This is the mathematical and statistical model referring to a physics theory. Originally, this was invented to predict the randomly floating atoms' movement. This is applied to financial modelling to predict the random walk movement of stock prices. Wiener process is derived from the model of Brownian motion to take an account of the variance varying differently from Brownian motion. The popular model is Ito Process invented by Prof. Kiyoshi Ito, Japanese mathematician, which is the derivation of Wiener process.
Ito-Process is more practical to explain in the financial market modelling because the future expected prices of financial assets often drift away from the current point.
The equation introduced here shows the differential of the variable in the time series. The drift coefficient denotes the lagging effect or the seasonal effect of the variable at a certain time period. The diffusion coefficient is simply the random walk part based on the normally distributed random variable while its theoretical explanation is far more complicated than explained here.
With Python, the diffusion is described as numpy.random.normal(0,SD) where SD stands for standard deviation or volatility in the financial modelling.
After plotting the time series, the expected value at the target time period tends to form a normal distribution. Then, the expected value is calculated under the given confidence interval with the confidence level (usually 95%) / the significance level (usually 5%). In case, it had better test the normality to avoid the bias.
Together with the unit root test assessing the stationarity of financial market and the capital asset pricing model indicating the assets' profitability and risk, the random walk model based on Ito process are popularly used in the financial market analyses.
