma;v^{n}}(s)-Z^{t,\gamma;v}(s)|^{2}+\frac{6}{\beta}|g(s,0,0,0,v^{n}(s))-g(s,0,0,0,v(s))|^{2}\big{]}e^{\beta(s-t)}ds,$

where and in the rest of this paper $C_{p}$ is a positive constant depending
only on the given parameters, whose values may be changed from line to line.
Let $\beta=2(1+\mu)+6\tilde{L}^{2}+1$, and then by the above inequality we
have

$\displaystyle\mathbb{E}|Y^{t,\gamma;v^{n}}(t)-Y^{t,\gamma;v}(t)|^{2}\leqslant$
$\displaystyle
C_{p}\mathbb{E}\|X^{t,\gamma;v^{n}}_{T}-X^{t,\gamma;v}_{T}\|_{\mathbb{C}}^{2}+C_{p}\mathbb{E}\|X^{t,\gamma;v^{n}}_{s}-X^{t,\gamma;v}_{s}\|_{\mathbb{C}}^{2}$
(5.5)
$\displaystyle+C_{p}\mathbb{E}\int_{t}^{T}|g(s,0,0,0,v^{n}(s))-g(s,0,0,0,v(s))|^{2}\big{]}ds.$

For the first and second terms on the right hand side of (5.5), by 2.2 we have

$\mathbb{E}\|X^{t,\gamma;v^{n}}_{T}-X^{t,\gamma;v}_{T}\|^{2}_{\mathbb{C}}+\mathbb{E}\|X^{t,\gamma;v^{n}}_{s}-X^{t,\gamma;v}_{s}\|^{2}_{\mathbb{C}}\leqslant
C_{p}\mathbb{E}\int_{t}^{T}|v^{n}(s)-v(s)|^{2}ds.$

For the third term on the right hand side of (5.5), it follows from the
dominated convergence theorem and (5.4) that

$\lim_{n\rightarrow\infty}\mathbb{E}\int_{t}^{T}\big{|}g(s,0,0,0,v^{n}(s))-g(s,0,0,0,v(s))\big{|}^{2}ds=0.$

Hence, taking the limits on both side of (5.5), we obtain

$\lim_{n\rightarrow\infty}\mathbb{E}|Y^{t,\gamma;v^{n}}(t)-Y^{t,\gamma;v}(t)|^{2}=0.$

Therefore, there is a subsequence of $\\{v^{n}\\}_{n=1}^{\infty}$, still
denoted by $\\{v^{n}\\}_{n=1}^{\infty}$ without loss of generality, such that

$\lim_{n\rightarrow\infty}Y^{t,\gamma;v^{n}}(t)=Y^{t,\gamma;v}(t)\quad\hbox{a.s.}$

In other words,

$\lim_{n\rightarrow\infty}J(t,\gamma,v^{n})=J(t,\gamma,v)\quad\hbox{a.s.}$

Now, by the arbitrariness of $v$ and the definition of essential supremum, we
obtain (5.3), and then (5.1) follows.

Step 2. We prove

$\mathop{\rm esssup}_{v\in\overline{{\cal U}}^{t}}J(t,\gamma;v)=\mathop{\rm
esssup}_{v\in{\cal U}^{t}}J(t,\gamma;v).$ (5.6)

Obviously,

$\mathop{\rm esssup}_{v\in\overline{{\cal
U}}^{t}}J(t,\gamma;v)\geqslant\mathop{\rm esssup}_{v\in{\cal
U}^{t}}J(t,\gamma;v).$ (5.7)

So we only need to prove the inverse inequality of (5.7). By Lemma 3.1 in
Chen–Wu [3], we have

$J(t,\gamma;v)=J(t,\gamma;\sum_{j=1}^{N}1_{A_{j}}v^{j})=\sum_{j=1}^{N}1_{A_{j}}J(t,\gamma;v^{j}),\quad\forall
v\in\overline{{\cal U}}^{t},$

where $v^{j}$ is $\mathscr{F}_{t}^{s}$-measurable for $j=1,2,\cdots,N$. It is
well known that $Y^{t,\gamma;v^{j}}(t)=J(t,\gamma;v^{j})$, as the solution of
BSDE, is deterministic in this case. Without lose of generality, assume

$J(t,\gamma;v^{1})\geqslant J(t,\gamma;v^{j}),\quad j=2,3,...,N.$

Then

$J(t,\gamma;v)\leqslant J(t,\gamma;v^{1})\leqslant\mathop{\rm
esssup}_{v\in{\cal U}^{t}}J(t,\gamma;v).$

Again, from the arbitrariness of $v$, we obtain

$\mathop{\rm esssup}_{v\in\overline{{\cal
U}}^{t}}J(t,\gamma;v)\leqslant\mathop{\rm esssup}_{v\in{\cal
U}^{t}}J(t,\gamma;v).$

Hence the desired result (5.6) holds.

