Solvability of Backward Stochastic Differential Equations with Quadratic Growth

Solvability of multidimensional quadratic BSDEs

We consider multidimensional quadratic BSDEs with bounded and unbounded terminal conditions. We provide sufficient conditions which guarantee existence and uniqueness of solutions. In particular, these conditions are satisfied if the terminal condition or the dependence in the system are small enough.

Solvability of a class of mean-field BSDEs with quadratic growth

In this paper, we study the multi-dimensional mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. Under small terminal value, the existence and uniqueness are proved for the multi-dimensional situation when the generator f(t,Y,E[Y],Z,E[Z]) is of quadratic growth with respect to the last four items, using some new methods. Besides, a kind of comparison theorem is obtained.

On the solvability of forward-backward stochastic differential equations driven by Teugels Martingales

We deal with a class of fully coupled forward-backward stochastic differential equations (FBSDE for short), driven by Teugels martingales associated with some Lévy process. Under some assumptions on the derivatives of the coefficients, we prove the existence and uniqueness of a global solution on an arbitrarily large time interval. Moreover, we establish stability and comparison theorems for the solutions of such equations. Note that the present work extends known results by Jianfeng Zhang (Discrete Contin. Dyn. Syst. Ser. B 6 (2006), no. 4, 927--940), proved for FBSDEs driven by a Brownian mo…

Nonlinear Backward Stochastic Evolutionary Equations Driven by a Space-Time White Noise

We study the well solvability of nonlinear backward stochastic evolutionary equations driven by a space-time white noise. We first establish a novel a priori estimate for solution of linear backward stochastic evolutionary equations, and then give an existence and uniqueness result for nonlinear backward stochastic evolutionary equations. A dual argument plays a crucial role in the proof of these results. Finally, an example is given to illustrate the existence and uniqueness result.

Fractional backward stochastic differential equations with delayed generator

In this paper, we focus on the solvability of a class of fractional backward stochastic differential equations (BSDEs, for short) with delayed generator. In this class of equations, the generator includes not only the values of the solutions of the present but also the past. Under Lipschitz condition, the existence and uniqueness of such BSDEs are established. A comparison theorem for this class of BSDEs is also obtained.

Solvability conditions for indefinite linear quadratic optimal stochastic control problems and associated stochastic Riccati equations

A linear quadratic optimal stochastic control problem with random coefficients and indefinite state/control weight costs is usually linked to an indefinite stochastic Riccati equation (SRE) which is a matrix-valued quadratic backward stochastic differential equation along with an algebraic constraint involving the unknown. Either the optimal control problem or the SRE is solvable only if the given data satisfy a certain structure condition that has yet to be precisely defined. In this paper, by introducing a notion of subsolution for the SRE, we derive several novel sufficient conditions for t…

Weak Closed-Loop Solvability of Stochastic Linear Quadratic Optimal Control Problems of Markovian Regime Switching System

In this paper, we investigate the open-loop and weak closed-loop solvabilities of stochastic linear quadratic (LQ, for short) optimal control problem of Markovian regime switching system. Interestingly, these two solvabilities are equivalent. We first provide an alternative characterization of the open-loop solvability of the LQ problem using the perturbation approach. Then, we study the weak closed-loop solvability of the LQ problem of Markovian regime switching system, and establish the equivalent relationship between open-loop and weak closed-loop solvabilities. Finally, we present an examp…

Anticipated backward stochastic differential equations with quadratic growth

In this paper, we study the solvability of anticipated backward stochastic differential equations (BSDEs, for short) with quadratic growth for one-dimensional case and multi-dimensional case. In these BSDEs, the generator, which is of quadratic growth in Z, involves not only the present information of solution (Y, Z) but also its future one. The existence and uniqueness of such BSDEs, under different conditions, are derived for several terminal situations, including small terminal value, bounded terminal value and unbounded terminal value.

