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.