Finally, (5.1) and (5.6) lead to

$\mathop{\rm esssup}_{v\in{\cal U}}J(t,\gamma;v)=\mathop{\rm
esssup}_{v\in{\cal U}^{t}}J(t,\gamma;v),$ (5.8)

which implies that the value function $u$ defined in (3.1) is a deterministic
function. ∎

###### Proof of 3.2.

By 2.3, for any $t\in[0,T]$, $\gamma,\bar{\gamma}\in\mathbb{C}$ and $v\in{\cal
U}$, we have

$\displaystyle|J(t,\gamma;v)|=|Y^{t,\gamma;v}(t)|\leqslant
C\big{(}1+\|\gamma\|_{\mathbb{C}}\big{)}$ (5.9)

and

$\displaystyle|J(t,\gamma;v)-J(t,\bar{\gamma};v)|\leqslant
C\|\gamma-\bar{\gamma}\|_{\mathbb{C}}.$ (5.10)

On the other hand, for arbitrary $\varepsilon>0$, there exist
$v,\bar{v}\in{\cal U}$ such that

$J(t,\gamma;v)\leqslant u(t,\gamma)\leqslant J(t,\gamma;\bar{v})+\varepsilon\
\ \ {\rm and}\ \ \ J(t,\bar{\gamma};\bar{v})\leqslant
u(t,\bar{\gamma})\leqslant J(t,\bar{\gamma};v)+\varepsilon,$ (5.11)

which combining (5.9) implies

$\displaystyle-C\big{(}1+\|\gamma\|_{\mathbb{C}}\big{)}\leqslant
J(t,\gamma;v)\leqslant u(t,\gamma)\leqslant
J(t,\gamma;\bar{v})+\varepsilon\leqslant
C\big{(}1+\|\gamma\|_{\mathbb{C}}\big{)}+\varepsilon.$

By the arbitrariness of $\varepsilon$, (3.2) holds. For (3.3), using (5.11)
again, we have

$\displaystyle J(t,\gamma;v)-J(t,\bar{\gamma},v)-\varepsilon\leqslant
u(t,\gamma)-u(t,\bar{\gamma})\leqslant
J(t,\gamma;\bar{v})-J(t,\bar{\gamma};\bar{v})+\varepsilon,$

which combining (5.10) implies that

$\displaystyle\displaystyle|u(t,\gamma)-u(t,\bar{\gamma})|\leqslant\max\\{|J(t,\gamma;v)-J(t,\bar{\gamma},v)|,\
|J(t,\gamma;\bar{v})-J(t,\bar{\gamma};\bar{v})|\\}+\varepsilon\leqslant
C\|\gamma-\bar{\gamma}\|_{\mathbb{C}}+\varepsilon.$

Then (3.3) follows from the arbitrariness of $\varepsilon$. ∎

###### Proof of 3.3.

The proof is divided into three steps.

Step 1. We consider a simple $\mathbb{C}$-valued random variable $\Upsilon\in
L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$ with a form below

$\Upsilon=\sum^{N}_{i=1}\gamma^{i}I_{A_{i}},$ (5.12)

where $\gamma^{i}\in\mathbb{C}$ with $1\leqslant i\leqslant N$,
$N\in\mathbb{N}$ and $\\{A_{i}\\}^{N}_{i=1}$ is a partition of
$(\Omega,\mathscr{F}_{t})$. For each $i=1,2,...,N,$ denote by
$(Y^{t,\gamma^{i};v}(\cdot),Z^{t,\gamma^{i};v}(\cdot))$ the adapted solution
of BSDE

$\displaystyle Y^{t,\gamma^{i};v}(s)=$ $\displaystyle\
\Phi(X_{T}^{t,\gamma^{i};v})+\int_{s}^{T}g(r,X_{r}^{t,\gamma^{i};v},Y^{t,\gamma^{i};v}(r),Z^{t,\gamma^{i};v}(r),v(r))dr$
(5.13)
$\displaystyle-\int_{s}^{T}Z^{t,\gamma^{i};v}(r)dW(r),\quad~{}s\in[t,T].$

Multiplying by $I_{A_{i}}$ on both side of BSDE (5.13) and merging the
corresponding terms, we get

$\displaystyle\ \sum_{i=1}^{N}1_{A_{i}}Y^{t,\gamma^{i};v}(s)$ $\displaystyle=$
$\displaystyle\Phi\left(\sum_{i=1}^{N}1_{A_{i}}X_{T}^{t,\gamma^{i};v}\right)+\int_{s}^{T}g\left(r,\sum_{i=1}^{N}1_{A_{i}}X_{r}^{t,\gamma^{i};v},\sum_{i=1}^{N}1_{A_{i}}Y^{t,\gamma^{i};v}(r),\sum_{i=1}^{N}1_{A_{i}}Z^{t,\gamma^{i};v}(r),v(r)\right)dr$
$\displaystyle\ \ \ \ \
-\int_{s}^{T}\sum_{i=1}^{N}1_{A_{i}}Z^{t,\gamma^{i};v}(r)dW(r),\quad~{}s\in[t,T].$

