The Lp Cauchy sequence for one-dimensional BSDEs with linear growth generators
Yuki Izumi
本文通过证明逼近序列是$L^p$意义下的Cauchy序列，直接证明了一维倒向随机微分方程的$L^p(p>1)$解的存在性.
In this paper, the existence of $L^p(p>1)$ solutions for one-dimensional backward stochastic differential equations will be shown directly by proving that an approximation sequence is a Cauchy one in the $L^p$ sense.
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In this paper, we establish representation theorems for generators of backward stochastic differential equations (BSDEs in short), whose generators are monotonic and convex growth in $y$ and quadratic growth in $z$. We also obtain a converse comparison theorem for such BSDEs.
In this paper, we establish representation theorems for generators of backward stochastic differential equations (BSDEs in short) in probability spaces with general filtration from the perspective of transposition solutions of BSDEs. As applications, we give a converse comparison theorem for generators of BSDEs and also some characterizations to positive homogeneity, independence of y, subadditivity and convexity of generators of BSDEs. Then, we extend concepts of g-expectations and conditional g-expectations to the probability spaces with general filtration and investigate their properties.
Representation theorem for generators of BSDEs driven by G-Brownian motion and its applications
我们得到了由G-布朗运动驱动的倒向随机微分方程生成元的一个表示定理，并利用这个表示定理得到了G-BSDEs的一个逆比较定理以及由G-BSDEs生成的非线性期望的一些等价结果.
In this paper, we establish a general representation theorem for generator of backward stochastic differential equation (BSDE), whose generator has a quadratic growth in $z$. As some applications, we obtain a general converse comparison theorem of such quadratic BSDEs and uniqueness theorem, translation invariance for quadratic $g$-expectation.
In this paper, we establish a local representation theorem for generators of reflected backward stochastic differential equations (RBSDE), whose generators are continuous with linear growth. It generalizes some known representation theorems for generators of backward stochastic differential equations (BSDE). As some applications, a general converse comparison theorem for RBSDE is obtained and some properties of RBSDE are discussed.
In [8] we established existence and uniqueness of solutions of backward stochastic differential equations in L^p under a monotonicity condition on the generator and in a general filtration. There was a mistake in the case 1 \textless{} p \textless{} 2. Here we give a corrected proof. Moreover the quasi-left continuity condition on the filtration is removed.
In this article, we deal with a multiple dimensional coupled Markovian BSDEs system with stochastic linear growth generators with respect to volatility processes. An existence result is provided by using approximation techniques.
In this paper, we study multi-dimensional reflected backward stochastic differential equations with diagonally quadratic generators. Using the comparison theorem for diagonally quadratic BSDEs which is established recently in [14], we obtain the existence and uniqueness of a solution by a penalization method. Moreover, we provide a comparison theorem.
The aim of this paper is to study an optimal stopping problem for dynamic risk measures induced by backward stochastic differential equations with jumps and delayed generator. Firstly, we connect the value function of this problem to reflected BSDEs with jump and delayed generator. Furthermore, after establishing existence and uniqueness result for this reflected BSDE, we use its to address through a mixed/optimal stopping game problem for the previous dynamic risk measure in ambiguity case.
In this paper we first prove a general representation theorem for generators of backward stochastic differential equations (BSDEs for short) by utilizing a localization method involved with stopping time tools and approximation techniques, where the generators only need to satisfy a weak monotonicity condition and a general growth condition in $y$ and a Lipschitz condition in $z$. This result basically solves the problem of representation theorems for generators of BSDEs with general growth generators in $y$. Then, such representation theorem is adopted to prove a probabilistic formula, in vis…
Minimal Supersolutions of BSDEs with Lower Semicontinuous Generators
研究了一类带生成元的倒向随机微分方程的极小上解的存在性和唯一性，该方程的生成元是联合下连续的，由控制变量的仿射函数下有界，并满足一个特殊的归一化性质.
