Quadratic BSDEs with Singular Generators and Unbounded Terminal Conditions: Theory and Applications

Abstract

We investigate a class of quadratic backward stochastic differential equations (BSDEs) with generators that are singular in y. First, we establish the existence of solutions and a comparison theorem, thereby extending the existing results in the literature. Furthermore, we analyze the stability properties, derive the Feynman–Kac formula, and prove the uniqueness of viscosity solutions for the corresponding singular semi-linear partial differential equations (PDEs). Finally, we demonstrate applications in the context of robust control linked to stochastic differential utility and the certainty equivalent based on g-expectation. In these applications, the quadratic coefficients in the generators, respectively, quantify ambiguity aversion and absolute risk aversion.

1. Introduction

This paper focuses on a one-dimensional backward stochastic differential equation (BSDE):

$$Y_t=\xi +{\int }_t^{T}f(u,Y_u,Z_u)du-{\int }_t^T{Z}_{u}d{W}_u,t\in [0,T],$$

(1)

where W denotes a standard d-dimensional Brownian motion defined on a filtered probability space $(\mathsf{\Omega },\mathcal{F},{\left({\mathcal{F}}_s\right)}_{s\in [0,T]},\mathbb{P})$. Here, ${\left({\mathcal{F}}_s\right)}_{s\in [0,T]}$ is the $\mathbb{P}$-completion of the filtration generated by W. The terminal condition $\xi$ is an ${\mathcal{F}}_T$-measurable $\mathbb{R}$-valued random variable, and the generator $f:[0,T]\times \mathsf{\Omega }\times \mathbb{R}\times {\mathbb{R}}^{1\times d}\to \mathbb{R}$ is a progressively measurable process. Nonlinear BSDEs were first pioneered under the Lipschitz condition by [1]. BSDEs with generators exhibiting quadratic growth in the variable z have been extensively researched, with significant contributions from [2,3,4,5,6] and others since [7] first investigated the case of bounded terminal conditions. This paper characterizes a specific category of BSDEs with the form of generators given by $g\left(y\right){\left|z\right|}^2$, which, to our knowledge, first appeared in [8]. The existence and uniqueness results for the case where g is globally integrable on $\mathbb{R}$ were demonstrated by [9], through the employment of Itô-Krylov’s formula and a “domination method” (see Lemma 1) derived from the existence result for reflected BSDEs obtained in [10]. Another intriguing case is $f(y,z)={\left|z\right|}^2/y$, meaning that the generator f is singular at $y=0$. In this instance, ref. [11] assumed that the generator f satisfied

$$0\le f(s,\omega ,y,z)\le a_s+b_{s}y+{\gamma }_{s}z+{\displaystyle \frac{\delta }{2y}}{\left|z\right|}^2,(s,\omega ,y,z)\in [0,T]\times \mathsf{\Omega }\times (0,+\infty )\times {\mathbb{R}}^{1\times d},$$

(2)

for some processes $a,b,\gamma$ and a constant $\delta$. By utilizing the domination method, they established the existence of solutions in ${\mathcal{S}}^p\times {\mathcal{L}}^2$. Furthermore, they proved a uniqueness result for bounded solutions using techniques from convex duality. In this paper, we prove that solutions exist in ${\mathcal{S}}^p\times {\mathcal{M}}^q$, and we establish a comparison theorem for $L^p$ solutions through $\theta$-techniques (see [3]). Moreover, our approach allows for a straightforward extension of these results to more generalized BSDEs with generators satisfying

$$0\le f(s,\omega ,y,z)\le a_s+b_s\varphi \left(y\right)+{\gamma }_s\left|z\right|+{\displaystyle \frac{{\left|z\right|}^2}{2\psi \left(y\right)}},(s,\omega ,y,z)\in [0,T]\times \mathsf{\Omega }\times (0,+\infty )\times {\mathbb{R}}^{1\times d},$$

under appropriate assumptions. Recent advances in this field include works by [12,13]. Ref. [12] pioneered the study of geometric BSDEs and included an investigation into a class of BSDEs with generators that satisfy the following structural condition:

$$0\le f(s,\omega ,y,z)\le a_{s}y+b_{s}y|ln\left(y\right)|+{\lambda }_{s}z+{\displaystyle \frac{\delta }{y}}{\left|z\right|}^2,(s,\omega ,y,z)\in [0,T]\times \mathsf{\Omega }\times (0,+\infty )\times {\mathbb{R}}^{1\times d},$$

where the processes $a,b$, and $\lambda$ must be bounded for uniqueness and stability. In contrast, our methodology employs a series of auxiliary functions (see Remark 2) leading to a non-trivial generalization of the related results obtained in [12], as shown in Remark 3. Furthermore, ref. [13] investigated BSDEs and reflected BSDEs with generators exhibiting one-sided growth in y and general quadratic growth in z. However, the assumptions in [13] cannot be directly applied to the equations presented in this study, and we provide further conclusions about the integrability of the solutions. Additionally, we derive the stability property for $L^p$ solutions and the Feynman–Kac formula within our framework, and we verify the uniqueness of viscosity solutions to the related singular semi-linear quadratic PDEs.

More importantly, this class of quadratic BSDEs with singularities and related partial differential equations (PDEs) has significant applications in fields such as physics, quantitative finance, and biology (see [11,14,15,16]). For instance, in finance the conditional generalized entropic risk measure (see [17]) can be expressed via Y, the value process of a solution to such a BSDE. Ref. [11] illustrated that Y can be ordinally equivalent to stochastic differential utility of the Epstein–Zin type with relative risk aversion represented by $0\le \delta \ne 1$ in (2). In Section 5.1, we demonstrate that Y can represent stochastic differential utility concerning robustness, while $1/\psi \left(y\right)$, the coefficient of ${\left|z\right|}^2/2$, quantifies the level of ambiguity aversion. In Example 2, we show that Y can depict the certainty equivalent of the terminal value based on g-expectation, while $1/\psi \left(y\right)$ represents the absolute risk aversion coefficient.

The structure of this paper is as follows. Section 2 and Section 3 introduce the notations, present the existence results, and establish the comparison theorem. In Section 4, we prove a stability result for the $L^p$ solutions, derive the Feynman–Kac formula within our theoretical framework, and verify the uniqueness of viscosity solutions to the associated PDE. Having established the theoretical foundations, Section 5 presents an application linking robust control to stochastic differential utility, slightly generalizing the results of [18,19], while including an example of the certainty equivalent based on g-expectation.

2. Notations and Existence Results

This section establishes the existence results for a class of quadratic BSDEs with singular generators. First, we introduce the notations used in this paper. We say a process or random variable satisfies some property if this property holds except on a $\mathbb{P}$-null subset of $\mathsf{\Omega }$, and, thus, we occasionally omit the notation $\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.$. We define ${\mathbb{E}}_t\left[·\right]:=\mathbb{E}\left[·\mid {\mathcal{F}}_t\right]$ as the conditional mathematical expectation with respect to ${\mathcal{F}}_t$. The set composed of stopping times $\tau$ satisfying $0\le \tau \le T$ is represented by ${\mathcal{T}}_{0,T}$. If N is an adapted and càdlàg process, we define $N^{*}:={sup}_{t\in [0,T]}\left|N_t\right|$. We recall that class $\left(D\right)$ comprises progressively measurable processes X satisfying that $\{X_{\tau }\mid \tau \in {\mathcal{T}}_{0,T}\}$ is uniformly integrable. Occasionally, we denote the stochastic integral ${\int }_0^{·}{Z}_{s}d{W}_s$ by $Z\circ W$ and denote by $\mathcal{E}\left(X\right)$ the stochastic exponential of a one-dimensional local martingale X. For any real number $p\ge 1$ and subsets $U\subset \mathbb{R},$$V\subset {\mathbb{R}}^n$, let us define the following spaces and notations.

${\mathcal{C}}^p\left(U\right)$: the space of functions from U to $\mathbb{R}$ having a continuous p-th derivative.

$L_{loc}^p\left(U\right)$: the space of locally $L^p$ integrable functions from U to $\mathbb{R}$.

${\mathcal{W}}_{p,loc}^2\left(U\right)$: the Sobolev space of functions $h:U\mapsto \mathbb{R}$, such that h and its generalized derivatives $h^{\prime }$ and $h^{″}$ belong to $L_{loc}^p\left(U\right)$.

$\mathcal{S}:=\mathcal{S}\left(\mathbb{R}\right)$: the space of adapted and continuous processes valued in $\mathbb{R}$.

${\mathbb{L}}^p:={\mathbb{L}}^p(\mathsf{\Omega },{\mathcal{F}}_T,\mathbb{P};\mathbb{R})$: the space of random variables $\eta$ fulfilling $\eta$ takes values in $\mathbb{R}$ and $\mathbb{E}\left[{\left|\eta \right|}^p\right]<+\infty$, with $\eta$ measurable with respect to ${\mathcal{F}}_T$.

${\mathcal{S}}^{\infty }:={\mathcal{S}}^{\infty }\left(\mathbb{R}\right)$: the space of bounded processes in $\mathcal{S}$.

${\mathcal{S}}^p:={\mathcal{S}}^p\left(\mathbb{R}\right)$: the space of all processes $Y\in \mathcal{S}$, such that ${\parallel Y\parallel }_{{\mathcal{S}}^p}:={\left(\mathbb{E}\left[\underset{0\le t\le T}{sup}{\left|Y_t\right|}^p\right]\right)}^{{\displaystyle \frac{1}{p}}}$$<+\infty$.

${\mathcal{L}}^2:={\mathcal{L}}^2\left({\mathbb{R}}^{1\times d}\right)$: the space of ${\mathbb{R}}^{1\times d}$-valued processes N satisfying N is progressively measurable and $\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.$, ${\int }_0^T{\left|N_s\right|}^{2}ds<\infty$.

${\mathcal{M}}^p:={\mathcal{M}}^p\left({\mathbb{R}}^{1\times d}\right)$: the space of progressively measurable processes G fulfilling that G is valued in ${\mathbb{R}}^{1\times d}$ and ${\parallel G\parallel }_{{\mathcal{M}}^p}:={\left(\mathbb{E}\left[{\left({\int }_0^T{\left|G_s\right|}^{2}ds\right)}^{p/2}\right]\right)}^{1/p}<+\infty$.

$\mathcal{K}$: the set of $\mathbb{R}$-valued processes K satisfying $K_0=0$, K is continuous, non-decreasing, progressively measurable, and $\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.$, $K_T<+\infty$.

$BMO$: the space of all uniformly integrable martingales N satisfying

$${\parallel N\parallel }_{BMO}:=\underset{\tau \in {\mathcal{T}}_{0,T}}{sup}{∥\mathbb{E}\left[|N_T-N_{\tau }||{\mathcal{F}}_{\tau }\right]∥}_{\infty }<\infty .$$

For any function $u:V\mapsto \mathbb{R},$ we define $D_V^{2,\pm }u$ as a mapping from V to the subsets of ${\mathbb{R}}^n\times S\left(n\right)$, where $S\left(n\right)$ is the set of real symmetric $n\times n$ matrices. For $\widehat{x}\in V$, we define the second-order “superjet” (see [20]) of u at $\widehat{x}$ as

$$\begin{array}{cc}& D_V^{2,+}u\left(\widehat{x}\right):=\left\{(p,K)\in {\mathbb{R}}^n\times S\left(n\right)\mid \right.\hfill \\ & u\left(x\right)\le u\left(\widehat{x}\right)+p^T(x-\widehat{x})+{\displaystyle \frac{1}{2}}{(x-\widehat{x})}^{T}K(x-\widehat{x})+o(|x-\widehat{x}{|}^2)\mathrm{as}V\ni x\to \widehat{x}\},\hfill \end{array}$$

the second-order “subjet” of u at $\widehat{x}$ as

$$\begin{array}{cc}& D_V^{2,-}u\left(\widehat{x}\right):=\left\{(p,K)\in {\mathbb{R}}^n\times S\left(n\right)\mid \right.\hfill \\ & u\left(x\right)\ge u\left(\widehat{x}\right)+p^T(x-\widehat{x})+{\displaystyle \frac{1}{2}}{(x-\widehat{x})}^{T}K(x-\widehat{x})+o(|x-\widehat{x}{|}^2)\mathrm{as}V\ni x\to \widehat{x}\},\hfill \end{array}$$

and their closures as

$$\begin{array}{cc}& {\overline{D}}_V^{2,\pm }u\left(\widehat{x}\right):=\left\{(p,K)\in {\mathbb{R}}^n\times S\left(n\right)\mid \exists x_n\in V\mathrm{and}(p_n,K_n)\in D_V^{2,\pm }u\left(x_n\right),\right.\hfill \\ & \mathrm{such}\mathrm{that}\left.(x_n,u\left(x_n\right),p_n,K_n)\to (\widehat{x},u\left(\widehat{x}\right),p,K)\mathrm{as}n\to \infty \right\}.\hfill \end{array}$$

We sometimes describe the BSDE with generator f and terminal condition $\xi$ as BSDE$(\xi ,f)$, rather than (1), for notational simplicity. A solution of BSDE$(\xi ,f)$ is defined as a pair of processes $(Y,Z):={\left\{(Y_t,Z_t)\right\}}_{t\in [0,T]}\in \mathcal{S}\times {\mathcal{L}}^2$ fulfilling that $\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.$, (1) holds and ${\int }_0^T\left|f(u,Y_u,Z_u)\right|du<\infty$. Furthermore, we designate $(Y,Z):={\left\{(Y_t,Z_t)\right\}}_{t\in [0,T]}$ as an $L^p$ solution, provided that $(Y,Z)\in {\mathcal{S}}^p\times {\mathcal{M}}^q$ for some $p>1,q>1$. Now, we establish the existence theorem for BSDE (1).

Theorem 1. 

Assume BSDE (1) satisfies that $\xi >0$, f is continuous in $(y,z)$ for each fixed $(s,\omega )$, and for any $(s,\omega ,y,z)\in [0,T]\times \mathsf{\Omega }\times (0,+\infty )\times {\mathbb{R}}^{1\times d},$

$$0\le f(s,\omega ,y,z)\le a_s+b_{s}y+{\gamma }_s\left|z\right|+{\displaystyle \frac{\delta }{2y}}{\left|z\right|}^2,$$

(3)

where $\delta \ge 0$ is a constant, and $a,b$, and γ are non-negative and progressively measurable processes. Then, BSDE (1) admits a solution $(Y,Z)$, such that $Y\in {\mathcal{S}}^{2p(1+\delta )}$ for some $p>1$ if

$$\mathbb{E}\left[exp\left\{{\displaystyle \frac{p}{p-1}}{\int }_0^T{\gamma }_s^{2}ds+2p\left(\delta +1\right){\int }_0^T{b}_{s}ds\right\}{\left(1+{\xi }^{1+\delta }+{\int }_0^T{a}_s^{\delta +1}ds\right)}^{2p}\right]<+\infty .$$

Moreover, $Z\in {\mathcal{M}}^{2p}$ if $0<\delta \ne 1$.

To prove Theorem 1, we recall the domination method, which was first employed by [21] and further developed in [9,11,22].

Lemma 1 

([22], Lemma 3.1). We say that the BSDE$(\xi ,f)$ satisfies the domination conditions if there exist two BSDEs, $({\xi }^1,f^1)$ and $({\xi }^2,f^2)$, such that

(1)

${\xi }^1\le \xi \le {\xi }^2$;

(2)

BSDE$({\xi }^1,f^1)$ and BSDE$({\xi }^2,f^2)$ have solutions $(Y^1,Z^1)$ and $(Y^2,Z^2)$ in $\mathcal{S}\times {\mathcal{L}}^2$, respectively, such that $Y_s^1\le Y_s^2,\forall s\in [0,T]$. For every $(s,\omega )\in [0,T]\times \mathsf{\Omega }$, $y\in [Y_s^1\left(\omega \right),Y_s^2\left(\omega \right)]$, and $z\in {\mathbb{R}}^d$, the following inequalities hold:

$$f^1(s,\omega ,y,z)\le f(s,\omega ,y,z)\le f^2(s,\omega ,y,z),$$

and

$$\left|f(s,\omega ,y,z)\right|\le {\eta }_s\left(\omega \right)+{\displaystyle \frac{{\zeta }_s\left(\omega \right)}{2}}{\left|z\right|}^2,$$

where η and ζ are two ${\mathcal{F}}_t$-adapted processes satisfying that ζ is continuous and $\forall \omega ,$${\int }_0^T\left|{\eta }_t\left(\omega \right)\right|dt<+\infty .$

Assume that the BSDE$(\xi ,f)$ satisfies the domination conditions and f is continuous in $(y,z)$ for each fixed $(s,\omega )$. Then, the BSDE$(\xi ,f)$ admits at least one solution $(Y,Z)$, such that

$$Y_s^1\le Y_s\le Y_s^2,\forall s\in [0,T],$$

(4)

and a maximal and a minimal solution exist among all the solutions satisfying (4).

