First-hitting-time model

About: First-hitting-time model is a research topic. Over the lifetime, 2722 publications have been published within this topic receiving 53045 citations.

Method for Random Vibration of Hysteretic Systems

Abstract: Based on a Markov-vector formulation and a Galerkin solution procedure, a new method of modeling and solution of a large class of hysteretic systems (softening or hardening, narrow or wide-band) under random excitation is proposed. The excitation is modeled as a filtered Gaussian shot noise allowing one to take the nonstationarity and spectral content of the excitation into consideration. The solutions include time histories of joint density, moments of all order, and threshold crossing rate; for the stationary case, autocorrelation, spectral density, and first passage time probability are also obtained. Comparison of results of numerical example with Monte-Carlo solutions indicates that the proposed method is a powerful and efficient tool.

On the Time Value of Ruin

Abstract: This paper studies the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transforms, which can naturally be interpreted as discounting. Hence the classical risk theory model is generalized by discounting with respect to the time of ruin. We show how to calculate an expected discounted penalty, which is due at ruin and may depend on the deficit at ruin and on the surplus immediately before ruin. The expected discounted penalty, considered as a function of the initial surplus, satisfies a certain renewal equation, which has a probabilistic interpretation. Explicit answers are obtained for zero initial surplus, very large initial surplus, and arbitrary initial surplus if the claim amount distribution is exponential or a mixture of exponentials. We generalize Dickson’s formula, which expresses the joint distribution of the surplus immediately prior to and at ruin in terms of the probability of ult...

First passage time approach to diffusion controlled reactions

Abstract: Association reactions involving diffusion in one, two, and three‐dimensional finite domains governed by Smoluchowski‐type equations (e.g., interchain reaction of macromolecules, ligand binding to receptors, repressor–operator association of DNA strand) are shown to be often well described by first‐order kinetics and characterized by an average reaction (passage) time τ. An inhomogeneous differential equation is derived which, for problems with high symmetry, yields τ by simple quadrature without taking recourse to detailed cumbersome time‐dependent solutions of the original Smoluchowski equation. The cases of diffusion and nondiffusion controlled processes are included in the treatment. For reaction processes involving free diffusion and intramolecular chain motion, the validity of the passage time approximation is analyzed.

Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process.

Abstract: An analytic expression for characteristic function defining a truncated L\'evy flight is derived. It is shown that the characteristic function yields results in agreement with recent simulations of truncated L\'evy flights by Mantegna and Stanley [Phys. Rev. Lett. 73, 2946 (1994)]. With the analytic expression for the characteristic function, the convergence of the L\'evy process towards the Gaussian is demonstrated without simulations. In the calculation of first return probability the simulations are replaced by numerical integration using simple quadratures.

The Brownian Movement and Stochastic Equations

Abstract: The irregular movements of small particles immersed in a liquid, caused by the impacts of the molecules of the liquid, were described by Brown in 1828.1 Since 1905 the Brownian movement has been treated statistically, on the basis of the fundamental work of Einstein and Smoluchowski. Let x(t) be the x-coordinate of a particle at time t. Einstein and Smoluchowski treated x(t) as a chance variable. They found the distribution of x(t) x(O) to be Gaussian, with mean 0 and variance a I t l, where a is a positive constant which can be calculated from the physical characteristics of the moving particles and the given liquid. More exactly, such a family of chance variables {x(t) } is now described as the family of chance variables determining a temporally homogeneous differential stochastic process: the distribution of x(s + t) x(t) is Gaussian, with mean 0, variance a I t , and if t1 < < tn.