This paper considers a class of scalar backward stochastic differential equations (BSDEs) with $L\exp(μ\sqrt{2\log(1+L)})$-integrable terminal values. We associate these BSDEs with BSDEs with integrable parameters through Girsanov change. Using this technique, we prove uniqueness, comparisons and stability for them under an extended monotonicity condition (more precisely one sided Osgood condition).
We give necessary and sufficient condition for existence and uniqueness of $\mathbb{L}^{p}$-solutions of reflected BSDEs with continuous barrier, generator monotone with respect to $y$ and Lipschitz continuous with respect to $z$, and with data in $\mathbb{L}^{p}$, $p\ge 1$. We also prove that the solutions may be approximated by the penalization method.
A note on convergence rate for reflected BSDEs with quadratic generators by penalization method
本文利用BMO鞅技术证明了带二次生成元的反射倒向随机微分方程与其惩罚方程的收敛速度为惩罚参数的1/2阶，并将结果应用于欧拉折线法数值逼近。
In this paper, we first study one-dimensional quadratic backward stochastic differential equations driven by $G$-Brownian motions ($G$-BSDEs) with unbounded terminal values. With the help of a $θ$-method of Briand and Hu [4] and nonlinear stochastic analysis techniques, we propose an approximation procedure to prove existence and uniqueness result when the generator is convex (or concave) and terminal value is of exponential moments of arbitrary order. Finally, we also establish the well-posedness of multi-dimensional G-BSDEs with diagonally quadratic generators.
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.
We consider BSDEs with two reflecting irregular barriers. We give necessary and sufficient conditions for existence and uniqueness of $\mathbb{L}^{p}$ solutions for equations with generators monotone with respect to $y$ and Lipschitz continuous with respect to $z$, and with data in $\mathbb{L}^{p}$ spaces for $p\ge 1$. We also prove that the solutions can be approximated via penalization method.
$L^p$ solutions of multidimensional BSDEs with weak monotonicity and general growth generators
在弱单调性条件下，结合生成元$g$在$y$中的一般增长条件，建立了多维倒向随机微分方程（BSDES）的$L^p\ (p>1)$解的存在唯一性.然后，我们综述了与弱单调性条件密切相关的几个条件，并对它们进行了有效的比较。最后，我们提出并证明了这类倒向随机微分方程的一个稳定性定理和一个$L^p\ (p>1)$解的比较定理。
The present paper is devoted to the study of the well-posedness of a type of BSDEs with triangularly quadratic generators. This work is motivated by the recent results obtained by Hu and Tang [14] and Xing and Žitković [28]. By the contraction mapping argument, we first prove that this type of triangularly quadratic BSDEs admits a unique local solution on a small time interval whenever the terminal value is bounded. Under additional assumptions, we build the global solution on the whole time interval by stitching local solutions. Finally, we give solvability results when the generators have pa…
This paper is devoted to the $L^p$ ($p>1$) solutions of one-dimensional backward stochastic differential equations (BSDEs for short) with general time intervals and generators satisfying some non-uniform conditions in $t$ and $ω$. An existence and uniqueness result, a comparison theorem and an existence result for the minimal solutions are respectively obtained, which considerably improve some known works. Some classical techniques used to deal with the existence and uniqueness of $L^p$ ($p>1$) solutions of BSDEs with Lipschitz or linear-growth generators are also developed in this paper.
The present paper is devoted to investigating the existence and uniqueness of solutions to a class of non-Lipschitz scalar valued backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs). In fact, when the generators are Lipschitz continuous in $y$ and uniformly continuous in $z$, we construct the unique solution to such equations by monotone convergence argument. The comparison theorem and related Feynman-Kac formula are stated as well.
This paper is devoted to proving a general invariant representation theorem for generators of general time interval backward stochastic differential equations, where the generator $g$ has a quadratic growth in the unknown variable $z$ and satisfies some stochastic growth conditions in the unknown variable $y$. This unifies and strengthens some known results. And, a natural and innovative idea is used to prove the representation theorem.