The proof of Theorem 1 relies on the following technical lemma:

Lemma 2. 

Let $\zeta >0$ be an ${\mathcal{F}}_T$-measurable $\mathbb{R}$-valued random variable, and let γ be a non-negative and progressively measurable process. Then, the following BSDE,

$$Y_t=\zeta +{\int }_t^T{\gamma }_s\left|Z_s\right|ds-{\int }_t^T{Z}_{s}d{W}_s,$$

(5)

admits a solution $\left(Y,Z\right)$ satisfying for some $p>1,$

$$\forall t\in [0,T],{\left(\mathbb{E}\left[\zeta \mid {\mathcal{F}}_t\right]\right)}^p\le Y_t^p\le \mathbb{E}\left[exp\left\{{\displaystyle \frac{p}{2(p-1)}}{\int }_t^T{\gamma }_s^{2}ds\right\}{\zeta }^p\mid {\mathcal{F}}_t\right],$$

if

$$\mathbb{E}\left[exp\left\{{\displaystyle \frac{p}{2(p-1)}}{\int }_0^T{\gamma }_s^{2}ds\right\}{\zeta }^p\right]<+\infty .$$

Proof. 

Consider the following BSDEs:

$${\overline{Y}}_t={\zeta }^p+{\int }_t^T\left({\gamma }_s\left|{\overline{Z}}_s\right|-{\displaystyle \frac{p-1}{2p}}{\displaystyle \frac{{\left|{\overline{Z}}_s\right|}^2}{{\overline{Y}}_s}}\right)ds-{\int }_t^T{\overline{Z}}_{s}d{W}_s$$

(6)

and

$$y_t={\zeta }^p-{\int }_t^T{\displaystyle \frac{p-1}{2p}}{\displaystyle \frac{{\left|z_s\right|}^2}{y_s}}ds-{\int }_t^T{z}_{s}d{W}_s.$$

(7)

Since the BSDE (7) has a solution $(y_t,z_t)$ denoted by $y_t={\left(\mathbb{E}\left[\zeta \mid {\mathcal{F}}_t\right]\right)}^p$, we infer from Young’s inequality and Lemma 1 that BSDE (6) admits a solution $\left(\overline{Y},\overline{Z}\right)$ satisfying ${\overline{Y}}_t\ge {\left(\mathbb{E}\left[\zeta \mid {\mathcal{F}}_t\right]\right)}^p$ if the BSDE

$$Y_t^{\prime }={\zeta }^p+{\int }_t^T{\displaystyle \frac{p{\gamma }_s^2}{2(p-1)}}{Y}_s^{\prime }ds-{\int }_t^T{Z}_s^{\prime }d{W}_s$$

(8)

admits a solution satisfying $Y_t^{\prime }\ge y_t$. It is clear from the integrability assumption that (8) admits a solution $\left(Y^{\prime },Z^{\prime }\right)$ given by

$$Y_t^{\prime }=\mathbb{E}\left[exp\left\{{\displaystyle \frac{p}{2(p-1)}}{\int }_t^T{\gamma }_s^{2}ds\right\}{\zeta }^p\mid {\mathcal{F}}_t\right]\ge y_t.$$

Therefore, BSDE (6) admits a solution $\left(\overline{Y},\overline{Z}\right)$ satisfying ${\overline{Y}}_t\ge {\left(\mathbb{E}\left[\zeta \mid {\mathcal{F}}_t\right]\right)}^p$. Consequently, BSDE (5) admits a solution $\left(Y,Z\right)$ given by

$${\left(Y_t^{\prime }\right)}^{{\displaystyle \frac{1}{p}}}\ge Y_t={\left({\overline{Y}}_t\right)}^{\frac{1}{p}}\ge \mathbb{E}\left[\zeta \mid {\mathcal{F}}_t\right],$$

which completes the proof. □

Proof of Theorem 1. 

Without loss of generality, we assume $\delta >0.$ Consider the BSDEs

$$Y_t^U=\xi +{\int }_t^T{a}_s+b_s{Y}_s^U+{\gamma }_s\left|Z_s^U\right|+{\displaystyle \frac{\delta }{2Y_s^U}}{\left|Z_s^U\right|}^{2}ds-{\int }_t^T{Z}_s^{U}d{W}_s$$

(9)

and

$$Y_t^L=\xi -{\int }_t^T{Z}_s^{L}d{W}_s.$$

Since $\xi$ is integrable, the second BSDE has a solution $(Y^L,Z^L)$ denoted by $Y_t^L=\mathbb{E}\left[\xi \mid {\mathcal{F}}_t\right]$ according to Proposition 1.1 in [22]. Using Lemma 1 (with $Y_t^2=Y_t^U$ and $Y_t^1=Y_t^L$), we infer that BSDE (1) has a solution $(Y,Z)$ satisfying $Y_t\ge \mathbb{E}\left[\xi \mid {\mathcal{F}}_t\right]$ if the BSDE (9) admits a solution $(Y^U,Z^U)$ satisfying $Y_t^U\ge Y_t^L$.

Define an invertible function

$$u\left(y\right):={\displaystyle \frac{y^{1+\delta }}{1+\delta }},y>0.$$

Then, $u(·),u^{-1}(·)\in {\mathcal{C}}^2\left((0,+\infty )\right)$. By Itô’s formula and Lemma 1, the BSDE (9) admits a solution $(Y_t^U,Z_t^U)$ satisfying $Y_t^U\ge Y_t^L$ if and only if

$$Y_t^0=u\left(\xi \right)+{\int }_t^T\left(a_s{\left(\left(\delta +1\right)Y_s^0\right)}^{{\displaystyle \frac{\delta }{\delta +1}}}+b_s\left(\delta +1\right)Y_s^0+{\gamma }_s\left|Z_s^0\right|\right)ds-{\int }_t^T{Z}_s^{0}d{W}_s$$

(10)

admits a solution $(Y_t^0,Z_t^0)$, such that $Y_t^0\ge u\left(Y_t^L\right).$ So far, the proof is similar to that in [11]. The results in [23] prove that Equation (10) has a solution $(Y^0,Z^0)$, such that $Y_t^0\ge u\left(Y_t^L\right)$, while we continue to use Lemma 1. From Young’s inequality, Equation (10) has a solution $(Y^0,Z^0)$, such that $Y_t^0\ge u\left(Y_t^L\right)$ if

$$Y_t^1=u\left(\xi \right)+{\int }_t^T\left({\displaystyle \frac{a_s^{\delta +1}}{\delta +1}}+\left(\delta +b_s\left(\delta +1\right)\right)Y_s^1+{\gamma }_s\left|Z_s^1\right|\right)ds-{\int }_t^T{Z}_s^{1}d{W}_s$$

(11)

admits a solution $(Y^1,Z^1)$, such that $Y_t^1\ge u\left(Y_t^L\right).$ By using Itô’s formula and Lemma 1 again, this holds if

$$\begin{array}{c}\hfill Y_t^2=e^{\delta T+\left(\delta +1\right){\int }_0^T{b}_{s}ds}\left(u\left(\xi \right)+{\int }_0^T{\displaystyle \frac{a_s^{\delta +1}}{\delta +1}}ds\right)+{\int }_t^T{\gamma }_s\left|Z_s^2\right|ds-{\int }_t^T{Z}_s^{2}d{W}_s\end{array}$$

(12)

admits a solution $\left(Y^2,Z^2\right)$, such that $Y_t^2\ge e^{\delta t+\left(\delta +1\right){\int }_0^t{b}_{s}ds}\left(u\left(\mathbb{E}\left[\xi \mid {\mathcal{F}}_t\right]\right)+{\int }_0^t{\displaystyle \frac{a_s^{\delta +1}}{\delta +1}}ds\right),$ which holds if (12) admits a solution $(Y^2,Z^2)$, such that $Y_t^2\ge \mathbb{E}\left[Y_T^2\mid {\mathcal{F}}_t\right]$ according to Jensen’s inequality.

From Lemma 2 and the integrability assumption, (12) admits a solution $\left(Y^2,Z^2\right)$, such that

$$\mathbb{E}\left[Y_T^2\mid {\mathcal{F}}_t\right]\le Y_t^2\le {\left(\mathbb{E}\left[exp\left\{{\displaystyle \frac{p}{2(p-1)}}{\int }_t^T{\gamma }_s^{2}ds\right\}{\left(Y_T^2\right)}^p\mid {\mathcal{F}}_t\right]\right)}^{{\displaystyle \frac{1}{p}}}.$$

Going back to BSDE (1), we conclude that BSDE (1) has a solution $(Y,Z)$, such that

$$\mathbb{E}\left[\xi \mid {\mathcal{F}}_t\right]\le Y_t\le u^{-1}\left(Y_t^2\right).$$

By Doob’s $L^p$ inequality and Jensen’s inequality, we have $Y\in {\mathcal{S}}^{2p(1+\delta )}$.

Now, we prove $Z\in {\mathcal{M}}^{2p}.$ Assume that $\delta >1$. We use C to denote a generic positive constant throughout the proof and subsequent sections, which may change from line to line. We define a function:

$$v\left(y\right):=\left\{\begin{array}{c}{\displaystyle \frac{1}{\delta -1}}\left({\displaystyle \frac{y^{1+\delta }-1}{\delta +1}}-{\displaystyle \frac{y^2-1}{2}}\right),y>1,\hfill \\ {\displaystyle \frac{y^2-1}{2}}-(y-1),0<y\le 1.\hfill \end{array}\right.$$

It is clear that $v(·)\in {\mathcal{C}}^2\left((0,+\infty )\right)$. Applying Itô’s formula to $v\left(Y_t\right)$ derives the following:

$$v\left(Y_t\right)=v\left(Y_0\right)+{\int }_0^t\left(v^{\prime }\left(Y_r\right)\left(-f\left(r,Y_r,Z_r\right)\right)+{\displaystyle \frac{1}{2}}{v}^{″}\left(Y_r\right){\left|Z_r\right|}^2\right)dr+{\int }_0^t{v}^{\prime }\left(Y_r\right)Z_{r}d{W}_r.$$

By the Cauchy–Schwarz inequality and the BDG inequality, we have

$$\mathbb{E}\left[{\left({\int }_0^T{\left|Z_u\right|}^{2}du\right)}^p\right]\le C\mathbb{E}\left[1+\underset{0\le s\le T}{sup}{\left(Y_s\right)}^{2p(1+\delta )}+{\left({\int }_0^T\left(a_s+b_s\right)ds\right)}^{2p}+{\left({\int }_0^T{\gamma }_t^{2}dt\right)}^{p(1+\delta )}\right].$$

According to the integrability assumption, $Z\in {\mathcal{M}}^{2p}.$ The case of $0<\delta <1$ can be proved similarly by taking $v\left(y\right)={\displaystyle \frac{y^2}{2(1-\delta )}}$ for $y>0$. This completes the proof. □

Remark 1. 

We give an a priori estimate for solutions to the BSDE (1) according to the proof of Theorem 1. Assume the BSDE (1) satisfies that $\xi >0$, f is continuous in $(y,z)$ for each fixed $(s,\omega )$, and for any $(s,\omega ,y,z)\in [0,T]\times \mathsf{\Omega }\times (0,+\infty )\times {\mathbb{R}}^{1\times d},$

$$0\le f(s,\omega ,y,z)\le a_s+b_{s}y+{\gamma }_s\left|z\right|+{\displaystyle \frac{\delta }{2y}}{\left|z\right|}^2,$$

where $\delta \ge 0$ is a constant, and $a,b$, and γ are non-negative and progressively measurable processes. For notational simplicity, we let (for $p>1$)

$${\overline{A}}_t:=exp\left\{{\displaystyle \frac{p}{2(p-1)}}{\int }_0^t{\gamma }_s^{2}ds+p\delta t+p\left(\delta +1\right){\int }_0^t{b}_{s}ds\right\}.$$

If $(y,z)\in \mathcal{S}\times {\mathcal{L}}^2$ is a solution to BSDE (1), such that $y_t>0,t\in [0,T]$ and

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}{\overline{A}}_t{\left(y_t^{1+\delta }+{\int }_0^t{a}_s^{\delta +1}ds\right)}^p\right]<\infty ,$$

then we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_t\left[\xi \right]\le y_t\le & {\left[{\left({\mathbb{E}}_t\left[{\displaystyle \frac{{\overline{A}}_T}{{\overline{A}}_t}}{\left({\xi }^{1+\delta }+{\int }_0^T{a}_s^{\delta +1}ds\right)}^p\right]\right)}^{{\displaystyle \frac{1}{p}}}-{\int }_0^t{a}_s^{\delta +1}ds\right]}^{{\displaystyle \frac{1}{1+\delta }}}:={\overline{y}}_t\left(\xi \right).\hfill \end{array}$$

(13)

This result can be directly verified through Itô’s formula.

Remark 2. 

We extend Theorem 1 to a more generalized BSDE (1) with the generator satisfying that f is continuous in $(y,z)$ for each fixed $(s,\omega )$, and

$$0\le f(s,\omega ,y,z)\le a_s+b_s\varphi \left(y\right)+{\gamma }_s\left|z\right|+{\displaystyle \frac{{\left|z\right|}^2}{2\psi \left(y\right)}},(s,\omega ,y,z)\in [0,T]\times \mathsf{\Omega }\times (0,+\infty )\times {\mathbb{R}}^{1\times d},$$

(14)

where $a,b$, and γ are non-negative and progressively measurable processes with $a>0\mathrm{or}b>0$; $\varphi :\left[0,\left.+\infty \right)\right.\mapsto \left[0,\left.+\infty \right)\right.$ is convex and increasing and has a continuous first derivative on $(0,+\infty );\varphi \left(x\right)>0,\forall x>0.\psi :\left[0,\left.+\infty \right)\right.\mapsto \left[0,\left.+\infty \right)\right.$ satisfies $\psi \left(0\right)=0$; $\varphi /\psi$ is increasing, and $1/\psi \left(x\right)$ is continuous on $(0,+\infty )$. For example, one can take the following functions, ϕ and ψ:

• $\varphi \left(y\right)=\psi \left(y\right)=exp\left(y\right),y>0;\varphi \left(0\right)=\psi \left(0\right)=0.$
• $\varphi \left(y\right)=\psi \left(y\right)=y^2,y\ge 0.$
• $\varphi \left(y\right)=y,\psi \left(y\right)=\sqrt{y},y\ge 0.$

To present our results, we define auxiliary functions:

$$\widehat{u}\left(y\right):={\int }_1^{y}exp\left({\int }_1^x{\displaystyle \frac{dr}{\psi \left(r\right)}}\right)dx,y>0.$$

$$\tilde{u}\left(y\right):={\int }_1^{y}exp\left({\int }_1^x{\displaystyle \frac{{\mathbb{1}}_{r>0}}{\psi \left(r\right)}}dr\right)dx,y>0.$$

$$\widehat{H}\left(y\right):={\int }_2^{2+y}{\displaystyle \frac{dr}{\varphi \left({\widehat{u}}^{-1}\left(r\right)\right){\widehat{u}}^{\prime }\left({\widehat{u}}^{-1}\left(r\right)\right)}},y>-2.$$

$$K\left(x\right):={\int }_0^{x}exp\left(-{\int }_1^z{\displaystyle \frac{{\mathbb{1}}_{r>0}}{\psi \left(r\right)}}dr\right)dz,x\ge 0,$$

$$\widehat{v}\left(y\right):={\int }_0^{y}K\left(x\right)exp\left({\int }_1^x{\displaystyle \frac{{\mathbb{1}}_{r>0}}{\psi \left(r\right)}}dr\right)dx,y>0.$$

For $p>1,0\le s\le T$ and $y\in (0,+\infty )$, we define

$$\mathsf{\Phi }(s,y;a_{·},b_{·},{\gamma }_{·}):=exp\left\{{\displaystyle \frac{p}{2(p-1)}}{\int }_0^s{\gamma }_r^{2}ds\right\}{\left({\widehat{H}}^{-1}\left(\widehat{H}\left({\left(\widehat{u}\left(y\right)\right)}^{+}\right)+{\int }_0^s\left({\displaystyle \frac{a_r}{\varphi \left(1\right)}}+b_r\right)dr\right)\right)}^p.$$

$$X_t:={\widehat{u}}^{-1}\left({\left\{\mathbb{E}\left[\mathsf{\Phi }(T,\xi ;a_{·},b_{·},{\gamma }_{·})\mid {\mathcal{F}}_t\right]\right\}}^{1/p}\right).$$

We conclude the following results:

• If $\xi >0$ and $\mathbb{E}\left[{\mathsf{\Phi }}^k(T,\xi ;a_{·},b_{·},{\gamma }_{·})\right]<+\infty$ for some $k>1$ then BSDE (1) has a solution $(Y,Z)$ satisfying $\widehat{u}\left(Y\right)\in {\mathcal{S}}^{kp}$;
• If $\xi >0$, ${\left({\mathbb{1}}_{x>0}/\psi \left(x\right)\right)}_{x\in \mathbb{R}}$ is locally integrable on $\mathbb{R}$, $\mathbb{E}\left[{\left({\int }_0^T{\gamma }_t^{2}dt\right)}^p\right]<+\infty$, $\mathbb{E}\left[{\varphi }^p\left({\widehat{X}}^{*}\right)\right]<+\infty$, and $\mathbb{E}\left[{\left({\widehat{X}}^{*}{\widehat{u}}^{\prime }\left({\widehat{X}}^{*}\right)\right)}^{2p}\right]<+\infty ,$ then BSDE (1) has a solution $(Y,Z)$ satisfying $\widehat{u}\left(Y\right)\in {\mathcal{S}}^{2p}$ and $Z\in {\mathcal{M}}^p$.

