Construction of continuous time Markovian arrival processes

Markovian arrival processes were introduced by Neuts in 1979 (Neuts 1979) and have been used extensively in the stochastic modeling of queueing, inventory, reliability, risk, and telecommunications systems. In this paper, we introduce a constructive approach to define continuous time Markovian arrival processes. The construction is based on Poisson processes, and is simple and intuitive. Such a construction makes it easy to interpret the parameters of Markovian arrival processes. The construction also makes it possible to establish rigorously basic equations, such as Kolmogorov differential equations, for Markovian arrival processes, using only elementary properties of exponential distributions and Poisson processes. In addition, the approach can be used to construct continuous time Markov chains with a finite number of states     

On optimal mean-field control problem of mean-field forward-backward stochastic system with jumps under partial information

This paper considers the problem of partially observed optimal control for forward-backward stochastic systems driven by Brownian motions and an independent Poisson random measure with a feature that the cost functional is of mean-field type. When the coefficients of the system and the objective performance functionals 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. The authors also investigate the mean-field type optimal control problem for the system driven by mean-field type forward-backward stochastic differential equations (FBSDEs in short) with jumps, where the coefficients contain not only the state process but also its expectation under partially observed information. The maximum principle is established using convex variational technique. An example is given to illustrate the obtained results.     

Backward linear-quadratic stochastic optimal control and nonzero-sum differential game problem with random jumps

This paper studies the existence and uniqueness of solutions of fully coupled forward-backward stochastic differential equations with Brownian motion and random jumps. The result is applied to solve a linear-quadratic optimal control and a nonzero-sum differential game of backward stochastic differential equations. The optimal control and Nash equilibrium point are explicitly derived. Also the solvability of a kind Riccati equations is discussed. All these results develop those of Lim, Zhou (2001) and Yu, Ji (2008).     

Recent progress on controllability/observability for systems governed by partial differential equations

<正> The main purpose of this paper is to overview some recent methods and results on controllability/observability problems for systems governed by partial differential equations.First,the authorsreview the theory for linear partial differential equations,including the iteration method for the nullcontrollability of the time-invariant heat equation and the Rellich-type multiplier method for the exactcontrollability of the time-invariant wave equation,and especially a unified controllability/observabilitytheory for parabolic and hyperbolic equations based on a global Carleman estimate.Then,the authorspresent sharp global controllability results for both semi-linear parabolic and hyperbolic equations,based on linearization approach,sharp observability estimates for the corresponding linearized systemsand the fixed point argument.Finally,the authors survey the local null controllability resultfor a class of quasilinear parabolic equations based on the global Carleman estimate,and the localexact controllability result for general hyperbolic equations based on a new unbounded perturbationtechnique.     

Vulnerable European Call Option Pricing Based on Uncertain Fractional Differential Equation

This paper presents two new versions of uncertain market models for valuing vulnerable European call option. The dynamics of underlying asset, counterparty asset, and corporate liability are formulated on the basis of uncertain differential equations and uncertain fractional differential equations of Caputo type, respectively, and the solution to an uncertain fractional differential equation of Caputo type is presented by employing the Mittag-Leffler function and α-path. Then, the pricing formul...     

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.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for stochastic systems with Poisson jumps

This paper discusses the H _∞ control problem for a class of linear stochastic systems driven by both Brownian motion and Poisson jumps. The authors give the basic theory about stabilities for such systems, including internal stability and external stability, which enables to prove the bounded real lemma for the systems. By means of Riccati equations, infinite horizon linear stochastic state-feedback H _∞ control design is also extended to such systems.     

