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 $$h/2$$ h / 2 -stage of trapezoidal rule, an $$h$$ h -stage martingale, followed by another $$h/2$$ h / 2 -stage of trapezoidal rule. In $$n$$ n time steps, an $$h/2$$ h / 2 -stage deterministic step follows another $$n-1$$ n - 1 times. Each of these adjacent pairs may be combined into a single $$h$$ h -stage, effectively producing a two-stage method with partial overlap between successive time steps