dXt dt = f(Xt, t)dt+ g(Xt, t)dWt (1) where Xt = X(t) is the realization of a stochastic process or random variable. f(Xt, t) is called the drift coefficient, that is the deterministic part of the SDE characterizing the local trend. g(Xt, t) denotes the diffusion coefficient, that is the stochastic part which influences the average size of the fluctuations of X. The fluctuations themselves origi...

In this study, we give a characterization of all torsion units which are in the unit group of ZS3 integral group ring of symmetric group S3, and classify conjugate classes of these units. We used the group of all doubly stochastic matrices in GL(3,Z) in this classification. The investigation of torsion units is not restricted with this study, and the classification of torsion units of bigger or...

We demonstrate the relevance of complex Gaussian stochastic processes to the stochastic state vector description of non-Markovian open quantum systems. These processes express the general Feynman-Vernon path integral propagator for open quantum systems as the classical ensemble average over stochastic pure state propagators in a natural way. They are the coloured generalization of complexWiener...

This paper presents a stochastic model of fatigue-induced crack propagation in metallic materials. The crack growth rate predicted by the model is guaranteed to be non-negative. The model structure is built upon the underlying principle of Karhunen±Lo eve expansion and does not require solutions of stochastic di€erential equations in either Wiener integral or Itô integral setting. As such this ...

The adaptive stabilization scheme based on tuning function for stochastic nonlinear systems with stochastic integral input-to-state stability SiISS inverse dynamics is investigated. By combining the stochastic LaSalle theorem and small-gain type conditions on SiISS, an adaptive output feedback controller is constructively designed. It is shown that all the closed-loop signals are bounded almost...

In this paper, we consider a class of impulsive stochastic Volterra-Levin equations. By establishing a new integral inequality, some sufficient conditions for the existence and global attractivity of periodic solution for impulsive stochastic Volterra-Levin equations are given. Our results imply that under the appropriate linear periodic impulsive perturbations, the impulsive stochastic Volterr...

We study the limit of functionals of stochastic processes for which an homogenization result holds. All these functionals involve stochastic integrals. Among them, we consider more particularly the Lévy area and those giving the solutions of some SDEs. The main question is to know whether or not the limit of the stochastic integrals is equal to the stochastic integral of the limit of each of it...

Uncertain calculus deals with the integral and differential of some uncertain processes. So far, uncertain integrals have been defined with respect to Liu process, renewal process, finite variation process, and multiple Liu processes. This paper presents an uncertain integral of a matrix of uncertain processes with respect to multi-dimensional Liu process, and verifies its linearity property. T...

We derive bridges from general multidimensional linear non time-homogeneous processes using only the transition densities of the original process giving their integral representations (in terms of a standard Wiener process) and so-called anticipative representations. We derive a stochastic differential equation satisfied by the integral representation and we prove a usual conditioning property ...

Proof. The stochastic differential equation (2) follows directly from the (multivariate) Itô theorem, using the fact that Zt has the form Zt = u(Mt , [M ]t ). Every Itô integral process V ·M is a local martingale provided that M is a local martingale and V 2 L§(M), so for each n <1 the stopped process Z øn is a local martingale, where øn is the first time t such that [M ]t = n, and so it follow...