Fully Coupled Forward-Backward Stochastic Functional Differential Equations and Applications to Quadratic Optimal Control

In this paper, we consider the fully coupled forward-backward stochastic functional differential equations (FBSFDEs) with stochastic functional differential equations as the forward equations and the generalized anticipated backward stochastic differential equations as the backward equations. We will prove the existence and uniqueness theorem for FBSFDEs. As an application, we deal with a quadratic optimal control problem for functional stochastic systems, and get the explicit form of the optimal control by virtue of FBSFDEs.

一类具有对角二次增长奇异生成元的多维倒向随机微分方程及其应用

A Class of Multi-dimensional Backward Stochastic Differential Equations with Singular Generators exhibiting Diagonally Quadratic Growth and Applications

本文研究具有对角二次增长奇异生成元和无界终端条件的多维倒向随机微分方程，推广了已有结果，并给出其在最优投资决策中的应用实例。

Backward SDEs with superquadratic growth

In this paper, we discuss the solvability of backward stochastic differential equations (BSDEs) with superquadratic generators. We first prove that given a superquadratic generator, there exists a bounded terminal value, such that the associated BSDE does not admit any bounded solution. On the other hand, we prove that if the superquadratic BSDE admits a bounded solution, then there exist infinitely many bounded solutions for this BSDE. Finally, we prove the existence of a solution for Markovian BSDEs where the terminal value is a bounded continuous function of a forward stochastic differentia…

On well-posedness of forward-backward SDEs-A unified approach

In this paper, we study the well-posedness of the Forward-Backward Stochastic Differential Equations (FBSDE) in a general non-Markovian framework. The main purpose is to find a unified scheme which combines all existing methodology in the literature, and to address some fundamental longstanding problems for non-Markovian FBSDEs. An important device is a decoupling random field that is regular (uniformly Lipschitz in its spatial variable). We show that the regulariy of such decoupling field is closely related to the bounded solution to an associated characteristic BSDE, a backward stochastic Ri…

Multi-Dimensional Backward Stochastic Differential Equations of Diagonally Quadratic generators

The paper is concerned with adapted solution of a multi-dimensional BSDE with a "diagonally" quadratic generator, the quadratic part of whose $i$th component only depends on the $i$th row of the second unknown variable. Local and global solutions are given. In our proofs, it is natural and crucial to apply both John-Nirenberg and reverse Hölder inequalities for BMO martingales.

对角生成元的正倒向随机微分方程全局可解性的统一方法

A unified approach to global solvability for FBSDEs with diagonal generators

本文针对对角生成元的FBSDEs，在单调性条件下提出统一方法，证明解耦场的全局存在性，并推广到Lipschitz情形，给出$L^p$估计，回答了Yong的问题，应用于随机控制和博弈。

Quadratic backward stochastic differential equations driven by $G$-Brownian motion: discrete solutions and approximation

In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the $G$-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces.

由$G$-布朗运动驱动的无限时域二次倒向随机微分方程

Infinite horizon quadratic backward stochastic differential equations driven by $G$-Brownian motion

该研究证明了带二次生成元的$G$-布朗运动驱动无限时域$G$-BSDEs的适定性，构造了无界系数线性$G$-BSDEs显式解并采用二次假设下线性化方法，同时建立了两类时域下的比较定理。

Backward doubly stochastic differential equations and SPDEs with quadratic growth

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

Differentiability of quadratic forward-backward SDEs with rough drift

In this paper, we consider quadratic forward-backward SDEs (QFBSDEs), for {which} the drift in the forward equation does not satisfy the standard globally Lipschitz condition and the driver of the backward system {possesses} nonlinearity of type $f(|y|)|z|^2,$ where $f$ is any locally integrable function. We prove both the Malliavin and classical derivative of the QFBSDE and provide representations of these processes. We study a numerical approximation of this system in the sense of \cite{ImkDosReis} in which the authors assume that the drift is Lipschitz and the driver of the BSDE is quadrati…

Backward SPDEs with non-local in time and space boundary conditions

We study linear backward stochastic partial differential equations of parabolic type with special boundary condition that connect the terminal value of the solution with a functional over the entire past solution. Uniqueness, solvability and regularity results for the solutions are obtained.

