Introduction to the numerical analysis of stochastic delay differential equations

We consider the problem of the numerical solution of stochastic delay differential equations of Itô formdX(t)=f(X(t),X(t−τ))dt+g(X(t),X(t−τ))dW(t), t∈[0,T]and X(t)=Ψ(t) for t∈[−τ,0], with given f,g, Wiener noise W and given τ>0, with a prescribed initial function Ψ. We indicate the nature of the equations of interest and give a convergence proof for explicit single-step methods. Some illustrative numerical examples using a strong Euler–Maruyama scheme are provided.