Dynamic portfolio choice under the time-varying, jumps, and knight uncertainty of asset return process

By introducing a stochastic element to the double-jump diffusion framework to measure the Knight uncertainty of asset return process,this paper builds the model of dynamic portfolio choice, which maximizes the expected utility of terminal portfolio wealth.Through specifying the state function of uncertainty-aversion,it utilizes the max-min method to derive the analytical solution of the model to study the effect of the time-varying,jumps,and Knight uncertainty of asset return process on dynamic portfolio choice and their interactions.Results of comparative analysis show:The time-varying results in positive or negative intertemporal hedging demand of portfolio,which depends on the coefficient of investor’s risk aversion and the correlation coefficient between return shift and volatility shift;the jumps in asset return overall reduce investor’s demand for the risky asset,which can be enhanced or weakened by the jumps in volatility;due to the existing of the Knight uncertainty,the investor avoids taking large position on risky asset,and the resulting is the improving of portfolio’s steady and immunity.At last,an empirical study is done based on the samples of Shanghai Exchange Composite Index monthly return data from January 1997 to December 2009,which not only tests the theoretical analysis but also demonstrates that the proposed method in the paper is useful from the aspect of portfolio’s equivalent utility.     

非完全市场衍生资产的相关定价法研究

主要研究非完全市场条件下欧式衍生资产的定价问题．利用Hillbert空间的投影理论，首先将Luenberger提出的相关定价法及Bertsimas等人提出的e-套利定价法与向量空间的投影问题联系起来，证明相关定价法与e-套利定价法的一致性及最相关资产的存在性问题，然后利用随机动态规划法研究离散时闻和连续时闻情形欧式衍生资产的风险对冲策略和定价问题，最后得到确定欧式衍生资产最优风险对冲策略所满足的偏微分方程及相应的近似定价。该微分方程在完全市场条件下与Black-Scholes方程完全一致．     

基于BSSDEs的一般跳过程的可违约期权的定价模型

本文采用均值-方差对冲方法, 对具有一般跳过程, 存在违约风险的期权定价做了深入研究, 首先建立了基于违约过程的半鞅的鞅表示定理, 其次定义最优方差鞅测度并构建两个倒向半鞅随机微分方程, 然后找出使成本函数最小的最优投资策略, 从而给出其定价公式. 本文的主要贡献在于首次给出了可违约半鞅过程的倒向随机半鞅微分方程, 并且给出了具有一般跳过程的可违约期权的定价公式, 具有一定的理论意义.     

Robust Designs Through Risk Sensitivity: An Overview

This is an overview paper on the relationship between risk-averse designs based on exponential loss functions with or without an additional unknown(adversarial) term and some classes of stochastic games. In particular, the paper discusses the equivalences between risk-averse controller and filter designs and saddle-point solutions of some corresponding risk-neutral stochastic differential games with different information structures for the players. One of the by-products of these analyses is that risk-averse controllers and filters(or estimators) for control and signal-measurement models are robust,through stochastic dissipation inequalities, to unmodeled perturbations in controlled system dynamics as well as signal and the measurement processes. The paper also discusses equivalences between risk-sensitive stochastic zero-sum differential games and some corresponding risk-neutral three-player stochastic zero-sum differential games, as well as robustness issues in stochastic nonzero-sum differential games with finite and infinite populations of players, with the latter belonging to the domain of mean-field games.     

A Mean-Field Optimal Control for Fully Coupled Forward-Backward Stochastic Control Systems with Lévy Processes

This paper is concerned with a class of mean-field type stochastic optimal control systems, which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales associated to Lévy processes. In these systems, the coefficients contain not only the state processes but also their marginal distribution, and the cost function is of mean-field type as well. The necessary and sufficient conditions for such optimal problems are obtained. Furthermore, the applications to the linear quadratic stochastic optimization control problem are investigated.

     

Controllability and Observability of Stochastic Singular Systems in Banach Spaces

Journal of Systems Science and Complexity - Exact (approximate) controllability and exact (approximate) observability of stochastic singular systems in Banach spaces are discussed. Firstly, the...     