By the existence and uniqueness of BSDE, it yields that

$Y^{t,\Upsilon;v}(s)=\sum_{i=1}^{N}1_{A_{i}}Y^{t,\gamma^{i};v}(s)\ \ \ {\rm
and}\ \ \
Z^{t,\Upsilon;v}(s)=\sum_{i=1}^{N}1_{A_{i}}Z^{t,\gamma^{i};v}(s),\quad~{}s\in[t,T].$

Then, by the definition of $J(\cdot)$, we obtain

$Y^{t,\Upsilon;v}(t)=\sum_{i=1}^{N}1_{A_{i}}Y^{t,\gamma^{i};v}(t)=\sum_{i=1}^{N}1_{A_{i}}J(t,\gamma^{i};v)=J(t,\sum_{i=1}^{N}1_{A_{i}}\gamma^{i};v)=J(t,\Upsilon;v).$

So the conclusion holds for the simple case.

Step 2. For any bounded $\Phi\in L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$,
by 2.1 there is a sequence of simple $\mathbb{C}$-valued random variable

$\Upsilon^{m}=\sum_{i=1}^{m}1_{A_{i}}\gamma^{i},$ (5.14)

which converges to $\Phi$ in $L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$. Here
$\gamma^{i}\in\mathbb{C}$ with $1\leqslant i\leqslant m$, $m\in\mathbb{N}$ and
$\\{A_{i}\\}^{m}_{i=1}$ is a partition of $(\Omega,\mathscr{F}_{t})$. By 2.3,
we have

$\mathbb{E}|Y^{t,\Phi;v}(t)-Y^{s,\Upsilon^{m};v}(t)|^{2}\leqslant
C\mathbb{E}\|\Phi-\Upsilon^{m}\|^{2}_{\mathbb{C}}\rightarrow 0\ \ \text{ as }\
m\rightarrow\infty.$

On the other hand, for the cost functional $J$ defined in (1.3), by (5.10) we
have

$\mathbb{E}|J(t,\Phi;v)-J(t,\Upsilon^{m};v)|\leqslant
C\mathbb{E}\|\Phi-\Upsilon^{m}\|^{2}_{\mathbb{C}}\rightarrow 0\ \ \text{ as }\
m\rightarrow\infty,$

which combining the result $Y^{t,\Upsilon^{m};v}(t)=J(t,\Upsilon^{m};v)$
implies that the conclusion holds in this case.

Step 3. For any $\Psi\in L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$, we define
$\Phi^{m}=(\Psi_{t}(r)\wedge m)\vee(-m)$ for $m\in\mathbb{N}$,
$r\in[t-\delta,t]$. Then $\Phi^{m}\in
L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$ is bounded and

$\mathbb{E}|Y^{t,\Psi;v}(t)-Y^{s,\Phi^{m};v}(t)|^{2}\leqslant
C\mathbb{E}\|\Psi-\Phi^{m}\|^{2}_{\mathbb{C}}\rightarrow 0\ \ \text{ as }\
m\rightarrow\infty.$

The desired result follows by a similar argument as Step 2. ∎

###### Proof of 3.4.

The proof is divided into three steps.

Step 1. For the simple $\mathbb{C}$-valued random variable $\Upsilon$ as
(5.12), (3.4) follows from

$Y^{t,\Upsilon;v}(t)=\sum_{i=1}^{N}1_{A_{i}}Y^{t,\gamma^{i};v}(t)=\sum_{i=1}^{N}1_{A_{i}}J(t,\gamma^{i};v)\leqslant\sum_{i=1}^{N}1_{A_{i}}u(t,\gamma^{i})=u(t,\Upsilon),\quad\forall
v\in{\cal U}.$

To prove (3.5), note that for each $\gamma^{i}$, by (5.8) there exists an
admissible control $v_{i}\in\mathcal{U}^{t}$ such that

$u(t,\gamma^{i})\leqslant Y_{t}^{t,\gamma^{i};v_{i}}+\varepsilon.$

Taking $v=\sum\limits_{i=1}^{N}v_{i}I_{A_{i}}\in\mathcal{U}$ we have

$Y_{t}^{t,\Upsilon;v}+\varepsilon=\sum_{i=1}^{N}(Y_{t}^{t,\gamma^{i};v_{i}}+\varepsilon)I_{A_{i}}\geqslant\sum_{i=1}^{N}u(t,\gamma^{i})I_{A_{i}}=u(t,\Upsilon).$

So (3.5) also holds for the simple $\mathbb{C}$-valued random variable.

