ON SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS,WITH UNBOUNDED STOPPING TIMES AS TERMINAL AND WITH NON-LIPSCHITZ COEFFICIENTS,AND PROBABILISTIC INTERPRETATION OF QUASI-LINEAR ELLIPTIC TYPE INTEGRO-DIFFERENTIAL EQUATIO

The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi_linear elliptic type integro_differential equations is obtained.

A Mean-Field Stochastic Maximum Principle for Optimal Control of Forward-Backward Stochastic Differential Equations with Jumps via Malliavin Calculus

This paper considers a mean-field type stochastic control problem where the dynamics is governed by a forward and backward stochastic differential equation (SDE) driven by Lévy processes and the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.

Controlled Mean-Field Backward Stochastic Differential Equations with Jumps Involving the Value Function

This paper discusses mean-field backward stochastic differentiM equations （mean-field BS- DEs） with jumps and a new type of controlled mean-field BSDEs with jumps, namely mean-field BSDEs with jumps strongly coupled with the value function of the associated control problem. The authors first prove the existence and the uniqueness as well as a comparison theorem for the above two types of BSDEs. For this the authors use an approximation method. Then, with the help of the notion of stochastic backward semigroups introduced by Peng in 1997, the authors get the dynamic programming principle （DPP） for the value functions. Furthermore, the authors prove that the value function is a viscosity solution of the associated nonlocal Hamilton-Jacobi-Bellman （HJB） integro-partial differential equation, which is unique in an adequate space of continuous functions introduced by Barles, et al. in 1997.

The Link between Stochastic Differential Equations with Non-Markovian Coefficients and Backward Stochastic Partial Differential Equations

In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extension of such kind of link in Markovian framework to non-Markovian framework.Different from Markovian framework, where the corresponding partial differential equation is deterministic, the backward stochastic partial differential equation here has a pair of adapted solutions, and thus the link has a much different form. Moreover, two examples are given to demonstrate the applications of the derived link.

Relationship Between General MP and DPP for the Stochastic Recursive Optimal Control Problem with Jumps

This paper is concerned with the relationship between general maximum principle and dynamic programming principle for the stochastic recursive optimal control problem with jumps,where the control domain is not necessarily convex.Relations among the adjoint processes,the generalized Hamiltonian function and the value function are proven,under the assumption of a smooth value function and within the framework of viscosity solutions,respectively.Some examples are given to illustrate the theoretical results.

An Accurate Numerical Scheme for Mean-Field Forward and Backward SDEs with Jumps

In this work,we propose an explicit second order scheme for decoupled mean-field forward backward stochastic differential equations with jumps.The sta-bility and the rigorous error estimates are presented,which show that the proposed scheme yields a second order rate of convergence,when the forward mean-field stochastic differential equation is solved by the weak order 2.0 Itˆo-Taylor scheme.Numerical experiments are carried out to verify the theoretical results.

Path-dependent backward stochastic Volterra integral equations with jumps,differentiability and duality principle

We study the existence and uniqueness of a solution to path-dependent backward stochastic Volterra integral equations(BSVIEs)with jumps,where path-dependence means the dependence of the free term and generator of a path of a c`adl`ag process.Furthermore,we prove path-differentiability of such a solution and establish the duality principle between a linear path-dependent forward stochastic Volterra integral equation(FSVIE)with jumps and a linear path-dependent BSVIE with jumps.As a result of the duality principle we get a comparison theorem and derive a class of dynamic coherent risk measures based on path-dependent BSVIEs with jumps.

BSDEs with Jumps and Path-Dependent Parabolic Integro-differential Equations

This paper deals with backward stochastic differential equations with jumps,whose data(the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonlinear path-dependent parabolic integrodifferential equations, and then obtains a new type of nonlinear Feynman-Kac formula related to such BSDEs with jumps under some regularity conditions.