These results can be verified using the same method as in Theorem 1. It is worth noting that we have checked that $\widehat{u}(·)\in {\mathcal{C}}^1\left((0,+\infty )\right)\cap {\mathcal{W}}_{1,loc}^2\left((0,+\infty )\right)$, which is critical for applying Itô–Krylov’s formula. The functions ${\widehat{u}}^{-1}(·),\tilde{u}(·),{\tilde{u}}^{-1}(·),\widehat{H}(·),{\widehat{H}}^{-1}(·)$, and $\widehat{v}(·)$ also satisfy this property.

3. Comparison Theorem

To obtain the uniqueness result, we assume a convexity condition in $y,z$ so as to exploit the $\theta$-technique proposed by [3], which has proved to be convenient for dealing with quadratic generators.

Theorem 2. 

Assume that BSDE(${\xi }^1,f^1$) and BSDE(${\xi }^2,f^2$) satisfy that $\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.$, $0<{\xi }^1\le {\xi }^2$, and $d\mathbb{P}\otimes du\text{-}\mathrm{a}.\mathrm{s}.,f^1(u,Y_u^2,Z_u^2)\le f^2(u,Y_u^2,Z_u^2)$. Moreover, assume that $f^1(t,·,·)$ is convex for any $t\in [0,T]$, and for any $(s,\omega ,y,z)\in [0,T]\times \mathsf{\Omega }\times (0,+\infty )\times {\mathbb{R}}^{1\times d},$

$$0\le f^1(s,\omega ,y,z)\le a_s+b_{s}y+{\gamma }_s\left|z\right|+{\displaystyle \frac{\delta }{2y}}{\left|z\right|}^2,$$

where $\delta \ge 0$ is a constant, and $a,b$, and γ are non-negative and progressively measurable processes satisfying for some $p>1,$

$$\mathbb{E}\left[{\left({\int }_0^T{a}_s^{2+\delta }ds\right)}^{2p}\right]<+\infty ,\mathrm{and}\mathbb{E}\left[exp\left\{{\displaystyle \frac{p}{p-1}}{\int }_0^T{\gamma }_s^{2}ds+2p(2+\delta ){\int }_0^T{b}_{t}dt\right\}\right]<+\infty .$$

Let $(Y^1,Z^1)$ and $(Y^2,Z^2)$ be solutions to BSDE(${\xi }^1,f^1$) and BSDE(${\xi }^2,f^2$), respectively, such that both $Y^1$ and $Y^2$ belong to ${\mathcal{S}}^{2p(2+\delta )}$; then, $\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.$, $\forall 0\le t\le T,Y_t^1\le Y_t^2$.

Proof. 

For any given $\theta \in (0,1)$, set

$${\mathsf{\Delta }}_{\theta }{Y}_t:={\displaystyle \frac{Y_t^1-\theta Y_t^2}{1-\theta }},{\mathsf{\Delta }}_{\theta }{Z}_t:={\displaystyle \frac{Z_t^1-\theta Z_t^2}{1-\theta }}\mathrm{and}{\mathsf{\Delta }}_{\theta }\xi :={\displaystyle \frac{{\xi }^1-\theta {\xi }^2}{1-\theta }}.$$

Suppose $L_{·}^0\left({\mathsf{\Delta }}_{\theta }Y\right)$ represents the local time of ${\mathsf{\Delta }}_{\theta }Y$ at 0. We obtain by Tanaka’s formula that

$$\begin{array}{cc}\hfill {\left({\mathsf{\Delta }}_{\theta }{Y}_t\right)}^{+}=& {\left({\mathsf{\Delta }}_{\theta }\xi \right)}^{+}+{\int }_t^T{\mathbb{1}}_{{\mathsf{\Delta }}_{\theta }{Y}_u>0}{\displaystyle \frac{1}{1-\theta }}\left[f^1\left(u,Y_u^1,Z_u^1\right)-\theta f^2\left(u,Y_u^2,Z_u^2\right)\right]du\hfill \\ \hfill -& {\int }_t^T{\mathbb{1}}_{{\mathsf{\Delta }}_{\theta }{Y}_u>0}{\mathsf{\Delta }}_{\theta }{Z}_{u}d{W}_u-{\displaystyle \frac{1}{2}}{\int }_t^{T}d{L}_u^0\left({\mathsf{\Delta }}_{\theta }Y\right),\hfill \end{array}$$

where

$$\begin{array}{cc}& {\mathbb{1}}_{{\mathsf{\Delta }}_{\theta }{Y}_u>0}{\displaystyle \frac{1}{1-\theta }}\left[f^1\left(u,Y_u^1,Z_u^1\right)-\theta f^2\left(u,Y_u^2,Z_u^2\right)\right]\hfill \\ \hfill =& {\mathbb{1}}_{{\mathsf{\Delta }}_{\theta }{Y}_u>0}{\displaystyle \frac{1}{1-\theta }}\left[f^1\left(u,Y_u^1,Z_u^1\right)-\theta f^1\left(u,Y_u^2,Z_u^2\right)+\theta f^1\left(u,Y_u^2,Z_u^2\right)-\theta f^2\left(u,Y_u^2,Z_u^2\right)\right]\hfill \\ \hfill \le & {\mathbb{1}}_{{\mathsf{\Delta }}_{\theta }{Y}_u>0}\left[a_u+b_u{\mathsf{\Delta }}_{\theta }{Y}_u+{\gamma }_u\left|{\mathsf{\Delta }}_{\theta }{Z}_u\right|+{\displaystyle \frac{\delta {\left|{\mathsf{\Delta }}_{\theta }{Z}_u\right|}^2}{2{\mathsf{\Delta }}_{\theta }{Y}_u}}\right].\hfill \end{array}$$

For ease of notation, let us denote

$$\begin{array}{cc}& C_t:=exp\left\{{\displaystyle \frac{1}{2(p-1)}}{\int }_0^t{\gamma }_s^{2}ds+(2+\delta ){\int }_0^t{b}_{s}ds+(1+\delta )t\right\},\hfill \\ & P_t^0:=C_t\left({\displaystyle \frac{{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_t\right)}^{+}\right)}^{2+\delta }}{2+\delta }}+{\int }_0^t{\displaystyle \frac{a_s^{2+\delta }}{2+\delta }}ds\right),\hfill \\ & G_t:={\left(P_t^0\right)}^p,\mathrm{and}{R}_t:=C_{t}p{\left(P_t^0\right)}^{p-1}{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_t\right)}^{+}\right)}^{1+\delta }{\mathbb{1}}_{{\mathsf{\Delta }}_{\theta }{Y}_t>0}{\mathsf{\Delta }}_{\theta }{Z}_t.\hfill \end{array}$$

Due to $P_t^0\ge 0$ and $p>1$, we cannot directly apply Itô’s formula to $G_t$. Instead, we invoke Lemma 3.1 in [24], which provides an Itô’s-type formula for $p\ge 1,y\to {\left|y\right|}^p$ based on the technique introduced in [25]. By applying Itô’s formula to $P_t^0$ and using Lemma 3.1 in [24], we deduce that

$$G_t\le G_T-{\int }_t^T{R}_{s}d{W}_s.$$

Let us define ${\sigma }_n^t:=inf\left\{s>t:{\int }_t^s{\left|R_u\right|}^{2}du>n\right\}\wedge T,t\in [0,T],n\in \mathbb{N}.$ Therefore, $G_t\le \mathbb{E}\left[G_{{\sigma }_n^t}\mid {\mathcal{F}}_t\right].$ Noting that ${\left({\mathsf{\Delta }}_{\theta }\xi \right)}^{+}\le {\xi }^1$, the dominated convergence theorem gives

$${\left({\displaystyle \frac{{\left(Y_t^1-\theta Y_t^2\right)}^{+}}{1-\theta }}\right)}^{p(2+\delta )}\le {\mathbb{E}}_t\left[C_T^p{\left({\left({\xi }^1\right)}^{2+\delta }+{\int }_0^T{a}_s^{2+\delta }ds\right)}^p\right].$$

Moving $(1-\theta )$ to the right-hand side and sending $\theta \to 1$, we complete the proof. □

Corollary 1. 

There is only one solution $(Y,Z)\in {\mathcal{S}}^{2p(2+\delta )}\times {\mathcal{M}}^{2p}$ for BSDE$(\xi ,f)$, if $\xi >0,f(t,·,·)$ is convex for any $t\in [0,T]$, and for any $(s,\omega ,y,z)\in [0,T]\times \mathsf{\Omega }\times (0,+\infty )\times {\mathbb{R}}^{1\times d},$

$$0\le f(s,\omega ,y,z)\le a_s+b_{s}y+{\gamma }_s\left|z\right|+{\displaystyle \frac{\delta }{2y}}{\left|z\right|}^2,$$

where $0\le \delta \ne 1$ is a constant, $a,b$, and γ are non-negative and progressively measurable processes, and for some $p>1$,

$$\mathbb{E}\left[exp\left\{{\displaystyle \frac{p}{p-1}}{\int }_0^T{\gamma }_s^{2}ds+2p(2+\delta ){\int }_0^T{b}_{t}dt\right\}{\left(1+{\xi }^{2+\delta }+{\int }_0^T{a}_s^{2+\delta }ds\right)}^{2p}\right]<+\infty .$$

More precisely,

$$\begin{array}{cc}& \mathbb{E}\left[\underset{0\le t\le T}{sup}{Y}_t^{2p(2+\delta )}+{\left({\int }_0^T{\left|Z_u\right|}^{2}du\right)}^p\right]\hfill \\ \hfill \le & C\mathbb{E}\left[exp\left\{{\displaystyle \frac{p}{p-1}}{\int }_0^T{\gamma }_s^{2}ds+2p(2+\delta ){\int }_0^T{b}_{t}dt\right\}{\left(1+{\xi }^{2+\delta }+{\int }_0^T{a}_s^{2+\delta }ds\right)}^{2p}\right],\hfill \end{array}$$

where C is a constant depending on $T,p$, and $\delta .$

Remark 3. 

For the more generalized BSDEs in Remark 2, a similar comparison theorem holds. In fact, let $(Y^1,Z^1)$ and $(Y^2,Z^2)$ be solutions to BSDE(${\xi }^1,f^1$) and BSDE(${\xi }^2,f^2$), respectively, fulfilling for a given $p>1,$

$$\forall \lambda >0,\mathbb{E}\left[\underset{0\le t\le T}{sup}\mathsf{\Phi }(t,\lambda Y_t^i;a_{·},b_{·},{\gamma }_{·})\right]<+\infty ,i=1,2.$$

Assume $d\mathbb{P}\otimes du\text{-}\mathrm{a}.\mathrm{s}.,f^1(u,Y_u^2,Z_u^2)\le f^2(u,Y_u^2,Z_u^2)$, and $\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.$, $0<{\xi }^1\le {\xi }^2$. If $f^1$ satisfies (14), ${\left({\mathbb{1}}_{x>0}/\psi \left(x\right)\right)}_{x\in \mathbb{R}}$ is locally integrable on $\mathbb{R}$ and $f^1$ is convex in $(y,z)$ for all $t\in [0,T]$ then $\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.$, $\forall 0\le t\le T,Y_t^1\le Y_t^2$.

Moreover, if BSDE($\xi ,f$) satisfies (14), ${\left({\mathbb{1}}_{x>0}/\psi \left(x\right)\right)}_{x\in \mathbb{R}}$ is locally integrable on $\mathbb{R}$, f is convex in $(y,z)$ for all $t\in [0,T]$, for some $p>1$,

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}{\varphi }^{p/2}\left(X_t\right)\right]<+\infty ,$$

and for any $\lambda \ge 1$,

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}exp\left\{{\displaystyle \frac{p}{2(p-1)}}{\int }_0^t{\gamma }_s^{2}ds\right\}{\left({\widehat{H}}^{-1}\left(\widehat{H}\left(\lambda X_t{\widehat{u}}^{\prime }\left(\lambda X_t\right)\right)+{\int }_0^t\left({\displaystyle \frac{a_s}{\varphi \left(1\right)}}+b_s\right)ds\right)\right)}^p\right]<\infty ,$$

then BSDE($\xi ,f$) has a unique solution $(Y,Z)$ in ${\mathcal{S}}^p\times {\mathcal{M}}^{p/2}$.

Corollary 2 

(BMO property). Assume that BSDE($\xi ,f$) satisfies that $\xi >0$ is bounded, (3) holds with $0\le \delta \ne 1,{\int }_0^T{a}_s^{2+\delta }ds,{\int }_0^T{b}_{s}ds$, and ${\int }_0^T{\gamma }_s^{2}ds$ are bounded. If $f(t,·,·)$ is convex for any $t\in [0,T]$ then there is only one solution $(Y,Z)$ for BSDE($\xi ,f$) fulfilling that Y is bounded and $Z\circ W\in BMO$.

Proof. 

Corollary 1 provides the existence, uniqueness of solution $(Y,Z)$, and presents the boundedness of $Y_t$. To prove that $Z\circ W\in BMO$, we apply Itô’s formula to $v\left(Y_t\right)$, where $v(·)$ is defined in the proof of Theorem 1. Taking the conditional expectation and employing a classical localization argument alongside the dominated convergence theorem yields the desired result. □

Remark 4. 

It is clear that for the more generalized BSDEs in Remark 2, Corollary 2 still holds.

