I'm writing an overview that is more economic than mathematical and I want to explain shortly the stochastic differential equation of Geometric Brownian Motion as simple and clear as possible
$$dS_t = S_t \mu dt + S_t \sigma dW_t $$
(I have already described the idea of Efficient market and why the Brownian motion is used).
So I wrote: In finance time series are often modeled as combination of drift (non-random trend) and volatility. This stochastic differential equation has two addends, the first controls the trend and is not random and the second is responsible for random fluctuations around the trend curve. Namely, after the very short time interval $dt$, the value of the stock price process $S_t$ will change by the small amount that is normally distributed with mean $ S_t \mu dt$ and variance $(S_t \sigma)^2dt$.
The last words were taken from Wikipedia article about SDEs and the problem is that I'm actually unsure in what is said there:
A heuristic (but very helpful) interpretation of the stochastic differential equation is that in a small time interval of length $\delta$ the stochastic process $X_t$ changes its value by an amount that is normally distributed with expectation $\mu(X_t, t)\delta$ and variance $\sigma(X_t, t)^2 \delta$ and is independent of the past behavior of the process. This is so because the increments of a Wiener process are independent and normally distributed.
How can this value be independent from the history of $X_t$ if both mean $\mu(X_t, t) \delta$ and variance $\sigma(X_t, t)^2 \delta$ depend on $X_t$? However, I understand that if this is not so, it wouldn't be Brownian motion.