生成元一致连续的多维倒向随机微分方程的$L^p$解

在生成元~$g$~的第~$i$~个分量~$g_i(t,y,z)$~仅仅依赖于矩阵~$z$~的第~$i$ 行的条件下, Hamad\`{e}ne于2003年证明了生成元一致连续的倒向随机微分 方程的~$L^2$~解的存在性, 其~$L^2$~解的唯一性由范胜君等于2010年得到. 本文进一步地证明了该类倒向随机微分 方程的~$L^p\ (p>1)$~解的存在唯一性.     

One-dimensional BSDEs with finite and infinite time horizons

This paper is devoted to solving one-dimensional backward stochastic differential equations (BSDEs), where the time horizon may be finite or infinite and the assumptions on the generator g are not necessary to be uniform on t. We first show the existence of the minimal solution for this kind of BSDEs with linear growth generators. Then, we establish a general comparison theorem for solutions of this kind of BSDEs with weakly monotonic and uniformly continuous generators. Finally, we give an existence and uniqueness result for solutions of this kind of BSDEs with uniformly continuous generators.     

具有一致连续生成元和可积参数的倒向随机微分方程

本文建立了具有可积参数的一维倒向随机微分方程(BSDE) 的一个新的存在唯一性结果, 其中BSDE 的生成元g 关于y 满足Osgood 条件且关于z 是α-Hölder (0 < α < 1) 连续的.     

倒向随机微分方程Lp解的性质

2003年Briand et al等在很一般的假设下建立了倒向随机微分方程（BSDEs）L^p解的存在唯一性定理.本文在此基础上得到了这种假设下一维BSDEs的L^p解的几个连续性质.     

BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces

This paper is devoted to real valued backward stochastic differential equations (BSDEs for short) with generators which satisfy a stochastic Lipschitz condition involving BMO martingales. This framework arises naturally when looking at the BSDE satisfied by the gradient of the solution to a BSDE with quadratic growth in Z$Z$. We first prove an existence and uniqueness result from which we deduce the differentiability with respect to parameters of solutions to quadratic BSDEs. Finally, we apply these results to prove the existence and uniqueness of a mild solution to a parabolic partial differential equation in Hilbert space with nonlinearity having quadratic growth in the gradient of the solution.     

Existence,Uniqueness and Approximation for L~p Solutions of Reflected BSDEs with Generators of One-sided Osgood Type

We prove several existence and uniqueness results for L~p(p 1) solutions of reflected BSDEs with continuous barriers and generators satisfying a one-sided Osgood condition together with a general growth condition in y and a uniform continuity condition or a linear growth condition in z.A necessary and sufficient condition with respect to the growth of barrier is also explored to ensure the existence of a solution. And, we show that the solutions may be approximated by the penalization method and by some sequences of solutions of reflected BSDEs. These results are obtained due to the development of those existing ideas and methods together with the application of new ideas and techniques, and they unify and improve some known works.     

具有一般终端时刻和非一致线性增长生成元的倒向随机微分方程的$L^p$解

In this paper, we establish the existence of the minimal L~p(p 1) solution of backward stochastic differential equations(BSDEs) where the time horizon may be finite or infinite and the generators have a non-uniformly linear growth with respect to t. The main idea is to construct a sequence of solutions {(Y~n, Z~n)} which is a Cauchy sequence in S~p× M~p space, and finally we prove {(Y~n, Z~n)} converges to the L~p(p 1) solution of BSDEs.     

Existence,uniqueness and approximation for <Emphasis Type="Italic">L</Emphasis><Superscript>p</Superscript> solutions of reflected BSDEs with generators of one-sided Osgood type

We prove several existence and uniqueness results for L ^p (p > 1) solutions of reflected BSDEs with continuous barriers and generators satisfying a one-sided Osgood condition together with a general growth condition in y and a uniform continuity condition or a linear growth condition in z. A necessary and sufficient condition with respect to the growth of barrier is also explored to ensure the existence of a solution. And, we show that the solutions may be approximated by the penalization method and by some sequences of solutions of reflected BSDEs. These results are obtained due to the development of those existing ideas and methods together with the application of new ideas and techniques, and they unify and improve some known works.     

Lp solutions of backward stochastic differential equations

In this paper, we are interested in solving backward stochastic differential equations (BSDEs for short) under weak assumptions on the data. The first part of the paper is devoted to the development of some new technical aspects of stochastic calculus related to BSDEs. Then we derive a priori estimates and prove existence and uniqueness of solutions in L^p p>1, extending the results of El Karoui et al. (Math. Finance 7(1) (1997) 1) to the case where the monotonicity conditions of Pardoux (Nonlinear Analysis; Differential Equations and Control (Montreal, QC, 1998), Kluwer Academic Publishers, Dordrecht, pp. 503–549) are satisfied. We consider both a fixed and a random time interval. In the last section, we obtain, under an additional assumption, an existence and uniqueness result for BSDEs on a fixed time interval, when the data are only in L¹.     

Reflected and Doubly Reflected BSDEs for Levy Processes： Solutions and Comparison

In this paper we study reflected and doubly reflected backward stochastic differential equations （BSDEs, for short） driven by Teugels martingales associated with L~vy process satisfying some moment condi- tions and by an independent Brownian motion. For BSDEs with one reflecting barrier, we obtain a comparison theorem using the Tanaka-Meyer formula. For BSDEs with two reflecting barriers, we first prove the existence and uniqueness of the solutions under the Mokobodski＇s condition by using the Snell envelope theory and then we obtain a comparison result.     

