where W is a Wiener process, Z is a Hölder continuous process with Hölder exponent greater than 1/2, the coefficients a, b, c depend on the past of the process X . The integral with respect to W is understood in the usual Itô sense, while the one with respect to Z is understood in the pathwise sense. (A precise definition of all objects is given in Section 2.) We will call this equation a mixed...

The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one. The scheme has been extended to the stochastic delay differential equations but the analysis of the convergence is technically complicated due to anticipative integrals in the remainder terms. This paper employs an elementary method to derive the Milstein...

The paper is concerned with a mixed stochastic delay differential equation involving both a Wiener process and a γ-Hölder continuous process with γ > 1/2 (e.g. a fractional Brownian motion with Hurst parameter greater than 1/2). It is shown that its solution depends continuously on the coefficients and the initial data. Two applications of this result are given: the convergence of solutions to ...

Stochastic delay differential equations allow the stochastic differential equations to incorporate past data segments. They are widely used to solve systems having delay feedbacks. In the linear Gaussian case, the differential equations can be solved by computing the covariance functions through the double Laplace transform. It turns out that the solution can be decomposed into the sum of indep...

substantiation of the averaging method for differential equations with maxima is presented. two theorems on substantiates for differential equations with maxima are established.

This paper is concerned with the exponential mean-square stabilisation of hybrid stochastic differential equations (also known as stochastic differential equations with Markovian switching) by delay feedback controls. Although the stabilisation by nondelay feedback controls for such equations has been discussed by several authors, there is so far little on the stabilisation by delay feedback co...