I am currently reading Anderson & Kurtz's Stochastic Analysis of Biochemical Systems. In this book, one studies chemical reaction networks and also how to simulate these with MATLAB. I know, that there is also a package in the R Project called GillespieSSA that can be used for those kind of simulations, too. So far, I was able to implement some of the algorithms. However, I am wondering, how I could compare my simulated graphs to an actual SDE. To be more precise: I have the assumption that one graph (or one species in terms of chemical reaction networks) should be following a specific SDE
$$dX_t=\mu dt+\sigma dW_t$$
How can I check, that my simulation is in fact 'solving' this SDE? Do I need to work with the expectation and variance of the simulation over time and compare it to some theoretical value? I have the feeling, that there must be a way to compare a computational value to a theoretical value, to verify, that my simulation is a possible candidate for a solution of this SDE, so to speak.
I'd be happy for any hint or help! Any book or paper recommendation are very welcome, too!