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León JA, Peralta L. Some Feller and Osgood type criteria for semilinear stochastic differential equations. STOCH DYNAM 2017. [DOI: 10.1142/s0219493717500113] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
By means of Itô’s formula and a comparison theorem for integral equations, we study the blow up in finite time of semilinear stochastic differential equations of the form [Formula: see text] Here, [Formula: see text] is non-negative and non-decreasing by components, [Formula: see text] is a predictable and continuous process, [Formula: see text] is an [Formula: see text]-Brownian motion and [Formula: see text] is an [Formula: see text]-measurable random variable. The results of this paper can be seen as extensions of the Feller and Osgood criteria.
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Affiliation(s)
- Jorge A. León
- Departamento de Control Automático, CINVESTAV-IPN, Apartado Postal 14-740 Ciudad de México 07000, Mexico
| | - Liliana Peralta
- Centro de Investigación en Matemáticas, UAEH, Carretera Pachuca-Tulancingo km 4.5 Pachuca, Hidalgo 42184, Mexico
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On Two Damage Accumulation Models and Their Size Effects. J Appl Probab 2007. [DOI: 10.1017/s0021900200002746] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
Two cumulative damage models are considered, the inverse gamma process and a composed gamma process. They can be seen as ‘continuous’ analogues of Poisson and compound Poisson processes, respectively. For these models the first passage time distribution functions are derived. Inhomogeneous versions of these processes lead to models closely related to the Weibull failure model. All models show interesting size effects.
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