Comparison of two numerical methods for stochastic delay differential equations and the relationship between bifurcation approximations and step length

We give introductions to delay differential equations, stochastic differential equations, numerical approximations, Brownian motion and Ito calculus, stability and bifurcation points and Lyapunov exponents. Using these methods we replicate the calculations in the paper by Neville J. Ford & Stewart J. Norton, entitled Noise induced changes to the behaviour of semi implicit Euler methods for stochastic delay differential equations undergoing bifurcation . We present our results that correspond to some of the tables and equations presented in their paper. We then apply the same methodology using a Milstein numerical method with the same parameters and random distributions and compare these results with our ndings from the Euler-Maruyama scheme. We fi nd that the Milstein scheme exhibits the same relational behaviours between the bifurcation approximations from the Lyapunov exponents and step length as was presented in Ford and Norton's paper for the Euler-Maruyama scheme, we also find that the Milstein scheme maintains its greater accuracy up to and including the bifurcation approximation