Consider $\{X_t , t \geq 0 \}$ real valued diffusion satisfying $$ d X_t = b(X_t) d t + \sigma (X_t) d W_t, \quad X_0 = x \in \mathbb{R} $$ where $b, \sigma$ are well-behaved functions and $W$ is a real valued Wiener process.
For the probabilistic numerical scheme, we consider a time step $\delta t \triangleq \frac{T}{N}$ and a sequence of times $t_n \triangleq n \delta t$ for $n = 0 .. N_{\delta}$ where $T$ is a fixed time and $N_{\delta t}$ is an integer. We construct inductively $N_{\delta t}$ random variables $\{ X_n, 1 \leq n \leq N_{\delta t} \}$. Here, for each $1 \leq n \leq N_{\delta t}$, $X_n$ is an approximation of $X_{t_n}$.
Then for any well-behaved function $f$ on $\mathbb{R}$, we proceed with the following approximation $u_M^{\delta t} \approx u$ where $$ u_M^{\delta t} \triangleq \frac{1}{M} \sum_{m=1}^M f(X_{N_{\delta t}}^m) \quad \mbox{and} \quad u \triangleq \mathbb{E} f(X_T). $$ and $\{ X^m, m = 1 ... M \}$ is an i.i.d. sequence of trajectories produced by the numerical scheme.
Define $\mathcal{E}_M^{\delta t} \triangleq |u_M^{\delta t} - u|$.
question: How to empirically (and smartly) estimate the rate of convergence of $\mathcal{E}_M^{\delta t}$ to $0$ with respect $\delta t$ and $M$ knowing that we can only compute $u_M^{\delta t}$ for a bunch of values of $\delta t$ and $M$? It includes the Monte Carlo and the the corresponding numerical scheme errors.
I hope it is a fine question.
