Set-valued Stochastic Integrals Research Articles

Filippov's Solution and Finite-Time Stability of Stochastic Systems for Discontinuous Control

In this paper, a suite of theoretic tools is provided for discontinuous control design and finite-time stability analysis of a class of stochastic differential systems. The notion of Filippov&#x0027;s solutions for stochastic differential systems is proposed, and the corresponding solution existence problem is explored. The classical It&#x00F4; differentiation formula is generalized for quasi-<inline-formula><tex-math notation="LaTeX">$C_{0}^{2}(\mathbb {R}^{n},\mathbb {R})$</tex-math></inline-formula>-class functions along Filippov&#x0027;s solutions of stochastic differential systems, and two involved set-valued stochastic integrals are introduced with a study on their properties. Some finite-time stability results of stochastic differential systems are revealed with Filippov&#x0027;s solutions, and one of them is applied to neural synchronization, together with case simulations.

Set-Valued Functions of Bounded Generalized Variation and Set-Valued Young Integrals

The paper deals with some properties of set-valued functions having bounded Riesz p-variation. Set-valued integrals of Young type for such multifunctions are introduced. Selection results and properties of such set-valued integrals are discussed. These integrals contain as a particular case set-valued stochastic integrals with respect to a fractional Brownian motion, and therefore, their properties are crucial for the investigation of solutions to stochastic differential inclusions driven by a fractional Brownian motion.

Selection Properties and Set-Valued Young Integrals of Set-Valued Functions

The paper deals with some selection properties of set-valued functions and different types of set-valued integrals of a Young type. Such integrals are considered for classes of Hölder continuous or with bounded Young p-variation set-valued functions. Two different cases are considered, namely set-valued functions with convex values and without convexity assumptions. The integrals contain as a particular case set-valued stochastic integrals with respect to a fractional Brownian motion, and therefore, their properties are crucial for the investigation of solutions to stochastic differential inclusions driven by a fractional Brownian motion.

Weak compactness of weak solutions sets to stochastic differential inclusions

The main result of the paper deals with weak compactness in distribution of weak solutions sets to stochastic differential inclusions defined by set-valued stochastic integrals presented in the paper [Kisielewicz, M., Michta, M. (2017). Integrably bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449(2):1892–1910]. Up to now only stochastic functional inclusions (see [Kisielewicz, M. (2013). Stochastic Differential and Inclusions Applications. New York: Springer]) have been investigated. The introductory parts of the paper deal with some properties of set-valued stochastic integrals and selected notions dealing with weak convergence in distribution of sequences of continuous stochastic processes.

Integrable boundedness of set-valued stochastic integrals

The paper deals with integrable boundedness of Itô set-valued stochastic integrals defined in 2003 by E.J. Jung and J.H. Kim in the paper [2]. Unfortunately integrable boundedness of integrals defined in the paper [2], has not been solved there. The problem was partially solved by M. Michta in the paper [8]. In the present paper it is proved that set-valued stochastic integrals defined in the paper [2] are integrably bounded if and only if they are single valued. Such result was obtained in the author's paper [7] by the additional assumption (see [7], Th. 8) that some Hypothesis B is satisfied.

Selection theorems for set-valued stochastic integrals

The paper is devoted to some selection theorems for set-valued stochastic integrals considered in papers by Kisielewicz et al. In particular, the Itô set-valued stochastic integrals are considered for absolutely summable countable subsets of the space of all square integrable -nonanticipative matrix-valued stochastic processes. Such integrals are integrably bounded. Selection theorems, presented in the paper, cannot be considered for integrals defined by Jung and Kim in their paper for -nonanticipative integrably bounded set-valued mappings , because such integrals are not in the general case integrably bounded.

Approximation theorems for set-valued stochastic integrals

ABSTRACTThe article is devoted to new properties of Aumann, Lebesgue, and Itô set-valued stochastic integrals considered in papers [1,2]. 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.

Integrably bounded set-valued stochastic integrals

The paper is devoted to properties of Aumann and Itô set-valued stochastic integrals, defined as some set-valued random variables. In particular the problem of integrable boundedness of the generalized Itô set-valued stochastic integrals is considered. Unfortunately, Itô set-valued stochastic integrals, defined by E.J. Jung and J.H. Kim in the paper [5], are not in general integrably bounded (see [8,15]). Therefore, in the present paper we consider generalized Itô set-valued stochastic integrals (see [10,11]) defined for absolutely summable and countable subsets of the space IL2(IR+×Ω,ΣIF,IRd×m) of all square integrable IF-nonanticipative matrix-valued stochastic processes. Such integrals are integrably bounded and possess properties needed in the theory of set-valued stochastic equations.

Set-Valued Stochastic Equation with Set-Valued Square Integrable Martingale

In this paper, we shall introduce the stochastic integral of a stochastic process with respect to set-valued square integrable martingale. Then we shall give the Aumann integral measurable theorem, and give the set-valued stochastic Lebesgue integral and set-valued square integrable martingale integral equation. The existence and uniqueness of solution to set-valued stochastic integral equation are proved. The discussion will be useful in optimal control and mathematical finance in psychological factors.