具有双重均值反射的均值场反向随机微分方程

Mean Field Backward Stochastic Differential Equations with Double Mean Reflections

本文研究了具有双重均值反射的均值场反向随机微分方程的可解性与唯一性，通过压缩映射、惩罚方法和固定点论证等方法，探讨了生成元和约束依赖于分布的MFBSDEs的数学性质。

A New Result for Second Order BSDEs with Quadratic Growth and its Applications

In this paper, we study a class of second order backward stochastic differential equations (2BSDEs) with quadratic growth in coefficients. We first establish solvability for such 2BSDEs and then give their applications to robust utility maximization problems.

Second Order Backward Stochastic Differential Equations with Quadratic Growth

We extend the wellposedness results for second order backward stochastic differential equations introduced by Soner, Touzi and Zhang \cite{stz} to the case of a bounded terminal condition and a generator with quadratic growth in the $z$ variable. More precisely, we obtain uniqueness through a representation of the solution inspired by stochastic control theory, and we obtain two existence results using two different methods. In particular, we obtain the existence of the simplest purely quadratic 2BSDEs through the classical exponential change, which allows us to introduce a quasi-sure version …

具有特殊结构的对角二次生成元的多维倒向随机微分方程

Multi-dimensional Backward Stochastic Differential Equations of Diagonally Quadratic Generators with a Special Structure

该研究针对一类带对角二次生成元的多维倒向随机微分方程，通过新的先验估计证明了非对角元素增长足够小时方程的唯一适定性，并给出了三角型对角二次生成元下的可解性结果。

Multi-dimensional Mean-field Type Backward Stochastic Differential Equations with Diagonally Quadratic Generators

In this paper, we study the multi-dimensional backward stochastic differential equations (BSDEs) whose generator depends also on the mean of both variables. When the generator is diagonally quadratic, we prove that the BSDE admits a unique local solution with a fixed point argument. When the generator has a logarithmic growth of the off-diagonal elements (i.e., for each $i$, the $i$-th component of the generator has a logarithmic growth of the $j$-th row $z^j$ of the variable $z$ for each $j \neq i$), we give a new apriori estimate and obtain the existence and uniqueness of the global solution…

Linear--Quadratic control for a class of stochastic Volterra equations: solvability and approximation

We provide an exhaustive treatment of Linear-Quadratic control problems for a class of stochastic Volterra equations of convolution type, whose kernels are Laplace transforms of certain signed matrix measures which are not necessarily finite. These equations are in general neither Markovian nor semimartingales, and include the fractional Brownian motion with Hurst index smaller than $1/2$ as a special case. We establish the correspondence of the initial problem with a possibly infinite dimensional Markovian one in a Banach space, which allows us to identify the Markovian controlled state varia…

Multi-Dimensional Super-Linear Backward Stochastic Volterra Integral Equations

In this paper, a systematic investigation is carried out for the general solvability of multi-dimensional backward stochastic Volterra integral equations (BSVIEs) with the generators being super-linear in the adjustment variable $Z$. Two major situations are discussed: (i) When the free term is bounded with the dependence of the generator on $Z$ being of ``diagonally strictly'' quadratic growth and being sub-quadratically coupled with off-diagonal components; (ii) When the free term is unbounded having exponential moments of arbitrary order with the dependence of the generator on $Z$ being dia…

Lipschitz-quadratic Regularization for Quadratic Semimartingale BSDEs

We refine the solvability of quadratic semimartingale BSDEs by employing a Lipschitz-quadratic regularization procedure. In the first step, we prove an existence and uniqueness result for a class of Lipschitz-quadratic BSDEs. A corresponding stability theorem and a Lipschitz-quadratic regularization are developed to solve quadratic BSDEs. The advantage of our approach is that much weaker conditions ensure the existence and uniqueness results.

