Euler-Maruyama approximation of backward doubly stochastic differential delay equations

In this paper, we attempt to introduce a new numerical approach to solve backward doubly stochastic differential delay equation ( shortly-BDSDDEs ). In the beginning, we present some assumptions to get the numerical scheme for BDSDDEs, from which we prove important theorem. We use the relationship between backward doubly stochastic differential delay equations and stochastic controls by interpreting BDSDDEs as some stochastic optimal control problems, to solve the approximated BDSDDEs and we prove that the numerical solutions of backward doubly stochastic differential delay equation converge to the true solution under the Lipschitz condition.

Carathéodory’s approximate solution to stochastic differential delay equation

The main aim of this paper is to discuss Carath?odory?s and Euler-Maruyama?s approximate solutions to stochastic differential delay equation. To make the theory more understandable, we impose the non-uniform Lipschitz condition and non-linear growth condition.

DYNAMICS OF DETERMINISTIC AND STOCHASTIC EVOLUTIONARY GAMES WITH MULTIPLE DELAYS

In the present paper, we study the effect of time delays in evolutionary games with one population of users and two strategies. The case where the delays, corresponding to different strategies, are not the same is considered. The local stability of the stationary state for the replicator dynamics is analyzed. We show that there is Hopf bifurcation. The stability of the bifurcating periodic solutions is determined by using the center manifold theorem and normal form theory. The stochastic evolutionary game with delay is taken into consideration. We also study the behavior of the first and second solution moments for linear stochastic differential delay equation in the presence of white and colored noise. The last part of the paper includes numerical simulations and conclusions.

Weak approximation of stochastic differential delay equations for bounded measurable function

In this paper we study the weak approximation problem of $E[\phi (x(T))] $ by $E[\phi (y(T))] $, where $x(T)$ is the solution of a stochastic differential delay equation and $y(T)$ is defined by the Euler scheme. For $\phi \in { C}_{b}^{3} $, Buckwar, Kuske, Mohammed and Shardlow (‘Weak convergence of the Euler scheme for stochastic differential delay equations’, LMS J. Comput. Math. 11 (2008) 60–69) have shown that the Euler scheme has weak order of convergence $1$. Here we prove that the same results hold when $\phi $ is only assumed to be measurable and bounded under an additional non-degeneracy condition.