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Abstract: The aim of this paper is to investigate how the correlation properties of astationary Markovian stochastic processes affect the First Passage Timedistribution. First Passage Time issues are a classical topic in stochasticprocesses research. They also have relevant applications, for example, in manyfields of finance such as the assessment of the default risk for firms- assets.By using some explicit examples, in this paper we will show that the tail ofthe First Passage Time distribution crucially depends on the correlationproperties of the process and it is independent from its stationarydistribution. When the process includes an infinite set of time-scales boundedfrom above, the FPTD shows tails modulated by some exponential decay. In thecase when the process is power-law correlated the FPTD shows power-law tails1-t^alfa+1-2 and therefore the moments of the FPTD are finite only whenn< alfa-1-2. We will also show that such power-law behaviour is not merelydue to the fact that the process includes an infinite and unbounded set oftime-scales. Rather, the time-scale must enter the FPTD with weights that mustbe distributed according to a power-law for large time-scales values. Finally,we will give a general result connecting the FPTD of an additive stochasticprocesses xt to the FPTD of a generic process yt related by a coordinatetransformation y=fx to the first one.



Autor: S. Micciché

Fuente: https://arxiv.org/







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