Step 2. For any $\Psi\in L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$, we
combine Steps 2-3 in Lemma 3.3 by diagonal rule to get a sequence of simple
$\mathbb{C}$-valued random variables $\Upsilon^{m}$ with a form as (5.14),
$m\in\mathbb{N}$, which converges to $\Psi$ in the space
$L^{2}_{\mathbb{F}}(\Omega;\mathbb{C}_{t})$. Hence, by 2.3 and 3.2, we have
for any $v\in{\cal U}$,

$\lim_{m\rightarrow\infty}Y^{t,\Upsilon^{m};v}(t)=Y^{t,\Phi;v}(t)\ \text{ a.s.
}\quad\hbox{ and }\quad\lim_{m\rightarrow\infty}u(t,\Upsilon^{m})=u(t,\Phi)\
\text{ a.s., }$

which combining $Y^{t,\Upsilon^{m};v}(t)\leqslant u(t,\Upsilon^{m})$,
$m=1,2,\cdots$, implies that (3.4) holds.

To prove (3.5), note that for arbitrary $\varepsilon>0$, there exists
$m\in\mathbb{N}$ such that

$\mathbb{E}\|\Psi-\Upsilon^{m}\|^{2}_{\mathbb{C}}\leqslant\frac{\varepsilon}{3C}.$

By 2.3 and 3.2 again, it yields that for any $u\in{\cal U}$,

$|Y^{t,\Psi;v}(t)-Y^{t,\Upsilon^{m};v}(t)|\leqslant\frac{\varepsilon}{3}\qquad\hbox{and}\qquad|u(t,\Psi)-u(t,\Upsilon^{m})|\leqslant\frac{\varepsilon}{3}.$
(5.15)

Then for each $\gamma^{i}$, by (5.8) there is an admissible control
$v^{i}\in{\cal U}^{t}$ such that

$u(t,\gamma^{i})\leqslant Y^{t,\gamma^{i};v^{i}}(t)+\frac{\varepsilon}{3}.$

Now, we set $v=\sum\limits_{i=1}^{m}1_{A_{i}}v^{i}$, and by (5.15) it yields
that

$\displaystyle Y^{t,\Psi;v}(t)$
$\displaystyle\geqslant-|Y^{t,\Upsilon^{m};v}(t)-Y^{t,\Psi,v}(t)|+Y^{t,\Upsilon^{m};v}(t)$
$\displaystyle\geqslant-\frac{\varepsilon}{3}+\sum_{i=1}^{m}1_{A_{i}}Y^{t,\gamma^{i};v^{i}}(t)$
$\displaystyle\geqslant-\frac{\varepsilon}{3}+\sum_{i=1}^{m}1_{A_{i}}\big{[}u(t,\gamma^{i})-\frac{\varepsilon}{3}\big{]}$
$\displaystyle=-\frac{2}{3}\varepsilon+\sum_{i=1}^{m}1_{A_{i}}u(t,\gamma^{i})$
$\displaystyle=-\frac{2}{3}\varepsilon+u(t,\Upsilon^{m})\geqslant-\varepsilon+u(t,\Psi),$

which implies that (3.5) holds. ∎

Next, we give the detailed proofs to the lemmas in Subsection 3.2.

###### Proof of 3.9.

Note that $\int_{\mathbb{R}}\rho_{n}(a)da=1$. Then we have

$g_{n}(t,\gamma,y,v)-g(t,\gamma,y,v)=\int_{\mathbb{R}}(g(t,\gamma,y-a,v)-g(t,\gamma,y,v))\rho_{n}(a)da.$

On the other hand, for each compact set
$K\subset[0,T]\times\mathbb{C}\times\mathbb{R}\times U$, there is a compact
set $\widetilde{K}$ such that $(t,\gamma,y-a,v)\in\widetilde{K}$ for any
$(t,\gamma,y,v)\in K$ and $a\in[-1,1].$ Notice that since $g(t,\gamma,y,v)$ is
continuous with respect to $(t,y,v)$ and Lipschitz continuous with respect to
$\gamma$, $g(t,\gamma,y,v)$ is continuous and uniformly continuous with
respect to $(t,\gamma,y,v)$ in the compact set $\widetilde{K}$. Therefore, for
arbitrary $\varepsilon>0$, as $n$ is sufficiently large, we have

$\displaystyle\sup_{(t,\gamma,y,v)\in K}|g_{n}(t,\gamma,y,v)-g(t,\gamma,y,v)|$
$\displaystyle\leqslant\sup_{(t,\gamma,y,v)\in
K}\int_{|a|\leqslant\frac{1}{n}}|g(t,\gamma,y-a,v)-g(t,\gamma,y,v)|\rho_{n}(a)da$
$\displaystyle\leqslant\varepsilon\int_{|a|\leqslant\frac{1}{n}}\rho_{n}(a)da=\varepsilon.$