4. Stability Result and Application to PDEs

4.1. Stability Result

The following result establishes a stability result for the solution to BSDE$(\xi ,f)$. To be more detailed, suppose that f is a generator satisfying (3) with coefficient processes $a_{·},b_{·},$ and ${\gamma }_{·}$, and $f(t,·,·)$ is convex for any $t\in [0,T]$. For each $n\ge 1,f_n$ is a generator fulfilling (3) with coefficient processes $a_{·}^n,b_{·}^n,$ and ${\gamma }_{·}^n$, and $f_n(t,·,·)$ is convex for any $t\in [0,T]$; $0\le \delta \ne 1$ is the constant in (3). For simplicity of representation, we denote for $p>1,$

$$A_t:=exp\left\{{\displaystyle \frac{p}{p-1}}{\int }_0^t{\gamma }_s^{2}ds+p(2+\delta ){\int }_0^t{b}_{s}ds\right\},$$

and

$$A_t^n:=exp\left\{{\displaystyle \frac{p}{p-1}}{\int }_0^t{\left({\gamma }_s^n\right)}^{2}ds+p(2+\delta ){\int }_0^t{b}_s^{n}ds\right\}.$$

Assume that $\xi$ and ${\left({\xi }_n\right)}_{n\ge 1}$ are strictly positive terminal conditions, such that

$$\mathbb{E}\left[A_T^4{\left(1+{\xi }^{2+\delta }+{\int }_0^T{a}_s^{2+\delta }ds\right)}^{4p}\right]+\underset{n\ge 1}{sup}\mathbb{E}\left[{\left(A_T^n\right)}^4{\left(1+{\left({\xi }_n\right)}^{2+\delta }+{\int }_0^T{\left(a_s^n\right)}^{2+\delta }ds\right)}^{4p}\right]<+\infty .$$

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Owing to Corollary 1, let $(Y,Z)\in {\mathcal{S}}^{4p(2+\delta )}\times {\mathcal{M}}^{4p}$ solve BSDE ($\xi ,f$), and let $(Y^n,Z^n)\in {\mathcal{S}}^{4p(2+\delta )}\times {\mathcal{M}}^{4p}$ solve BSDE (${\xi }_n,f_n$) for each $n\ge 1$.

Theorem 3. 

If ${\xi }_n\to \xi$ in $L^{2p(\delta +2)}$ and ${\int }_0^T|f_n(r,Y_r,Z_r)-f(r,Y_r,Z_r)|dr\to 0$ in $L^{2p(\delta +2)}$ as $n\to \infty$ then $Y^n$ converges to Y in ${\mathcal{S}}^{4p}$ and $Z^n$ converges to Z in ${\mathcal{M}}^{2p}$.

As a preliminary step, we establish the following lemma:

Lemma 3. 

There exists a constant $C>0$ independent of n, such that

$$\begin{array}{cc}& \mathbb{E}\left[{\left({\int }_0^{T}f(s,Y_s{Z}_s)+f_n(s,Y_s^n,Z_s^n)ds\right)}^{2p}\right]\hfill \\ \hfill \le & C\mathbb{E}\left[A_T^4{\left(1+{\xi }^{2+\delta }+{\int }_0^T{a}_s^{2+\delta }ds\right)}^{4p}\right]+C\underset{n\ge 1}{sup}\mathbb{E}\left[{\left(A_T^n\right)}^4{\left(1+{\left({\xi }_n\right)}^{2+\delta }+{\int }_0^T{\left(a_s^n\right)}^{2+\delta }ds\right)}^{4p}\right].\hfill \end{array}$$

Proof. 

Since f and $f_n$ satisfy (3), we have

$$\begin{array}{cc}& \mathbb{E}\left[{\left({\int }_0^{T}f(s,Y_s{Z}_s)+f_n(s,Y_s^n,Z_s^n)ds\right)}^{2p}\right]\hfill \\ \hfill \le & C\mathbb{E}\left[{\left({\int }_0^T{a}_s+a_s^n+b_s{Y}_s+b_s^n{Y}_s^n+{\gamma }_s\left|Z_s\right|+{\gamma }_s^n\left|Z_s^n\right|+{\displaystyle \frac{\delta {\left|Z_s\right|}^2}{2Y_s}}+{\displaystyle \frac{\delta {\left|Z_s^n\right|}^2}{2Y_s^n}}ds\right)}^{2p}\right]\hfill \\ \hfill \le & C\left\{\mathbb{E}\left[A_T+{\left({\int }_0^T{a}_{s}ds\right)}^{2p}+\underset{0\le s\le T}{sup}{Y}_s^{4p(2+\delta )}+{\left({\int }_0^T{\left|Z_s\right|}^{2}ds\right)}^{2p}+{\left({\int }_0^T{\displaystyle \frac{\delta {\left|Z_s\right|}^2}{Y_s}}ds\right)}^{2p}\right]\right.\hfill \\ \hfill +& \left.\underset{n\ge 1}{sup}\mathbb{E}\left[A_T^n+{\left({\int }_0^T{a}_s^{n}ds\right)}^{2p}+\underset{0\le s\le T}{sup}{\left(Y_s^n\right)}^{4p(2+\delta )}+{\left({\int }_0^T{\left|Z_s^n\right|}^{2}ds\right)}^{2p}+{\left({\int }_0^T{\displaystyle \frac{\delta {\left|Z_s^n\right|}^2}{Y_s^n}}ds\right)}^{2p}\right]\right\}.\hfill \end{array}$$

From Theorem 1, we have

$$\begin{array}{cc}\hfill \mathbb{E}\left[{\left({\int }_0^T{\left|Z_s\right|}^{2}ds\right)}^{2p}\right]\le & C\mathbb{E}\left[A_T+{\left({\int }_0^T{a}_{s}ds\right)}^{4p}+\underset{0\le s\le T}{sup}{Y}_s^{4p(2+\delta )}\right]\hfill \\ \hfill \le & C\mathbb{E}\left[A_T^4{\left(1+{\xi }^{2+\delta }+{\int }_0^T{a}_s^{2+\delta }ds\right)}^{4p}\right],\hfill \end{array}$$

and, similarly,

$$\underset{n\ge 1}{sup}\mathbb{E}\left[{\left({\int }_0^T{\left|Z_s^n\right|}^{2}ds\right)}^{2p}\right]\le C\underset{n\ge 1}{sup}\mathbb{E}\left[{\left(A_T^n\right)}^4{\left(1+{\left({\xi }_n\right)}^{2+\delta }+{\int }_0^T{\left(a_s^n\right)}^{2+\delta }ds\right)}^{4p}\right].$$

To estimate $\mathbb{E}\left[{\left({\int }_0^T{\displaystyle \frac{\delta {\left|Z_s\right|}^2}{Y_s}}ds\right)}^{2p}\right]$, we define for $\delta >0,$

$$I\left(y\right):=\left\{\begin{array}{cc}{\displaystyle \frac{y^{\delta +1}}{\delta (\delta +1)}}-{\displaystyle \frac{y}{\delta }}-{\displaystyle \frac{\delta }{\delta +1}},\hfill & y\ge 1,\hfill \\ ylny-y,\hfill & 0<y<1.\hfill \end{array}\right.$$

Then, $I(·)\in {\mathcal{C}}^2\left((0,+\infty )\right)$. Applying Itô’s formula to $I\left(Y_t\right)$ and noting the fact that ${\left(lnx\right)}^2{\mathbb{1}}_{0<x\le 1}\le {\displaystyle \frac{1}{x}}{\mathbb{1}}_{0<x\le 1}$, we obtain

$$\begin{array}{cc}\hfill \mathbb{E}\left[{\left({\int }_0^T{\displaystyle \frac{{\left|Z_s\right|}^2}{Y_s}}ds\right)}^{2p}\right]\le & C\mathbb{E}\left[A_T+{\left({\int }_0^T{a}_{s}ds\right)}^{2p(2+\delta )}+\underset{0\le s\le T}{sup}{\left(Y_s\right)}^{4p(2+\delta )}+{\left({\int }_0^T{\left|Z_s\right|}^{2}ds\right)}^{2p}\right]\hfill \\ \hfill \le & C\mathbb{E}\left[A_T^4{\left(1+{\xi }^{2+\delta }+{\int }_0^T{a}_s^{2+\delta }ds\right)}^{4p}\right].\hfill \end{array}$$

Analogously,

$$\underset{n\ge 1}{sup}\mathbb{E}\left[{\left({\int }_0^T{\displaystyle \frac{{\left|Z_s^n\right|}^2}{Y_s^n}}ds\right)}^{2p}\right]\le C\underset{n\ge 1}{sup}\mathbb{E}\left[{\left(A_T^n\right)}^4{\left(1+{\left({\xi }_n\right)}^{2+\delta }+{\int }_0^T{\left(a_s^n\right)}^{2+\delta }ds\right)}^{4p}\right].$$

Therefore,

$$\begin{array}{cc}& \mathbb{E}\left[{\left({\int }_0^{T}f(s,Y_s{Z}_s)+f_n(s,Y_s^n,Z_s^n)ds\right)}^{2p}\right]\hfill \\ \hfill \le & C\mathbb{E}\left[A_T^4{\left(1+{\xi }^{2+\delta }+{\int }_0^T{a}_s^{2+\delta }ds\right)}^{4p}\right]+C\underset{n\ge 1}{sup}\mathbb{E}\left[{\left(A_T^n\right)}^4{\left(1+{\left({\xi }_n\right)}^{2+\delta }+{\int }_0^T{\left(a_s^n\right)}^{2+\delta }ds\right)}^{4p}\right].\hfill \end{array}$$

We complete the proof. □

Proof of Theorem 3. 

First, we need to verify that ${\left\{\underset{0\le t\le T}{sup}{|Y_t^n-Y_t|}^{4p}\right\}}_{n\ge 1}$ is uniformly integrable. According to Corollary 1,

$$\underset{n\ge 1}{sup}\mathbb{E}\left[\underset{t\in [0,T]}{sup}{|Y_t^n-Y_t|}^{4p(2+\delta )}\right]\le C\underset{n\ge 1}{sup}\mathbb{E}\left[\underset{t\in [0,T]}{sup}{\left|Y_t^n\right|}^{4p(2+\delta )}\right]+C\mathbb{E}\left[\underset{t\in [0,T]}{sup}{\left|Y_t\right|}^{4p(2+\delta )}\right]<\infty ;$$

thus, ${\left\{\underset{t\in [0,T]}{sup}{|Y_t^n-Y_t|}^{4p}\right\}}_{n\ge 1}$ is uniformly integrable by de la Vallée Poussin’s criterion.

Our task now is to prove that $\mathrm{as}n\to \infty ,{sup}_{0\le t\le T}|Y_t^n-Y_t|\to 0\mathrm{in}\mathbb{P}.$ For any given $0\le s\le T,\theta \in (0,1)$ and $n\ge 1,$ we define the following notation:

$$\begin{array}{cc}& {\mathsf{\Delta }}_{\theta }\xi :={\displaystyle \frac{\xi -\theta {\xi }_n}{1-\theta }},{\mathsf{\Delta }}_{\theta }{Y}_s:={\displaystyle \frac{Y_s-\theta Y_s^n}{1-\theta }},{\mathsf{\Delta }}_{\theta }{Z}_s:={\displaystyle \frac{Z_s-\theta Z_s^n}{1-\theta }},\hfill \\ & \mathsf{\Delta }{f}_s:=f(s,Y_s,Z_s)-f_n(s,Y_s,Z_s),\hfill \\ & R_s:={\displaystyle \frac{{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_s\right)}^{+}\right)}^{2+\delta }}{2+\delta }}+{\int }_0^s{\displaystyle \frac{{\left(a_r^n\right)}^{2+\delta }}{2+\delta }}dr,\hfill \\ & D_s^n:=exp\left\{{\displaystyle \frac{p}{p-1}}{\int }_0^s{\left({\gamma }_r^n\right)}^{2}dr+p(2+\delta ){\int }_0^s{b}_r^{n}dr+(1+\delta )sp\right\}.\hfill \end{array}$$

Applying the Tanaka–Meyer–Itô formula and Lemma 3.1 in [24], we obtain from assumptions on $f_n$ and Young’s inequality that

$$\begin{array}{cc}\hfill R_t^p{D}_t^n& \le R_T^p{D}_T^n+{\int }_t^{T}p{D}_s^n{R}_s^{p-1}{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_s\right)}^{+}\right)}^{1+\delta }{\displaystyle \frac{\left|\mathsf{\Delta }{f}_s\right|}{1-\theta }}\hfill \\ & -{\displaystyle \frac{p(p-1)}{4}}{D}_s^n{\mathbb{1}}_{R_s>0}{R}_s^{p-2}{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_s\right)}^{+}\right)}^{2+2\delta }{\left|{\mathsf{\Delta }}_{\theta }{Z}_s\right|}^{2}ds\hfill \\ & +{\int }_t^{T}p{D}_s^n{R}_s^{p-1}{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_s\right)}^{+}\right)}^{1+\delta }{\mathsf{\Delta }}_{\theta }{Z}_{s}d{W}_s,0\le t\le T.\hfill \end{array}$$

Let

$$N_t:={\int }_0^{t}p{D}_s^n{R}_s^{p-1}{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_s\right)}^{+}\right)}^{1+\delta }{\mathsf{\Delta }}_{\theta }{Z}_{s}d{W}_s.$$

Due to (15) and the integrability properties of $Y,Z$, and $Z^n$, we derive that N is a martingale. Thus,

$$\begin{array}{cc}& {\displaystyle \frac{p(p-1)}{4}}\mathbb{E}\left[{\int }_t^T{D}_s^n{\mathbb{1}}_{R_s>0}{R}_s^{p-2}{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_s\right)}^{+}\right)}^{2+2\delta }{\left|{\mathsf{\Delta }}_{\theta }{Z}_s\right|}^{2}ds\right]\hfill \\ \hfill \le & \mathbb{E}\left[R_T^p{D}_T^n\right]+\mathbb{E}\left[{\int }_t^{T}p{D}_s^n{R}_s^{p-1}{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_s\right)}^{+}\right)}^{1+\delta }{\displaystyle \frac{\left|\mathsf{\Delta }{f}_s\right|}{1-\theta }}ds\right],\hfill \end{array}$$

and it follows from the BDG inequality that

$$\begin{array}{c}\hfill \mathbb{E}\left[\underset{0\le t\le T}{sup}{R}_t^p{D}_t^n\right]\le \mathbb{E}\left[R_T^p{D}_T^n\right]+\mathbb{E}\left[{\int }_0^{T}p{D}_s^n{R}_s^{p-1}{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_s\right)}^{+}\right)}^{1+\delta }{\displaystyle \frac{\left|\mathsf{\Delta }{f}_s\right|}{1-\theta }}ds\right]+c\left(1\right)\mathbb{E}\left[{\langle N\rangle }_T^{1/2}\right].\end{array}$$

(16)

Then, Young’s inequality yields that

$$\begin{array}{cc}\hfill & c\left(1\right)\mathbb{E}\left[{\langle N\rangle }_T^{1/2}\right]\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}\mathbb{E}\left[\underset{0\le t\le T}{sup}{R}_t^p{D}_t^n\right]+{\displaystyle \frac{2p{c}^2\left(1\right)}{p-1}}\left\{\mathbb{E}\left[R_T^p{D}_T^n\right]+\mathbb{E}\left[{\int }_0^{T}p{D}_s^n{R}_s^{p-1}{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_s\right)}^{+}\right)}^{1+\delta }{\displaystyle \frac{\left|\mathsf{\Delta }{f}_s\right|}{1-\theta }}ds\right]\right\}.\hfill \end{array}$$

(17)

By substituting (17) to (16) and denoting $C_1:=2(1+{\displaystyle \frac{2p{c}^2\left(1\right)}{p-1}})$, we have

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}{R}_t^p{D}_t^n\right]\le C_1\mathbb{E}\left[R_T^p{D}_T^n+{\int }_0^{T}p{D}_s^n{R}_s^{p-1}{\left({\left({\mathsf{\Delta }}_{\theta }{Y}_s\right)}^{+}\right)}^{1+\delta }{\displaystyle \frac{\left|\mathsf{\Delta }{f}_s\right|}{1-\theta }}ds\right].$$

(18)

Analogously, employing Young’s inequality to the last term in (18) yields

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}{R}_t^p{D}_t^n\right]\le C\mathbb{E}\left[R_T^p{D}_T^n+{\left({\int }_0^T{\left(D_s^n\right)}^{{\displaystyle \frac{1}{p(2+\delta )}}}{\displaystyle \frac{\left|\mathsf{\Delta }{f}_s\right|}{1-\theta }}ds\right)}^{p(2+\delta )}\right].$$

