Stochastic recursive inclusions with non-additive iterate-dependent Markov noise

In this paper we study the asymptotic behaviour of stochastic approximation schemes with set-valued drift function and non-additive iterate-dependent Markov noise. We show that a linearly interpolated trajectory of such a recursion is an asymptotic pseudotrajectory for the flow of a limiting differential inclusion obtained by averaging the set-valued drift function of the recursion w.r.t. the stationary distributions of the Markov noise. The limit set theorem by Benaim is then used to characterize the limit sets of the recursion in terms of the dynamics of the limiting differential inclusion. We then state two variants of the Markov noise assumption under which the analysis of the recursion is similar to the one presented in this paper. Scenarios where our recursion naturally appears are presented as applications. These include controlled stochastic approximation, subgradient descent, approximate drift problem and analysis of discontinuous dynamics all in the presence of non-additive iterate-dependent Markov noise.     

On exact solutions to epidemic dynamic models

In this study, we address an SIR (susceptible-infected-recovered) model that is given as a system of first order differential equations and propose the SIR model on time scales which unifies and extends continuous and discrete models. More precisely, we derive the exact solution to the SIR model and discuss the asymptotic behavior of the number of susceptibles and infectives. Next, we introduce an SIS (susceptible-infected-susceptible) model on time scales and find the exact solution. We solve the models by using the Bernoulli equation on time scales which provides an alternative method to the existing methods. Having the models on time scales also leads to new discrete models. We illustrate our results with examples where the number of infectives in the population is obtained on different time scales.     

Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

We study global and local stabilities of the stationary zero solution to certain infinite-dimensional stochastic differential equations. The stabilities are in terms of fractional powers of the linear part of the drift. The abstract results are applied to semilinear stochastic partial differential equations with non-Lipschitzian drift terms and, in particular, to some specific models of population dynamics. We also expose the stabilizing effect of noise on the otherwise unstable zero solution

As a basic tool we use the Forward Inequality, a generalization of Kolmogorov's forward equation; it is an application of Lyapunov's second method with a sequence of Lyapunov functionals     

Jump Phenomena and Bifurcations in Stochastic Vehicle-Road Dynamics

When vehicles ride on uneven roads, they are excited to vertical random vibrations whose stationary rms-values (root-mean-square) strongly depend on the velocity of the vehicle. To investigate this vibration behavior, it is appropriate to introduce road models in way domain which are based on the theory of stochastic differential equations and transformed from way to time by means of velocity-dependent way and noise increments. The random base excitations by roads are applied to nonlinear quarter car models. They lead to stationary rms-values of the vertical vehicle vibrations which become resonant for critical velocities and show jump phenomena similar to those of the Duffing oscillator under harmonic excitations. In the stochastic case, jump phenomena are only observable for narrow-banded road excitations. They vanish for increasing car damping and excitation bandwidth. For efficient simulations of the road-vehicle model, the n state equations are utilized to derive n(n + 1)/2 stochastic covariance equations. For small step sizes, their numerical mean square solutions coincide with the nonlinear results of fix-point iterations obtained when the noise terms of the covariance equations are omitted. It can easily be shown, that this deterministic approach leads to the correct stationary covariances in the linear case. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equations for second moments of solutions of a system of linear differential equations with random semi-Markov coefficients and random input

We derive equations that determine second moments of a random solution of a system of Itô linear differential equations with coefficients depending on a finite-valued random semi-Markov process. We obtain necessary and sufficient conditions for the asymptotic stability of solutions in the mean square with the use of moment equations and Lyapunov stochastic functions.

     

Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: itô's case

In this article we prove new results concerning the long-time behaviour of random fields that are solutions in some generalized sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random fields eventually converge with probability. one to a global attractor represented by a single random variable whose properties we investigate in detail. We analyze the partial differential equations of this article in light Itô's stochastic calculus and thereby obtain stabilization and stability results which are substantially different from our earlier results concerning their interpretation in the sense of Stratonovitch. In particular, the asymptotic properties of the random fields that we investigate here exhibit no recurrence and oscillatory properties     

The Burgers equation with a random force and a general model for directed polymers in random environments

Summary. The study of the Burgers equation with a random force leads via a Hopf-Cole type transformation to a stochastic heat equation having a white noise with spatial parameters type potential. The latter can be studied by means of a general model of directed polymers in random environments with two point random potentials. These models exhibit a Gaussian behavior at large times and have certain stationary distributions which yield the corresponding results for the above stochastic heat and Burgers equations. Received: 18 July 1995 / In revised form: 5 August 1995     

Innovations algorithm asymptotics for periodically stationary time series with heavy tails

The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper we compute the asymptotic distribution for these estimates in the case where the underlying noise sequence has infinite fourth moment but finite second moment. In this case, the sample covariances on which the innovations algorithm are based are known to be asymptotically stable. The asymptotic results developed here are useful to determine which model parameters are significant. In the process, we also compute the asymptotic distributions of least squares estimates of parameters in an autoregressive model.     

Coloured noise for dispersion of contaminants in shallow waters

In this article, we explore the application of a set of stochastic differential equations called particle model in simulating the advection and diffusion of pollutants in shallow waters. The Fokker–Planck equation associated with this set of stochastic differential equations is interpreted as an advection–diffusion equation. This enables us to derive an underlying particle model that is exactly consistent with the advection–diffusion equation. Still, neither the advection–diffusion equation nor the related traditional particle model accurately takes into account the short-term spreading behaviour of particles. To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this article. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection–diffusion equation.     