So the desired conclusion is derived. ∎

###### Proof of 3.10.

Firstly, it is easy to check that the smootherized generator $g_{n}$ satisfies
the assumptions (A3)-(A6). Then, using Itô’s lemma to
$|Y^{t,\gamma,n;v}(s)-Y^{t,\gamma;v}(s)|^{2}e^{\beta(s-t)}$ on the interval
$[t,T]$, we have for any $(t,\gamma)\in K$,

$\displaystyle\mathbb{E}[|Y^{t,\gamma,n;v}(t)-Y^{t,\gamma;v}(t)|^{2}]+\mathbb{E}\int_{t}^{T}\left[\beta\left|Y^{t,\gamma,n;v}(s)-Y^{t,\gamma;v}(s)\right|^{2}+\left|Z^{t,\gamma,n;v}(s)-Z^{t,\gamma;v}(s)\right|^{2}\right]e^{\beta(s-t)}ds$
$\displaystyle=$ $\displaystyle
2\mathbb{E}\int_{t}^{T}\big{[}Y^{t,\gamma,n;v}(s)-Y^{t,\gamma;v}(s)\big{]}\cdot\big{[}g_{n}(s,X_{s}^{t,\gamma;v},Y^{t,\gamma,n;v}(s),v(s))-g(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))\big{]}e^{\beta(s-t)}ds$
$\displaystyle=$ $\displaystyle
2\mathbb{E}\int_{t}^{T}\big{[}Y^{t,\gamma,n;v}(s)-Y^{t,\gamma;v}(s)\big{]}\cdot\big{[}g_{n}(s,X_{s}^{t,\gamma;v},Y^{t,\gamma,n;v}(s),v(s))-g_{n}(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))\big{]}e^{\beta(s-t)}ds$
$\displaystyle+\mathbb{E}\int_{t}^{T}2\big{[}Y^{t,\gamma,n;v}(s)-Y^{t,\gamma;v}(s)\big{]}\cdot\big{[}g_{n}(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))-g(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))\big{]}e^{\beta(s-t)}ds$
$\displaystyle\leqslant$ $\displaystyle
2\mathbb{E}\int_{t}^{T}\mu|Y^{t,\gamma,n;v}(s)-Y^{t,\gamma;v}(s)|^{2}e^{\beta(s-t)}ds+\mathbb{E}\int_{t}^{T}\Big{[}\frac{\beta}{2}|Y^{t,\gamma,n;v}(s)-Y^{t,\gamma;v}(s)|^{2}$
$\displaystyle\quad+\frac{2}{\beta}\big{|}g_{n}(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))-g(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))\big{|}^{2}\Big{]}e^{\beta(s-t)}ds,$

where in the last inequality we have used Hölder and inequality (A5) satisfied
by $g_{n}$. Now, taking $\beta=4\mu+2$ we have

$\displaystyle\mathbb{E}[|Y^{t,\gamma,n;v}(t)-Y^{t,\gamma;v}(t)|^{2}]$
$\displaystyle\leqslant$ $\displaystyle
C_{p}\mathbb{E}\int_{t}^{T}\big{|}g_{n}(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))-g(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))\big{|}^{2}ds$
$\displaystyle=$ $\displaystyle
C_{p}\mathbb{E}\int_{t}^{T}|g_{n}-g|^{2}1_{\\{\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}\geqslant
N\\}\cup\\{\sup\limits_{s\in[t,T]}|Y^{t,\gamma;v}(s)|\geqslant N\\}\\}}ds$
$\displaystyle+C_{p}\mathbb{E}\int_{t}^{T}|g_{n}-g|^{2}1_{\\{\\{\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}<N\\}\cap\\{\sup\limits_{s\in[t,T]}|Y^{t,\gamma;v}(s)|<N\\}\\}}ds$
$\displaystyle\leqslant$ $\displaystyle
C_{p}\mathbb{E}\int_{t}^{T}|g_{n}-g|^{2}1_{\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}\geqslant
N\\}}ds+C_{p}\mathbb{E}\int_{t}^{T}|g_{n}-g|^{2}1_{\\{\sup\limits_{s\in[t,T]}|Y^{t,\gamma;v}(s)|\geqslant
N\\}}ds$
$\displaystyle+C_{p}\mathbb{E}\int_{t}^{T}|g_{n}-g|^{2}1_{\\{\\{\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}<N\\}\cap\\{\sup\limits_{s\in[t,T]}|Y^{t,\gamma;v}(s)|<N\\}\\}}ds.$

