# 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,