On solutions of stochastic differential equations with parameters modeled by random sets

Schmelzer, Bernhard

doi:10.1016/j.ijar.2010.08.006

Cited by 21 publications

(12 citation statements)

References 6 publications

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“…Thus, the right-hand side of (22) coincides with the right-hand side of (20). Formula (21) can be obtained by applying (18) to ϕ.…”

Section: Proofmentioning

confidence: 97%

“…Note that the values of X are closed subsets of E n , more precisely closed cylindrical sets, and not elements of F n . One can prove ( [20]) that X is Effros-measurable by using selections and the so-called Fundamental measurability theorem for multifunctions ( [2,8]). Consequently, the map…”

Section: The Multivariate Casementioning

confidence: 99%

“…Let ψ : P n → [0, 1] be a multivariate belief function. Then (20) is the unique multivariate basic probability assignment such that…”

Section: Proofmentioning

confidence: 99%

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Characterizing joint distributions of random sets by multivariate capacities

Schmelzer

2012

International Journal of Approximate Reasoning

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By the Choquet theorem, distributions of random closed sets can be characterized by a certain class of set functions called capacity functionals. In this paper a generalization to the multivariate case is presented, that is, it is proved that the joint distribution of finitely many random sets can be characterized by a multivariate set function being completely alternating in each component, or alternatively, by a capacity functional defined on complements of cylindrical sets. For the special case of finite spaces a multivariate version of the Moebius inversion formula is derived. Furthermore, we use this result to formulate an existence theorem for set-valued stochastic processes.

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“…Thus, the right-hand side of (22) coincides with the right-hand side of (20). Formula (21) can be obtained by applying (18) to ϕ.…”

Section: Proofmentioning

confidence: 97%

Section: The Multivariate Casementioning

confidence: 99%

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Characterizing joint distributions of random sets by multivariate capacities

Schmelzer

2012

International Journal of Approximate Reasoning

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“…Under certain conditions (see [1,2]) that are fulfilled for linear systems of SDEs almost all of its sample functions x(·, ·, ν w ) are continuous with respect to (t, a).…”

Section: Set-valued Stochastic Processesmentioning

confidence: 99%

“…Roughly speaking, the aleatory uncertainty of the earthquake record can be described stochastically, whereas the epistemic parameter uncertainty can be accounted for by means of random sets. A framework that combines stochastic differential equations with random set coefficients, resulting in set-valued stochastic processes, has been developed by the first author [1,2]. The purpose of this paper is to present the concept and demonstrate how it can produce a realistic assessment of the performance of TMDs under parameter uncertainty.…”

Section: Introductionmentioning

confidence: 99%

Efficiency of tuned mass dampers with uncertain parameters on the performance of structures under stochastic excitation

Schmelzer

Oberguggenberger

Adam

2010

Proceedings of the Institution of Mechanical Engineers, Part O:

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Tuned mass dampers serve the purpose of damping vibrations of structures such as earthquake-induced vibrations. In their design, two types of uncertainty are relevant: the stochastic excitation (e.g. earthquake record) and the inherent uncertainty of internal parameters of the devices themselves. This paper presents a new framework that admits the combination of stochastic processes and interval-type parameter uncertainty, modelled by random sets. The approach is applied to show how the efficiency of tuned mass dampers can be realistically assessed in the presence of uncertainty.

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2014

Introduction to Imprecise Probabilities

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On solutions of stochastic differential equations with parameters modeled by random sets

Cited by 21 publications

References 6 publications

Characterizing joint distributions of random sets by multivariate capacities

Characterizing joint distributions of random sets by multivariate capacities

Efficiency of tuned mass dampers with uncertain parameters on the performance of structures under stochastic excitation

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