Scalar BSDEs of iterated-logarithmically sublinear generators with integrable terminal values
本研究针对带可积参数的标量倒向随机微分方程，证明了当生成元关于第二未知变量具有迭代对数一致连续性时可积适应解的存在唯一性，改进了已有相关结果。
In [3], the authors proved that uniqueness holds among solutions whose exponentials are $L^p$ with $p$ bigger than a constant $γ$ ($p\textgreater{}γ$). In this paper, we consider the critical case: $p=γ$. We prove that the uniqueness holds among solutions whose exponentials are $L^γ$ under the additional assumption that the generator is strongly convex.
Quadratic BSDEs with convex generators and unbounded terminal conditions
在以前的工作中，我们证明了一个存在性结果的BSDES与二次发电机关于变量z和无界终端条件。然而，在这项工作中没有唯一性结果。本文的主要目的就是填补这一空白。为了得到这类倒向随机微分方程的比较定理，我们假设生成元关于变量z是凸的。在这种凸性假设下，我们也能够证明一个稳定性结果的精神，先验估计中所述的文章N。El Karoui，S. Peng和M.- C.奎内斯有了这些工具，我们可以推导出非线性Feynman-Kac公式。
We analyze multidimensional BSDEs in a filtration that supports a Brownian motion and a Poisson random measure. Under a monotonicity assumption on the driver, the paper extends several results from the literature. We establish existence and uniqueness of solutions in $L^p$ provided that the generator and the terminal condition satisfy appropriate integrability conditions. The analysis is first carried out under a deterministic time horizon, and then generalized to random time horizons given by a stopping time with respect to the underlying filtration. Moreover, we provide a comparison principl…
Multi-dimensional reflected BSDEs driven by $G$-Brownian motion with diagonal generators
本文研究由 $G$-Brownian 运动驱动且生成元为对角型的多维反射 $G$-BSDEs 的适定性，利用惩罚法与 Picard 迭代证明解的存在唯一性，并揭示其与完全非线性 PDE 系统障碍问题的联系。
Existence, uniqueness and comparison theorem on unbounded solutions of general time interval BSDEs with sub-quadratic generators
本文研究一般时间区间上具有次二次生成元的一维BSDEs，在时变增长条件下证明无界解的存在性、唯一性与比较定理，并提出若干首次探索的唯一性充分条件。
Uniqueness of adapted solutions to scalar BSDEs with Peano-type generators
本文研究终端值几乎必然为正时具有Peano型生成元的标量BSDE适应解的唯一性，通过最优随机控制验证和变量替换约化为凸二次BSDE两种方法，在特殊情形下给出精确结果。
Existence and stability of measure solutions for BSDE with generators of quadratic growth
重点是发电机与二次增长的控制变量，我们认为测量解决方案的Battlefield，一个解决方案的概念对应的概念，风险中性措施在数学金融。在测度解方面，求解一个贝叶斯问题归结为关于一个底层过滤的鞅表示。与等价于历史度量的度量相关的度量解提供了经典解。我们推导出的情况下，在生成函数是连续的，至多线性增长的控制变量（对应于发电机的最多二次增长在通常意义上），并与随机约束的时间参数的随机积分是一个BMO鞅的测量解决方案的存在性。我们的主要工具包括测度解序列的稳定性，通过测度的弱收敛得到了其极限解。
On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions
在以前的工作中，P.Briand和Y. Hu证明了允许所有指数矩的解的唯一性。本文证明了在允许某些给定指数矩的解之间存在唯一性。这些指数矩是自然的，因为它们由存在定理给出。由于这个唯一性结果，我们可以加强由P.Briand和Y.胡
In this paper we prove some uniqueness results for quadratic backward stochastic differential equations without any convexity assumptions on the generator. The bounded case is revisited while some new results are obtained in the unbounded case when the terminal condition and the generator depend on the path of a forward stochastic differential equation. Some of these results are based on strong estimates on $Z$ that are interesting on their own and could be applied in other situations.
