Svetlana Janković

One linear analytic approximation for stochastic integrodifferential equations

Abstract This article concerns the construction of approximate solutions for a general stochastic integrodifferential equation which is not explicitly solvable and whose coefficients functionally depend on Lebesgue integrals and stochastic integrals with respect to martingales. The approximate equations are linear ordinary stochastic differential equations, the solutions of which are defined on sub-intervals of an arbitrary partition of the time interval and connected at successive division points. The closeness of the initial and approximate solutions is measured in the L p -th norm, uniformly on the time interval. The convergence with probability one is also given.

An analytic approximation of solutions of stochastic differential equations

This paper is devoted to the construction of an approximate solution of the stochastic differential equation of the Ito type, defined on a partition of the time-interval. The coefficients of the equation by their Taylor series up to arbitrary derivatives are approximated. The closeness of the original and approximate solutions is measured in the sense of the LP-norm and with probability one.

On a general decay stability of stochastic Cohen–Grossberg neural networks with time-varying delays

Abstract To the best of our knowledge, there are only few results on general decay stability applied to stochastic neural networks. For stochastic Cohen–Grossberg neural networks with time-varying delays, we study in the present paper both the pth moment ( p ⩾ 2 ) and almost sure stability on a general decay rate and partly generalize and improve some known results referring to the exponential stability. We also extend the usual notion on a general decay function, which allows us to study both the pth moment and almost sure stability even if the exponential stability cannot be shown. Some examples are presented to support and illustrate the theory.

On perturbed nonlinear Itô type stochastic integrodifferential equations

In this paper we consider the solution of the stochastic nonlinear integrodifferential equation of the Ito type with small perturbations, by comparing it with the solution of the corresponding unperturbed equation of the equal type. We investigate the closeness in the (2m)th moment sense of these solutions on finite fixed intervals or on intervals whose length tends to infinity as small perturbations tend to zero.

Neutral stochastic functional differential equations with additive perturbations

The paper deals with the solution to the neutral stochastic functional differential equation whose coefficients depend on small perturbations, by comparing it with the solution to the corresponding unperturbed equation of the equal type. We give conditions under which these solutions are close in the (2m)th mean, on finite time-intervals and on intervals whose length tends to infinity as small perturbations tend to zero.

On some stability problems of impulsive stochastic Cohen–Grossberg neural networks with mixed time delays

Abstract This paper covers the topic of both the p th moment ( p ⩾ 2 ) and almost sure stability of impulsive stochastic Cohen–Grossberg neural networks with mixed time delays. We partially use a known result on exponential stability of impulsive stochastic functional differential systems, based on the Razumikhin type technique, and extend it to the case of stochastic neural networks using the Lyapunov function method and a Gronwall type inequality. Additionally, we consider the stability with respect to a general decay function which includes exponential, but also more general lower rate decay functions as the polynomial and the logarithmic ones. This fact gives us the opportunity to study general decay almost sure stability, even when the exponential one cannot be discussed. Suitable examples which support the theory are also presented.

GENERALIZED STOCHASTIC PERTURBATION-DEPENDING DIFFERENTIAL EQUATIONS

The paper is devoted to the generalized stochastic differential equations of the Itoˆ type whose coefficients are additionally perturbed and dependent on a small parameter. Their solutions are compared with the solutions of the corresponding unperturbed equations. We give conditions under which the solutions of these equations are close in the (2m)-th moment sense on finite intervals or on intervals whose length tends to infinity as the small parameter tends to zero. We also give the degree of the closeness of these solutions.The paper is devoted to the generalized stochastic differential equations of the Itoˆ type whose coefficients are additionally perturbed and dependent on a small parameter. Their solutions are compared with the solutions of the corresponding unperturbed equations. We give conditions under which the solutions of these equations are close in the (2m)-th moment sense on finite intervals or on intervals whose length tends to infinity as the small parameter tends to zero. We also give the degree of the closeness of these solutions.

The Razumikhin approach on general decay stability for neutral stochastic functional differential equations

Abstract The topic of the paper is both the pth moment and almost sure stability on a general decay rate for neutral stochastic functional differential equations, by applying the Razumikhin approach. This concept is extended to neutral stochastic differential delay equations. The results obtained in the paper are more general and they may be specialized on the exponential, polynomial or logarithmic stability. Moreover, some neutral stochastic functional differential equations which are not pth moment or almost surely exponentially stable, could be stable with respect to a certain lower decay rate. In that sense, some nontrivial examples are presented to justify and illustrate the usefulness of the theory. More precisely, one can say anything about both the pth moment and almost sure exponential stability, although the solutions are pth moment and almost surely polynomially or logarithmically stable.

On a class of backward stochastic Volterra integral equations

Abstract In this paper, under some restrictions of the time interval, we compare a class of backward stochastic Volterra integral equations with the corresponding simpler one; to be precise, we give the relations between their solutions under global and local Lipschitz conditions on their generator functions. Using these relations, it could be easier to study solutions of more complex equations, where coefficients in backward integrals could be treated as perturbations.

On a class of backward doubly stochastic differential equations

In this paper, a new class of backward doubly stochastic differential equations is studied. This type of equations has a more general form of the forward Ito integrals compared to the ones which have been studied until now. We conclude that unique solutions of these equations can be represented with the help of solutions of the corresponding backward doubly stochastic differential equations, considered earlier in paper [5] by Pardoux and Peng. Some comparison theorems are also given, as well as a probabilistic interpretation for solutions of the corresponding quasilinear stochastic partial differential equations.