Amplitude equations for SPDEs with cubic nonlinearities

For a quite general class of stochastic partial differential equations with cubic nonlinearities, we derive rigorously amplitude equations describing the essential dynamics using the natural separation of timescales near a change of stability. Typical examples are the Swift–Hohenberg equation, the Ginzburg–Landau (or Allen–Cahn) equation and some model from surface growth. We discuss the impact of degenerate noise on the dominant behaviour, and see that additive noise has the potential to stabilize the dynamics of the dominant modes. Furthermore, we discuss higher order corrections to the amplitude equation.     

Stationary Conjugation of Flows for Parabolic SPDEs with Multiplicative Noise and Some Applications

Abstract

The purpose of this paper is to transform a nonlinear stochastic partial differential equation of parabolic type with multiplicative noise into a random partial differential equation by using a bijective random process. A stationary conjugation is constructed, which is of interest for asymptotic problems. The conjugation is used here to prove the existence of the stochastic flow, the perfect cocycle property and the existence of the random attractor, all nontrivial properties in the case of multiplicative noise.     

Measure attractors and random attractors for stochastic partial differential equations

In the theory of stochastic differential equations we can distinguish between two kinds of attractors. The first one is the attractor (measure attractor) with respect to the Markov semigroup generated by a stochastic differential equation. The second meaning of attractors (random attractors) is to be understood with respect to each trajectory of the random equation. The aim of this paper is to bring together the two meanings of attractors. In particular, we show the existence of measure attractors if random attractors exist. We can also show the uniqueness of the stationary distributions of the stochastic Navier-Stokes equation if the viscosity is large     

Asymptotic Behavior of Solutions of Some Difference Equations Defined by Weakly Dependent Random Vectors

In this article, we consider not only stochastic differential equations driven by the Wiener process but also by processes with stationary increments from the view points of time series analysis for mathematical finance. Corresponding to Black-Scholes type stochastic differential equations, we consider difference equations defined by weakly dependent sequence of random vectors and examine the asymptotic behavior of their solutions.     

Parametric inference for discrete observations of diffusion processes with mixed effects

Stochastic differential equations with mixed effects provide means to model intra-individual and inter-individual variability in repeated experiments leading to longitudinal data. We consider N i.i.d. stochastic processes defined by a stochastic differential equation with linear mixed effects which are discretely observed. We study a parametric framework with distributions leading to explicit approximate likelihood functions and investigate the asymptotic behavior of estimators under the asymptotic framework : the number N of individuals (trajectories) and the number n of observations per individual tend to infinity within a fixed time interval. The estimation method is assessed on simulated data for various models.     

Stochastic functional differential equations modelling materials with selective recall

We study a problem in stochastic functional differential equations which, in addition to a standard one-one-parameter noise term involves a random perturbation of the memory. This problem can also be regarded as a first order hyperbolic system of stochastic partial differential equations with given initial data and nonlocal boundary data. Existence and uniqueness of a solution is established and the generator of the associated Markov process is analyzed. Thereafter, for two model problems arising from first- and second-order integro-differential equations suggested by physical applications we establish asymptotic stability in probability of the associated stochastic processes.     

Stochastic partial differential equations for some measure-valued diffusions

Summary We consider two classes of measure-valued diffusion processes; measure-valued branching diffusions and Fleming-Viot diffusion models. When the basic space is R ¹, and the drift operator is a fractional Laplacian of order 1<α≦2, we derive stochastic partial differential equations based on a space-time white noise for these two processes. The former is the expected one by Dawson, but the latter is a new type of stochastic partial differential equation.     

Forward and backward stochastic differential equations with normal constraints in law

In this paper we investigate the well-posedness of backward or forward stochastic differential equations whose law is constrained to live in an a priori given (smooth enough) set and which is reflected along the corresponding “normal” vector. We also study the associated interacting particle system reflected in mean field and asymptotically described by such equations. The case of particles submitted to a common noise as well as the asymptotic system is studied in the forward case. Eventually, we connect the forward and backward stochastic differential equations with normal constraints in law with partial differential equations stated on the Wasserstein space and involving a Neumann condition in the forward case and an obstacle in the backward one.     

Bifurcation analysis of a population model and the resulting SIS epidemic model with delay

This paper deals with the model for matured population growth proposed in Cooke et al. [Interaction of matiration delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999) 332–352] and the resulting SIS epidemic model. The dynamics of these two models are still largely undetermined, and in this paper, we perform some bifurcation analysis to the models. By applying the global bifurcation theory for functional differential equations, we are able to show that the population model allows multiple periodic solutions. For the SIS model, we obtain some local bifurcation results and derive formulas for determining the bifurcation direction and the stability of the bifurcated periodic solution.     

On anticipated backward stochastic differential equations with Markov chain noise

In 2013, Lu and Ren considered anticipated backward stochastic differential equations driven by finite state, continuous time Markov chain noise and established the existence and uniqueness of the solutions of these equations and a scalar comparison theorem. In this article, we provide an estimate for their solutions and study the duality between these equations and stochastic differential delayed equations with Markov chain noise. Finally, we derive another comparison theorem for these solutions depending only on the two drivers.