Since ${sup}_{n\ge 1}\mathbb{E}\left[{\left(D_T^n\right)}^4{\left(1+{\int }_0^T{\left(a_s^n\right)}^{2+\delta }ds\right)}^{4p}\right]<+\infty$, we deduce from Jensen’s inequality and Young’s inequality that

$$\begin{array}{cc}& \mathbb{E}\left[\underset{0\le t\le T}{sup}{\left(Y_t-\theta Y_t^n\right)}^{+}\right]\hfill \\ \hfill \le & (1-\theta )C{\left\{1+\mathbb{E}\left[{\left({\displaystyle \frac{{(\xi -\theta {\xi }_n)}^{+}}{1-\theta }}\right)}^{2p(2+\delta )}+{\left({\int }_0^T{\displaystyle \frac{\left|\mathsf{\Delta }{f}_s\right|}{1-\theta }}ds\right)}^{2p(2+\delta )}\right]\right\}}^{{\displaystyle \frac{1}{p(2+\delta )}}}.\hfill \end{array}$$

Interchanging $Y^n$ and Y and similar deductions then leads to

$$\begin{array}{cc}& \mathbb{E}\left[\underset{0\le t\le T}{sup}{\left(Y_t^n-\theta Y_t\right)}^{+}\right]\hfill \\ \hfill \le & (1-\theta )C{\left\{1+\mathbb{E}\left[{\left({\displaystyle \frac{{({\xi }_n-\theta \xi )}^{+}}{1-\theta }}\right)}^{2p(2+\delta )}+{\left({\int }_0^T{\displaystyle \frac{\left|\mathsf{\Delta }{f}_s\right|}{1-\theta }}ds\right)}^{2p(2+\delta )}\right]\right\}}^{{\displaystyle \frac{1}{p(2+\delta )}}}.\hfill \end{array}$$

Finally, it can be easily checked that

$$\begin{array}{cc}& \mathbb{E}\left[\underset{0\le t\le T}{sup}|Y_t^n-Y_t|\right]\hfill \\ \hfill \le & (1-\theta )C\left\{1+\mathbb{E}\left[{\left({\int }_0^T{\displaystyle \frac{\left|\mathsf{\Delta }{f}_s\right|}{1-\theta }}ds\right)}^{2p(2+\delta )}+{\left({\displaystyle \frac{|{\xi }_n-\xi |}{1-\theta }}\right)}^{2p(2+\delta )}+{\xi }^{2p(2+\delta )}+{\xi }_n^{2p(2+\delta )}\right]\right\}.\hfill \end{array}$$

Recall that $|\xi -{\xi }_n|$ and ${\int }_0^T\left|\mathsf{\Delta }{f}_s\right|ds$ converge to 0 in $L^{2p(2+\delta )}$ space as $n\to \infty ;$ consequently,

$$\underset{n\to \infty }{lim}\mathbb{E}\left[\underset{0\le t\le T}{sup}|Y_t^n-Y_t|\right]\le (1-\theta )C\left\{1+\mathbb{E}\left[{\xi }^{2p(2+\delta )}\right]\right\}.$$

By sending $\theta \to 1,$ we have ${sup}_{0\le t\le T}|Y_t^n-Y_t|$ converging to 0 in probability. Thus, we arrive at the conclusion that $\underset{n\to \infty }{lim}\mathbb{E}\left[{sup}_{0\le t\le T}{|Y_t^n-Y_t|}^{4p}\right]=0$.

Now, we prove that $\underset{n\to \infty }{lim}\mathbb{E}\left[{\left({\int }_0^T{|Z_t^n-Z_t|}^{2}dt\right)}^p\right]=0$. Applying Itô’s formula to ${|Y_t^n-Y_t|}^2$, we have

$$\begin{array}{cc}\hfill {\int }_0^T{|Z_t^n-Z_t|}^{2}dt\le & {\left({\xi }_n-\xi \right)}^2+2\underset{0\le t\le T}{sup}|Y_t^n-Y_t|{\int }_0^T\left(f(s,Y_s{Z}_s)+f_n(s,Y_s^n,Z_s^n)\right)ds\hfill \\ \hfill -& {\int }_0^{T}2\left(Y_s^n-Y_s\right)\left(Z_s^n-Z_s\right)d{W}_s.\hfill \end{array}$$

By Hölder’s inequality and the BDG inequality, we have

$$\begin{array}{cc}& \mathbb{E}\left[{\left({\int }_0^T{|Z_t-Z_t^n|}^{2}dt\right)}^p\right]\hfill \\ \hfill \le & C\left\{\mathbb{E}\left[|{\xi }_n{-\xi |}^{2p}\right]+\mathbb{E}\left[\underset{0\le t\le T}{sup}{|Y_t^n-Y_t|}^{2p}\right]\right.\hfill \\ \hfill +& \left.{\left(\mathbb{E}\left[\underset{0\le t\le T}{sup}{|Y_t^n-Y_t|}^{2p}\right]\right)}^{{\displaystyle \frac{1}{2}}}{\left(\mathbb{E}\left[{\left({\int }_0^T\left(f(s,Y_s{Z}_s)+f_n(s,Y_s^n,Z_s^n)\right)ds\right)}^{2p}\right]\right)}^{{\displaystyle \frac{1}{2}}}\right\}.\hfill \end{array}$$

From Lemma 3 and (15), it follows that

$$\begin{array}{cc}& \mathbb{E}\left[{\left({\int }_0^T{|Z_t-Z_t^n|}^{2}dt\right)}^p\right]\hfill \\ \hfill \le & C\left\{\mathbb{E}\left[|{\xi }_n{-\xi |}^{2p}\right]+{\left(\mathbb{E}\left[\underset{0\le t\le T}{sup}{|Y_t^n-Y_t|}^{2p}\right]\right)}^{{\displaystyle \frac{1}{2}}}+\mathbb{E}\left[\underset{0\le t\le T}{sup}{|Y_t^n-Y_t|}^{2p}\right]\right\}.\hfill \end{array}$$

Note that the generic constant C is independent of $n.$ Consequently, we infer that $Z^n$ converges to Z in ${\mathcal{M}}^{2p}$. Now, the proof is complete. □

4.2. Application to Singular Quadratic PDEs

In this subsection, we investigate singular quadratic PDEs associated with the BSDEs in our study. To be specific, we prove the related nonlinear Feynman–Kac formula, as well as the uniqueness of viscosity solutions. Consider the semi-linear PDE for $z\in {\mathbb{R}}^n,0\le s\le T,$

$$-\mathcal{L}u(s,z)-{\partial }_{s}u(s,z)-f\left(s,z,u(s,z),\left({\left({\partial }_{z}u\right)}^{\top }\sigma \right)(s,z)\right)=0,u(T,·)=\mathsf{\Psi }(·),$$

(19)

in which $\mathcal{L}$ refers to the infinitesimal generator of $X^{t^{\prime },x^{\prime }}$, the solution to the following SDE for each fixed $(t^{\prime },x^{\prime })\in [0,T]\times {\mathbb{R}}^n,$

$$X_t=x^{\prime }+{\int }_{t^{\prime }}^{t}B\left(r,X_r\right)dr+{\int }_{t^{\prime }}^t\sigma \left(r,X_r\right)d{W}_r,t^{\prime }\le t\le T,X_t=x^{\prime },0\le t\le t^{\prime }.$$

(20)

In addition, let $(Y^{t^{\prime },x^{\prime }},Z^{t^{\prime },x^{\prime }})$ solve the BSDE

$$Y_s=\mathsf{\Psi }\left(X_T^{t^{\prime },x^{\prime }}\right)+{\int }_s^{T}f(r,X_r^{t^{\prime },x^{\prime }},Y_r,Z_r)dr-{\int }_s^T{Z}_{r}d{W}_r,0\le s\le T.$$

(21)

Our first goal is to verify that (19) has a viscosity solution defined by

$$u(s,x):=Y_s^{s,x},s\in 0,\le s\le T,x\in {\mathbb{R}}^n.$$

(22)

First, we retrospect the definition of viscosity solutions to (19).

Definition 1. 

A function u that is continuous on $[0,T]\times {\mathbb{R}}^n$ is defined as a viscosity subsolution (resp. supersolution) to (19) if for each $(s,x)\in \left[0,T\right)\times {\mathbb{R}}^n$ and $\varpi \in {\mathcal{C}}^{1,2}([0,T]\times {\mathbb{R}}^n)$ satisfying $(s,x)$ is a local minimum (resp. maximum) of $\varpi -u$,

$$-{\partial }_s\varpi (s,x)-\mathcal{L}\varpi (s,x)-f\left(s,x,\varpi (s,x),\left({\left({\partial }_x\varpi \right)}^{\top }\sigma \right)(s,x)\right)\le 0\left(\mathrm{resp}.\ge 0\right)$$

as well as

$$\forall x\in {\mathbb{R}}^n,u(T,x)\le \mathsf{\Psi }\left(x\right)\left(\mathrm{resp}.\ge \right).$$

Furthermore, u is termed as a viscosity solution if it is a viscosity supersolution as well as a viscosity subsolution.

Now, we give our assumptions.

(A.1) $B:[0,T]\times {\mathbb{R}}^n\mapsto {\mathbb{R}}^n$ as well as $\sigma :[0,T]\times {\mathbb{R}}^n\mapsto {\mathbb{R}}^{n\times d}$ are continuous functions fulfilling for all $(s,z,z^1,z^2)\in [0,T]\times {\mathbb{R}}^n\times {\mathbb{R}}^n\times {\mathbb{R}}^n,$

$$\left|B\right(s,0\left)\right|+\left|\sigma \right(s,z\left)\right|\le J$$

and

$$|B(s,z^1)-B(s,z^2)|+|\sigma (s,z^1)-\sigma (s,z^2)|\le J|z^1-z^2|$$

are met for some constant $J>0$.

(A.2) $f:[0,T]\times {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^{1\times d}\mapsto \mathbb{R}$ and $\mathsf{\Psi }:{\mathbb{R}}^n\mapsto (0,+\infty )$ are continuous functions fulfilling for all $(s,x,y,z)\in [0,T]\times {\mathbb{R}}^n\times (0,+\infty )\times {\mathbb{R}}^{1\times d}$,

$$\begin{array}{cc}& (y,z)\mapsto f(s,y,z)\mathrm{is}\mathrm{convex},\hfill \\ & \mathsf{\Psi }\left(x\right)\le k\left(1+{\left|x\right|}^q\right),\hfill \\ \hfill \mathrm{and}& 0\le f(s,x,y,z)\le k\left(1+{\left|x\right|}^q+y+\left|z\right|\right)+{\displaystyle \frac{\delta {\left|z\right|}^2}{2y}}\hfill \end{array}$$

are met for some constants $0\le \delta \ne 1,q\ge 1$ and $k\ge 0.$

Given (A.1) and constants $p\ge 1,t^{\prime }\in [0,T]$ and $x^{\prime }\in {\mathbb{R}}^n$, there exists a unique solution $X^{t^{\prime },x^{\prime }}\in {\mathcal{S}}^p\left({\mathbb{R}}^n\right)$ to SDE (20). Consequently, we infer from (A.2) together with Corollary 1 that BSDE (21) admits only one solution, $(Y,Z)\in {\mathcal{S}}^p\times {\mathcal{M}}^p,\forall p\ge 1.$

Theorem 4. 

Assuming that conditions (A.1) and (A.2) are met, the function u defined in (22) is a viscosity solution for (19) and fulfills the following inequality:

$$0<u(t^{\prime },x^{\prime })\le C\left(1+{\left|x^{\prime }\right|}^q\right),\forall t^{\prime }\in [0,T],x^{\prime }\in {\mathbb{R}}^n.$$

(23)

Proof. 

Let us first prove the continuity of the map $(t,x)\mapsto u(t,x)$. Consider a sequence ${(t_n,x_n)}_{n\ge 1}$ converging to $(t,x)$ as $n\to +\infty .$ We begin by observing that

$$\forall p>0,\underset{n\to \infty }{lim}\mathbb{E}\left[{|\mathsf{\Psi }\left(X_T^{t_n,x_n}\right)-\mathsf{\Psi }\left(X_T^{t,x}\right)|}^p\right]=0,$$

which follows directly from $\underset{n\to \infty }{lim}\mathbb{E}\left[{|X_T^{t_n,x_n}-X_T^{t,x}|}^p\right]=0,\forall p>0$ and the Vitali convergence theorem. Now, we establish that for any $p>0,$

$$\underset{n\to \infty }{lim}\mathbb{E}\left[{\left({\int }_0^T\left|f(s,X_s^{t_n,x_n},Y_s^{t,x},Z_s^{t,x})-f(s,X_s^{t,x},Y_s^{t,x},Z_s^{t,x})\right|ds\right)}^p\right]=0.$$

(24)

Since f is continuous and $\mathbb{E}\left[\underset{n\ge 1}{sup}\underset{0\le s\le T}{sup}{\left|X_s^{t_n,x_n}\right|}^p\right]\le C{e}^{Cp\left|x\right|},\forall p>0$ (see [26]), the dominated convergence theorem gives that

$$\underset{n\to \infty }{lim}{\int }_0^T\left|f(s,X_s^{t_n,x_n},Y_s^{t,x},Z_s^{t,x})-f(s,X_s^{t,x},Y_s^{t,x},Z_s^{t,x})\right|ds=0,\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}..$$

Note the fact that

$$\begin{array}{cc}& \underset{n\ge 1}{sup}\mathbb{E}\left[{\left({\int }_0^T\left|f(s,X_s^{t_n,x_n},Y_s^{t,x},Z_s^{t,x})-f(s,X_s^{t,x},Y_s^{t,x},Z_s^{t,x})\right|ds\right)}^p\right]\hfill \\ \hfill \le & C\left\{1+{\left|x\right|}^{q(p\vee 2)}+\underset{n\ge 1}{sup}{\left|x_n\right|}^{q(p\vee 2)}+\mathbb{E}\left[{\left({\int }_0^T{\displaystyle \frac{\delta {\left|Z_s^{t,x}\right|}^2}{Y_s^{t,x}}}ds\right)}^p\right]+\mathbb{E}\left[\underset{0\le s\le T}{sup}{\left(Y_s^{t,x}\right)}^{p(2+\delta )}\right]\right\}.\hfill \end{array}$$

From Lemma 3, we obtain that for any $p>0$ and $\delta >0,$

$$\mathbb{E}\left[{\left({\int }_0^T{\displaystyle \frac{{\left|Z_s^{t,x}\right|}^2}{Y_s^{t,x}}}ds\right)}^p\right]\le C\left\{1+\mathbb{E}\left[\underset{0\le s\le T}{sup}{\left(Y_s^{t,x}\right)}^{2p(2+\delta )}\right]+\mathbb{E}\left[{\left({\int }_0^T{\left|Z_s^{t,x}\right|}^{2}ds\right)}^p\right]\right\}.$$

Thus, (24) follows immediately from the Vitali convergence theorem. The integrable condition (15) still holds, and the continuity of u is derived from Theorem 3. The growth of u follows routinely from (A.2) and Corollary 1. Finally, we prove that u defined in (22) is a viscosity solution to PDE (19) via a classical approximation technique together with the stability results of viscosity solutions (see [26,27]). For $n\ge 1,(t^{\prime },x^{\prime })\in [0,T]\times {\mathbb{R}}^n$, $(t,x,y,z)\in [0,T]\times {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^{1\times d}$, we define

$${\sigma }_n:=inf\left\{s>0:{\left|X_s^{t^{\prime },x^{\prime }}\right|}^q>n\right\}\wedge T,\mathrm{and}$$

$$f^n\left(t,x,y,z\right):={\mathbb{1}}_{t\le {\sigma }_n}inf\left\{f\left(t,x,p,q\right)\vee 0+n|p-y|+n|q-z|:(p,q)\in \mathbb{Q}\times {\mathbb{Q}}^{1\times d}\right\}.$$

By [28], each $f^n$ is uniformly Lischitz continuous in $(y,z)$, and ${\left\{f^n\right\}}_{n\ge 1}$ is increasing and converges uniformly on compact sets to $f\vee 0.$ Moreover, for any $(t^{\prime },x^{\prime })\in [0,T]\times {\mathbb{R}}^n$ and $(t,x,y,z)\in [0,T]\times {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^{1\times d}$, (A.2) implies the following inequalities:

• $\left|f^n\left(t,X_t^{t^{\prime },x^{\prime }},y,z\right)\right|\le k(2+n)+n\left(1+\left|y\right|+\left|z\right|\right);$
• $0\le f^n\left(t,x,y,z\right)\le f\left(t,x,y,z\right)\vee 0;$
• If $y>0,0\le f^n\left(t,x,y,z\right)\le f\left(t,x,y,z\right)\le k\left(1+{\left|x\right|}^q+y+\left|z\right|\right)+{\displaystyle \frac{\delta {\left|z\right|}^2}{2y}}.$

For any fixed $l>1$, we set ${\xi }^l:=\left({\displaystyle \frac{1}{l}}\right)\vee \mathsf{\Psi }\left(X_T^{t^{\prime },x^{\prime }}\right)$, and we consider the BSDE,

$$y_t^{n,l}={\xi }^l+{\int }_t^T{f}^n\left(s,X_s^{t^{\prime },x^{\prime }},y_s^{n,l},z_s^{n,l}\right)ds-{\int }_t^T{z}_s^{n,l}d{W}_s,t\in [0,T].$$

(25)

By [25], (A.1), and (A.2), this BSDE admits a unique solution $(y^{n,l},z^{n,l})\in {\mathcal{S}}^p\times {\mathcal{M}}^p,\forall p\ge 1.$ From Remark 1, we have that for any $l,n>1,$

$${\mathbb{E}}_t\left[{\xi }^l\right]\le y_t^{n,l}\le {\overline{y}}_t\left({\xi }^l\right),$$

(26)

where ${\overline{y}}_t\left({\xi }^l\right)$ is defined in (13).

