Martingale representation theorem for set-valued martingales

The present paper contains a martingale representation theorem for set-valued martingales defined on a filtered probability space with a filtration generated by a Brownian motion. It is proved that such type martingales can be defined by some generalized set-valued stochastic integrals with respect to a given Brownian motion. The main result of the paper is preceded by short part devoted to the definition and some properties of generalized set-valued stochastic integrals.     

Set-valued stochastic integral equations driven by martingales

We consider a notion of set-valued stochastic Lebesgue–Stieltjes trajectory integral and a notion of set-valued stochastic trajectory integral with respect to martingale. Then we use these integrals in a formulation of set-valued stochastic integral equations. The existence and uniqueness of the solution to such the equations is proven. As a generalization of set-valued case results we consider the fuzzy stochastic trajectory integrals and investigate the fuzzy stochastic integral equations driven by bounded variation processes and martingales.     

On Set-Valued Stochastic Integrals

We define a set-valued stochastic integral with respect to a 1-dimensional Brownian motion. The paper develops multivalued analogs to the theory of singlevalued stochastic integrals. It is expected that these results will be useful to study set-valued and fuzzy stochastic analysis.     

Approximation theorems for set-valued stochastic integrals

The article is devoted to new properties of Aumann, Lebesgue, and Itô set-valued stochastic integrals considered in papers [1 Kisielewicz, M. (2014). Properties of generalized set-valued stochastic integrals. Discuss. Math. (DICO) 34:131–147. [Google Scholar],2 Kisielewicz, M., Michta, M. (2017). Integrably bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449:1893–1910.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, it contains some approximation theorems for Aumann and Itô set-valued stochastic integrals. Hence, in particular, it follows that Aumann and Lebesgue set-valued stochastic integrals cover a.s., both for measurable and IF-nonanticipative integrably bounded set-valued stochastic processes.     

Strong solution of Itô type set-valued stochastic differential equation

In this paper, we shall firstly illustrate why we should introduce an It5 type set-valued stochastic differential equation and why we should notice the almost everywhere problem. Secondly we shall give a clear definition of Aumann type Lebesgue integral and prove the measurability of the Lebesgue integral of set-valued stochastic processes with respect to time t. Then we shall present some new properties, especially prove an important inequality of set-valued Lebesgue integrals. Finally we shall prove the existence and the uniqueness of a strong solution to the It5 type set-valued stochastic differential equation.     

Stochastic integral with respect to set-valued square integrable martingales

In this paper, we shall firstly illustrate why we should consider integral of a stochastic process with respect to a set-valued square integrable martingale. Secondly, we shall prove the representation theorem of set-valued square integrable martingale. Thirdly, we shall give the definition of stochastic integral of a stochastic process with respect to a set-valued square integrable martingale and the representation theorem of this kind of integrals. Finally, we shall prove that the stochastic integral is a set-valued sub-martingale.     

有界可料过程关于集值平方可积鞅的集值随机积分

本文定义了一类有界可料过程关于集值平方可积鞅的集值随机积分,并研究了集植随机积分的性质。此为建立集值随机分析的理论奠定了基础。     

集值Lebesgue—Stieltjes积分

本文首先刻划了B(R_ )上的集值测度,其次建立了(R_ B(R_ ))上的集值Lebesgue-Stieltjes积分.最后,进—步建立了集值随机Lebesgue-Stietjes积分的理论.     

Solvability of Langevin differential inclusions with set-valued diffusion terms on Riemannian manifolds

The existence of weak solution is proved for a Langevin type second-order stochastic differential inclusion on a complete Riemannian manifold, having both drift and diffusion terms set-valued. The construction of solution involves integral operators with Riemannian parallel translation and a special sequence of continuous ?-approximations for an upper semicontinuous set-valued mapping with convex bounded closed values, that is proved to converge point-wise to a Borel measurable selection.     

The Itô-Clifford integral

We present a theory of non-commutative stochastic integration analogous to the Itô-theory. It is shown that Wick products of Fermi fields define martingales and that stochastic integrals with respect to these are defined for adapted (operatorvalued) square-integrable integrands. For square-integrable martingales associated with an arbitrary probability gage space a stochastic integral is defined, and a Doob-Meyer decomposition for supermartingales obtained.     

