On solutions to fuzzy stochastic differential equations with local martingales

This paper deals with a class of fuzzy stochastic differential equations (FSDEs) driven by a continuous local martingale under the Lipschitzian condition. Such equations can be useful in modeling hybrid systems, where the phenomena are simultaneously subjected to two kinds of uncertainties: randomness and fuzziness. The solutions of the FSDEs are the fuzzy stochastic processes, and their uniqueness is considered to be in a strong sense. Thus, the existence and uniqueness of solutions to the FSDEs under the Lipschitzian condition is first proven. Moreover, some asymptotic properties of the solutions to the FSDEs are investigated. Finally, an illustrating example on the interest term model is provided.     

On partial-information optimal singular control problem for mean-field stochastic differential equations driven by Teugels martingales measures

This paper is concerned with partial-information mixed optimal stochastic continuous–singular control problem for mean-field stochastic differential equation driven by Teugels martingales and an independent Brownian motion, where the Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes. The control variable has two components; the first being absolutely continuous, and the second singular. Partial-information necessary and sufficient conditions of optimal continuous–singular control for these mean-field models are investigated. As an illustration, this paper studies a partial-information linear quadratic control problem of mean-field type involving continuous–singular control.     

随机限制市场中不定权益的对冲成本

本文是2004年陈典发工作([1])的继续,我们研究随机限制市场中欧式不定权益的对冲问题,获得了它们的上对冲成本以及对冲交易策略的风险溢价表达式。     

On optimal control of mean-field stochastic systems driven by Teugels martingales via derivative with respect to measures

ABSTRACT

This paper deals with partial information stochastic optimal control problem for general controlled mean-field systems driven by Teugels martingales associated with some Lévy process having moments of all orders, and an independent Brownian motion. The coefficients of the system depend on the state of the solution process as well as of its probability law and the control variable. We establish a set of necessary conditions in the form of Pontryagin maximum principle for the optimal control. We also give additional conditions, under which the necessary optimality conditions turn out to be sufficient. The proof of our result is based on the derivative with respect to the probability law by applying Lions derivatives and a corresponding Itô formula. As an application, conditional mean-variance portfolio selection problem in incomplete market, where the system is governed by some Gamma process is studied to illustrate our theoretical results.     

连续参数集值鞅的正则性

给出了连续参数集值正则鞅的几个充要条件，并给出了连续参数集值正则鞅的鞅选择定理及表示定理。     

广义Brownian Sheet的鞅刻画

在文献［１］的基础上给出了广义ＢｒｏｗｎｉａｎＳｈｅｅｔ鞅刻画：两参数连续适应过程（Ｂ２）ｚ∈Ｒ２＋为广义ＢｒｏｗｎｉａｎＳｈｅｅｔ的充要条件是（Ｂ２）ｚ∈Ｒ２＋为Ｆｏｕｑｕｅ意义下的两参数局部鞅，且存在Ｒ２＋上的Ｌ—Ｓ测度ｄ，（ｄ《λ，λ为Ｒ２＋上的Ｌｅｂｅｓｇｕｅ测度）使得（Ｂ２ｚ－ｄ（Ｚ）ｚ∈Ｒ２＋为Ｆｏｕｑｕｅ意义下的两参数局部鞅，其中以ｄ（ｚ）＝ｄ（（０，ｚ］）。     

On uniform integrability and asymptotically risk-efficient sequential estimation

In the theory of asymptotically risk-efficient sequential estimation, one encounters uniform integrability of both positive and negative powers of stopping rules and moment convergence of randomly stopped statistics. We describe a simple approach to obtain these uniform integrability and moment convergence results, not only in the classical setting of i.i.d. observations but also for much more general stochastic sequences. We also use this approach to establish asymptotic risk efficiency of sequential estimators of means of stochastic sequences and to derive asymptotic approximations for the mean squared errors of ratio estimators.     

关于停点的停止定理

研究了停点的停止定理，推广了中科院数学所陈培德、法国数学家Mazziotto.G、以色列数学家Merzbach．E等人的结果，得到了上鞅（鞅）停止定理和可料形式停止定理；在两停点可比较条件下，得到了上鞅（鞅）加强的停止定理和加强的可料形式停止定理。     

局部平方可积鞅的泛函重对数律

采用离散化方法和截尾技巧，由鞅的指数不等式和Skorohod嵌入定理，得到局部平方可积鞅在Kolmogorov条件下的泛函重对数律。     

Necessary and sufficient conditions for optimal control of stochastic systems associated with Lévy processes

The paper is concerned with a stochastic optimal control problem where the controlled systems are driven by Teugel’s martingales and an independent multi-dimensional Brownian motion. Necessary and sufficient conditions for an optimal control of the control problem with the control domain being convex are proved by the classical method of convex variation, and the coefficients appearing in the systems are allowed to depend on the control variables. As an application, the linear quadratic stochastic optimal control problem is studied.