由$G$-布朗运动驱动的具有二次生成函数的双反射反向随机微分方程

Doubly Reflected Backward SDEs Driven by $G$-Brownian Motion with Quadratic Generator

本文研究由$G$-布朗运动驱动的双反射反向随机微分方程在二次生成函数下的性质，证明了解的存在性和唯一性，并通过罚函数反射建立其与具有双障碍的非线性偏微分方程的关系。

Robustness of quadratic hedging strategies in finance via backward stochastic differential equations with jumps

We consider a backward stochastic differential equation with jumps (BSDEJ) which is driven by a Brownian motion and a Poisson random measure. We present two candidate-approximations to this BSDEJ and we prove that the solution of each candidate- approximation converges to the solution of the original BSDEJ in a space which we specify. We use this result to investigate in further detail the consequences of the choice of the model to (partial) hedging in incomplete markets in finance. As an application, we consider models in which the small variations in the price dynamics are modeled with a Poi…

Multi-dimensional backward stochastic differential equations of diagonally quadratic generators: the general result

This paper is devoted to a general solvability of a multi-dimensional backward stochastic differential equation (BSDE) of a diagonally quadratic generator $g(t,y,z)$, by relaxing the assumptions of \citet{HuTang2016SPA} on the generator and terminal value. More precisely, the generator $g(t,y,z)$ can have more general growth and continuity in $y$ in the local solution; while in the global solution, the generator $g(t,y,z)$ can have a skew sub-quadratic but in addition "strictly and diagonally" quadratic growth in the second unknown variable $z$, or the terminal value can be unbounded but the g…

Recursive Utility Processes, Dynamic Risk Measures and Quadratic Backward Stochastic Volterra Integral Equations

For an $\cF_T$-measurable payoff of a European type contingent claim, the recursive utility process/dynamic risk measure can be described by the adapted solution to a backward stochastic differential equation (BSDE). However, for an $\cF_T$-measurable stochastic process (called a position process, not necessarily $\dbF$-adapted), mimicking BSDE's approach will lead to a time-inconsistent recursive utility/dynamic risk measure. It is found that a more proper approach is to use the adapted solution to a backward stochastic Volterra integral equation (BSVIE). The corresponding notions are called …

具有二次增长的平均场倒向随机微分方程与非局部偏微分方程

Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth

本文研究二次增长的一维平均场BSDE，证明解的存在唯一性与比较定理，给出粒子系统收敛速率，并通过其为非局部PDE粘性解提供概率表示，得到PDE解的存在唯一性。

A Global Stochastic Maximum Principle for Forward-Backward Stochastic Control Systems with Quadratic Generators

We study a stochastic optimal control problem for forward-backward control systems with quadratic generators. In order to establish the first and second-order variational and adjoint equations, we obtain a new estimate for one-dimensional linear BSDEs with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales (BMO-martingales for short) and prove the solvability for a class of multi-dimensional BSDEs with this type. Finally, a new global stochastic maximum principle is deduced.

Large Deviation Principle for Backward Stochastic Differential Equations with a stochastic Lipschitz condition on $z$

In this paper, a probabilistic interpretation for the viscosity solution of a parabolic partial differential equation is obtained by virtue of the solution of a class of quadratic backward stochastic differential equations (BSDEs, for short). Furthermore, we prove the convergence and the large deviation principle for the solution of this class of quadratic BSDEs, which is associated with a family of Markov processes with the diffusion coefficients that tend to be zero.

Backward stochastic partial differential equations with quadratic growth

This paper is concerned with the existence and uniqueness of weak solutions to the Cauchy-Dirichlet problem of backward stochastic partial differential equations (BSPDEs) with nonhomogeneous terms of quadratic growth in both the gradient of the first unknown and the second unknown. As an example, we consider a non-Markovian stochastic optimal control problem with cost functional formulated by a quadratic BSDE, where the corresponding value function satisfies the above quadratic BSPDE.

具有交互式二次生成元的多维非马尔可夫倒向随机微分方程

Multi-dimensional non-Markovian backward stochastic differential equations of interactively quadratic generators

本文研究了具有交互式二次生成元的多维非马尔可夫倒向随机微分方程的一般可解性，提出了生成元的一般结构，建立了局部和全局有界及无界解的存在唯一性结果，改进了现有工作并部分回答了Jackson2023SPA提出的问题。

Weighted Bounded Mean Oscillation applied to Backward Stochastic Differential Equations

We deduce conditional $L_p$-estimates for the variation of a solution of a BSDE. Both quadratic and sub-quadratic types of BSDEs are considered, and using the theory of weighted bounded mean oscillation we deduce new tail estimates for the solution $(Y,Z)$ on subintervals of $[0,T]$. Some new results for the decoupling technique introduced in \cite{jossain} are obtained as well and some applications of the tail estimates are given.