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.
Weighted solutions of random time horizon BSDEs with stochastic monotonicity and general growth generators and related PDEs
研究具有随机终端时间、随机单调性及一般增长生成元的多维 BSDEs，在无有限矩限制下通过创新分析技术建立了加权解的存在唯一性，统一强化了现有结果，并给出连续依赖性、稳定性定理及 PDEs 的非线性 Feynman-Kac 公式。
$L^p$ $(p\geq 1)$ solutions of multidimensional BSDEs with monotone generators in general time intervals
本文主要研究用$L^p$ $(p\geq 1)$求解一般时间区间的多维倒向随机微分方程。当生成元在$y$中单调且在$z$中Lipschitz连续且关于$t$非一致时，利用卷积和弱收敛的方法研究了$L^p$ $(p>1)$解的存在唯一性.然后我们得到了$L^1$解的存在性和唯一性，并给出了生成元在$z$中关于$t$非一致次线性增长的附加假设.
In this paper, we give several new results on solvability of a quadratic BSDE whose generator depends also on the mean of both variables. First, we consider such a BSDE using John-Nirenberg's inequality for BMO martingales to estimate its contribution to the evolution of the first unknown variable. Then we consider the BSDE having an additive expected value of a quadratic generator in addition to the usual quadratic one. In this case, we use a deterministic shift transformation to the first unknown variable, when the usual quadratic generator depends neither on the first variable nor its mean,…
The purpose of this note is to propose a new approach for the probabilistic interpretation of Hamilton-Jacobi-Bellman equations associated with stochastic recursive optimal control problems, utilizing the representation theorem for generators of backward stochastic differential equations. The key idea of our approach for proving this interpretation consists of transmitting the signs between the solution and generator via the identity given by representation theorem. Compared with existing methods, our approach seems to be more applicable for general settings. This can also be regarded as a new…
We establish a general existence and uniqueness result of $L^1$ solution for a multidimensional backward stochastic differential equation (BSDE for short) with generator $g$ satisfying a one-sided Osgood condition as well as a general growth condition in $y$, and a Lipschitz condition together with a sublinear growth condition in $z$, which improves some existing results. In particular, we put forward and prove a stability theorem of the $L^1$ solutions for the first time. A new type of $L^1$ solution is also investigated. Some delicate techniques involved in the relationship between convergen…
On Malliavin's differentiability of BSDE with time delayed generators driven by Brownian motions and Poisson random measures
研究了由布朗运动和Poisson随机测度驱动的具有时滞生成元的倒向随机微分方程的解，这两个测度构成了Levy过程的两个分量.在这种新型方程中，生成器可以依赖于解的过去值，通过将它们以时滞反馈回动态。对于这样的时滞时滞Bynomial，我们证明了解的存在性和唯一性，只要我们限制在一个足够小的时间范围内或发电机具有一个足够小的Lipschitz常数。我们研究微分变分或Malliavin意义上，并推导出方程的Malliavin梯度过程所满足的。在所选择的随机基础上，这解决了光滑度相对于我们的Levy过程中的经典Malliavin衍生希尔伯特空间值的随机变量的连续部分，以及相对于纯跳跃组件，它采取的形式的增量商算子相关的皮卡德差分算子。
Multidimensional BSDEs with uniformly continuous generators and general time intervals
研究了一类具有一般时间区间的多维倒向随机微分方程，其中生成元在$(y,z)$中关于$t$非一致连续.通过建立一般时间区间的确定性倒向微分方程的一些结果，利用Girsanov定理和卷积技巧，得到了这类倒向随机微分方程解的存在唯一性的一个新结果，将Hamadene（2003）和Fan，Jiang，Tian（2011）的结果推广到一般时间区间的情形.