Maximum principle for partially-observed optimal control problems of stochastic delay systems

This paper is concerned with partially-observed optimal control problems for stochastic delay systems. Combining Girsanov’s theorem with a standard variational technique, the authors obtain a maximum principle on the assumption that the system equation contains time delay and the control domain is convex. The related adjoint processes are characterized as solutions to anticipated backward stochastic differential equations in finite-dimensional spaces. Then, the proposed theoretical result is applied to study partially-observed linear-quadratic optimal control problem for stochastic delay system and an explicit observable control variable is given.     

Stochastic differential equations and stochastic linear quadratic optimal control problem with Lévy processes

In this paper, the authors first study two kinds of stochastic differential equations (SDEs) with Lévy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations (BSDEs) driven by Lévy processes, the authors proceed to study a stochastic linear quadratic (LQ) optimal control problem with a Lévy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results. This work was supported by the National Basic Research Program of China (973 Program) under Grant No. 2007CB814904, the Natural Science Foundation of China under Grant No. 10671112 and Shandong Province under Grant No. Z2006A01, and Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20060422018.     

控制系统中随机信号的仿真与分析

对仿真中用到的各种随机信号的数学模型进行了讨论。定量分析了有限带宽白噪声模块BLWN，给出了设定谊模块参数的关系式．对有色噪声的仿真进行了定量的分析和验证。分析了指数相关随机过程的数学模型，指出仿真时可用均匀分布来代替泊松分布，并给出了相应的M函数文件。文中还讨论了仿真中用BLWN模块来近似给出指数相关随机过程的方法。     

巨灾死亡率债券定价模型研究

巨灾死亡率债券是以规避巨灾死亡率风险为目的的金融衍生产品,此类债券的给付与死亡率指数相关联.利用含Poisson频率的跳跃-扩散过程刻画随机死亡率指数发生跳跃的频率与幅度,应用共同单调理论处理不同地区死亡率指数之间的相关性问题.进一步运用王氏变换(Wang transform)方法,在不完全市场中给出巨灾死亡率债券的定价模型.最后,依据我国人口死亡率数据进行实证研究.     

A Class of Jump-Diffusion Stochastic Differential System Under Markovian Switching and Analytical Properties of Solutions

Our article discusses a class of Jump-diffusion stochastic differential system under Markovian switching(JD-SDS-MS). This model is generated by introducing Poisson process and Markovian switching based on a normal stochastic differential equation. Our work dedicates to analytical properties of solutions to this model. First, we give some properties of the solution, including existence,uniqueness, non-negative and global nature. Next, boundedness of first moment of the solution to this model is considered. Third, properties about coefficients of JD-SDS-MS is proved by using a right continuous markov chain. Last, we study the convergence of Euler-Maruyama numerical solutions and apply it to pricing bonds.     

突发事件、灵活性与资源配置

以几何布朗运动和泊松过程为基础,建立了一个可以灵活描述环境不确定性的复合随机过程,在此基础上,运用随机最优控制模型对资产配置问题进行了求解.研究结论表明:①如果稀有事件可以运用泊松过程描述,那么资源配置计划可以运用不动点定理求解;②如果某种资产或某个部门面临的稀有事件会导致其价值降低,那么稀有事件不仅仅会影响在这种资产或部门上的资源投入,还会影响到与这种资产或部门相关的其他资产或部门的资源投入.最后,就金融机构和制造业企业的资源配置问题进行了讨论.结果表明,对于金融机构而言灵活的描述环境的不确定性可以提高资产配置的效率,制造业企业可以通过灵活的资产配置策略有效的对不确定性做出积极反应.     

基于已实现二阶矩预测的期货套期保值策略及对股指期货的应用

资产收益的跳跃行为给套期保值决策带来了挑战. 提出了考虑跳跃、基于预测的VecHAR-RVRCOV-J模型, 首次将高频数据中蕴含的跳跃信息引入套期保值决策, 对期货和现货收益率的已实现二阶矩做异质滞后阶向量自回归, 构造动态套期保值比率的预测统计量. 实证应用中以沪深300股指期货及沪深300指数为对象构建套期保值策略, 在样本内和样本外的综合套保绩效考核上, 新模型优于常用的二元GARCH模型.