For simplicity, we define

$\displaystyle\displaystyle J_{1}\triangleq$ $\displaystyle\
\mathbb{E}\int_{t}^{T}\big{|}g_{n}(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))-g(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))\big{|}^{2}1_{\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}\geqslant
N\\}}ds,$ $\displaystyle\displaystyle J_{2}\triangleq$ $\displaystyle\
\mathbb{E}\int_{t}^{T}\big{|}g_{n}(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))-g(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))\big{|}^{2}1_{\\{\sup\limits_{s\in[t,T]}|Y^{t,\gamma;v}(s)|\geqslant
N\\}}ds,$ $\displaystyle\displaystyle J_{3}\triangleq$ $\displaystyle\
\mathbb{E}\int_{t}^{T}\big{|}g_{n}(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))-g(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))\big{|}^{2}$
$\displaystyle\qquad\quad\cdot
1_{\\{\\{\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}<N\\}\cap\\{\sup\limits_{s\in[t,T]}|Y^{t,\gamma;v}(s)|<N\\}\\}}ds.$

Next we will deal with $J_{1}$, $J_{2}$ and $J_{3}$ in turn.

First, for $J_{1}$, by (A6) we have

$\displaystyle\sup_{(t,\gamma)\in K}J_{1}$
$\displaystyle\leqslant\displaystyle$ $\displaystyle\sup_{(t,\gamma)\in
K}2\mathbb{E}\int_{t}^{T}\Big{(}\big{|}g_{n}(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))\big{|}^{2}+\big{|}g(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))\big{|}^{2}\Big{)}1_{\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}\geqslant
N\\}}ds$ $\displaystyle\leqslant\displaystyle$
$\displaystyle\sup_{(t,\gamma)\in
K}C_{p}\mathbb{E}\int_{t}^{T}\Big{(}1+\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}^{2}+|Y^{t,\gamma;v}(s)|^{2p}\Big{)}1_{\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}\geqslant
N\\}}ds$ $\displaystyle\leqslant\displaystyle$
$\displaystyle\sup_{(t,\gamma)\in
K}C_{p}\Big{[}\mathbb{E}\int_{t}^{T}\Big{(}1+\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}^{4}+|Y^{t,\gamma;v}(s)|^{4p}\Big{)}ds\Big{]}^{\frac{1}{2}}\cdot\Big{[}\sup_{(t,\gamma)\in
K}\mathbb{P}(\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}\geqslant
N)\Big{]}^{\frac{1}{2}}.$

By Chebychev’s inequality and 2.2, we have for any $N>0$,

$\mathbb{P}(\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}\geqslant
N)\leqslant\frac{1}{N^{2}}\mathbb{E}\|X_{s}^{t,\gamma;v}\|^{2}_{\mathbb{C}}\leqslant\frac{C_{p}}{N^{2}}(1+\|\gamma\|^{2}_{\mathbb{C}}).$

Note that the initial value $\gamma$ is bounded, so for arbitrary $\delta>0$,
when $N$ is sufficiently large, it yields that

$\sup_{(t,\gamma)\in K}\mathbb{P}(\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}\geqslant
N)\leqslant\delta.$ (5.16)

In addition, Propositions 2.1 and 2.2 imply that

$\sup_{(t,\gamma)\in
K}\Big{[}\mathbb{E}\int_{t}^{T}\Big{(}1+\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}^{4}+|Y^{t,\gamma;v}(s)|^{4p}\Big{)}ds\Big{]}^{\frac{1}{2}}\leqslant\infty.$

This, together with (5.16), indicates that for arbitrary $\varepsilon>0$,
there is a sufficiently large $N_{1}$ such that when $N\geqslant N_{1}$, we
have for all $n\in\mathbb{N}$,

$J_{1}\leqslant\varepsilon,\quad\hbox{uniformly in $K$}.$

For $J_{2}$, 2.3 implies

$\mathbb{E}\Big{[}\sup_{s\in[t,T]}|Y^{t,\gamma;v}(s)|^{2p}\Big{]}\leqslant
C_{p}(1+\|\gamma\|_{\mathbb{C}}^{2p}).$

Similar to the above discussion for $J_{1}$, the Chebychev’s inequality leads
to that for arbitrary $\delta>0$,

$\sup_{(t,\gamma)\in K}\mathbb{P}(\sup_{s\in[t,T]}|Y^{t,\gamma;v}(s)|\geqslant
N)\leqslant\delta.$

Then there is a sufficiently large $N_{2}$ such that when $N\geqslant N_{1}$,
we have for all $n\in\mathbb{N}$,

