Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions

The classical existence-and-uniqueness theorem of the solution to a stochastic differential delay equation (SDDE) requires the local Lipschitz condition and the linear growth condition (see e.g. [11], [12] and [20]). The numerical solutions under these conditions have also been discussed intensively (see e.g. [4], [10], [13], [16], [17], [18], [21], [22] and [24]). Recently, Mao and Rassias [14] and [15] established the generalized Khasminskii-type existence-and-uniqueness theorems for SDDEs, where the linear growth condition is no longer imposed. These generalized Khasminskii-type theorems cover a wide class of highly nonlinear SDDEs but these nonlinear SDDEs do not have explicit solutions, whence numerical solutions are required in practice. However, there is so far little numerical theory on SDDEs under these generalized Khasminskii-type conditions. The key aim of this paper is to close this gap.     

Stochastic population dynamics under regime switching

In this paper we will develop a new stochastic population model under regime switching. Our model takes both white and color environmental noises into account. We will show that the white noise suppresses explosions in population dynamics. Moreover, from the point of population dynamics, our new model has more desired properties than some existing stochastic population models. In particular, we show that our model is stochastically ultimately bounded.     

Stochastic delay Lotka-Volterra model

We reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay Lotka-Volterra model but will also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the delay Lotka-Volterra model into the Itô form , and show that although the solution to the original delay equation may explode to infinity in a finite time, with probability one that of the associated stochastic delay equation does not. We also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix A.     

Stochastic differential delay equations of population dynamics

In this paper we stochastically perturb the delay Lotka-Volterra model

     

Stochastic population dynamics under regime switching II

This is a continuation of our paper [Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334 (2007) 69-84] on stochastic population dynamics under regime switching. In this paper we still take both white and color environmental noise into account. We show that a sufficient large white noise may make the underlying population extinct while for a relatively small noise we give both asymptotically upper and lower bound for the underlying population. In some special but important situations we precisely describe the limit of the average in time of the population.     

Quadratic covariation estimates in non-smooth stochastic calculus

Given a Brownian Motion W$W$, in this paper we study the asymptotic behavior, as ε→0$\epsilon \to 0$, of the quadratic covariation between f(εW)$f\left(\epsilon W\right)$ and W$W$ in the case in which f$f$ is not smooth. Among the main features discovered is that the speed of the decay in the case f∈C^α$f\in C^{\alpha }$ is at least polynomial in ε$\epsilon$ and not exponential as expected. We use a recent representation as a backward–forward Itô integral of [f(εW),W]$[f\left(\epsilon W\right),W]$ to prove an ε$\epsilon$-dependent approximation scheme which is of independent interest. We get the result by providing estimates to this approximation. The results are then adapted and applied to generalize the results of Almada Monter and Bakhtin (2011) and Bakhtin (2011) related to the small noise exit from a domain problem for the saddle case.     

Stochastic Analysis of the Fractional Brownian Motion

Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula, the Itô–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations.     

Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching

In this paper, we prove that a stochastic logistic population under regime switching controlled by a Markov chain is either stochastically permanent or extinctive, and we obtain the sufficient and necessary conditions for stochastic permanence and extinction under some assumptions. In the case of stochastic permanence we estimate the limit of the average in time of the sample path of the solution by two constants related to the stationary probability distribution of the Markov chain and the parameters of the subsystems of the population model. Finally, we illustrate our conclusions through two examples.     

Analysis of a delayed stochastic predator–prey model in a polluted environment

In this paper, we investigate two-species Lotka–Volterra delayed stochastic predator–prey systems, with and without pollution, denoted by (M)$(M)$ and (_M0)$(M_0)$, respectively. We show that there exists a unique non-negative solution in each system that is permanent in time average under certain conditions. Moreover, the non-permanence of model (M)$(M)$ is studied. Finally, computer simulations are carried out to verify our results.     

Nonlinear complementarity problem and systems of convex inequalities

LetH be a nonempty closed convex subset of a topological vector spaceE andF be a real-valued function onH × H Then, we prove that, under some conditions, there existsx ₀H such thatF(x ₀,y)0 for ally H. Furthermore, we obtain a necessary and sufficient condition that a finite system of convex inequalities is irreducibly inconsistent.This work was supported in part by the Matsunaga Research Grant. The author wishes to express his sincere thanks to Professor H. Umegaki for his invaluable suggestions and advice.     