Infinite horizon jump-diffusion forward-backward stochastic differential equations and their application to backward linear-quadratic problems

In this paper, we investigate infinite horizon jump-diffusion forward-backward stochastic differential equations under some monotonicity conditions. We establish an existence and uniqueness theorem, two stability results and a comparison theorem for solutions to such kind of equations. Then the theoretical results are applied to study a kind of infinite horizon backward stochastic linear-quadratic optimal control problems, and then differential game problems. The unique optimal controls for the control problems and the unique Nash equilibrium points for the game problems are obtained in closed…

具有或无反射的弱条件下的反向双随机微分方程

Backward doubly stochastic differential equations with or without reflection under weak conditions

本文在弱条件下研究反向双随机微分方程（BDSDEs）的well-posedness，探讨了生成元f在y和z上的增长条件对存在性、唯一性和比较原理的影响。

带奇异漂移的二次正倒向随机微分方程的时间离散化

Time discretization of Quadratic Forward-Backward SDEs with singular drifts

本文研究带奇异漂移的二次正倒向随机微分方程时间离散化的收敛速率，通过推广变分正则性、结合BMO过程刻画，实现接近1/2的误差控制，填补了该领域数值分析的关键空白。

Diagonally quadratic BSDE with oblique reflection and optimal switching

The present paper is devoted to the study of diagonally quadratic backward stochastic differential equation with oblique reflection. Using a penalization approach, we show the existence fo a solution by providing some delicated a priori estimates. We further obtain the uniqueness by verifying the first component of the solution is indeed the value of a switching probelm for quadratic BSDEs. Moreover, we provide an extension for the solvability and apply our results to study a risk-sensitive switching problem for functional stochastic differential equations.

On Quadratic BSDEs with Final Condition in L2

This thesis consists of three parts. In the first part, we study $\mathbb{L}^p$ solutions of a large class of BSDEs. Existence, comparison theorem, uniqueness and a stability result are proved. In the second part, we establish the solvability of quadratic semimartingale BSDEs. In contrast to current literature, we use Lipschitz-quadratic regularization and obtain the existence and uniqueness results with minimal assumptions. The third part is a brief summary of quadratic semimartingales and the monotone stability result. This provides an alternative proof of monotone stability result for quadr…

Quadratic BSDEs with rough drivers and $L^2$--terminal condition

In this paper, we study the existence and uniqueness of solutions to quadratic Backward Stochastic Differential Equations (QBSDEs for short) with rough driver and square integrable terminal condition. The main idea consists in using both Doss-Sussman and Zvonkin type transformations. As an application we study connection between QBSDEs and quadratic PDEs with rough drivers. We also obtain Backward Doubly SDEs and QBSDEs driven by Fractional Brownian with Hurst parameter greater than $\frac{1}{4}$ as particular cases of our QBSDEs with rough drivers. A probabilistic representation of a class of…

On forward and backward SPDEs with non-local boundary conditions

We study linear stochastic partial differential equations of parabolic type with non-local in time or mixed in time boundary conditions. The standard Cauchy condition at the terminal time is replaced by a condition that mixes the random values of the solution at different times, including the terminal time, initial time and continuously distributed times. For the case of backward equations, this setting covers almost surely periodicity. Uniqueness, solvability and regularity results for the solutions are obtained. Some possible applications to portfolio selection are discussed.

