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In this paper, we present a DI SIR epidemic model with two categories stochastic perturbations. The long time behavior of the two stochastic systems is studied. Mainly, we show how the solution goes around the infection-free equilibrium and the endemic equilibrium of deterministic system under different conditions.

In this paper, we discuss the two-group SIR model introduced by Guo, Li and Shuai [H.B. Guo, M.Y. Li, Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q. 14 (2006) 259–284], allowing random fluctuation around the endemic equilibrium. We prove the endemic equilibrium of the model with random perturbation is stochastic asymptotically stable in the large. In addition, the stability condition is obtained by the construction of Lyapunov function. Finally, numerical simulations are presented to illustrate our mathematical findings.

Volatility plays an important role in portfolio management and option pricing. Recently, there has been a growing interest in modeling volatility of the observed process by nonlinear stochastic process [S.J. Taylor, Asset Price Dynamics, Volatility, and Prediction, Princeton University Press, 2005; H. Kawakatsu, Specification and estimation of discrete time quadratic stochastic volatility models, Journal of Empirical Finance 14 (2007) 424–442]. In [H. Gong, A. Thavaneswaran, J. Singh, Filtering for some time series models by using transformation, Math Scientist 33 (2008) 141–147], we have studied the recursive estimates for discrete time stochastic volatility models driven by normal errors. In this paper, we study the recursive estimates for various classes of continuous time nonlinear non-Gaussian stochastic volatility models used for option pricing in finance.

We establish a scaling limit theorem for a large class of Dawson–Watanabe superprocesses whose underlying spatial motions are symmetric Hunt processes, where the convergence is in the sense of convergence in probability. When the underling process is a symmetric diffusion with C b 1 -coefficients or a symmetric Levy process on R d whose Levy exponent Ψ ( η ) is bounded from below by c | η | α for some c > 0 and α ∈ ( 0 , 2 ) when | η | is large, a stronger almost sure limit theorem is established for the superprocess. Our approach uses the principal eigenvalue and the ground state for some associated Schrodinger operator. The limit theorems are established under the assumption that an associated Schrodinger operator has a spectral gap.

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.

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 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.

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.

The main aim of this paper is to improve some results obtained by Mao [X. Mao, The LaSalle-type theorems for stochastic functional differential equations, Nonlinear Stud. 7 (2000) 307–328]. Our new theorems give better results while conditions imposed are much weaker than in the paper mentioned above. For example, we need only the local Lipschitz condition but neither the linear growth condition nor the bounded moment condition on the solutions. To guarantee the existence and uniqueness of the global solution to the underlying stochastic functional differential equation (SFDE) under the weaker conditions imposed in this paper, we establish a generalised existence-and-uniqueness theorem which covers a wider class of nonlinear SFDEs as demonstrated by the examples discussed in this paper. Moreover, from our improved results follow some new criteria on the stochastic asymptotic stability for SFDEs.

In this paper, we consider the linear stochastic heat equation with additive noise in dimension one. Then, using the representation of its solution X as a stochastic convolution of the cylindrical Brownian motion with respect to an operator-valued kernel, we derive Itô's- and Tanaka's-type formulae associated to X.

Given an integer m, a probability measure ν on [0,1], a process X and a real function g, we define the m-order ν-integral having as integrator X and as integrand g(X). In the case of the fractional Brownian motion B, for any locally bounded function g, the corresponding integral vanishes for all odd indices m>1/2H and any symmetric ν. One consequence is an Itô–Stratonovich type expansion for the fractional Brownian motion with arbitrary Hurst index 01/6.

This paper discusses a randomized nonautonomous logistic equation d N ( t ) = N ( t ) [ ( a ( t ) − b ( t ) N ( t ) ) d t + α ( t ) d B ( t ) ] , where B ( t ) is 1-dimensional standard Brownian motion. We show that E [ 1 N ( t ) ] has a unique positive T-periodic solution E [ 1 N p ( t ) ] provided a ( t ) , b ( t ) , and α ( t ) are continuous T-periodic functions, a ( t ) > 0 , b ( t ) > 0 and ∫ 0 T [ a ( s ) − α 2 ( s ) ] d s > 0 .

The main aim of this note is to improve some results obtained in the author's earlier paper (1999, J. Math. Anal. Appl.236, 350–369). From the improved result follow some useful criteria on the stochastic asymptotic stability and boundedness.

Let X t be a continuous local martingale satisfying X 0 =0 and K 1 q ( t )⩽〈 X 〉 t ⩽ K 2 q ( t ) a.s. for a nondecreasing function q with constants K 1 and K 2 . Define for a Borel function f, M t = ∫ 0 t f(X s ) d X s and M t ∗ = sup 0⩽s⩽t |M s | . If f is in L 2 and f ≠0 then for any slowly increasing function φ there exist two positive constants c and C such that for all stopping times T cEφ(M T ∗2 )⩽Eφ( q(T) )⩽C(Eφ(M T ∗2 )+1). Suppose that f 2 is even and ψ(x)= ∫ 0 x ( ∫ 0 y f 2 (t) d t) d y is moderate. If φ satisfies one of the 3 conditions: (i) φ is slowly increasing, (ii) φ is concave if f ∉ L 2 , and (iii) φ is moderate if ψ(x) is convex, then there exist two positive constants c and C such that for all stopping times T cEφ(M T ∗2 )⩽Eφ∘ψ( q(T) )⩽CEφ(M T ∗2 ). Define T r = inf {t>0; |M t |=r}, r>0 . The growth rate function of ET r γ can be found for appropriate γ , as an application of the above inequalities. The method of proving the main result also yields a similar type of two-sided inequality for the integrable Brownian continuous additive functional over all stopping times.

The main aim of this paper is to establish stochastic versions of the well-known LaSalle stability theorem. From these stochastic versions follow many classical results on stochastic stability. This shows clearly the power of our new results.

It is shown that the suitably normalized maximum likelihood estimates of the parameters of periodical type of some multidimensional stationary AR processes have exactly normal distribution. This provides a generalization of the well-known behaviour of the estimate of period of a complex 1-dimensional AR process (see [1–6]).

Different kinds of variations associated with a continuous two-parameter martingale bounded in L 4 are studied. As an application a “compact” Ito formula and a version of a two-parameter Tanaka formula are proved.