Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudostable and pseudo-unstable manifolds for a class of random partial differential equations and stochastic partial differential equations is shown. Unlike the invarian...

We discuss stochastic functional differential equation under regime switching dx t f xt, r t , t dt q r t x t dW1 t σ r t |x t |βx t dW2 t . We obtain unique global solution of this system without the linear growth condition; furthermore, we prove its asymptotic ultimate boundedness. Using the ergodic property of the Markov chain, we give the sufficient condition of almost surely exponentially ...

Introduction This chapter is concerned with continuous time processes, which are often modeled as a system of ordinary differential equations. These models assume that the observed dynamics are driven exclusively by internal, deterministic mechanisms. However, real biological systems will always be exposed to influences that are not completely understood or not feasible to model explicitly, and...

In this paper, we consider Internet models, which respond to a congestion signal from the network described by a stochastic and hybrid differential equation. We consider Internet networks with one source and r access links, as well as with r sources and one access link. We analyze the conditions for the existence of a solution and the algorithms needed to determine the solution. We carry out nu...

We introduce a stochastic version of the classical Perron’s method to construct viscosity solutions to linear parabolic equations associated to stochastic differential equations. Using this method, we construct easily two viscosity (sub and super) solutions that squeeze in between the expected payoff. If a comparison result holds true, then there exists a unique viscosity solution which is a ma...

We introduce a stochastic version of the classical Perron’s method to construct viscosity solutions to linear parabolic equations associated to stochastic differential equations. Using this method, we construct easily two viscosity (sub and super) solutions that squeeze in between the expected payoff. If a comparison result holds true, then there exists a unique viscosity solution which is a ma...

Non-parametric modeling is a method which relies heavily on data and motivated by the smoothness properties in estimating a function which involves spline and non-spline approaches. Spline approach consists of regression spline and smoothing spline. Regression spline with Bayesian approach is considered in the first step of a two-step method in estimating the structural parameters for stochasti...

For a given bivariate Lévy process (Ut, Lt)t≥0, necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation dVt = Vt− dUt + dLt are obtained. Neither strict positivity of the stochastic exponential of U nor independence of V0 and (U,L) are assumed and noncausal solutions may appear. The form of the stationary solution is determ...

A semi-classical non-Hamiltonian model of a spontaneous collapse of unstable quantum system is given. The time evolution of the system becomes non-Hamiltonian at random instants of transition of pure states to reduced ones, η 7→ Cη, given by a contraction C. The counting trajectories are assumed to satisfy the Poisson law. A unitary dilation of the concractive stochastic dynamics is found. In p...

In this research, Allan Variance analysis is used to identify the stochastic error sources existing in inertial sensors and to determine the corresponding noise parameters. According to the noise parameters, the power spectral density (PSD) function of the stochastic error sources can be determined. The differential equation descriptions for individual stochastic errors are then derived for two...