A maximum principle for forward-backward stochastic Volterra integral equations and applications in finance

This paper formulates and studies a stochastic maximum principle for forward-backward stochastic Volterra integral equations (FBSVIEs in short), while the control area is assumed to be convex. Then a linear quadratic (LQ in short) problem for backward stochastic Volterra integral equations (BSVIEs in short) is present to illustrate the aforementioned optimal control problem. Motivated by the technical skills in solving above problem, a more convenient and briefer method for the unique solvability of M-solution for BSVIEs is proposed. At last, we will investigate a risk minimization problem by …

由G-布朗运动驱动的带双重约束的二次倒向随机微分方程

Quadratic BSDEs with double constraints driven by G-Brownian motion

本文研究由G-布朗运动驱动的带双重均值反射的二次倒向随机微分方程的适定性，利用G-BMO鞅技术、不动点论证、Skorokhod问题及θ-方法，在有界和无界终端条件下建立了解的存在唯一性。

Strong solutions of forward-backward stochastic differential equations with measurable coefficients

This paper investigates solvability of fully coupled systems of forward-backward stochastic differential equations (FBSDEs) with irregular coefficients. In particular, we assume that the coefficients of the FBSDEs are merely measurable and bounded in the forward process. We crucially use compactness results from the theory of Malliavin calculus to construct strong solutions. Despite the irregularity of the coefficients, the solutions turn out to be differentiable, at least in the Malliavin sense and, as functions of the initial variable, in the Sobolev sense.

Infinite Horizon and Ergodic Optimal Quadratic Control for an Affine Equation with Stochastic Coefficients

We study quadratic optimal stochastic control problems with control dependent noise state equation perturbed by an affine term and with stochastic coefficients. Both infinite horizon case and ergodic case are treated. To this purpose we introduce a Backward Stochastic Riccati Equation and a dual backward stochastic equation, both considered in the whole time line. Besides some stabilizability conditions we prove existence of a solution for the two previous equations defined as limit of suitable finite horizon approximating problems. This allows to perform the synthesis of the optimal control.

带非线性Young驱动的倒向随机微分方程 II

Backward stochastic differential equations with nonlinear Young drivers II

本文研究无界驱动的带非线性Young积分的倒向随机微分方程适定性，提出新局部化方法拓展可解性，推导Young PDE的非线性Feynman-Kac公式，并给出非Lipschitz情形下有界域与全空间PDE问题的误差估计。

具有二次增长的多维预期倒向随机微分方程

Multi-dimensional anticipated backward stochastic differential equations with quadratic growth

本文放宽既有研究假设，将二次增长预期倒向随机微分方程的可解性研究从一维有界终值推广到多维有界/无界终值情形，建立了局部与全局解存在唯一性的三项新结果。

Backward stochastic differential equations with rough drivers

Backward stochastic differential equations (BSDEs) in the sense of Pardoux-Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci., 176, 200--217, 1992] provide a non-Markovian extension to certain classes of non-linear partial differential equations; the non-linearity is expressed in the so-called driver of the BSDE. Our aim is to deal with drivers which have very little regularity in time. To this end we establish continuity of BSDE solutions with respect to rough path metrics in the sense of Lyons […

Mean Field Linear-Quadratic-Gaussian (LQG) Games of Forward-Backward Stochastic Differential Equations

This paper studies a new class of dynamic optimization problems of large-population (LP) system which consists of a large number of negligible and coupled agents. The most significant feature in our setup is the dynamics of individual agents follow the forward-backward stochastic differential equations (FBSDEs) in which the forward and backward states are coupled at the terminal time. This current paper is hence different to most existing large-population literature where the individual states are typically modeled by the SDEs including the forward state only. The associated mean-field linear-…

关于风险敏感线性二次均场博弈的可解性

On the Solvability of Risk-Sensitive Linear-Quadratic Mean-Field Games

本文研究了由线性随机动力学和二次或指数二次成本函数描述的均场博弈，通过平方完成和测度变换方法显式求解玩家最优策略，不依赖传统方法，提供了风险敏感Riccati方程额外项的解释，并得出了均场均衡存在的充分条件。

Exponential Quadratic BSDEs with infinite activity Jumps

In this paper, we study a Backward Stochastic Differential Equation with Jumps (BSDEJs in short) where the jumps have infinite activity. Following a forward approach based on Exponential Quadratic semimartingale, we prove the existence of solution of Quadratic BSDEJs with unbounded terminal condition and quadratic growth in z.