正倒向随机微分方程的适应解及其比较定理

研究了一类正倒向随机微分方程的适应解，其中正向方程不需要满足非退化条件，我们证明了在某些单调条件下，正倒向随机微分方程存在唯一的适应解，并给出了该正倒向随机微分方程的比较定理。     

正倒向随机微分方程解的比较定理

讨论了正倒向随机微分方程解的比较问题.阐述了正倒向随机微分方程在随机最优控制、现代金融理论中的广泛而深刻的应用, 对于一类正倒向随机微分方程, 利用Ito公式、停时等随机分析方法,通过构造辅助正倒向随机微分方程,得到了正倒向随机微分方程解的比较定理.     

倒向随机Volterra积分方程适应解的表示

倒向随机Volterra积分方程可以看作(确定性)Volterra积分方程和倒向随机微分方程的推广,在随机最优控制理论和数学金融学中有诸多应用.本文利用正倒向随机微分方程适应解表示的思想,得到所研究的一类倒向随机Volterra积分方程适应解的表示.这样的结果对研究适应解的正则性以及数值计算有重要的意义.     

带吸收系数的正倒向随机微分方程的可解性

利用叠代估计方法研究带吸收系数的正倒向随机微分方程的可解性,在正向随机微分方程的扩散系数可以退化的情形下,证明了适应解的存在性和唯-性,也研究这类正倒向随机微分方程与偏微分方程的联系.     

非Lipschitz条件下的包含下微分算子的带跳倒向随机微分方程(英文)

本文在非Lipschitz系数下,考虑了一类多值的倒向随机微分方程.利用极大单调算子的Yosida估计和倒向随机微分方程在非Lipschitz条件下解的存在唯一性,获得了多值带跳的倒向随机微分方存在唯一解的结论.     

关于反射倒向随机微分方程的解的一些性质(英文)

研究了关于反射倒向随机微分方程的解的一些性质.同时在适当的条件下建立了关于反射倒向随机微分方程生成元的一个唯一性定理和一个逆比较定理.     

正倒向随机微分方程理论基础及相关应用

本文从随机微分方程和倒向随机微分方程基本理论和应用背景谈起,结合随机最优控制理论和金融市场中的期权定价理论导出完全耦合的正倒向随机微分方程的形式.进而从该类方程的可解性这一角度出发,对已有的理论方法进行分析和探讨,引入一种非马尔科夫框架下保证解的存在唯一性的“统一框架”方法,给出比较定理、解的高维估计等重要性质,并联系相关偏微分方程系统给出其概率解释.对实际中应用广泛的线性正倒向随机微分方程引入了一种线性变换的方法作为“统一框架”方法的重要补充和完善,使得正倒向随机微分方程的应用更加广泛.     

基于超前倒向随机微分方程的动态风险度量

本文研究了基于超前倒向随机微分方程的时间相容的过程的动态凸(一致性)风险度量的问题.利用对超前倒向随机微分方程生成元的适当假设,建立超前倒向随机微分方程生成元与过程的动态凸(一致性)风险度量的对应模型,证明了超前倒向随机微分方程的解可以定义时间相容的过程的风险度量.得到了基于超前倒向随机微分方程的风险度量,推广了基于倒向随机微分方程的动态风险度量.由于超前倒向随机微分方程生成元中包含当前时刻和未来时刻的解,因此本文的结论对风险的预测更加可靠.     

随机Lipschitz条件下一般时间终端的多维BSDE解的存在唯一性

某个新的范数空间上利用压缩映像原理,在生成元g满足对二元组■均不一致的随机Lipschitz条件下,证明了一般时间终端多维倒向随机微分方程(BSDE)解的存在唯一性,作为推论得到了该类倒向随机微分方程解的递归迭代序列的收敛性.     

一类非Lipschitz条件的Backward SDE适应解的存在唯一性

本文中，我们在非Lipschitz条件下证明了倒向随机微分方程的局部与整体适应解的存在唯一性，推广了PaLrdoux-Peng定理．     

Curve following in illiquid markets

In this article the problem of curve following in an illiquid market is addressed. The optimal control is characterised in terms of the solution to a coupled FBSDE involving jumps via the technique of the stochastic maximum principle. Analysing this FBSDE, we further show that there are buy and sell regions. In the case of quadratic penalty functions the FBSDE admits an explicit solution which is determined via the four step scheme. The dependence of the optimal control on the target curve is studied in detail.     

Risk‐based Decisions on the Asset Structure of a Bank under Partial Economic Information

We present a model of a bank's dynamic asset management problem in the case of partially observed future economic conditions and with regulatory requirements governing the level of risk taken. The result is an optimal control problem with a random terminal condition arising from the partial observation of a parameter of a maximized functional. The Stochastic Maximum Principle reduces the problem to finding a solution to a Forward Backward Stochastic Differential Equation (FBSDE). As optimization usually implies the dependence of the forward equation on solutions of the backward equation we allow the drift and diffusion of the forward part to be functions of the solution of the backward equation. The necessary conditions for the existence of solutions of FBSDE in such a form are derived. A numerical scheme is then implemented to solve a particular case.     

