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On stochastic differential equations with reflecting boundary condition in convex domains

Łaukajtys W.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 4

445-455

Let D be an open convex set in R^d and let F be a Lipschitz operator defined on the space of adapted cadlag processes. We show that for any adapted process H and any semimartingale Z there exists a unique strong solution of the following stochastic differential equation (SDE) with reflection on the boundary of D: Xt=Ht + integral of (F(X)s-, dZs] on the interval [0, t]+Kt, t belongs to R+. Our proofs are based on new a priori estimates for solutions of the deterministic Skorokhod problem.

Approximation of the Zakai Equation in a Nonlinear Filtering Problem With Delay

Twardowska K., Marnik T., Pasławska-Południak M.

International Journal of Applied Mathematics and Computer Science

2003

Vol. 13, no 2

151-160

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.

Cellular Automata and Many-Particle Systems Modeling Aggregation Behavior Among Populations

Morale D.

International Journal of Applied Mathematics and Computer Science

2000

Vol. 10, no 1

157-173

A cellular automaton model is presented in order to describe mutual interactions among the individuals of a population due to social decisions.The scheme is used for getting qualitative results, comparable to field experiments carried out on a population of ants which present an aggregative behavior. We also present a second description of a biological spatially structured population of N individuals by a system of stochastic differential equations of Ito type. A 'law of large numbers' to a continuum dynamics described by an integro-differential equation is given.