Euler-Maruyama method

For $d{X}_t=a(X_t)dt+\sigma (X_t)d{W}_t$, the Euler-Maruyama method advanceswhere $\Delta W_n\sim N(0,\Delta t)$.

For a scalar stochastic differential equation, the Milstein method adds the quadratic-variation correctionUnder standard smoothness assumptions it has strong order of convergence one and weak order of convergence one.