Take the tour ×  Let $X$ be a continuous 1-d diffusion:  $$ dX_t
= a(X_t)dt + b(X_t)dW_t, X_0 = x. $$ W is a standard Brownian
Motion and $a(\cdot)$ and $b(\cdot)$ can have nice regularity
properties.  Let $Z^n_1,Z^n_2,\ldots$ be i.i.d. standard normal
random variables. The Euler discretization $X^n$ with step size
$\frac{1}{n}$ is the process given by  $$ X^n_\frac{k}{n} =
X^n_\frac{k-1}{n} + \frac{1}{n}\cdot a(X^n_\frac{k-1}{n}) +
\frac{1}{\sqrt{n}}b(X^n_\frac{k-1}{n}) \cdot Z^n_k, X^n_0 = x.
$$  A (at least to me) reasonable way to embed $X$ and
$X^1,X^2,\ldots$ on the same probability space is by taking the
$Z_i$ generated by the increments of the Brownian Motion $W$:
$Z^n_i = \sqrt{n} \cdot (W_\frac{i}{n} - W_\frac{i-1}{n})$.  The
reason why I want all process defined on the same space is that
I want convergence rates, and I have heard that weak convergence
results aren't appropriate for this, although I don't really
understand why.  It's believable that in some sense $X^n
\rightarrow X$. There are sever