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Approximation of the Zakai equation in a nonlinear filtering problem with delay

Krystyna Twardowska, Tomasz Marnik, Monika Pasławska-Południak (2003)

International Journal of Applied Mathematics and Computer Science

A nonlinear filtering problem with delays in the state and observation equations is considered. The unnormalized conditional probability density of the filtered diffusion process satisfies the so-called Zakai equation and solves the nonlinear filtering problem. We examine the solution of the Zakai equation using an approximation result. Our theoretical deliberations are illustrated by a numerical example.

Asymptotic and exponential decay in mean square for delay geometric Brownian motion

Jan Haškovec (2022)

Applications of Mathematics

We derive sufficient conditions for asymptotic and monotone exponential decay in mean square of solutions of the geometric Brownian motion with delay. The conditions are written in terms of the parameters and are explicit for the case of asymptotic decay. For exponential decay, they are easily resolvable numerically. The analytical method is based on construction of a Lyapunov functional (asymptotic decay) and a forward-backward estimate for the square mean (exponential decay).

Blow-up results for some reaction-diffusion equations with time delay

Hongliang Wang, Yujuan Chen, Haihua Lu (2012)

Annales Polonici Mathematici

We discuss the effect of time delay on blow-up of solutions to initial-boundary value problems for nonlinear reaction-diffusion equations. Firstly, two examples are given, which indicate that the delay can both induce and prevent the blow-up of solutions. Then we show that adding a new term with delay may not change the blow-up character of solutions.

Control of a class of chaotic systems by a stochastic delay method

Lan Zhang, Cheng Jian Zhang, Dongming Zhao (2010)

Kybernetika

A delay stochastic method is introduced to control a certain class of chaotic systems. With the Lyapunov method, a suitable kind of controllers with multiplicative noise is designed to stabilize the chaotic state to the equilibrium point. The method is simple and can be put into practice. Numerical simulations are provided to illustrate the effectiveness of the proposed controllable conditions.

Existence and controllability of fractional-order impulsive stochastic system with infinite delay

Toufik Guendouzi (2013)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

This paper is concerned with the existence and approximate controllability for impulsive fractional-order stochastic infinite delay integro-differential equations in Hilbert space. By using Krasnoselskii's fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of impulsive fractional stochastic system under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided...

Existence of solutions and approximate controllability of impulsive fractional stochastic differential systems with infinite delay and Poisson jumps

Chinnathambi Rajivganthi, Krishnan Thiagu, Palanisamy Muthukumar, Pagavathigounder Balasubramaniam (2015)

Applications of Mathematics

The paper is motivated by the study of interesting models from economics and the natural sciences where the underlying randomness contains jumps. Stochastic differential equations with Poisson jumps have become very popular in modeling the phenomena arising in the field of financial mathematics, where the jump processes are widely used to describe the asset and commodity price dynamics. This paper addresses the issue of approximate controllability of impulsive fractional stochastic differential...

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