Numerically Measuring Euler–Maruyama Mean-Square Convergence
Summary
The document asks how to verify the Euler–Maruyama method’s stated mean-square convergence order numerically. It proposes estimating the root mean square error across simulated paths at each time point, taking the maximum over the grid, and comparing results across step sizes. A fine-grid Euler–Maruyama path serves as a proxy for the unknown exact solution, with coarse runs driven by the same Brownian paths.
The question reports that this maximum-in-time measure does not show the expected convergence, while averaging absolute endpoint errors appears to produce the expected half-order rate. It raises a useful distinction between estimating an L2 error and averaging absolute errors, which targets a different quantity. The excerpt does not include the accepted explanation or resolve the discrepancy, so it offers a diagnostic question rather than a validated numerical procedure. Its fine-grid reference is also an approximation, not the exact solution.
Key ideas
- Mean-square error is estimated by averaging squared pathwise errors and then taking the square root.
- A maximum over time points differs from measuring error only at the terminal time.
- Using the same Brownian paths for fine and coarse simulations supports pathwise error comparisons.
- Averaging absolute errors estimates a different error measure from the L2 norm.
- The document reports conflicting numerical outcomes but does not provide a resolution.
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# Verify mean-square convergence of the Euler-maruyama scheme numerically
# Verify mean-square convergence of the Euler-maruyama scheme numerically
I have a question about the order of convergence of the Euler-Maruyama scheme and how one verifes this numerically.
I have read that the Euler-Maruyama scheme is mean-squared convergent of order 1/2 where (I think) mean-square convergence is defined by:
$$ \max_{n=1\dots N} \lVert X(t_n) -Y_n \rVert_{L_2} \leq C h^{1/2}, $$ where $ \lVert Z \rVert_{L_2} = \left( \mathbb{E}|Z|^2\right)^{1/2}$, $X(t_n)$ is the solution to an SDE at timepoint $t_n$ and $Y_n$ is the Euler-Maruyama approximation at $t_n$.
I want to verify this numerically and my approach is to approximate the norm with a Monte Carlo simulation by simulating $M$ sample paths and computing
$$ \lVert X(t_n) -Y_n \rVert_{L_2} \approx \left( \frac{1}{M} \sum_{j=1}^M |X^j(t_n) -Y^j_n|^2 \right)^{1/2} $$
($j$ is the number of the simulated sample path) for a sequence of different grids and for each grid take the maximum of the Monte Carlo means. Instead of using the true solution to the SDE I calculate the Euler-Maruyama solution on a very fine grid and use this as the reference solution. I then simulate the EM-approximation on coarser grids (but using the driver process that I used for the reference solution). This however gives me strange results (the convergence plot doesn't indicate any convergence).
It seems to work (I get order of convergence $1/2$) if I instead take the $|X^j(T) - Y^j_N|$ (the absolute value of the difference at the end point for each sampel path $j$) and then take the avergage of these for each grid. My question is if I have misunderstood something or my approach is incorrect?
Thank you and kind regards,Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.