On a class of stochastic differential equations arising from the stochastic approximation theory

The paper dealt with generalized stochastic approximation procedures of Robbins-Monro type. We consider these procedures as strong solutions of some stochastic differential equations with respect to semimartingales and investigate their almost sure convergence and mean square convergence     

On the support of solutions to stochastic differential equations with path-dependent coefficients

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron–Martin space under the flow of mild solutions to a system of path-dependent ordinary differential equations. Our result extends the Stroock–Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on functional Itô calculus.     

Stability analysis of a stochastic logistic model with nonlinear diffusion term

This paper presents an asymptotic analysis of a stochastic logistic population model with nonlinear diffusion term. The classical probability method is applied to obtain the criteria of asymptotic behavior for the considered model. The numerical simulations validate the efficiency of the theory analysis.     

Nonlinear stochastic integration with a non-smooth family of integrators

We consider nonlinear stochastic integrals of Itô-type w.r.t. a family of semimartingales which depend on a spatial parameter. These integrals were introduced by Carmona/Nualart, Kunita, and Le Jan. The extension of the elementary nonlinear integral is based on the condition that the semimartingale kernel has nice continuity properties in the spatial parameter. We investigate the case that continuity is not available and suggest different directions of generalization. This brings us beyond the case that any integral can be approximated by integrals with integrands taking only finitely many values.     

Nonlinear flows of stochastic linear delay equations

The sample non-linearity of the classical Wiener integral as a function of continuous integrands is pointed out. As an application it is shown that the solution of a linear stochastic delay equation is an almost surely non-linear function of the initial trajectory segment.     

A note on a derivation method for SDE models: Applications in biology and viability criteria

We discuss a method, which was popularized by E. J. Allen and that is frequently used in applications to construct SDE models. The derivation procedure is based on information about the elementary processes involved in the dynamics and their corresponding probabilities. We formulate criteria for the viability of the resulting models. In particular, explicit necessary and sufficient conditions are deduced for the non-negativity and/or boundedness of solutions. Moreover, we show that the class of deterministic models for which the construction leads to an admissible SDE extension is strongly limited. Several examples are presented to illustrate the implications of our results.     

Positive and unbounded solution of stochastic delayed evolution equations

This article is concerned with the blowup phenomenon of stochastic delayed evolution equations. We first establish the sufficient condition to ensure the existence of a unique nonnegative solution of stochastic parabolic equations. Then the problem of blow-up solutions in mean L^q-norm, q ? 1, in a finite time is considered. The main aim in this article is to investigate the effect of time delay and stochastic term. A new result shows that the stochastic delayed term can induce singularities.     

Right-continuous solutions of systems of stochastic integral equations

Unique solutions are shown to exist for systems of stochastic integral equations which allow right-continuous semimartingales (also known as quasimartingales) as differentials.     

The interrelation between stochastic differential inclusions and set-valued stochastic differential equations

In this paper we connect the well established theory of stochastic differential inclusions with a new theory of set-valued stochastic differential equations. Solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty, bounded, convex and closed subsets of the space L²$L^2$ consisting of square integrable random vectors. We show that for the solution X$X$ to a set-valued stochastic differential equation corresponding to a stochastic differential inclusion, there exists a solution x$x$ for this inclusion that is a _‖⋅‖L2${\Vert \cdot \Vert }_{L^2}$-continuous selection of X$X$. This result enables us to draw inferences about the reachable sets of solutions for stochastic differential inclusions, as well as to consider the viability problem for stochastic differential inclusions.     

Multi-Step Maruyama Methods for Stochastic Delay Differential Equations

Abstract

In this article the numerical approximation of solutions of Itô stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their 𝕃 _p -consistency, numerical 𝕃 _p -stability and 𝕃 _p -convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency.     

Large deviations for stochastic nonlinear beam equations

We establish a large deviation principle for the solutions of stochastic partial differential equations for nonlinear vibration of elastic panels (also called stochastic nonlinear beam equations).     

Some properties of stochastic differential equations driven by the G-Brownian motion

In this paper, we study the property of continuous dependence on the parameters of stochastic integrals and solutions of stochastic differential equations driven by the G-Brownian motion. In addition, the uniqueness and comparison theorems for those stochastic differential equations with non-Lipschitz coefficients are obtained.     

The asymptotic behaviors of a stochastic social epidemics model with multi-perturbation

Alcohol abuse is a major social problem, which is often called social epidemic, for the some similarities to the classical infectious diseases. In this paper, we formulated a new stochastic alcoholism model based on the deterministic model proposed in \cite{Wangxy}, with the mortalities of all populations as well as the contact infected coefficient are all perturbed. Based on this model, we investigate the long-term stochastic dynamics behaviors of two equilibria of the corresponding deterministic model and point out the effect of random disturbance on the stability of the system. Finally, we carry out numerical simulations to support our theoretical results.     

Derivation of Several SDE Systems in One- and Two-Locus Population Genetics

A direct approach is described for deriving stochastic differential equations (SDEs) for the dynamics of evolving populations. Itô SDEs are presented and compared for populations of haploid and diploid individuals with one or more alleles at one locus undergoing pure genetic drift. The results agree with previous investigations in mathematical genetics using diffusion approximations. Furthermore, a stochastic differential equation model is derived for diploid populations with two alleles at two loci. The derived SDE systems provide unifying, consistent models.     

Efficient stochastic model for the point kinetics equations

A new stochastic model for the point kinetics equations with I-delayed neutron precursor groups is presented. In this stochastic model, the point kinetics equations are separated into three terms: prompt neutrons, delayed neutrons and external neutrons source. The matrix form of the efficient stochastic model is solved by a semi-analytical method. The semi-analytical method is based on the exponential function of the coefficient matrix. The eigenvalues of the coefficient matrix and Gaussian elimination are used to calculate this exponential function. The mean and standard deviation of neutron and precursor populations of the efficient stochastic model with step, ramp, and sinusoidal reactivities are computed. The results of the efficient stochastic model are compared with the results of Allen's stochastic model for the point kinetics equations. This comparison confirms that the efficient stochastic model is an accurate model compared with the deterministic point kinetics equations. This stochastic model is efficient to study the natural behavior of neutron and precursor populations in the nuclear reactor dynamics.     

Some exact Wick type stochastic generalized Boussinesq equation solutions

The resolution of the stochastic generalized Boussinesq equation driven by a white noise is undertaken. Explicit solutions are found thanks to a white noise functional approach and the F-expansion method. Among these solutions, periodic and solitonic ones are pointed out.     

Concentration inequalities for stochastic differential equations of pure non-Poissonian jumps

We provide concentration inequalities for solutions to stochastic differential equations of pure not-necessarily Poissonian jumps. Our proofs are based on transportation cost inequalities for square integrable functionals of point processes with stochastic intensity and elements of stochastic calculus with respect to semi-martingales. We apply the general results to solutions of stochastic differential equations driven by renewal and non-linear Hawkes point processes.     

Strong solution of the stochastic Burgers equation

This work introduces a pathwise notion of solution for the stochastic Burgers equation, in particular, our approach encompasses the Cole–Hopf solution. The developments are based on regularization arguments from the theory of distributions.