Real-valued Choquet integrals for set-valued mappings

In this paper a new kind of real-valued Choquet integrals for set-valued mappings is introduced, and some elementary properties of this kind of Choquet integrals are studied. Convergence theorems of a sequence of Choquet integrals for set-valued mappings are shown. However, in the case of the monotone convergence theorem of the nonincreasing sequence of Choquet integrals for set-valued mappings, we point out that the integrands must be closed. Specially, this kind of real-valued Choquet integrals for set-valued mappings can be regarded as the Choquet integrals for single-valued functions.     

Approximating stationary points of stochastic optimization problems in Banach space

In this paper, we present a uniform strong law of large numbers for random set-valued mappings in separable Banach space and apply it to analyze the sample average approximation of Clarke stationary points of a nonsmooth one stage stochastic minimization problem in separable Banach space. Moreover, under Hausdorff continuity, we show that with probability approaching one exponentially fast with the increase of sample size, the sample average of a convex compact set-valued mapping converges to its expected value uniformly. The result is used to establish exponential convergence of stationary sequence under some metric regularity conditions.     

Set-Valued Stochastic Integrals and Equations with Respect to Two-Parameter Martingales

This article is concerned with notions of set-valued stochastic integrals driven by two-parameter martingales and increasing processes. We investigate their main properties and we consider next multivalued stochastic integral equations in the plane. We establish the existence and uniqueness of solutions to such equations as well as their additional properties.     

实值可料过程关于模糊集值Wiener过程的伊藤积分

通过利用水平集和承集将实值可料过程关于模糊集值W iener过程的伊藤积分与实值可料过程关于可积有界紧凸集值W iener过程的伊藤积分联系起来,给出相应的定义和性质,并初步探讨了该伊藤积分在随机微分方程方面的应用。     

Stochastic calculus for Markov processes associated with non-symmetric Dirichlet forms

Nakao’s stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting.It? ’s formula in terms of the extended stochastic integrals is obtained.     

不等式约束随机非线性规划的扰动Karush-Kuhn-Tucker系统的解路径

论文聚焦概率测度发生扰动时的随机非线性规划的稳定性分析的研究.目标函数的Lipschitz连续性和可行集值映射的度量正则性条件可保证最优解集合的外半连续性和最优值的Lipschitz连续性.更重要地,本文证明了,如果原问题的极小点处线性无关约束规范和强二阶充分性条件成立,那么存在一Lipschitz连续的解路径满足扰动问题的Karush-Kuhn-Tucker条件.     

A Wong-Zakai-type theorem for certain discontinuous semimartingales

Solutions of stochastic differential equations having differentials of bounded variation processes on the right-hand side can be defined by means of Lebesgue-Stieltjes integrals or by continuous extension of Stieltjes integrals. Both solutions are compared here and formulas that extend the Wong-Zakai theorem are obtained.     

在Pbkc（c[0,1]）与Pbkc（Lp[0,1]）取值的集值随机变量（1）

对拟连续测度空间(G,β,u)的一致有界等度连续函数族,通过包含关系,取凸包和闭包,构造了在Pbkc(c[0,1])与Pbkc(Lp[0,1])取值的集值随机变量及连续的集值映射,深化了集值随机过程理论研究.     

ONE LINEAR ANALYTIC APPROXIMATION FOR STOCHASTIC INTEGRODIFFERENTIAL EQUATIONS

This article concerns the construction of approximate solutions for a general stochastic integrodifferential equation which is not explicitly solvable and whose coeffcients functionally depend on Lebesgue integrals and stochastic integrals with respect to martingales. The approximate equations are linear ordinary stochastic differential equations, the solutions of which are defined on sub-intervals of an arbitrary partition of the time interval and connected at successive division points. The closeness of the initial and approximate solutions is measured in the L^p-th norm, uniformly on the time interval. The convergence with probability one is also given.     

Stochastic integrals in Riemann manifolds

Stochastic integrals are constructed with values in a compact Riemann manifold from a continuous martingale integrator that is given in the tangent space of the initial point of the stochastic integral and from a stochastic tensor field of linear endomorphisms of the tangent bundle. The integrals that are formed are continuous processes that suitably preserve the martingale property. These stochastic integrals should be useful for the applications of a stochastic calculus in Riemann manifolds.