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.