On weak solutions of forward�Cbackward SDEs

In this paper we continue exploring the notion of weak solution of forward?Cbackward stochastic differential equations (FBSDEs) and associated forward?Cbackward martingale problems (FBMPs). The main purpose of this work is to remove the constraints on the martingale integrands in the uniqueness proofs in our previous work (Ma et?al. in Ann Probab 36(6):2092?C2125, 2008). We consider a general class of non-degenerate FBSDEs in which all the coefficients are assumed to be essentially only bounded and uniformly continuous, and the uniqueness is proved in the space of all the square integrable adapted solutions, the standard solution space in the FBSDE literature. A new notion of semi-strong solution is introduced to clarify the relations among different definitions of weak solution in the literature, and it is in fact instrumental in our uniqueness proof. As a by-product, we also establish some a priori estimates of the second derivatives of the solution to the decoupling quasilinear PDE.     

Forward–backward SDEs with jumps and classical solutions to nonlocal quasilinear parabolic PDEs

We obtain an existence and uniqueness theorem for fully coupled forward–backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of Ladyzhenskaya et al. (1968) to non-local PDEs of this class. Namely, we obtain the existence and uniqueness of a classical solution to both the Cauchy problem and the initial–boundary value problem for non-local quasilinear parabolic second-order PDEs.     

A Note on FBSDE characterization of mean exit times

In this Note, we present a new explicit characterization for a mean exit time problem recently treated by the author, in form of a quadratic Forward–Backward Stochastic Differential Equation (FBSDE) with a random terminal time. An a priori estimate and a uniqueness result for such a type of FBSDE are also proved, under certain conditions. To cite this article: C. Makasu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Forward-backward SDEs with discontinuous coefficients

In this paper, we are interested in the well-posedness of a class of fully coupled forward-backward SDE (FBSDE) in which the forward drift coefficient is allowed to be discontinuous with respect to the backward component of the solution. Such an FBSDE is motivated by a practical issue in regime-switching term structure interest rate models, and the discontinuity makes it beyond any existing framework of FBSDEs. In a Markovian setting with non-degenerate forward diffusion, we show that a decoupling function can still be constructed and that it is a Sobolev solution to the corresponding quasilinear PDE. As a consequence we can then argue that the FBSDE admits a weak solution in the sense of [1 Antonelli, F., Ma, J. (2003). Weak solutions of forward-backward SDE’s. Stochastic Analysis and Applications 21(3):493–514.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 2 Ma, J., Zhang, J., Zheng, Z. (2008). Weak solutions for backward stochastic differential equations, A martingale approach. The Annals of Probability 36(6):2092–2125.[Crossref], [Web of Science ®] , [Google Scholar]]. In the one-dimensional case, we further prove that the weak solution of the FBSDE is actually strong, and it is pathwisely unique. Our approach does not use the well-known Yamada–Watanabe Theorem, but instead follows the idea of Krylov for SDEs with measurable coefficients.     

Terminal-Dependent Statistical Inferences for FBSDE

Backward stochastic differential equation (BSDE) has been well studied and widely applied in mathematical finance. The main difference from the original stochastic differential equation (OSDE) is that the BSDE is designed to depend on a terminal condition, which plays key roles in certain financial and ecological circumstances. However, to the best of our knowledge, the terminal-dependent statistical inference for such model has not been explored in the existing literature. This article proposes two terminal-dependent estimation methods via terminal control variable the integral form models of forward-backward stochastic differential equation (FBSDE). We take these measures because the resulting models contain terminal condition as model variable, and therefore, the corresponding estimators inherit the terminal-dependent characteristic. In this article, the FBSDE is first rewritten as regression versions and then two semi-parametric estimation approaches are proposed. Because of the control variable and integral form, our regression versions are more complex than the classical ones, and the inference methods are somewhat different from which designed for the OSDE. Even so, the statistical properties of the terminal-dependent methods are similar to the classical ones. Simulations are conducted to demonstrate finite sample behaviors.     

On Solutions of BSDE with Two Barriers Under Non-Lipschitz Condition

In this paper,we derive the existence and uniqueness theorem for the adapted solution to backward stochastic differential equations with two barriers under non-Lipschitz condition via penalization method.     

Solving forward-backward stochastic differential equations explicitly — a four step scheme

Summary In this paper we investigate the nature of the adapted solutions to a class of forward-backward stochastic differential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration, via a direct Four Step Scheme. Using this scheme, we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic PDE system. Moreover the uniqueness of the adapted solutions (over an arbitrary time duration), as well as the continuous dependence of the solutions on the parameters, can all be proved within this unified framework. Some special cases are studied separately. In particular, we derive a new form of the integral representation of the Clark-Haussmann-Ocone type for functionals (or functions) of diffusions, in which the conditional expectation is no longer needed.Supported in part by U.S. NSF grant# DMS-9301516Supported in part by U.S. NSF grant # DMS-9103454Supported in part by NSF of China and Fok Ying Tung Education Foundation; part of this work was performed while visiting the IMA, University of Minnesota, Minneapolis, MN 55455     

Stochastic Navier-Stokes equations

We construct a solution to stochastic Navier-Stokes equations in dimension n4 with the feedback in both the external forces and a general infinite-dimensional noise. The solution is unique and adapted to the Brownian filtration in the 2-dimensional case with periodic boundary conditions or, when there is no feedback in the noise, for the Dirichlet boundary condition. The paper uses the methods of nonstandard analysis.The research of this author was supported by an SERC Grant.