Random Integral Equations with Applications to Stochastic Systems

Tsokos, Chris P.; Padgett, W. J.

doi:10.1007/bfb0059959

Cited by 67 publications

(71 citation statements)

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“…These equations generalize the classical single-valued stochastic differential equations [47,48], random differential equations [54], deterministic set-valued and fuzzy differential equations [31,32]. Our analysis concerns the equations driven by semimartingales that constitute the largest class of integrators with respect to which stochastic integrals can be reasonably defined.…”

Section: Discussionmentioning

confidence: 99%

Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

Malinowski

2014

Open Mathematics

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Abstract:We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L 2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A; M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari's inequality.

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Section: Discussionmentioning

confidence: 99%

Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

Malinowski

2014

Open Mathematics

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“…The reader can see the detailed results in the monographs [17,26,27], the papers [24] and the references therein.…”

Section: Preliminariesmentioning

confidence: 99%

“…Hence, the study of the fractional differential equations with random parameters seem to be a natural one. We refer the reader to the monographs [3,17,26,27], the papers [6-9, 12, 14] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On fractional random differential equations with delay

Vu¹,

Phung²,

Phương³

2016

OpMath

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“…Random differential equations, as natural extensions of deterministic ones, arise in many applications and have been investigated by many mathematicians. We refer the reader to the monographs [18,31,43]. The initial value problems for fractional differential with random parameters have been studied by Lupulescu and Ntouyas [33].…”

Section: Introductionmentioning

confidence: 99%