Stochastic Differential Equations (SDEs)

Both interpretations define how the stochastic integral is evaluated, but differ in the midpoint convention used in their Riemann-sum approximations. The practical consequence is that Itô's lemma (the chain rule for Itô SDEs) contains an extra quadratic variation term absent in ordinary calculus, while Stratonovich SDEs satisfy the standard chain rule. Physics conventions often favour Stratonovich because physical derivations using white-noise limits naturally arrive at it; finance and probability theory predominantly use Itô. The two are mathematically equivalent but interconverted by a drift correction — the analyst must pick one convention and be consistent throughout.

Euler-Maruyama is the simplest choice and sufficient when only distributional (weak) accuracy is needed, or when the diffusion coefficient does not depend on the state. Milstein adds a correction term that accounts for the derivative of the diffusion with respect to the state, raising strong convergence order from 0.5 to 1.0 — making individual paths more accurate at a given step size. When the diffusion is state-dependent and you need accurate individual trajectories (for example, in variance-reduction or path-dependent option pricing), Milstein is the better default.

Strong convergence measures how close individual simulated paths are to the true solution — it is a pathwise criterion. Weak convergence measures how accurately the distribution of the solution (means, variances, option prices) is reproduced — it is a distributional criterion. A scheme can have excellent weak convergence while individual paths diverge substantially, and vice versa. If you need accurate distributions (e.g., computing expected payoffs), weak convergence is the relevant standard. If you need realistic individual trajectories (e.g., particle tracking, Monte Carlo with path-dependent payoffs), strong convergence matters.

The correlation structure is encoded in the diffusion matrix. To generate correlated Brownian increments, apply the Cholesky decomposition of the covariance matrix to independent standard-normal increments at each time step. This transforms uncorrelated noise into correlated noise that matches the specified joint covariance, ensuring the simulated paths respect the true joint dynamics of the multidimensional system.