Set-valued and fuzzy stochastic differential equations in M-type 2 Banach spaces

In this paper we study set-valued stochastic differential equations in M-type 2 Banach spaces. Their drift terms and diffusion terms are assumed to be set-valued and single-valued respectively. These coefficients are considered to be random which makes the equations to be truely nonautonomous. Firstly we define set-valued stochastic Lebesgue integral in a Banach space. This integral is a set-valued random variable. We state its properties such as additivity with respect to the interval of integration, continuity as a function of the upper limit of integration, integrable boundedness. The existence and uniqueness of solution to set-valued differential equations in M-type 2 Banach space is obtained by a method of successive approximations. We show that the approximations are uniformly bounded and converge to the unique solution. A distance between $n$th approximation and exact solution is estimated and a continuous dependence of solution with respect to the data of the equation is proved. Finally, we construct a fuzzy stochastic Lebesgue integral in a Banach space and examine fuzzy stochastic differential equations in M-type 2 Banach spaces. We investigate properties like those in set-valued cases. All the results are achieved without assumption on separability of underlying sigma-algebra.

Boundedness of set-valued stochastic integrals

Boundedness of set-valued stochastic integrals

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.

Remarks on unboundedness of set-valued Itô stochastic integrals

The paper deals with set-valued stochastic integrals which are defined as a random multifunctions. Integrable boundedness of a such integral is one of the most important features in potential applications. Unfortunately, up to now there were no correct proofs of such property. Surprisingly, also a negative answer to this problem is still not explained correctly. Hence the problem seems to be still undetermined. We shall show that in general the answer is negative. We shall provide several relatively simple examples both in convex and nonconvex-valued case. As a consequence, we will show that under some restrictions on the class of selections of the integrand the integrable boundedness of set-valued Itô's integral is equivalent to its single valuedness.

Properties of generalized set-valued stochastic integrals

Properties of generalized set-valued stochastic integrals

Some properties of set-valued stochastic integrals of multiprocesses with finite Castaing representations

The paper contains new properties of set-valued stochastic integrals defined as multifunctions with subtrajectory integrals equal to closed decomposable hulls of functional set-valued integrals defined in the author paper [8]. In particular, it is proved that such defined integrals for set-valued predictable square integrably bounded processes having finite Castaing representations are square integrably bounded. Up to now this property has not been proved. Unfortunately, in the general case the above boundedness problem is still open.

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.

On a new set-valued stochastic integral with respect to semimartingales and its applications

We present a new approach to a concept of a set-valued stochastic integral with respect to semimartingales. Such an integral, called set-valued stochastic up-trajectory integral, is compatible with the decomposition of the semimartingale. Some properties of this integral are stated. We show applicability of the new integral in set-valued stochastic integral equations driven by multidimensional semimartingales. The uniqueness theorem is presented. Then we extend the notion of the set-valued stochastic up-trajectory integral to definition of a fuzzy stochastic up-trajectory integral with respect to semimartingales. A result on uniqueness of a solution to fuzzy stochastic integral equations incorporating the new fuzzy stochastic up-trajectory integral driven by the multidimensional semimartingale is stated.

Set-Valued Stochastic Integrals with Respect to Finite Variation Processes

In a Euclidean space Rd, the Lebesgue-Stieltjes integral of set-valued stochastic processes with respect to real valued finite variation process is defined directly by employing all integrably bounded selections instead of taking the decomposable closure appearing in some existed references. We shall show that this kind of integral is measurable, continuous in t under the Hausdorff metric and L2-bounded.

Set-valued stochastic integrals with respect to Poisson processes in a Banach space

In a separable Banach space X, at first we study X-valued stochastic integrals with respect to the Poisson random measure N(dsdz) and the compensated Poisson random measure N∼(dsdz) generated by a stationary Poisson stochastic process p. When the characteristic measure ν of p is finite, both N(dsdz) and N∼(dsdz) are of finite variation a.s. Then the set-valued integrals with respect to the Poisson random measure and the compensated Poisson random measure are integrably bounded. The set-valued integral with respect to the compensated Poisson random measure is a right continuous (under Hausdorff metric) set-valued martingale.

Fuzzy set-valued stochastic Lebesgue integral

This paper studies Lebesgue integral of a fuzzy closed set-valued stochastic process with respect to the time t. Firstly, a progressively measurable fuzzy closed set-valued stochastic process is discussed and an almost everywhere problem in the former Aumann type Lebesgue integral of the level-set process is pointed out. Secondly, a new definition of the Lebesgue integral by decomposable closure is given, focusing on Aumann representation theorem, representation theorem and property of convexity. It is proved that the fuzzy closed set-valued stochastic Lebesgue integral is a fuzzy closed set-valued stochastic process which is widely used in the fuzzy world with randomness. Finally, the fuzzy closed set-valued stochastic Lebesgue integral in Lp-space is studied, especially on an almost everywhere problem.