Give the Milstein correction to the Euler-Maruyama scheme for dX = a(X) dt + b(X) dW, state the strong and weak convergence orders of both schemes, and say when Euler already attains strong order 1.

Give the Milstein correction to the Euler-Maruyama scheme for dX = a(X) dt + b(X) dW, state the strong and weak convergence orders of both schemes, and say when Euler already attains strong order 1.

Approach: Expand b(X_s) around the left endpoint using Ito, keep the leading extra term involving the double stochastic integral, and identify when its coefficient vanishes.

The Milstein scheme adds (1/2) b(X_n) b'(X_n)((dW)^2 - h) to the Euler step, giving X_{n+1} = X_n + a h + b dW + (1/2) b b'(dW^2 - h); Euler-Maruyama has strong convergence order 0.5 and weak convergence order 1, Milstein has strong order 1 and weak order 1. The extra term comes from applying Ito to b(X_s) inside the stochastic integral over one step, which produces the double integral ∫∫ dW dW = ((dW)^2 - h)/2 as the leading correction. Euler already attains strong order 1 for additive noise, meaning b constant, since then b' = 0 and the correction vanishes identically. Because both schemes share weak order 1, plain Monte Carlo pricing of a European payoff gains nothing from Milstein, and the higher strong order matters for path dependent payoffs, for multilevel Monte Carlo where the variance between levels is what is being reduced, and for pathwise sensitivity estimators.

Follow-up: Why does Milstein lose its advantage in multiple dimensions with non-commuting noise, and what has to be simulated to recover it?

Key concepts: Milstein scheme, strong convergence order, weak convergence order, additive noise.