Search results for " Stochastic calculus"

Direct Derivation of Corrective Terms in SDE Through Nonlinear Transformation on Fokker–Planck Equation

2004

This paper examines the problem of probabilistic characterization of nonlinear systems driven by normal and Poissonian white noise. By means of classical nonlinear transformation the stochastic differential equation driven by external input is transformed into a parametric-type stochastic differential equation. Such equations are commonly handled with Ito-type stochastic differential equations and Ito's rule is used to find the response statistics. Here a different approach is proposed, which mainly consists in transforming the Fokker–Planck equation for the original system driven by external input, in the transformed probability density function of the new state variable. It will be shown …

Stochastic Kinetics with Wave Nature

2003

We consider stochastic second-order partial differential equations. We indroduce a noisy non-linear wave equation and discuss its connections, in particular via the Lorentz transformation, with known stochastic models.

Statistic moments of the total energy of potential systems and application to equivalent non-linearization

2000

In this paper some properties of the total energy moments of potential systems, subjected to external white noise processes, are shown. Potential systems with a polynomial form of energy-dependent damping have been considered. It is shown that the analytical relations between the statistical moments of the energy associated with such systems can be obtained with the aid of the standard Ito calculus. Furthermore, it is shown that, for the stationary case, these analytical relations are very useful for the application of the equivalent non-linearization technique.

A new stochastic representation for the decay from a metastable state

2002

Abstract We show that a stochastic process on a complex plane can simulate decay from a metastable state. The simplest application of the method to a model in which the approach to equilibrium occurs through transitions over a potential barrier is discussed. The results are compared with direct numerical simulations of the stochastic differential equations describing system's evolution. We have found that the new method is much more efficient from computational point of view than the direct simulations.

The Master Equation

2009

The Langevin Equation

2009

Equivalent Non-Linearization of Hysteretic Systems by Means of RPS

2018

BackgroundThe analysis of elastoplastic systems with hardening (Bouc-Wen systems) under stochastic (seismic) loads needs the evaluation of higher order statistics even in the simplest case of normal distributed input. ObjectiveIn this paper, a non-linearization technique is proposed in order to evaluate the moments of any order of the response. MethodThis technique is developed by means of a nonlinear class of systems whose statistics are a priori known. The parameters of such systems can be chosen in such a way that the two systems are equivalent in a wide sense. Result & ConclusionIn the paper, the strategy to obtain the equivalence and the reliability of the results are discussed.

Experimental Studies of Noise—Induced Phenomena in a Tunnel Diode

2007

Noise induced phenomena are investigated in a physical system based on a tunnel diode. The stochastic differential equation describing this physical system is analog to the Langevin equation of an overdamped Brownian particle diffusing in a nonlinear potential. This simple and versatile physical system allows a series of experiments testing and clarifying the role of the noise and of its correlation in the stochastic dynamics of bistable or metastable systems. Experimental investigations of stochastic resonance, resonant activation and noise enhanced stability are discussed.

Stochastic Differential Calculus

1993

In many cases of engineering interest it has become quite common to use stochastic processes to model loadings resulting from earthquake, turbulent winds or ocean waves. In these circumstances the structural response needs to be adequately described in a probabilistic sense, by evaluating the cumulants or the moments of any order of the response (see e.g. [1, 2]). In particular, for linear systems excited by normal input, the response process is normal too and the moments or the cumulants up to the second order fully characterize the probability density function of both input and output processes. Many practical problems involve processes which are approximately normal and the effect of the…

Set-valued stochastic integral equations driven by martingales

2012

Abstract We consider a notion of set-valued stochastic Lebesgue–Stieltjes trajectory integral and a notion of set-valued stochastic trajectory integral with respect to martingale. Then we use these integrals in a formulation of set-valued stochastic integral equations. The existence and uniqueness of the solution to such the equations is proven. As a generalization of set-valued case results we consider the fuzzy stochastic trajectory integrals and investigate the fuzzy stochastic integral equations driven by bounded variation processes and martingales.