We show existence of a unique solution and a comparison theorem for a one-dimensional backward stochastic differential equation with jumps that emerge from a Lévy process. The considered generators obey a time-dependent extended monotonicity condition in the y-variable and have linear time-dependent growth. Within this setting, the results generalize those of Royer (2006), Yin and Mao (2008) and, in the $L^2$-case with linear growth, those of Kruse and Popier (2016). Moreover, we introduce an approximation technique: Given a BSDE driven by Brownian motion and Poisson random measure, we conside…
We first establish the existence of an unbounded solution to a backward stochastic differential equation (BSDE) with generator $g$ allowing a general growth in the state variable $y$ and a sub-quadratic growth in the state variable $z$, like $|z|^α$ for some $α\in (1,2)$, when the terminal condition satisfies a sub-exponential moment integrability condition like $\exp\left(μL^{2/α^*}\right)$ for the conjugate $α^*$ of $α$ and a positive parameter $μ>μ_0$ with a certain value $μ_0$, which is clearly weaker than the usual $\exp(μL)$ integrability and stronger than $L^p\ (p>1)$ integrability. The…
Quadratic BSDEs with Singular Generators and Unbounded Terminal Conditions: Theory and Applications
本文研究生成元关于 $y$ 奇异的二次 BSDE，建立解的存在性、比较、稳定性及相关 PDE 粘性解唯一性，并应用于稳健控制中的效用与确定性等价问题。
Anticipated BSDEs driven by fractional Brownian motion with time-delayed generator
本文研究由分数布朗运动驱动且生成子含时滞的预期倒向随机微分方程，利用不动点定理证明了解的存在唯一性，并建立了解的比较定理。
Reflected BSDE driven by a marked point process with a convex/concave generator
本文研究由标记点过程驱动且生成元为凸/凹函数的反射倒向随机微分方程，利用不动点、$θ$-方法和截断技术证明其在无界终端条件与障碍下的适定性，并应用于美式期权定价。
In this paper, we study multidimensional generalized BSDEs that have a monotone generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. First, we prove the existence and uniqueness of $\mathbb{L}^p(p\ge2)$-solutions in the case of a fixed terminal time under suitable $p$-integrability conditions on the data. Then, we extend these results to the case of a random terminal time. Furthermore, we provide a comparison result in dimension $1$.
This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator $g$ satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable $y$, and a stochastic-Lipschitz condition in the state variable $z$. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in \citet{XiaoFan2017Stochastics} and \citet{LiuLiFan2019CAM}. In particular, we put forward and prove…
BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces
本文研究了由圆柱Wiener过程驱动的具有二次生成元的实值倒向随机微分方程解的可微性.这个问题的主要新颖之处在于这样一个事实，即梯度方程的二次BMO有发电机满足随机利普希茨条件涉及BMO鞅。我们展示了非线性Kolmogorov方程的一些应用。
We prove a uniqueness result of the unbounded solution for a quadratic backward stochastic differential equation whose terminal condition is unbounded and whose generator $g$ may be non-Lipschitz continuous in the state variable $y$, non-convex (non-concave) in the state variable $z$, and instead satisfies a strictly quadratic condition and an additional assumption. The key observation is that if the generator is strictly quadratic, then the quadratic variation of the first component of the solution admits an exponential moment. Typically, a Lipschitz perturbation of some convex (concave) func…
Feynman-Kac formula for BSDEs with jumps and time delayed generators associated to path-dependent nonlinear Kolmogorov equations
本文提出了一种将具有跳跃和时间延迟生成函数的BSDEs解与路径依赖非线性柯尔莫戈罗夫方程解关联的非线性Feynman-Kac公式，并应用于研究大型投资者问题的扩展。
We study Backward Stochastic Differential Equations on a probability space equipped with a Brownian filtration. We assume that the terminal value and the generator at zero are merely integrable. Moreover, the generator is assumed to be non-increasing with respect to the value variable (with no restrictions on the growth) and Lipschitz continuous, with sublinear growth, with respect to the control variable. We provide a priori estimate and stability result for solutions to the aforementioned BSDEs.
