ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES

In this paper, we focus on anticipated backward stochastic Volterra integral equations(ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps.Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.

EXISTENCE OF SOLUTION AND APPROXIMATE CONTROLLABILITY OF A SECOND-ORDER NEUTRAL STOCHASTIC DIFFERENTIAL EQUATION WITH STATE DEPENDENT DELAY

This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanam’s novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.

BACKWARD STOCHASTIC DIFFERENTIAL EQUATION WITH RANDOM MEASURES

Backward stochastic differential equations (BSDE) are discussed in many papers. However, in those papers, only Brownian motion and Poisson process are considered. In this paper, we consider BSDE driven by continuous local martingales and random measures.

Forward-backward doubly stochastic differential equations and related stochastic partial differential equations

The notion of bridge is introduced for systems of coupled forward-backward doubly stochastic differential equations (FBDSDEs). It is proved that if two FBDSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBDSDEs. Finally, the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential equations (SPDEs) combined with algebra equations is given. One distinctive character of this result is that the forward component of the FBDSDEs is coupled with the backward variable.

Backward Doubly Stochastic Differential Equations with Jumps and Stochastic Partial Differential-Integral Equations

Backward doubly stochastic differential equations driven by Brownian motions and Poisson process(BDSDEP) with non-Lipschitz coeffcients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations(SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.

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

This paper discusses mean-field backward stochastic differential equations(mean-field BSDEs) 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 Maximum Principle for One Kind of Stochastic Optimization Problem and Application in Dynamic Measure of Risk

The authors get a maximum principle for one kind of stochastic optimization problem motivated by dynamic measure of risk. The dynamic measure of risk to an investor in a financial market can be studied in our framework where the wealth equation may have nonlinear coefficients.

Maximum principle for optimal control of neutral stochastic functional differential systems

This paper is concerned with optimal control of neutral stochastic functional differential equations(NSFDEs). The Pontryagin maximum principle is proved for optimal control, where the adjoint equation is a linear neutral backward stochastic functional equation of Volterra type(VNBSFE). The existence and uniqueness of the solution are proved for the general nonlinear VNBSFEs. Under the convexity assumption of the Hamiltonian function, a sufficient condition for the optimality is addressed as well.

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.

风险资产的一般定价模型

研究了连续时间下风险资产的一般定价模型.利用时间-风险折现方法实现了风险资产在实际测度下的定价,并给出其具体的价格表达式.

带跳的倒向重随机微分方程的比较定理

在Lipschitz条件下,利用Gronwall不等式、Young不等式和Ito^公式等,得到了带跳的倒向重随机微分方程解的比较定理,说明了带跳的倒向重随机微分方程的系数和终端值越大,其解越大.

基于基差风险模型带红利支付的最优交易策略

在部分信息下研究"基差风险模型"中引入红利率影响因素时,投资人终端净财富期望指数效用最大化问题.先建立部分信息下的金融市场框架和带有红利支付的"基差风险模型",再应用半鞅理论和倒向随机微分方程,求解价值过程和最优交易策略精确解.

时间依赖的关卡期权定价

讨论了一种新型期权———关卡期权的定价问题.一般关于关卡期权的讨论往往只涉及比较简单的情况,即期权障碍是恒定不变的,但实际上期权障碍是会随时间而变化的,文中在关卡期权的关卡值关于时间依赖的假设下,借助倒向随机微分方程方法和等价鞅方法,推导出一种欧式下降敲出看涨关卡期权的定价公式.

三叉树模型下标的资产期权定价

讨论了一般三叉树模型期权定价公式 ,它是二叉树模型的推广。证明了三叉树模型下期权价格所满足的方程是Black

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 ;

反射型的带跳倒向双重随机微分方程(英文)

证明了反射型的带跳倒向双重随机微分方程的解的存在唯一性.主要方法是Snell包和不动点定理.

局部Lipschitz条件下的带跳倒向重随机微分方程

在局部Lipschitz条件下,利用Gronwall不等式、Holder不等式和Ito公式等,得到了任意给定时间区间上,布朗运动和泊松过程混合驱动的倒向重随机微分方程解的存在唯一性结果,从而推广了谷艳玲以及孙晓君和卢英的相关结果.

白噪声和泊松随机测度驱动的倒向重随机微分方程

本文研究了一类由白噪声和泊松随机测度驱动的倒向重随机微分方程,并建立了此类方程解的定义以及Yamada-Watanabe定理.

实际概率测度下一般风险资产的定价模型

本文利用倒向随机微分方程研究了连续时间下基于可交易证券的风险资产定价模型。解决了如何在实际测度及不知无风险利率的情况下对一般(可能不可交易)资产进行定价的问题。与此同时,得到了远期测度与实际测度之间的关系,此关系不依赖于风险中性测度。

一类由广义g-期望诱导的相干风险度量及其表示定理

本文通过广义g-期望引入了一类风险度量ρg,并在:g=-rtg-μ|z|时给出了相干风险度量ρg的表示定理。