The Cauchy problem of Backward Stochastic Super-Parabolic Equations with Quadratic Growth

The paper is devoted to the Cauchy problem of backward stochastic superparabolic equations with quadratic growth.We prove two Ito formulas in the whole space.Furthermore,we prove the existence of weak solutions for the case of onedimensional state space,and the uniqueness of weak solutions without constraint on the state space.

基于投资理论的保险定价公式

在保险公司是风险中性的假设下 ,运用倒向随机微分方程的理论 ,研究了保险公司在风险投资框架下的保险定价问题。首先 ,建立了保险定价问题的线性正倒向随机微分方程数学模型 ;然后 ,根据一类特殊线性倒向随机微分方程的显式解 ,推出了由风险投资确定的保险定价公式 ;最后 ,进行了算例分析。

倒向随机微分方程解的比较定理(英文)

毛学荣新近将彭实戈和Pardoux关于倒向随机微分方程解的存在性定理推广到非Lipschitz系数情景．此文将彭实戈的比较定理推广到这一情形．主要工具是Tanaka－Meer公式，Davis不等式和Bihari不等式．

多维带跳倒向双重随机微分方程解的性质

本文研究一类多维带跳倒向双重随机微分方程,给出了It(?)公式在带跳倒向双重随机积分情形下的推广形式,同时运用推广形式的It(?)公式,在Lipschitz条件下证明了方程解的存在性和唯一性。

收益流不连续时项目最佳投资时机分析

讨论了当项目收益流为不连续随机过程时,投资时机的选择问题.把投资机会看作一种实物期权,并用B rown运动和Poisson跳过程分别刻画收益流受到的连续性随机扰动和随机冲击(引起收益流不连续的随机事件),证明在此情况下可利用收益流当前值的临界值作为投资时机选择的依据.进而,用带跳反射倒向随机微分方程方法解出这一临界值.

倒向随机微分方程及其应用

本文将介绍一类新的方程：倒向随机微分方程．为便于理解，我们将首先通过与常微分方程和经典的随机微分方程（Ｉｔ．ｏ方程）的对比．并通过数理经济和数学金融学中的一个典型的例子来引入倒向随机微分方程．然后给出解的存在唯一性定理和比较定理．并介绍非线性Ｆｅｙｎｍａｎ－Ｋａｃ公式，它给出了倒向随机微分方程的解与一大类常见的非线性偏微分方程（组）的解之间的对应关系，从而为将来利用Ｍｏｎｔé－Ｃａｒｌｏ型的随机计算方法计算大量的偏微分方程开辟了新的途径．

无界停时带跳倒向随机微分方程解的存在唯一性(Ⅰ)

研究以无界停时为终端的带跳倒向随机微分方程在李氏条件下解的存在唯一性，其解存在的空间与终端为有界停时的情形不同．

具有不同借贷利率的BlackScholes模型

在不同借贷利率以及股票的期望收益率、波动率和红利率都随时间变化（非随机）情形下，利用倒向随机微分方程及Ｆｅｙｎｍ ａｎＫａｃ公式得到欧式看涨和看跌期权定价公式．由此可看出借贷利率各自对期权价格的影响，并得到欧式看涨和看跌期权的平价关系．

关于系数平方增长的带跳BSDE的解(Ⅰ)

讨论了系数关于q为平方增长,p和-y为指数增长的带跳倒向随机微分方程(BSDE)解的存在性,以及有这种系数的反射BSDE解的存在性。

关于g-上鞅的上穿不等式和强g-上鞅(Ⅰ)

推广了无穷时间水平带跳倒向随机微分方程(BSDE)解的比较定理,并用这种带跳BSDE定义了g＿鞅与g＿上鞅,证明了g＿上鞅的上穿不等式。

非Lipschitz条件下的带跳的倒向随机微分方程

证明了带跳的倒向随机微分方程在某种非Lipschitz条件下的适应解的存在唯一性 ;

带扰动倒向随机微分方程解的存在唯一性

讨论了带扰动PBSDEx(t)+∫1t[f(x(s))+F(x(s))]ds+∫1t[g(x(s))+G(x(s))+y(s)]dW(s)=X解的存在唯一性问题.