A weak second-order split-step method for numerical simulations of stochastic differential equations

In an analogy from symmetric ordinary differential equation numerical integrators, we derive a three-stage, weak 2nd-order procedure for Monte-Carlo simulations of Itô stochastic differential equations. Our composite procedure splits each time step into three parts: an \(h/2\) -stage of trapezoidal rule, an \(h\) -stage martingale, followed by another \(h/2\) -stage of trapezoidal rule. In \(n\) time steps, an \(h/2\) -stage deterministic step follows another \(n-1\) times. Each of these adjacent pairs may be combined into a single \(h\) -stage, effectively producing a two-stage method with partial overlap between successive time steps.