Exponential stability of equidistant Euler–Maruyama approximations of stochastic differential delay equations

Our aim is to study under what conditions the exact and numerical solution (based on equidistant nonrandom partitions of integration time-intervals) to a stochastic differential delay equation (SDDE) share the property of mean-square exponential stability. Our approach is trying to avoid the use of Lyapunov functions or functionals. We show that under a global Lipschitz assumption an SDDE is exponentially stable in mean square if and only if for some sufficiently small stepsize Δ$\Delta$ the Euler–Maruyama (EM) method is exponentially stable in mean square. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same “if and only if” result. The important feature of this result is that it transfers the asymptotic problem into a finite-time convergence problem. Replacing the exact and EM numerical solution with stochastic processes, we also discuss whether a family of stochastic processes share the stability property. This new approach allows us to discuss (i) whether a family of SDDEs share the stability property, and (ii) whether an SDDE with variable time lag shares stability property with the modified EM method. As another application of this general approach we consider a linear SDDE with variable time lag and establish an “if and only if” result. It should also be mentioned that the questions addressed, results proved, as well as style of analysis borrow heavily from [14] but the computations involved to cope with time delay are nontrivial.     

Stochastic comparisons of series and parallel systems with randomized independent components

Consider a series or parallel system of independent components and assume that the components are randomly chosen from two different batches, with the components of the first batch being more reliable than those of the second. In this note it is shown that the reliability of the system increases, in usual stochastic order sense, as the random number of components chosen from the first batch increases in increasing convex order. As a consequence, we establish a result analogous to the Parrondo’s paradox, which shows that randomness in the number of components extracted from the two batches improves the reliability of the series system.     

On the asymptotic distribution of a unit root test against ESTAR alternatives

We derive the null distribution of the nonlinear unit root test proposed in Kapetanios et al. [Kapetanios, G., Shin, Y., Snell, A., 2003. Testing for a unit root in the nonlinear STAR framework. Journal of Econometrics 112, 359-379] when nonzero means or both means and deterministic trends are accounted for. Some discrepancies to claims in Kapetanios et al. are discussed.     

Existence of Mild Solutions to Stochastic Delay Evolution Equations with a Fractional Brownian Motion and Impulses

In this article, we study the existence of mild solutions to stochastic impulsive evolution equations with time delays, driven by fractional Brownian motion with the Hurst index H > 1/2 via a new fixed point analysis approach.     

On interpolation variants of Newton’s method for functions of several variables

A generalization of the variants of Newton’s method based on interpolation rules of quadrature is obtained, in order to solve systems of nonlinear equations. Under certain conditions, convergence order is proved to be 2d+1, where d is the order of the partial derivatives needed to be zero in the solution. Moreover, different numerical tests confirm the theoretical results and allow us to compare these variants with Newton’s classical method, whose convergence order is d+1 under the same conditions.     

An Uniqueness Result for a Class of Wiener-Space Valued Stochastic Differential Equations

We prove a generalization of Bismut-Itô-Kunita formula to infinite dimensions and derive an uniqueness result for Wiener space valued processes which holds for a special class of Bernstein processes.     

A stochastic model of AIDS and condom use

In this paper we introduce stochasticity into a model of AIDS and condom use via the technique of parameter perturbation which is standard in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as desired in any population dynamics. We also carry out a detailed analysis on asymptotic stability both in probability one and in pth moment. Our results reveal that a certain type of stochastic perturbation may help to stabilise the underlying system.     

Positive solutions of a nonlinear impulsive equation with periodic boundary conditions

In this paper, we investigate nonlinear second order differential equations subject to linear impulse conditions and periodic boundary conditions. Sign properties of an associated Green’s function are exploited to get existence results for positive solutions of the nonlinear boundary value problem with impulse. Upper and lower bounds for positive solutions are also given. The results obtained yield periodic positive solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.     

Efficient high-order methods based on golden ratio for nonlinear systems

We derive new iterative methods with order of convergence four or higher, for solving nonlinear systems, by composing iteratively golden ratio methods with a modified Newton’s method. We use different efficiency indices in order to compare the new methods with other ones and present several numerical tests which confirm the theoretical results.