Very nice observation. I don't know whether I have caught your point, but I try to explain it.

I may not answer your question directly. Instead, I try to clarify some concepts so that you may have a clear picture and solve your puzzle yourself.

1. You mentioned that you try to simulate the distribution of ΔS/S in a very short time interval. However, you cannot do it directly. Since we know that Brownian motion is a kind of "rough" function whose quadratic variation is not zero. If you simulate with random number directly, you actually ignore its quadratic term. You can solve dS=uSdt+gSdW, with some elementary stochastic calculus knowledge. And the solution is actually lnSt=lnS0+(u-1/2*g^2)t+g*Wt which means
ΔS/S=dlnS=(u-1/2*g^2)dt+g*dW. However, if you think ΔS/S~N(uΔt,(g^2)*Δt), this only holds when time interval is very large which means the stock price does not move continuously. Since we know that the Brownian motion is a continuous function, there may cause some problems when you simulate the stock price discretely. The fundamental difference is its mean.

2. Why we call dS=uSdt+gSdW a GBM, because the solution is in exponential form: that is St=S0*exp(...), which means that the volatility is the volatility of stock return not the stock price itself and the stock price is log normally distributed (such process prevent the stock price take a negative value)

3.If you apply the BM with a drift(generalized wiener process) to model the stock process
dS=adt+bdW, which means the stock price is normally distributed, you can integral it directly and find that S=at+bW. This is an early way to modeling stock process. You mentioned that "a" is a constant, but later you say a=Su. There is an obvious paradox here. S is stochastic therefore "a" is surly not a constant, which is contradictory to your previous assumption.

4. There are many ways to modeling stock processes, you can not always dig an intuitive meaning on each process. We just know that what parameters are responsible for mean and what are for variance. We don't say they are variance or they are mean. Just like we call b a variance or a volatility term, does not mean itself is variance or volatility. Or whether it is the volatility of the price or return. We just know that if b is bigger, the stock price will fluctuate more and so does the return.

If you want more detail or more specific question, add my qq.

感谢。。。。。。。。。。。。。

学习了

sg为dz（服从几何布朗运动，均值为0，方差为1）的系数，由于每个dt是相互独立的，所以可以求得ds的方差为dz的sg倍

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