Remarks on unboundedness of set-valued Itô stochastic integrals

Michta, Mariusz

doi:10.1016/j.jmaa.2014.11.041

Cited by 20 publications

(9 citation statements)

References 36 publications

(36 reference statements)

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“…Let us note also that the same argumentation applies when we choose any point of time ∈ instead of . Another remark is that it is easy to see that due to Corollary 3.9 (as well as Corollary 3.11 and Remark 3.12) in [55]) one can construct many other examples of nontrivial fuzzy integrands for which fuzzy stochastic Itô's integrals are not integrally bounded. This means that the fuzzy stochastic integral equation…”

Section: Fuzzy-valued Stochastic Equations With Fuzzy Stochastic Solumentioning

confidence: 99%

“…for all ∈ [0,1], where cl 2, dec( ) denotes a closed decomposable hull of a given set ⊂ 2, (see, e.g., [54] for details). Now, let us consider a special one-dimensional case and the mapping ℎ : × Ω → F ( 1 ) defined by ℎ( , ) fl for every ( , ) ∈ × Ω, where ∈ F ( 1 ) is such that ( ) = ( + 1)I [55] for such chosen mapping ℎ, it follows that the set cl 2,1 dec(∫ 0 [ℎ( )] ( )) is unbounded as a subset of the space 2,1 for every ∈ [0, 1). Therefore by (47) for every ∈ [0, 1) the set S 2 F ([(F) ∫ 0 ℎ ] , ) is unbounded in 2,1 too.…”

Section: Fuzzy-valued Stochastic Equations With Fuzzy Stochastic Solumentioning

confidence: 99%

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Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

Michta

2015

Mathematical Problems in Engineering

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The first aim of the paper is to present a survey of possible approaches for the study of fuzzy stochastic differential or integral equations. They are stochastic counterparts of classical approaches known from the theory of deterministic fuzzy differential equations. For our aims we present first a notion of fuzzy stochastic integral with a semimartingale integrator and its main properties. Next we focus on different approaches for fuzzy stochastic differential equations. We present the existence of fuzzy solutions to such equations as well as their main properties. In the first approach we treat the fuzzy equation as an abstract relation in the metric space of fuzzy sets over the space of square integrable random vectors. In the second one the equation is interpreted as a system of stochastic inclusions. Finally, in the last section we discuss fuzzy stochastic integral equations with solutions being fuzzy stochastic processes. In this case the notion of the stochastic Itô's integral in the equation is crisp; that is, it has single-valued level sets. The second aim of this paper is to show that there is no extension to more general diffusion terms.

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Section: Fuzzy-valued Stochastic Equations With Fuzzy Stochastic Solumentioning

confidence: 99%

Section: Fuzzy-valued Stochastic Equations With Fuzzy Stochastic Solumentioning

confidence: 99%

Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

Michta

2015

Mathematical Problems in Engineering

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“…Let us observe that boundedness of a set 1I [0,t] G is not sufficient for square integrable boundedness of a generalized set-valued integral t 0 GdB τ . Indeed, by virtue of results of [11] there exists an integrably bounded set-valued IF-nonanticipative process G = (G t ) t≥0 such that S IF (G) is a bounded subset of IL 2 (IR + × Ω, Σ IF , IR d×m ) and E t 0 G τ dB τ 2 = ∞. Taking G = S IF (G) we obtain for every fixed…”

Section: Integrable Boundedness Of Generalized Set-valued Integralsmentioning

confidence: 99%

“…Up to now we were not able to construct any example of set-valued stochastic processes, different on a singleton, having integrably bounded set-valued integrals defined in [4]. It was shown by M. Michta (see [11]) that in the general case set-valued stochastic integrals defined by E.J. Jung and J.H.…”

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Properties of generalized set-valued stochastic integrals

Kisielewicz

2014

Discuss. Math. Differential Inclusions, Control and Optimizatio

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The paper is devoted to properties of generalized set-valued stochastic integrals defined in [10]. These integrals generalize set-valued stochastic integrals defined by E.J. Jung and J.H. Kim in the paper [4]. Up to now we were not able to construct any example of set-valued stochastic processes, different on a singleton, having integrably bounded set-valued integrals defined in [4]. It was shown by M. Michta (see [11]) that in the general case set-valued stochastic integrals defined by E.J. Jung and J.H. Kim, are not integrably bounded. Generalized set-valued stochastic integrals, considered in the paper, are in some non-trivial cases square integrably bounded and can be applied in the theory of stochastic differential equations with set-valued solutions.t 0 GdB τ is understood as an F t -measurable set-valued random variable 132 M. Kisielewicz with values in the d-dimensional Euclidean space IR d and subtrajectory integrals S Ft ( t 0 GdB τ ) equal to dec J t (G). By J t we denote the Itô isometry defined on the space IL 2 (IR + × Ω, Σ IF , IR d×m ) by setting J t (g) = t 0 g τ dB τ for every g ∈ IL 2 (IR + × Ω, Σ IF , IR d×m ). Subtrajectory integrals S Ft ( t 0 GdB τ ) of

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Preliminaries

Kisielewicz¹

2020

Set-Valued Stochastic Integrals and Applications

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Remarks on unboundedness of set-valued Itô stochastic integrals

Cited by 20 publications

References 36 publications

Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

Properties of generalized set-valued stochastic integrals

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