对流扩散方程的解

关注如下的对流扩散方程 $$ u_{t}=\text{div}(|\nabla u^{m}|^{p-2}\nabla u^{m})+\sum_{i=1}^{N}\frac{\partial b_{i}(u^{m})}{\partial x_{i}} $$ 的初边值问题. 若 $p>1+\frac{1}{m}$, 通过考虑正则化问题的解 $u_{k}$, 利用 Moser 迭代技巧, 得到了$u_{k}$ 的 $L^{\infty}$ 模与 梯度 $\nabla u_{k}$ 的 $L^{p}$ 模的局部有界性. 利用紧致性定理, 得到了对流扩散方程本身解的存在性. 若 $p<1+\frac{1}{m},\ p>2$ 或者 $p=1+\frac{1}{m}$, 利用类似的方法可以得到解的存在性. 证明了解的唯一性, 同时讨论了正性和熄灭性等解的性质.     

Multidimensional BSDEs with weak monotonicity and general growth generators

This paper aims at solving a multidimensional backward stochastic differential equation （BSDE） whose generator g satisfies a weak monotonicity condition and a general growth condition in y. We first establish an existence and uniqueness result of solutions for this kind of BSDEs by using systematically the technique of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Then, we overview some assumptions related closely to the monotonieity condition in the literature and compare them in an effective way, which yields that our existence and uniqueness result really and truly unifies the Mao condition in y and the monotonieity condition with the general growth condition in y, and it generalizes some known results. Finally, we prove a stability theorem and a comparison theorem for this kind of BSDEs, which also improves some known results.     

L p (p>1) solutions for one-dimensional BSDEs with linear-growth generators

In this paper we study the existence and uniqueness of L ^p (p>1) solutions for one-dimensional backward stochastic differential equations (BSDEs in short) whose generator g and terminal condition ?? satisfy $\mathbf{E}[|\xi|^{p}+(\int_{0}^{T} |g(t,0,0)|\, \mathrm {d}t)^{p}]<+\infty$ . We get an existence result under the condition that g is continuous and of linear growth in (y,z). And, we also prove an existence and uniqueness result where g satisfies the Osgood condition in y and is uniformly continuous in z. Particularly, a comparison theorem for L ^p (p>1) solutions of BSDEs is obtained in this paper.     

Quadratic BSDEs with convex generators and unbounded terminal conditions

In Briand and Hu (Probab Theory Relat Fields 136(4):604–618, 2006), the authors proved an existence result for BSDEs with quadratic generators with respect to the variable z and with unbounded terminal conditions. However, no uniqueness result was stated in that work. The main goal of this paper is to fill this gap. In order to obtain a comparison theorem for this kind of BSDEs, we assume that the generator is convex with respect to the variable z. Under this assumption of convexity, we are also able to prove a stability result in the spirit of the a priori estimates stated in Karoui et al. (Math Finance 7(1):1–71, 1997). With these tools in hands, we can derive the nonlinear Feynman–Kac formula in this context.     

Existence and uniqueness result for multidimensional BSDEs with generators of Osgood type

This paper is interested in solving a multidimensional backward stochastic differential equation (BSDE) whose generator satisfies the Osgood condition in y and the Lipschitz condition in z. We establish an existence and uniqueness result of solutions for this kind of BSDEs, which generalizes some known results.     

Solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian Motions

The local existence and uniqueness of the solutions to backward stochastic differential equations(BSDEs, in short) driven by both fractional Brownian motions with Hurst parameter H ∈(1/2, 1) and the underlying standard Brownian motions are studied. The generalization of the It formula involving the fractional and standard Brownian motions is provided. By theory of Malliavin calculus and contraction mapping principle, the local existence and uniqueness of the solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian motions are obtained.     

BSDEs and SDEs with time-advanced and -delayed coefficients

ABSTRACT

This paper introduces a class of backward stochastic differential equations (BSDEs), whose coefficients not only depend on the value of its solutions of the present but also the past and the future. For a sufficiently small time delay or a sufficiently small Lipschitz constant, the existence and uniqueness of such BSDEs is obtained. As an adjoint process, a class of stochastic differential equations (SDEs) is introduced, whose coefficients also depend on the present, the past and the future of its solutions. The existence and uniqueness of such SDEs is proved for a sufficiently small time advance or a sufficiently small Lipschitz constant. A duality between such BSDEs and SDEs is established.     

Mean-Field Backward Stochastic Differential Equations Driven by Fractional Brownian Motion

In this paper, we study a new class of equations called mean-field backward stochastic differential equations(BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H 1/2. First, the existence and uniqueness of this class of BSDEs are obtained. Second, a comparison theorem of the solutions is established. Third, as an application, we connect this class of BSDEs with a nonlocal partial differential equation(PDE, for short), and derive a relationship between the fractional mean-field BSDEs and PDEs.     

Finite and infinite time interval BSDEs with non-Lipschitz coefficients

This paper aims at solving multidimensional backward stochastic differential equations (BSDEs) under weaker assumptions on the coefficients, considering both a finite and an infinite time interval. We establish a general existence and uniqueness result of the solutions to finite and infinite time interval BSDEs with non-Lipschitz coefficients, which generalizes the corresponding results in Mao (1995), Wang and Wang (2003), Wang and Huang (2009), Chen (1997) and Chen and Wang (2000).     

Markovian Quadratic BSDEs with an Unbounded Sub-quadratic Growth

This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations (BSDEs for short) with bounded terminal conditions. The generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable z. The existence and uniqueness results are given to these BSDEs. As an application, an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.