Additionally, ${\left\{y^{n,l}\right\}}_{n\ge 1}$ is increasing by the classical comparison theorem in [29]. For any $l>1$, we define $y_t^l:=\underset{n\ge 1}{sup}{y}_t^{n,l}$. Following [2], we will prove there exists an appropriate process $z^l$, such that $(y^l,z^l)$ solves the BSDE

$$y_t^l=\left({\displaystyle \frac{1}{l}}\right)\vee \mathsf{\Psi }\left(X_T^{t^{\prime },x^{\prime }}\right)+{\int }_t^{T}f\left(s,X_s^{t^{\prime },x^{\prime }},y_s^l,z_s^l\right)ds-{\int }_t^T{z}_s^{l}d{W}_s,t\in [0,T].$$

(27)

Then, it follows from Theorems 2 and 3 that $y_t^l$ decreasingly converges to the value process $Y_t$ of the solution to BSDE (21). By (13) and (26), it can be checked that $\underset{t\to T}{lim}{y}_t^l={\xi }^l.$ Thus, $\underset{t\to T}{lim}{y}_t^l=y_T^l.$ To apply the monotone stability theorem in [7], we define the stopping time for any $m>1,$

$${\tau }_m:=inf\left\{s>0:{\overline{y}}_s\left({\xi }^l\right)+{\left|X_s^{t^{\prime },x^{\prime }}\right|}^q>m\right\}\wedge T,$$

and the truncated function $\rho \left(y\right):={\displaystyle \frac{1}{l}}{\mathbb{1}}_{y<{\displaystyle \frac{1}{l}}}+y{\mathbb{1}}_{{\displaystyle \frac{1}{l}}\le y\le m}+m{\mathbb{1}}_{y>m},y\in \mathbb{R}.$ Then, $(y_t^{m,n,l},z_t^{m,n,l})$$:=(y_{t\wedge {\tau }_m}^{n,l},{\mathbb{1}}_{t\le {\tau }_m}{z}_t^{n,l})$ solves the BSDE,

$$y_t^{m,n,l}=y_{{\tau }_m}^{n,l}+{\int }_t^T{\mathbb{1}}_{s\le {\tau }_m}{f}^n\left(s,X_s^{t^{\prime },x^{\prime }},\rho \left(y_s^{m,n,l}\right),z_s^{m,n,l}\right)ds-{\int }_t^T{z}_s^{m,n,l}d{W}_s,t\in [0,T].$$

Since for any $(s,y,z)\in [0,T]\times \mathbb{R}\times {\mathbb{R}}^{1\times d},$

$$\begin{array}{cc}\hfill 0& \le {\mathbb{1}}_{s\le {\tau }_m}{f}^n\left(s,X_s^{t^{\prime },x^{\prime }},\rho \left(y\right),z\right)\hfill \\ & \le {\mathbb{1}}_{s\le {\tau }_m}f\left(s,X_s^{t^{\prime },x^{\prime }},\rho \left(y\right),z\right)\hfill \\ & \le {\mathbb{1}}_{s\le {\tau }_m}k(1+{\left|X_s^{t^{\prime },x^{\prime }}\right|}^q+\rho \left(y\right)+\left|z\right|)+{\displaystyle \frac{\delta {\left|z\right|}^2}{2\rho \left(y\right)}}\hfill \\ & \le k(1+m+m+{\displaystyle \frac{1}{2}})+{\displaystyle \frac{k+l\delta }{2}}{\left|z\right|}^2,\hfill \end{array}$$

${\mathbb{1}}_{s\le {\tau }_m}{f}^n\left(s,X_s^{t^{\prime },x^{\prime }},\rho \left(y\right),z\right)$ converges uniformly on compact sets to ${\mathbb{1}}_{s\le {\tau }_m}f\left(s,X_s^{t^{\prime },x^{\prime }},\rho \left(y\right),z\right)$ as $n\to \infty$, and $y_{{\tau }_m}^{n,l}({\displaystyle \frac{1}{l}}\le y_{{\tau }_m}^{n,l}\le m)$ increasingly converges to $\underset{n\ge 1}{sup}{y}_{{\tau }_m}^{n,l}=y_{{\tau }_m}^l$, ref. [7] gives that there exist processes $y_t^{m,l}:=\underset{n\ge 1}{sup}{y}_t^{m,n,l}=\underset{n\ge 1}{sup}{y}_{t\wedge {\tau }_m}^{n,l}=y_{t\wedge {\tau }_m}^l$ and $z_t^{m,l}:=\underset{n\to \infty }{lim}{z}_t^{m,n,l}$ (in ${\mathcal{M}}^2$), such that $\left(y_t^{m,l},z_t^{m,l}\right)$ solves the BSDE,

$$y_t^{m,l}=y_{{\tau }_m}^l+{\int }_t^T{\mathbb{1}}_{s\le {\tau }_m}f\left(s,X_s^{t^{\prime },x^{\prime }},\rho \left(y_s^{m,l}\right),z_s^{m,l}\right)ds-{\int }_t^T{z}_s^{m,l}d{W}_s,t\in [0,T].$$

Since $y_{t\wedge {\tau }_m}^{m+1,l}=y_{t\wedge {\tau }_m}^l=y_t^{m,l},z_t^{m+1,l}{\mathbb{1}}_{t\le {\tau }_m}=z_t^{m,l},$ we define

$$z_t^l:=z_t^{1,l}{\mathbb{1}}_{t\le {\tau }_1}+\sum _{m\ge 2}{z}_t^{m,l}{\mathbb{1}}_{{\tau }_{m-1}<t\le {\tau }_m}.$$

It is clear from [2] that $z^l\in {\mathcal{L}}^2.$ Note that for any $t\in [0,T],m,l>1,{\displaystyle \frac{1}{l}}\le y_{t\wedge {\tau }_m}^l\le m$; then, we arrive at

$$y_{t\wedge {\tau }_m}^l=y_{{\tau }_m}^l+{\int }_{t\wedge {\tau }_m}^{{\tau }_m}f\left(s,X_s^{t^{\prime },x^{\prime }},y_s^l,z_s^l\right)ds-{\int }_{t\wedge {\tau }_m}^{{\tau }_m}{z}_s^{l}d{W}_s,t\in [0,T].$$

Since $y_t^{m,l}$ is continuous on $[0,T]$ and ${\tau }_m\to T,$$y_t^l$ is continuous on $\left[0,\right.\left.T\right),$ we know that $\underset{t\to T}{lim}{y}_t^l=y_T^l$; thus, $y^l$ is continuous on $[0,T]$. When $m\to \infty ,$ we obtain $(y^l,z^l)$, which solves the BSDE (27). Moreover, in view of (26), Remark 1, and Theorem 1, we deduce that $(y^l,z^l)\in {\mathcal{S}}^p\times {\mathcal{M}}^p,\forall p\ge 1.$

For the PDE part, classical works [29,30] imply that for any $n,l>1,$$u^{n,l}(s,x):={\left(y_s^{n,l}\right)}^{s,x},s\in [0,T],x\in {\mathbb{R}}^n$ is a viscosity solution to the PDE,

$$-\mathcal{L}u(s,z)-{\partial }_{s}u(s,z)-f^n\left(s,z,u(s,z),\left({\left({\partial }_{z}u\right)}^{\top }\sigma \right)(s,z)\right)=0,u(T,·)=\left({\displaystyle \frac{1}{l}}\right)\vee \mathsf{\Psi }(·).$$

We know that $\left\{u^{n,l}\right\}$ increases pointwise to $u^l(s,x):={\left(y_s^l\right)}^{s,x}>0$ as $n\to \infty$, and Dini’s theorem ensures uniform convergence on compact sets of $[0,T]\times {\mathbb{R}}^n$. In addition, $f^n$ converges uniformly on compact sets to $0\vee f$. By (A.2) and the stability property of viscosity solutions (see, e.g., Theorem 1.7 in Chapter 5 of [31], or Proposition I.3 in [32]), $u^l$ is a viscosity solution to the PDE

$$-\mathcal{L}u(s,z)-{\partial }_{s}u(s,z)-f\left(s,z,u(s,z),\left({\left({\partial }_{z}u\right)}^{\top }\sigma \right)(s,z)\right)=0,u(T,·)=\left({\displaystyle \frac{1}{l}}\right)\vee \mathsf{\Psi }(·).$$

By Theorems 2 and 3, $\left\{u^l\right\}$ decreases pointwise to u defined in (22) as $l\to \infty$. Dini’s theorem again ensures that $\left\{u^l\right\}$ converges to u uniformly on compact sets of $[0,T]\times {\mathbb{R}}^n$. By the stability property of viscosity solutions again, we conclude that u defined in (22) is a viscosity solution to PDE (19). □

We further show the uniqueness of viscosity solutions to (19). Let us consider the following additional assumption.

(A.3) For any given $\tau >0$, there exists a function $m_{\tau }:\left.\left[0,+\infty \right.\right)\to \left.\left[0,+\infty \right.\right)$ satisfying $\underset{x\to 0}{lim}{m}_{\tau }\left(x\right)=0$. Moreover, assume that for any $\left|x^1\right|,\left|x^2\right|,\left|y\right|\le \tau ,z\in {\mathbb{R}}^{1\times d},$ and $0\le s\le T,$

$$|f(s,x^1,y,z)-f(s,x^2,y,z)|\le m_{\tau }\left(|x^1-x^2|\left(\left|z\right|+1\right)\right).$$

Theorem 5 (Uniqueness). 

PDE (19) has, at most, one viscosity solution that satisfies (23), provided that (A.1), (A.2), and (A.3) are met. In particular, for PDE (19) the unique viscosity solution fulfilling (23) is u defined in (22).

Our proof of Theorem 5 is in the spirit of [30]. Let us first present the following lemmas, which utilize the same notation as in the proof of Lemmas 3.7 and 3.8 in [30].

Lemma 4. 

Let u be a viscosity subsolution, let κ be a viscosity supersolution of PDE (19), and let both u and κ satisfy (23). Then, $h:=u-\kappa$ is a viscosity subsolution of

$$\left(-{\partial }_{t}h-\mathcal{L}h\right)h^{+}-k{h}^2-k\left|{\left({\partial }_{x}h\right)}^{\top }\sigma \right|\left|h\right|-\delta {\left|{\left({\partial }_{x}h\right)}^{\top }\sigma \right|}^2=0,(t,x)\in \left[0,\right.\left.T\right)\times {\mathbb{R}}^n,$$

(28)

where k and δ are constants in (A.2).

Proof. 

Let $(s_0,w_0)\in \left[0,\right.\left.T\right)\times {\mathbb{R}}^n$ as well as $\phi \in {\mathcal{C}}^{1,2}\left(\left[0,T\right]\times {\mathbb{R}}^n\right)$ satisfy that there is a solid neighborhood of $(s_0,w_0)$ denoted by $U(s_0,w_0)$, such that $0=h(s_0,w_0)-\phi (s_0,w_0)>h(s,w)-\phi (s,w)$ for each $(s,w)\in U(s_0,w_0)\setminus \left\{(s_0,w_0)\right\}$. Take a compact set $\mathcal{T}\times B\subset U(s_0,w_0)$, such that $(s_0,w_0)\in \mathcal{T}\times B$. Define a function ${\psi }_{\epsilon }:(0,T)\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}$,

$${\psi }_{\epsilon }(s,w,z):=u(s,w)-\kappa (s,z)-{\displaystyle \frac{{|w-z|}^2}{2\epsilon }}-\phi (s,w),$$

in which $\epsilon >0$ tends to zero. Owing to the upper semicontinuity of ${\psi }_{\epsilon }$, the maximum of ${\psi }_{\epsilon }$ on $\mathcal{T}\times B\times B$ is obtained at a point $(s_{\epsilon },w_{\epsilon },z_{\epsilon })\in \mathcal{T}\times B\times B$. It is not hard to derive that

(i)

$(s_{\epsilon },w_{\epsilon },z_{\epsilon })\to (s_0,w_0,w_0)$ as $\epsilon \to 0$;

(ii)

$\underset{\epsilon \to 0}{lim}{\displaystyle \frac{{|w_{\epsilon }-z_{\epsilon }|}^2}{\epsilon }}=0.$

Indeed, since

$$u(s_{\epsilon },w_{\epsilon })-\kappa (s_{\epsilon },z_{\epsilon })-{\displaystyle \frac{{|w_{\epsilon }-z_{\epsilon }|}^2}{2\epsilon }}-\phi (s_{\epsilon },w_{\epsilon })\ge u(s_0,w_0)-\kappa (s_0,w_0)-\phi (s_0,w_0),$$

it follows that ${|w_{\epsilon }-z_{\epsilon }|}^2\le 4\epsilon \underset{(s,w,z)\in \mathcal{T}\times B\times B}{max}|u(s,w)-\kappa (s,z)-\phi (s,w)|.$ Sending $\epsilon \to 0,$ we obtain that $w_{\epsilon }$ and $z_{\epsilon }$ converge to the same point. Let us assume $(s_{\epsilon },w_{\epsilon },z_{\epsilon })$ converges to $(\tilde{s},\tilde{w},\tilde{w})\in \mathcal{T}\times B\times B$ as $\epsilon \to 0$. Then, we have

$$u(\tilde{s},\tilde{w})-\kappa (\tilde{s},\tilde{w})-\underset{\epsilon \to 0}{lim}{\displaystyle \frac{{|w_{\epsilon }-z_{\epsilon }|}^2}{2\epsilon }}-\phi (\tilde{s},\tilde{w})\ge u(s_0,w_0)-\kappa (s_0,w_0)-\phi (s_0,w_0).$$

Note the fact that $h-\phi$ achieves its strict maximum in $U(s_0,w_0)$ at $(s_0,w_0)$; the previous inequality implies that both (i) and (ii) hold.

