Many important continuous-time Markov processes — for instance, the Ornstein-Uhlenbeck process and the Bessel processes — can be defined as solutions to stochastic differential equations with drift and diffusion coefficients that depend only on the current value of the process. The general form of such an equation (for a one-dimensional process with a one-dimensional driving Brownian motion) is...

Dynamic systems in many branches of science and industry are often perturbed by various types of environmental noise. Analysis of this class of models are very popular among researchers. In this paper, we present a method for approximating solution of fractional-order stochastic delay differential equations driven by Brownian motion. The fractional derivatives are considered in the Caputo sense...

Preface The purpose of these notes is to provide an introduction to to stochastic differential equations (SDEs) from applied point of view. Because the aim is in applications, much more emphasis is put into solution methods than to analysis of the theoretical properties of the equations. From pedagogical point of view the purpose of these notes is to provide an intuitive understanding in what S...

Invariant sets of dynamic systems play an important role in many situations when the dynamic behavior is constrained in some way. Knowing that a set in the state space of a system is invariant means that we have bounds on the behavior. We can verify that pre-specified bounds which originate from, for example, safety restrictions, physical constraints, or state-feedback magnitude bounds are not ...

The solvability of forward-backward stochastic differential equations with absorption coefficients is studied by the successive approximation method. The existence and uniqueness of an adapted solution are established for the equations which allow the diffusion in the forward stochastic differential equations to be degenerate. The authors also study their connection with partial differential eq...