# Reference request – From the stochastic process to its realization.

I am looking for rigorous results on the performance of stochastic processes in the sense of some limits, say, something in the following lines.
Consider a stochastic differential equation:
$$mathrm d X_t = alpha (X_t, t) mathrm d t + beta (X_t, t) mathrm d B_t, X_0 = Z$$
where $$B_t$$ It is a Brownian movement.
Under certain conditions similar to Lipschitz, we can build a random process $$W ^ h_t$$ By time discretization with the step. $$h$$ such that
$$mathbb E[|X_t – W^h_t|] le mathcal O (h ^ n)$$
for some natives $$n$$.

Now, we want to compute the realizations of $$W ^ h_t$$. In general, it is only possible to the extent that a pseudorandom generator provides.

Can we discuss the goodness of fit using a single realization?

Is there an algorithm that, provided with some parameters (for example, seed), calculate a "pseudorealization"? $$hat w ^ h (t)$$ Which "approximates" a true realization in terms of the empirical histogram or something like that?

I can not fully understand this step. Namely, if something can be a pirori said about the numerical solutions of the SDE in terms of (pseudo) realizations.