$J_{2}\leqslant\varepsilon,\quad\hbox{uniformly in $K$}.$

For $J_{3}$, denote $\tilde{N}=N_{1}\vee N_{2}$. For $v\in{\cal U}$, we have

$\displaystyle\displaystyle\sup_{(t,\gamma)\in K}J_{3}\leqslant$
$\displaystyle\ \mathbb{E}\int_{t}^{T}\sup_{(t,\gamma)\in
K}\big{|}g_{n}(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))-g(s,X_{s}^{t,\gamma;v},Y^{t,\gamma;v}(s),v(s))\big{|}^{2}$
$\displaystyle\qquad\quad\cdot
1_{\\{\\{\|X_{s}^{t,\gamma;v}\|_{\mathbb{C}}<\tilde{N}\\}\cap\\{\sup\limits_{s\in[t,T]}|Y^{t,\gamma;v}(s)|<\tilde{N}\\}\\}}ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\ \mathbb{E}\
\int_{t}^{T}\sup_{(t,\gamma,y)\in[0,T]\times
B^{\mathbb{C}}_{\tilde{N}}\times[-\tilde{N},\tilde{N}]}|g_{n}(t,\gamma,y,v)-g(t,\gamma,y,v)|^{2}ds,$

where $B^{\mathbb{C}}_{\tilde{N}}$ is the closed ball with the origin $0$ and
the radium $\tilde{N}$ in $\mathbb{C}$. By the dominated convergence theorem
and 3.9, we have, as $n$ is sufficiently large,

$J_{3}\leqslant\varepsilon,\quad\hbox{uniformly in $K$}.$

Therefore, due to the arbitrariness of $\varepsilon$, the conclusion follows
and the proof is complete. ∎

###### Proof of 3.11.

For arbitrary $\varepsilon>0$, $t\in[0,T]$ and $\gamma\in\mathbb{C}$, there is
a control $\bar{v}\in\mathcal{U}$ such that

$u(t,\gamma)<Y_{t}^{t,\gamma;\bar{v}}+\varepsilon.$

Hence we have

$u(t,\gamma)-u_{n}(t,\gamma)=u(t,\gamma)-\sup_{v\in\mathcal{U}}Y^{t,\gamma,n;v}(t)\leqslant
Y^{t,\gamma;\bar{v}}(t)+\varepsilon-Y^{t,\gamma,n;\bar{v}}(t).$ (5.17)

On the other hand, there is a control $\tilde{v}\in\mathcal{U}$ such that

$u_{n}(t,\gamma)\leqslant Y^{t,\gamma,n;\tilde{v}}(t)+\varepsilon,$

which gives

$u(t,\gamma)-u_{n}(t,\gamma)\geqslant
u(t,\gamma)-Y^{t,\gamma,n;\tilde{v}}(t)-\varepsilon\geqslant
Y^{t,\gamma;\tilde{v}}(t)-Y^{t,\gamma,n;\tilde{v}}(t)-\varepsilon.$ (5.18)

Combining (5.17) and (5.18), we obtain

$|u(t,\gamma)-u_{n}(t,\gamma)|\leqslant\mathbb{E}[|Y^{t,\gamma;\bar{v}}(t)-Y^{t,\gamma,n;\bar{v}}(t)|]+\mathbb{E}[|Y^{t,\gamma;\tilde{v}}(t)-Y^{t,\gamma,n;\tilde{v}}(t)|]+2\varepsilon.$

Note that $\bar{v}$ and $\tilde{v}$ are given admissible controls. Then by
3.10 it yields that for each compact set $K\subset[0,T]\times\mathbb{C}$,

$\lim_{n\rightarrow\infty}\sup_{(t,\gamma)\in
K}\mathbb{E}[|Y^{t,\gamma;\bar{v}}(t)-Y^{t,\gamma,n;\bar{v}}(t)|]+\mathbb{E}[|Y^{t,\gamma;\tilde{v}}(t)-Y^{t,\gamma,n;\tilde{v}}(t)|]=0.$

Finally, owing to the arbitrariness of $\varepsilon$, we obtain the uniform
convergence of the value functions $u_{n}$ to $u$ in $K$. ∎

###### Proof of 3.12.

For any $t\in[0,T],\ \gamma\in\mathbb{C},\ r\in\mathbb{R},\
p\in\mathbb{R}^{n},\ A\in\mathbb{S}^{n},\ v\in U$, we set

$\mathcal{A}=\langle
p,b(t,\gamma,v)\rangle+\frac{1}{2}Tr\left(\sigma(t,\gamma,v)\sigma^{*}(t,\gamma,v)A\right)+g_{n}(t,\gamma,r,v)$

and

$\mathcal{B}=\langle
p,b(t,\gamma,v)\rangle+\frac{1}{2}Tr\left(\sigma(t,\gamma,v)\sigma^{*}(t,\gamma,v)A\right)+g(t,\gamma,r,v).$

