Search

This paper is concerned with the exponential contraction and expansion of regime-switching diffusions. The focus is on the behavior of the distance from one solution to another solution, rather than with respect to some equilibrium point. The moment and almost sure exponential contraction and expansion are investigated. General criteria and easily verifiable conditions for the moment and almost sure exponential contraction and expansion are obtained. The contribution of the Markovian switching to the exponential contraction and expansion is revealed. Several examples together with numerical experiments are provided for demonstration. The convergence of contracting observers with noisy measurements under Markovian switching and the connection between exponential contraction and global attractivity of stochastic population systems are presented. [ABSTRACT FROM AUTHOR]

This work focuses on stochastic systems of weakly interacting particles containing different populations represented by multi-classes. The dynamics of each particle depends not only on the empirical measure of the whole population but also on those of different populations. The limits of such systems as the number of particles tends to infinity are investigated. We establish the existence, uniqueness, and basic properties of solutions to the limiting McKean-Vlasov equations of these systems and then obtain the rate of convergence of the sequences of empirical measures associated with the systems to their limits in terms of the pth Monge-Wasserstein distance. [ABSTRACT FROM AUTHOR]

Devoted to numerical solutions of stochastic differential equations (SDEs), this work constructs a sequences of re-embedded numerical solutions having the same distribution as those of the original SDE in a new probability space. It is shown that the re-embedded numerical solutions converge strongly to the solution of the SDE. Moreover, the rate of convergence is ascertained. The main theorem is obtained by deriving a number of technical lemmas, that are interesting in their own right. Different from the well-known results in numerical solutions of SDEs, in lieu of the usual Brownian motion increments in the algorithm, an easily implementable sequence of independent and identically distributed (i.i.d.) random variables is used. Being easier to implement compared to the construction of Brownian increments, such an i.i.d. sequence is preferable in the actual computation. As far as the convergence and uniform mean square error estimates are concerned, the use of the i.i.d. sequence does not introduce essential difficulties compared with that of the Brownian increments. Nevertheless, the analysis becomes much more difficult in the study of rates of convergence because one has to deal with the difference of the Brownian increments and the i.i.d. sequence in the almost sure sense. This paper presents a new approach to establishing rates of convergence. [ABSTRACT FROM PUBLISHER]