Comparison theorems for mean-field BSDEs whose generators depend on the law of the solution $(Y,Z)$
本文针对系数依赖于解过程$(Y,Z)$联合分布的一维均值场BSDEs，借助Malliavin calculus和BMO鞅论证建立了两个比较定理及一个强比较结果，同时比较了解的$Y$和$Z$分量。
This paper aims at solving a one-dimensional backward stochastic differential equation (BSDE for short) with only integrable parameters. We first establish the existence of a minimal $L^1$ solution for the BSDE when the generator $g$ is stronger continuous in $(y,z)$ and monotonic in $y$ as well as it has a general growth in $y$ and a sublinear growth in $z$. Particularly, the $g$ may be not uniformly continuous in $z$. Then, we put forward and prove a comparison theorem and a Levi type theorem on the minimal $L^1$ solutions. A Lebesgue type theorem on $L^1$ solutions is also obtained. Further…
We consider a backward stochastic differential equation with a generator that can be subjected to delay, in the sense that its current value depends on the weighted past values of the solutions, for instance a distorted recent average. Existence and uniqueness results are provided in the case of possibly infinite time horizon for equations with, and without reflection. Furthermore, we show that when the delay vanishes, the solutions of the delayed equations converge to the solution of the equation without delay. We argue that these equations are naturally linked to forward backward systems, an…
Finite and infinite time horizon for BSDE with Poisson jumps
本文研究了一类带跳的实值倒向随机微分方程的解，其中时域可以是有限的，也可以是无限的。在线性增长生成元下，证明了极小解的存在性。利用比较定理证明了当生成元一致连续且满足弱单调条件时，这类方程解的存在唯一性。
In this paper, we consider the backward stochastic differential equation (BSDE) with generator $f(y)|z|^2,$ where the function $f$ is defined on an open interval $D$ and locally integrable. The existence and uniqueness of bounded solutions and $L^p(p\geq1)$ solutions of such BSDEs are obtained. Some comparison theorems and a converse comparison theorem of such BSDEs are established. As an application, we give a probabilistic interpretation of viscosity solution of quadratic PDEs.
On the existence and uniqueness of unbounded solutions to quadratic BSDEs with convex generators
本文研究二次倒向随机微分方程无界解的存在唯一性，在放宽生成元连续性条件下推广已有结果，并给出显式解及比较定理。
p-integrable solutions to multidimensional BSDEs and degenerate systems of PDEs with logarithmic nonlinearities
研究了多维倒向随机微分方程，其中包含对数非线性项u log u.更精确地说，我们建立了多维倒向微分方程p-可积解（p > 1）的存在性、唯一性和稳定性，该方程具有p-可积终端条件和关于变量y和z的超线性增长生成元.这是用生成器f（y，z）来完成的，它既不能在变量y中局部单调，也不能在变量z中局部Lipschitz。此外，它不是一致连续的。作为应用，我们建立了具有超线性增长生成元和p-可积终端数据的可能退化半线性抛物型方程组Sobolev解的存在唯一性.例如，我们的结果涵盖了物理学中产生的某些偏微分方程（系统）。
Mean-field reflected BSDEs driven by a marked point process
本文研究由标记点过程驱动的平均场反射倒向随机微分方程，利用g-期望表示引理证明其在Lipschitz生成元条件下的存在唯一性，并通过θ-方法进一步建立指数增长生成元和无界终值情形的适定性。
A uniqueness theorem for solution of BSDEs
本文证明了：若$g$在$z$中一致连续，关于$(\oo,t)$一致连续，且与$y$无关，则以$g$为生成元的倒向随机微分方程（BDED）的解是唯一的.
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