Milstein Scheme

The Milstein scheme for simulating SDEs, a strong order 1 improvement on Euler-Maruyama.
Author

John Robin Inston

Published

September 25, 2026

1 Idea

The Milstein scheme simulates the solution of an SDE (see SDE Simulation) and improves on the Euler-Maruyama Method by keeping one more term of the stochastic Taylor expansion. Consider

\[ dX_t=f(t,X_t)\,dt+g(t,X_t)\,dW_t, \]

with Lipschitz coefficients \(f,g\) and Brownian Motion \(W\), simulated on the grid \(t_n=n\Delta t\) with \(\Delta W_n=W_{t_{n+1}}-W_{t_n}\). Euler-Maruyama freezes \(g\) at the left endpoint. Applying the Itô Formula to \(g\) instead shows the next-largest term is \(L^1g\int\!\!\int dW\,dW\), where \(L^1=g\,\partial_x\), and the double integral is explicit,

\[ \int_{t_n}^{t_{n+1}}\int_{t_n}^{s}dW_u\,dW_s =\tfrac{1}{2}\left((\Delta W_n)^2-\Delta t\right). \]

2 The Scheme

\[ X_{n+1}=X_n+f(t_n,X_n)\Delta t+g(t_n,X_n)\Delta W_n +\frac{1}{2}\,g(t_n,X_n)\,\partial_xg(t_n,X_n) \left((\Delta W_n)^2-\Delta t\right). \]

The extra term is a correction for the fact that the diffusion coefficient changes across the step, and it vanishes when \(g\) does not depend on \(x\) (additive noise), in which case Milstein coincides with Euler-Maruyama.

3 Example: Black-Scholes-Merton Model

For the BSM model under \(\mathbb{Q}\), \(f=rS\) and \(g=\sigma S\), so \(g\,\partial_xg=\sigma^2S\). Writing \(\Delta W=\sqrt{\Delta t}\,\xi\) with \(\xi\sim\mathcal{N}(0,1)\),

\[ S_{t+\Delta t}=S_t\left\{1+\sigma\sqrt{\Delta t}\,\xi +\Delta t\left[r+\tfrac{1}{2}\sigma^2(\xi^2-1)\right]\right\}. \]

This is easy to simulate with a loop, where h is the step size.

S <- S0
for (i in 2:length(t)) {
  xi <- rnorm(1, 0, 1)
  S[i] <- S[i-1] * (1 + sigma * sqrt(h) * xi +
            h * (r + 0.5 * sigma^2 * (xi^2 - 1)))
}

4 Convergence

To compare the two schemes one measures the error \(X_n\to X(t_n)\) in the strong sense, \(\mathbb{E}|X_n-X(t_n)|\), and in the weak sense, \(|\mathbb{E}\,\phi(X_n)-\mathbb{E}\,\phi(X(t_n))|\). Under the usual smoothness assumptions Milstein has strong order \(1\), against order \(1/2\) for Euler-Maruyama, while both have weak order \(1\). The comparison is carried out in SDE Simulation.

5 References

  1. Higham, D. J. and Kloeden, P. E. (2021). An Introduction to the Numerical Simulation of Stochastic Differential Equations. SIAM, Section 17.4.
  2. Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer.
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