Note that

$H_{n}-H=\sup_{v\in U}\mathcal{A}-\sup_{v\in U}\mathcal{B}\leqslant\sup_{v\in
U}(\mathcal{A}-\mathcal{B})=\sup_{v\in
U}\left(g_{n}-g\right)\leqslant\sup_{v\in U}\left|g_{n}-g\right|$

and

$H-H_{n}=\sup_{v\in U}\mathcal{B}-\sup_{v\in U}\mathcal{A}\leqslant\sup_{v\in
U}(\mathcal{B}-\mathcal{A})=\sup_{v\in
U}\left(g-g_{n}\right)\leqslant\sup_{v\in U}\left|g-g_{n}\right|,$

which implies

$\left|H_{n}-H\right|\leqslant\sup_{v\in
U}\left|g_{n}(t,\gamma,r,v)-g(t,\gamma,r,v)\right|.$

Note that for each compact set
$K\subset[0,T]\times\mathbb{C}\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{S}^{n}$,
and any $(t,\gamma,r,p,A)\in K$, $v\in U$, there is a compact set
$\widetilde{K}\in[0,T]\times\mathbb{C}\times\mathbb{R}\times U$ such that
$(t,\gamma,r,v)\in\widetilde{K}$. So by 3.9 we have

$\displaystyle\lim_{n\rightarrow\infty}\sup_{(t,\gamma,r,p,A)\in
K}\left|H_{n}(t,\gamma,r,p,A)-H(t,\gamma,r,p,A)\right|$
$\displaystyle\leqslant$
$\displaystyle\lim_{n\rightarrow\infty}\sup_{(t,\gamma,r,p,A)\in K}\sup_{v\in
U}\left|g_{n}(t,\gamma,r,v)-g(t,\gamma,r,v)\right|$ $\displaystyle\leqslant$
$\displaystyle\lim_{n\rightarrow\infty}\sup_{(t,\gamma,y,v)\in\widetilde{K}}\left|g_{n}(t,\gamma,y,v)-g(t,\gamma,y,v)\right|=0.$

Therefore, the uniform convergence of $H_{n}$ to $H$ in $K$ follows from
above. ∎

###### Proof of 3.13.

The proof is similar to the proof of Lemma 6.2 in Fleming–Soner [9]. Let
$\phi\in C^{1,2}_{\text{Lip}}([0,T]\times\mathbb{C};\mathbb{R})$ and
$(\bar{t},\bar{\gamma})\in[0,T]\times\mathbb{C}$ be a strict minimizer of
$\phi-u$ with $\phi(\bar{t},\bar{\gamma})=u(\bar{t},\bar{\gamma})$. On one
hand, due to that $u_{n}$ converges to $u$ uniformly in each compact subset of
their domain, there exists a sequence
$(t_{n},\gamma_{n})\rightarrow(\bar{t},\bar{\gamma})$ as $n\rightarrow\infty$,
such that $(t_{n},\gamma_{n})$ is a local minimum of $\phi-u_{n}$. On the
other hand, if we set
$\tilde{\phi}(t,\gamma)\triangleq\phi(t,\gamma)+u_{n}(t_{n},\gamma_{n})-\phi(t_{n},\phi_{n})$,
then $(t_{n},\gamma_{n})$ is a local minimum of $\tilde{\phi}-u_{n}$ and
$\tilde{\phi}(t_{n},\gamma_{n})=u_{n}(t_{n},\gamma_{n})$. Hence, from the
viscosity property of $u_{n}$, we have that

$\frac{\partial}{\partial
t}\tilde{\phi}(t_{n},\gamma_{n})+H_{n}\big{(}t_{n},\gamma_{n},\tilde{\phi}(t_{n},\gamma_{n}),D_{x}\tilde{\phi}(t_{n},\gamma_{n}),D_{x}^{2}\tilde{\phi}(t_{n},\gamma_{n})\big{)}\geqslant
0.$

Finally, leting $n\rightarrow\infty$ and using the uniform convergence of
$H^{n}$ to $H$, we conclude that $u$ is a viscosity subsolution of the
limiting equation (3.16). Similarly the supersolution property can be proved
similarly. ∎

##  6 Conclusion

In conclusion, we show that under the assumptions (A1)-(A6), the dynamic
programming principle for delayed stochastic recursive optimal control problem
holds under the non-Lipschitz circumstances in 3.5. Moreover, as proved in
3.14, the value function of the delayed stochastic recursive optimal control
problem is the viscosity solution of the corresponding HJB equation (3.9), in
the setting of non-Lipschitz generator $g(t,\gamma,y,v)$ of BSDE serving as
the recursive cost functional. However, when the generator $g$ also depends on
$z$, the second one of the solution pair, the result is still open, and we
hope to solve it in the near future. Finally, in Section 5, to demonstrate the
application of our theoretical study to mathematical finance models, a
consumption-investment problem under the delayed continuous-time Epstein-Zin
utility with a non-Lipschitz is presented and the HJB equation which the value
function satisfies is given.

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