It follows from Theorem 8.3 in [20] that there exist two triplets, $(0,p,X)\in {\overline{D}}^{2,+}(u-\phi )(s_{\epsilon },w_{\epsilon })$ and $(0,-p,-Z)\in {\overline{D}}^{2,+}(-\kappa )(s_{\epsilon },z_{\epsilon })$, such that

(1)

$p={\displaystyle \frac{w_{\epsilon }-z_{\epsilon }}{\epsilon }}$;

(2)

$-{\displaystyle \frac{3}{\epsilon }}\left(\begin{array}{cc}I& 0\\ 0& I\end{array}\right)\le \left(\begin{array}{cc}X& 0\\ 0& -Z\end{array}\right)\le {\displaystyle \frac{3}{\epsilon }}\left(\begin{array}{cc}I& -I\\ -I& I\end{array}\right).$

It follows that $({\phi }_s(s_{\epsilon },w_{\epsilon }),p+{\partial }_w\phi (s_{\epsilon },w_{\epsilon }),X+D^2\phi (s_{\epsilon },w_{\epsilon }))\in {\overline{D}}^{2,+}u(s_{\epsilon },w_{\epsilon })$ and $(0,p,Z)\in {\overline{D}}^{2,-}\kappa (s_{\epsilon },z_{\epsilon })$. Since u and $\kappa$ are, respectively, a viscosity subsolution and a viscosity supersolution of (19) satisfying (23), we deduce that

$$\begin{array}{cc}\hfill -& {\phi }_s(s_{\epsilon },w_{\epsilon })-{\displaystyle \frac{1}{2}}tr\left(\left(\sigma {\sigma }^{\top }\right)(s_{\epsilon },w_{\epsilon })X\right)-B^{\top }(s_{\epsilon },w_{\epsilon })p\hfill \\ \hfill -& \mathcal{L}\phi (s_{\epsilon },w_{\epsilon })-f\left(s_{\epsilon },w_{\epsilon },u(s_{\epsilon },w_{\epsilon }),{(p+{\partial }_w\phi (s_{\epsilon },w_{\epsilon }))}^{\top }\sigma (s_{\epsilon },w_{\epsilon })\right)\le 0\hfill \end{array}$$

and

$$-{\displaystyle \frac{1}{2}}tr\left(\left(\sigma {\sigma }^{\top }\right)(s_{\epsilon },z_{\epsilon })Z\right)-B^{\top }(s_{\epsilon },z_{\epsilon })p-f\left(s_{\epsilon },z_{\epsilon },\kappa (s_{\epsilon },z_{\epsilon }),p^{\top }\sigma (s_{\epsilon },z_{\epsilon })\right)\ge 0.$$

If $(e_1,e_2,...,e_n)$ is an orthonormal basis of ${\mathbb{R}}^n$ then we have

$$\begin{array}{cc}& tr\left({\left(\sigma \sigma \right)}^{\top }(s_{\epsilon },w_{\epsilon })X\right)-tr\left(\left(\sigma {\sigma }^{\top }\right)(s_{\epsilon },z_{\epsilon })Z\right)\hfill \\ \hfill =& \sum _{i=1}^n\left[{\left(\sigma (s_{\epsilon },w_{\epsilon })e_i\right)}^{\top }\left(X\sigma (s_{\epsilon },w_{\epsilon })e_i\right)-{\left(\sigma (s_{\epsilon },z_{\epsilon })e_i\right)}^{\top }\left(Z\sigma (s_{\epsilon },z_{\epsilon })e_i\right)\right]\le {\displaystyle \frac{3J^2}{\epsilon }}{|w_{\epsilon }-z_{\epsilon }|}^2.\hfill \end{array}$$

It follows from assumptions (A.1) and (A.3) that $p^{\top }\left(B(s_{\epsilon },w_{\epsilon })-B(s_{\epsilon },z_{\epsilon })\right)\le {\displaystyle \frac{J}{\epsilon }}{|w_{\epsilon }-z_{\epsilon }|}^2$, and there exists a large enough constant R, such that $\left|w_{\epsilon }\right|,\left|z_{\epsilon }\right|,u(s_{\epsilon },w_{\epsilon }),\kappa (s_{\epsilon },z_{\epsilon })\le R.$

For any fixed $0<\theta <1$, subtracting the viscosity inequalities for u and $\kappa$ yields

$$\begin{array}{cc}& -{\phi }_s(s_{\epsilon },w_{\epsilon })-\mathcal{L}\phi (s_{\epsilon },w_{\epsilon })\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}tr\left(\left(\sigma {\sigma }^{\top }\right)(s_{\epsilon },w_{\epsilon })X\right)-{\displaystyle \frac{1}{2}}tr\left(\left(\sigma {\sigma }^{\top }\right)(s_{\epsilon },z_{\epsilon })Z\right)+p^{\top }\left(B(s_{\epsilon },w_{\epsilon })-B(s_{\epsilon },z_{\epsilon })\right)\hfill \\ \hfill +& f\left(s_{\epsilon },w_{\epsilon },u(s_{\epsilon },w_{\epsilon }),{(p+{\partial }_w\phi (s_{\epsilon },w_{\epsilon }))}^{\top }\sigma (s_{\epsilon },w_{\epsilon })\right)\hfill \\ \hfill -& f\left(s_{\epsilon },z_{\epsilon },u(s_{\epsilon },w_{\epsilon }),{(p+{\partial }_w\phi (s_{\epsilon },w_{\epsilon }))}^{\top }\sigma (s_{\epsilon },w_{\epsilon })\right)\hfill \\ \hfill +& f\left(s_{\epsilon },z_{\epsilon },u(s_{\epsilon },w_{\epsilon }),{(p+{\partial }_w\phi (s_{\epsilon },w_{\epsilon }))}^{\top }\sigma (s_{\epsilon },w_{\epsilon })\right)-f\left(s_{\epsilon },z_{\epsilon },\kappa (s_{\epsilon },z_{\epsilon }),p^{\top }\sigma (s_{\epsilon },z_{\epsilon })\right)\hfill \\ \hfill \le & {\displaystyle \frac{3J^2+2J}{2\epsilon }}{|w_{\epsilon }-z_{\epsilon }|}^2+m_R\left(|w_{\epsilon }-z_{\epsilon }|\left(1+\left|{\left(p+{\partial }_w\phi (s_{\epsilon },w_{\epsilon })\right)}^{\top }\sigma (s_{\epsilon },w_{\epsilon })\right|\right)\right)\hfill \\ \hfill +& (1-\theta )f\left(s_{\epsilon },z_{\epsilon },{\displaystyle \frac{u(s_{\epsilon },w_{\epsilon })-\theta \kappa (s_{\epsilon },z_{\epsilon })}{1-\theta }},{\displaystyle \frac{{(p+{\partial }_w\phi (s_{\epsilon },w_{\epsilon }))}^{\top }\sigma (s_{\epsilon },w_{\epsilon })-\theta p^{\top }\sigma (s_{\epsilon },z_{\epsilon })}{1-\theta }}\right).\hfill \end{array}$$

Multiplying both sides by ${\left(u(s_{\epsilon },w_{\epsilon })-\theta \kappa (s_{\epsilon },z_{\epsilon })\right)}^{+}$, we have

$$\begin{array}{cc}& \left(-{\phi }_s(s_{\epsilon },w_{\epsilon })-\mathcal{L}\phi (s_{\epsilon },w_{\epsilon })\right){\left(u(s_{\epsilon },w_{\epsilon })-\theta \kappa (s_{\epsilon },z_{\epsilon })\right)}^{+}\hfill \\ \hfill \le & \left[{\displaystyle \frac{3J^2+2J}{2\epsilon }}{|w_{\epsilon }-z_{\epsilon }|}^2+m_R\left(|w_{\epsilon }-z_{\epsilon }|\left(1+\left|{\left(p+{\partial }_w\phi (s_{\epsilon },w_{\epsilon })\right)}^{\top }\sigma (s_{\epsilon },w_{\epsilon })\right|\right)\right)\right.\hfill \\ \hfill +& k(1-\theta )(1+{\left|z_{\epsilon }\right|}^q)+k{\left(u(s_{\epsilon },w_{\epsilon })-\theta \kappa (s_{\epsilon },z_{\epsilon })\right)}^{+}+k\left|\left({\left({\partial }_w\phi \right)}^{\top }\sigma \right)(s_{\epsilon },w_{\epsilon })\right|\hfill \\ \hfill +& \left.k{\displaystyle \frac{|w_{\epsilon }-z_{\epsilon }|}{\epsilon }}|\sigma (s_{\epsilon },w_{\epsilon })-\theta \sigma (s_{\epsilon },z_{\epsilon })|\right]{\left(u(s_{\epsilon },w_{\epsilon })-\theta \kappa (s_{\epsilon },z_{\epsilon })\right)}^{+}\hfill \\ \hfill +& \delta {\left|\left({\left({\partial }_w\phi \right)}^{\top }\sigma \right)(s_{\epsilon },w_{\epsilon })\right|}^2+\delta {\displaystyle \frac{{|w_{\epsilon }-z_{\epsilon }|}^2}{{\epsilon }^2}}{|\sigma (s_{\epsilon },w_{\epsilon })-\theta \sigma (s_{\epsilon },z_{\epsilon })|}^2.\hfill \end{array}$$

By sending $\theta \to 1$, we derive

$$\begin{array}{cc}& \left(-{\phi }_s(s_{\epsilon },w_{\epsilon })-\mathcal{L}\phi (s_{\epsilon },w_{\epsilon })\right){\left(u(s_{\epsilon },w_{\epsilon })-\kappa (s_{\epsilon },z_{\epsilon })\right)}^{+}\hfill \\ \hfill \le & \left\{{\displaystyle \frac{3J^2+2J}{2\epsilon }}{|w_{\epsilon }-z_{\epsilon }|}^2+m_R\left(|w_{\epsilon }-z_{\epsilon }|\left(1+\left|{\left(p+{\partial }_w\phi (s_{\epsilon },w_{\epsilon })\right)}^{\top }\sigma (s_{\epsilon },w_{\epsilon })\right|\right)\right)\right.\hfill \\ \hfill +& k|u(s_{\epsilon },w_{\epsilon })-\kappa (s_{\epsilon },z_{\epsilon })|+k\left|\left({\left({\partial }_w\phi \right)}^{\top }\sigma \right)(s_{\epsilon },w_{\epsilon })\right|\hfill \\ \hfill +& \left.k{\displaystyle \frac{|w_{\epsilon }-z_{\epsilon }|}{\epsilon }}|\sigma (s_{\epsilon },w_{\epsilon })-\sigma (s_{\epsilon },z_{\epsilon })|\right\}|u(s_{\epsilon },w_{\epsilon })-\kappa (s_{\epsilon },z_{\epsilon })|\hfill \\ \hfill +& \delta {\left|\left({\left({\partial }_w\phi \right)}^{\top }\sigma \right)(s_{\epsilon },w_{\epsilon })\right|}^2+\delta {\displaystyle \frac{{|w_{\epsilon }-z_{\epsilon }|}^2}{{\epsilon }^2}}{|\sigma (s_{\epsilon },w_{\epsilon })-\sigma (s_{\epsilon },z_{\epsilon })|}^2.\hfill \end{array}$$

By sending $\epsilon \to 0$, we have

$$\begin{array}{cc}& \left(-{\phi }_s(s_0,w_0)-\mathcal{L}\phi (s_0,w_0)\right){\left(\phi (s_0,w_0)\right)}^{+}\hfill \\ \hfill \le & \left\{k\left|\phi (s_0,w_0)\right|+k\left|\left({\left({\partial }_w\phi \right)}^{\top }\sigma \right)(s_0,w_0)\right|\right\}\left|\phi (s_0,w_0)\right|+\delta {\left|\left({\left({\partial }_w\phi \right)}^{\top }\sigma \right)(s_0,w_0)\right|}^2.\hfill \end{array}$$

Therefore, h is a viscosity subsolution of (28). □

The following lemma aims to establish a viscosity supersolution of (28).

Lemma 5. 

For any $A>0$ and sufficiently large C, the function

$$\mathcal{X}(s,w):=e^{\left(A+C(T-s)\right)\left(1+{\left|w\right|}^2\right)},(s,w)\in [s_1,T]\times {\mathbb{R}}^n,$$

where $s_1:=T-{\displaystyle \frac{A}{C}},$ satisfies for all $(s,w)\in [s_1,T]\times {\mathbb{R}}^n,$

$$-{\partial }_s\mathcal{X}(s,w)-\mathcal{L}\mathcal{X}(s,w)-k\mathcal{X}(s,w)-k\left|\left({\sigma }^{\top }{\partial }_w\mathcal{X}\right)(s,w)\right|-\delta {\displaystyle \frac{{\left|\left({\sigma }^{\top }{\partial }_w\mathcal{X}\right)(s,w)\right|}^2}{\mathcal{X}(s,w)}}>0.$$

Proof. 

Given $(s,w)\in [s_1,T]\times {\mathbb{R}}^n$, it is clear that ${\partial }_s\mathcal{X}(s,w)=-C\mathcal{X}(s,w)(1+{\left|w\right|}^2)$, ${\partial }_w\mathcal{X}(s,w)=2\left(C(T-s)+A\right)\mathcal{X}(s,w)w$, and $D^2\mathcal{X}(s,w)=2\left(C(T-s)+A\right)\mathcal{X}(s,w)I+4{\left(C(T-s)+A\right)}^2\mathcal{X}(s,w)w{w}^{\top }.$ Consequently,

$$\left|{\partial }_w\mathcal{X}(s,w)\right|\le 4A\mathcal{X}(s,w)\left|w\right|\mathrm{and}\left|D^2\mathcal{X}(s,w)\right|\le 4A\mathcal{X}(s,w)+16A^2\mathcal{X}(s,w){\left|w\right|}^2.$$

Finally, we conclude that

$$\begin{array}{cc}& -{\partial }_s\mathcal{X}(s,w)-\mathcal{L}\mathcal{X}(s,w)-k\mathcal{X}(s,w)-k\left|\left({\sigma }^{\top }{\partial }_w\mathcal{X}\right)(s,w)\right|-\delta {\displaystyle \frac{{\left|\left({\sigma }^{\top }{\partial }_w\mathcal{X}\right)(s,w)\right|}^2}{\mathcal{X}(s,w)}}\hfill \\ \hfill \ge & \mathcal{X}(s,w)\left[C(1+{\left|w\right|}^2)-J^2\left(2A+8A^2{\left|w\right|}^2\right)-4AJ(1+k+\left|w\right|)\left|w\right|-k-\delta 16A^2{J}^2{\left|w\right|}^2\right].\hfill \end{array}$$

We complete the proof by taking C large enough. □

Proof of Theorem 5. 

Assume that $u,\kappa$ are two viscosity solutions to PDE (19) satisfying (23). Define $h:=u-\kappa .$ In view of Lemma 5 and any fixed $\alpha >0$, define $N(s,w):=\left(h(s,w)-\alpha \mathcal{X}(s,w)\right)e^{ks},(s,w)\in \left[s_1,T\right]\times {\mathbb{R}}^n$. Since for some $A>0$, it holds that for each $s\in [s_1,T],$

$$\underset{\left|w\right|\to \infty }{lim}\left|h(s,w)\right|e^{-A{\left|w\right|}^2}=0,$$

there exists a compact set $E\in {\mathbb{R}}^n$ satisfying that $N(s,w)<0,\forall (s,w)\in [s_1,T]\times E^c$. Since N is continuous, $\underset{(s,w)\in [s_1,T]\times E}{max}N(s,w)$ is achieved at some point $(\overline{s},\overline{w})$. Suppose that $N(\overline{s},\overline{w})>0$. Then, the maximum of N on $[s_1,T]\times {\mathbb{R}}^n$ is obtained at $(\overline{s},\overline{w})$ and $\overline{s}<T.$ Let us introduce a function,

$$\phi (s,w):=\alpha \mathcal{X}(s,w)+\left(h(\overline{s},\overline{w})-\alpha \mathcal{X}(\overline{s},\overline{w})\right)e^{k\left(\overline{s}-s\right)},(s,w)\in [s_1,T]\times {\mathbb{R}}^n.$$

Then, $\phi (\overline{s},\overline{w})>0$ and $\left(h-\phi \right)(s,w)\le 0=\left(h-\phi \right)(\overline{s},\overline{w})$ for all $(s,w)\in [s_1,T]\times {\mathbb{R}}^n$. According to Lemma 4,

$$\left(\left(-{\partial }_s\phi -\mathcal{L}\phi \right)\phi \right)(\overline{s},\overline{w})-k{\phi }^2(\overline{s},\overline{w})-k\left|\left({\sigma }^{\top }{\partial }_w\phi \right)(\overline{s},\overline{w})\right|\phi (\overline{s},\overline{w})-\delta {\left|\left({\sigma }^{\top }{\partial }_w\phi \right)(\overline{s},\overline{w})\right|}^2\le 0,$$

while it follows from the definition of $\phi$ and Lemma 5 that

$$\begin{array}{cc}& \left(\left(-{\partial }_s\phi -\mathcal{L}\phi \right)\phi \right)(\overline{s},\overline{w})-k{\phi }^2(\overline{s},\overline{w})-k\left|\left({\sigma }^{\top }{\partial }_w\phi \right)(\overline{s},\overline{w})\right|\phi (\overline{s},\overline{w})-\delta {\left|\left({\sigma }^{\top }{\partial }_w\phi \right)(\overline{s},\overline{w})\right|}^2\hfill \\ \hfill \ge & \alpha h(\overline{s},\overline{w})\left[\left(-{\partial }_s\mathcal{X}-\mathcal{L}\mathcal{X}-k\mathcal{X}\right)(\overline{s},\overline{w})-k\left|\left({\sigma }^{\top }{\partial }_w\mathcal{X}\right)(\overline{s},\overline{w})\right|-\delta {\displaystyle \frac{{\left|\left({\sigma }^{\top }{\partial }_w\mathcal{X}\right)(\overline{s},\overline{w})\right|}^2}{\mathcal{X}(\overline{s},\overline{w})}}\right]\hfill \\ \hfill >& 0,\hfill \end{array}$$

which is a contradiction with the previous inequality. Therefore, $h(s,w)\le \alpha \mathcal{X}(s,w),$$\forall (s,w)\in [s_1,T]\times {\mathbb{R}}^n.$ Sending $\alpha \to 0$, we have that $u\le \kappa$ on $[s_1,T]\times {\mathbb{R}}^n$.

