Strong and Weak Convergence of SDE Schemes

You can't solve most stochastic differential equations (SDEs) exactly, so you simulate them on a computer using a discrete scheme like Euler-Maruyama, stepping forward in small time increments. Because the true solution involves genuine randomness, there are two very different questions you could ask about how good that simulation is, and confusing them leads to picking the wrong numerical scheme for the job.

The physical analogy is weather forecasting. You could ask "did my simulated storm track the exact path the real storm took, minute by minute?" — that's like asking whether individual simulated paths match individual true paths, which is called strong convergence. Or you could instead ask "does my simulation get the right distribution of possible storm outcomes — the right average path, the right spread?" without caring whether any single simulated run matches any single real run — that's weak convergence, and it only cares about matching statistics like the mean or a derivative price computed as an expectation.

Formally, strong convergence measures the expected size of the pathwise error, $\mathbb{E}[|X_T - \hat X_T|]$, between the true path $X$ and the simulated path $\hat X$ at the same random shocks; weak convergence instead measures the error in expectations of some function, $|\mathbb{E}[f(X_T)] - \mathbb{E}[f(\hat X_T)]|$, which can be small even when individual paths diverge wildly. Euler-Maruyama typically achieves strong order 0.5 (error shrinks like the square root of the step size) but weak order 1.0 (error shrinks proportionally to the step size) — a scheme can therefore look much better or much worse depending purely on which notion of convergence you check.

This distinction matters directly for pricing: computing an option price via Monte Carlo only needs weak convergence, since only the expectation matters, so a cheaper scheme with weak order 1 suffices; but hedging strategies or path-dependent risk measures that depend on matching actual trajectories need strong convergence, and typically require finer time steps for the same accuracy.