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We consider the Wiener integral with respect to a d -parameter Hermite process with Hurst multi-index H = (H 1 ,.. , H d) ∈ (1 2 , 1) d and we analyze the limit behavior in distribution of this object when the components of H tend to 1 and/or 1 2. As examples, we focus on the solution to the stochastic heat equation with additive Hermite noise and to the Hermite Ornstein-Uhlenbeck process. [ABSTRACT FROM AUTHOR]

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 I L 2 ( I R + × Ω , Σ I F , I R d × m ) of all square integrable I F -nonanticipative matrix-valued stochastic processes. Such integrals are integrably bounded and possess properties needed in the theory of set-valued stochastic equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: General conditions for normalized, time-scaled stochastic integrals of independently scattered Levy random measures to converge to a limit are described. The idea is to provide general conditions to bypass the use of characteristic functions, which can sometimes have tedious calculations, to simplify and shorten the proofs of convergence of infinitely divisible processes, which have previously been done on a case by case basis. It is of particular interest to study both small and large scale asymptotics, since there are many applications, such as modeling internet traffic. The purpose of this paper is to generalize previous results to the greater class of all stochastically continuous (or, more generally, separable in probability) infinitely divisible processes with no Gaussian part and to expand the results to the multi-dimensional case. [Copyright &y& Elsevier]

Abstract: In this paper we use a definition of the fractional stochastic integral given by Carmona et al. (2003) in [19] and develop a simple approximation method to study quasi-linear stochastic differential equations by fractional Brownian motion. We also propose a stochastic process, namely fractional semimartingale, to model for the noise driving in some financial models. [Copyright &y& Elsevier]

Abstract: We give a strong approximation of Rosenblatt process via transport processes and we give the rate of convergence. [Copyright &y& Elsevier]

Abstract: The present paper is devoted to properties of set-valued stochastic integrals defined as some special type of set-valued random variables. In particular, it is shown that if the probability base is separable or probability measure is nonatomic then defined set-valued stochastic integrals can be represented by a sequence of Itôʼs integrals of nonanticipative selectors of integrated set-valued processes. Immediately from Michaelʼs continuous selection theorem it follows that the indefinite set-valued stochastic integrals possess some continuous selections. The problem of integrably boundedness of set-valued stochastic integrals is considered. Some remarks dealing with stochastic differential inclusions are also given. [Copyright &y& Elsevier]

Abstract: We use a method developed in Carmona et al. (2003) to study the fractional geometric mean-reversion processes. Our obtained results hold for any . [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: We construct a family of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener–Itô integral with respect to the fractional Brownian motion. We assume that and we prove our approximation result for the integrands f in a rather general class. [Copyright &y& Elsevier]

Abstract: R-Boundedness is a randomized boundedness condition for sets of operators which in recent years has found many applications in the maximal regularity theory of evolution equations, stochastic evolution equations, spectral theory and vector-valued harmonic analysis. However, in some situations additional geometric properties such as Pisier''s property (α) are required to guaranty the R-boundedness of a relevant set of operators. In this paper we show that a weaker property called semi-R-boundedness can be used to avoid these geometric assumptions in the context of Schauder decompositions and the -calculus. Furthermore, we give weaker conditions for stochastic integrability of certain convolutions. [Copyright &y& Elsevier]

Abstract: We develop a theory of Malliavin calculus for Banach space-valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener–Itô isometry to Banach spaces. In the white noise case we obtain two sided -estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener–Itô chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer''s inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces. [Copyright &y& Elsevier]

Abstract: We introduce two types of the Stratonovich stochastic integrals for two-parameter processes, and investigate the relationship of these Stratonovich integrals and various types of Skorohod integrals with respect to a fractional Brownian sheet. By using this relationship, we derive a differentiation formula in the Stratonovich sense for fractional Brownian sheet through Itô formula. Also the relationship between the two types of the Stratonovich integrals will be obtained and used to derive a differentiation formula in the Stratonovich sense. In this case, our proof is based on the repeated applications of differentiation formulas in the Stratonovich form for one-parameter Gaussian processes. [Copyright &y& Elsevier]

Abstract: We prove a smoothing property and the irreducibility of transition semigroups corresponding to a class of semilinear stochastic equations on a separable Hilbert space H. Existence and uniqueness of invariant measures are discussed as well. [Copyright &y& Elsevier]

Abstract: In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let be a fractional Brownian sheet with Hurst parameters , and a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in , and four types of stochastic surface integrals: , , , , , . As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters . Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process. [Copyright &y& Elsevier]

In this note we shall prove that the stochastic integral with respect to a semimartingale can be defined by Riemann''s approach. However in this approach we use non-uniform meshes instead of the usual uniform meshes. [Copyright &y& Elsevier]

We derive estimates for the integral modulus of continuity of probability densities of infinitely divisible distributions. The paper is split into two parts. The first part deals with general infinitely divisible distributions. The second part is concerned with densities of random integrals with respect to a Lévy process. We will see major differences between integrals over compact and non-compact intervals. [ABSTRACT FROM AUTHOR]