Setting $s_2:={\left(s_1-{\displaystyle \frac{A}{C}}\right)}^{+}$, we obtain analogously that $u\le \kappa$ on $[s_2,s_1]\times {\mathbb{R}}^n$. Step by step, we prove that $u\le \kappa$ on $[0,T]\times {\mathbb{R}}^n$. Ultimately, interchanging u and $\kappa$, we obtain that $\kappa \le u$ on $[0,T]\times {\mathbb{R}}^n$. This completes the proof. □

5. Applications in Finance

5.1. Linking Robust Control to Stochastical Differential Utility

Since [33] pioneered the integration of robust control theory from engineering into economics, an increasing number of studies have investigated the impact of model uncertainty on portfolio choices and asset pricing (see [18,19,34,35,36]). In this paper, we exploit the methodology developed in [18], which showed the connection between a robust control criterion proposed by [34] and a form of the stochastic differential utility (SDU) of [8] without relying on any underlying dynamics. Let us consider a progressively measurable process, $F:[0,T]\times \mathsf{\Omega }\times \mathbb{R}\mapsto \mathbb{R}$, a continuous function $\psi :(0,+\infty )\mapsto (0,+\infty )$ together with a terminal value $\xi$. X represents the set of $N\in {\mathcal{L}}^2\left({\mathbb{R}}^{d\times 1}\right)$ fulfilling that $\mathcal{E}(N^{\top }\circ W)$ is a $\mathbb{P}$-martingale. For each $N\in X,E^N$ denotes the expectation under the probability measure $P^N$ defined by

$${\displaystyle \frac{d{P}^N}{d\mathbb{P}}}{|}_{{\mathcal{F}}_t}=\mathcal{E}{(N^{\top }\circ W)}_t.$$

We consider the following robust control criterion:

$${\widehat{V}}_t=\mathrm{ess}inf\left\{V_t^x:x\in \widehat{X}\right\},$$

(29)

where

$$V_t^x=E^x\left[\xi +{\int }_t^T\left(F(s,V_s^x)+{\displaystyle \frac{\psi (-V_s^x)}{2}}{\left|x_s\right|}^2\right)ds\mid {\mathcal{F}}_t\right],$$

(30)

and

$$\widehat{X}:=\left\{x\in X:\mathrm{a}.\mathrm{s}.P^x,V^x<0\mathrm{is}\mathrm{bounded}\right\}.$$

In the framework of [18], $\widehat{V}$ equals the SDU V that satisfies the BSDE,

$$V_t=\xi +{\int }_t^{T}F(r,V_r)-{\displaystyle \frac{{\left|M_r\right|}^2}{2\psi (-V_r)}}dr-{\int }_t^T{M}_{r}d{W}_r,t\in [0,T],$$

(31)

provided that in a properly defined space BSDE (31) has a unique solution, $(V,M)$. The following proposition fills this gap according to our existence and uniqueness results.

Proposition 1. 

Assume ξ and the generator of (31) satisfy the following conditions:

•

ξ is bounded and $\xi <-D$ for some constant $D>0$.

•

Given any fixed $(s,\omega )$, F is concave and continuous and satisfies

$$\forall y<0,-{\alpha }_s\left(\omega \right)-{\beta }_s\left(\omega \right)\varphi (-y)\le F(s,\omega ,y)\le 0,$$

in which α and β are non-negative processes, such that ${\int }_0^T{\alpha }_{s}ds$ and ${\int }_0^T{\beta }_{s}ds$ are bounded, and where $\varphi ,\psi :(0,+\infty )\mapsto (0,+\infty )$ satisfying $\varphi \in {\mathcal{C}}^1\left((0,+\infty )\right)$ is convex, ψ is concave, increasing, and continuous, $\varphi /\psi$ is increasing, and ${\left({\mathbb{1}}_{x>0}/\psi \left(x\right)\right)}_{x\in \mathbb{R}}$ is locally integrable on $\mathbb{R}$.

Then, BSDE (31) admits a unique negative solution $(V,M)$, such that $V\in {\mathcal{S}}^{\infty }\left(\mathbb{R}\right)$ and $M\circ W\in BMO$. With regard to the robust control criterion (29), if $\widehat{x}=-{\displaystyle \frac{1}{\psi (-V)}}{M}^{\top }$ then $\widehat{x}\in \widehat{X}$, and

$$\widehat{V}=V^{\widehat{x}}=V.$$

Proof. 

Clearly, $(V,M)$ solves (31) if and only if $(\overline{V},\overline{M}):=(-V,-M)$ solves

$${\overline{V}}_t=-\xi +{\int }_t^T\left(-F(s,-{\overline{V}}_s)+{\displaystyle \frac{{\left|{\overline{M}}_s\right|}^2}{2\psi \left({\overline{V}}_s\right)}}\right)ds-{\int }_t^T{\overline{M}}_{s}d{W}_s,t\in \left[0T\right],$$

which admits a unique solution according to Section 2 and Section 3. It remains to prove the last argument.

We first show that for any $x\in \widehat{X},V\le V^x.$ According to the martingale representation theorem, there exists $M^x\in {\mathcal{L}}^2$ fulfilling

$$V_t^x=\xi +{\int }_t^{T}F(s,V_s^x)+{\displaystyle \frac{\psi (-V_s^x)}{2}}{\left|x_s\right|}^{2}ds-{\int }_t^T{M}_s^{x}d{W}_s^x.$$

(32)

Note that (31) can be reformulated as

$$V_t=\xi +{\int }_t^{T}F(s,V_s)-{\displaystyle \frac{{\left|M_s\right|}^2}{2\psi (-V_s)}}-M_s{x}_{s}ds-{\int }_t^T{M}_{s}d{W}_s^x,t\in [0,T].$$

Combining dynamics for $V^x$ and V, we derive from Tanaka’s formula that for each fixed $0<\theta <1$,

$$\begin{array}{cc}& {\left(\theta V_t-V_t^x\right)}^{+}\hfill \\ \hfill =& (\theta -1)\xi +{\int }_t^T{\mathbb{1}}_{\theta V_s>V_s^x}\{\left[\theta F(s,V_s)-F(s,V_s^x)\right]-{\displaystyle \frac{\theta \psi (-V_s)}{2}}{|M_s{\displaystyle \frac{1}{\psi (-V_s)}}+x_s|}^2\hfill \\ \hfill +& {\displaystyle \frac{{\left|x_s\right|}^2}{2}}\left(\theta \psi (-V_s)-\psi (-V_s^x)\right)\}ds-{\int }_t^T{\mathbb{1}}_{\theta V_s>V_s^x}\left(\theta M_s-M_s^x\right)d{W}_s^x-{\displaystyle \frac{1}{2}}{\int }_t^{T}d{L}_s^0\left(\theta V-V^x\right)\hfill \\ \hfill \le & (\theta -1)\xi +{\int }_t^T{\mathbb{1}}_{\theta V_s>V_s^x}(1-\theta )\left({\alpha }_s+{\beta }_s\varphi \left({\displaystyle \frac{\theta V_s-V_s^x}{1-\theta }}\right)\right)ds-{\int }_t^T{\mathbb{1}}_{\theta V_s>V_s^x}\left(\theta M_s-M_s^x\right)d{W}_s^x.\hfill \end{array}$$

To eliminate $\varphi \left({\displaystyle \frac{\theta V_s-V_s^x}{1-\theta }}\right)$, we define

$$G\left(y\right):={\int }_2^{2+y}{\displaystyle \frac{dr}{\varphi \left(r\right)}},y>-2.$$

Obviously, $G\in {\mathcal{C}}^2\left((-2,+\infty )\right)$ and G is strictly increasing. Applying Itô’s formula to

$$P_t:=G^{-1}\left(G\left({\displaystyle \frac{{\left(\theta V_t-V_t^x\right)}^{+}}{1-\theta }}\right)+{\int }_0^t\left({\displaystyle \frac{{\alpha }_s}{\varphi \left(1\right)}}+{\beta }_s\right)ds\right)$$

and denoting

$$Q_t:={\displaystyle \frac{\varphi (2+P_t)}{\varphi (2+\frac{{\left(\theta V_t-V_t^x\right)}^{+}}{1-\theta })}}{\mathbb{1}}_{\theta V_t>V_t^x}{\displaystyle \frac{\theta M_t-M_t^x}{1-\theta }}$$

yield that

$$P_t\le G^{-1}\left(G\left(-\xi \right)+{\int }_0^T\left({\displaystyle \frac{{\alpha }_s}{\varphi \left(1\right)}}+{\beta }_s\right)ds\right)-{\int }_t^T{Q}_{s}d{W}_s^x.$$

Provided that $V^x,{\int }_0^T{\alpha }_{s}ds,$ and ${\int }_0^T{\beta }_{s}ds$ are bounded, it is apparent from a classical localization technique and the dominated convergence theorem that

$${\left(\theta V_t-V_t^x\right)}^{+}\le (1-\theta )E_t^x\left[G^{-1}\left(G\left(-\xi \right)+{\int }_0^T\left({\displaystyle \frac{{\alpha }_s}{\varphi \left(1\right)}}+{\beta }_s\right)ds\right)\right].$$

Sending $\theta \to 1,$ we obtain that for any $x\in \widehat{X}$, $V\le V^x$. Now, we declare that if $\widehat{x}=-{\displaystyle \frac{1}{\psi (-V)}}{M}^{\top }$ then $V^{\widehat{x}}\le V$, which can be verified through applying Tanaka’s formula to ${\left(\theta V^{\widehat{x}}-V\right)}^{+}$.

In conclusion, if $\widehat{x}\in \widehat{X}$ then $\widehat{V}=V^{\widehat{x}}=V$. Note that $V\in {\mathcal{S}}^{\infty }$ and $M\circ W\in BMO$; thus, $\widehat{x}\in X.$ Because of the boundness of $V,\xi ,{\int }_0^T{\alpha }_{s}ds$, and ${\int }_0^T{\beta }_{s}ds$, we have that $V^{\widehat{x}}$ is bounded a.s. $P^{\widehat{x}}$. This completes the proof. □

Example 1. 

As an example, consider the Epstein–Zin-type stochastic differential utility used in [19], which is

$$F(t,V_t^x)={\displaystyle \frac{1}{1-\rho }}\left\{{\displaystyle \frac{c_t^{1-\rho }}{{\left((1-\gamma )V_t^x\right)}^{\frac{\gamma -\rho }{1-\gamma }}}}-\eta (1-\gamma )V_t^x\right\},$$

where $\rho >1,\gamma >1$ and $\eta \ge 0.$

Suppose we plug this process back into (30). It is easy to check that Proposition 1 still holds when $\gamma <\rho ,\psi \left(x\right)\le {\displaystyle \frac{\gamma -1}{\rho -\gamma }}x,\forall x>0$ and ${\int }_0^T{c}_t^{1-\rho }dt$ is bounded. This result is a slight extension of that in [19].

5.2. Certainty Equivalent Based on g-Expectation

We take an example to show that the value process of the BSDE considered in this paper can represent a certainty equivalent of the terminal value based on g-expectation.

Example 2. 

Suppose that γ is a real-valued non-negative stochastic process, $g(t,z):={\gamma }_t\left|z\right|$ is a generator, and $\xi <0$ is a terminal condition. Consider a utility function u defined as

$$u\left(y\right):=e^{2\sqrt{-y}}\left({\displaystyle \frac{1}{2}}-\sqrt{-y}\right)-{\displaystyle \frac{1}{2}},y<0.$$

For $y<0,$ $u^{\prime }\left(y\right)=e^{2\sqrt{-y}},u^{″}\left(y\right)={\displaystyle \frac{1}{\sqrt{-y}}}{e}^{2\sqrt{-y}}.$ If

$$\xi <0,\mathbb{E}\left[exp\left\{{\displaystyle \frac{p}{p-1}}{\int }_0^T{\gamma }_s^{2}ds\right\}\left(1+e^{4p\sqrt{-\xi }}{\left(-\xi \right)}^p\right)\right]<+\infty \mathrm{for}\mathrm{some}p>1$$

then there is only one solution $(\overline{Y},\overline{Z})$ in ${\mathcal{S}}^{2p}\times {\mathcal{M}}^p$ satisfying the BSDE,

$${\overline{Y}}_t=u\left(\xi \right)+{\int }_t^T\left(-{\gamma }_s\left|{\overline{Z}}_s\right|\right)ds-{\int }_t^T{\overline{Z}}_{s}d{W}_s,t\in [0,T].$$

We denote the conditional g-expectation of $u\left(\xi \right)$ under ${\mathcal{F}}_t$ by ${\epsilon }_t^g\left[u\left(\xi \right)\right]:={\overline{Y}}_t$ (see more about g-expectation in [37]). The certainty equivalent of ξ at time $t\in [0,T]$ based on g-expectation is defined as

$$C_t\left(\xi \right):=u^{-1}\left({\epsilon }_t^g\left[u\left(\xi \right)\right]\right).$$

Putting $Z_t={\displaystyle \frac{1}{u^{\prime }\left(u^{-1}\left({\overline{Y}}_t\right)\right)}}{\overline{Z}}_t$, the pair $\left(C\left(\xi \right),Z\right)\in {\mathcal{S}}^{2p}\times {\mathcal{M}}^p$ is the solution to the BSDE,

$$Y_t=\xi -{\int }_t^T\left({\gamma }_r\left|Z_r\right|+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\left|Z_r\right|}^2}{\sqrt{-Y_r}}}\right)dr-{\int }_t^T{Z}_{r}d{W}_r,t\in [0,T],$$

in which ${\displaystyle \frac{1}{\sqrt{-y}}}$ represents the coefficient of absolute risk aversion of u at level y.

6. Discussion

In this article, we establish the existence and comparison theorem for $L^p$ solutions to a class of backward stochastic differential equations (BSDEs) with singular generators, extending the results of [11]. Additionally, we investigate the stability property, derive the Feynman–Kac formula, and provide the uniqueness of viscosity solutions to the associated partial differential equations (PDEs). More importantly, we demonstrate an application in the context of robust control linked to stochastic differential utility, slightly generalizing the results of [18,19]. Additionally, we present an example involving certainty equivalents. In our two applications, the quadratic coefficients in the generators, respectively, quantify ambiguity aversion and absolute risk aversion. Future research will investigate the analysis and applications of singular equations driven by other noise disturbance, such as the well-posedness and stabilities of mean-field equations driven by G-Brownian motion (see, e.g., [38,39]) and stabilization issues in stochastic nonlinear delay systems driven by Lévy processes (see, e.g., [40]).