The fluctuation for the stock market these days serves as satisfied materials for stochastics learning.
Originally, price of stock is concerned to fulfill the function: ds=udt+b*dW, where dW is a Wenier process.
However, in the process of huge simulation, S probably goes into negative, which contraticts the common
sense. Thus, a new formular is put forward.
dLn(S)=a*S*dt+bS*dW. this guarantees that S remains in the positive interval. But in the long run, S is gradually
increasing. Still, it's not satisfactory to depict the fluctuation phenomenon.
Then comes up with the dLn(s)=(v-a*S)dt+b*dW. The formular would adjust the trend in the following ways.
if S goes too high, say, above v/a, the coefficient of dt turns to negative, subsquently pulls down the trend;
vice vesa when the price is lower. This seems better. However, it implies that the voliatility constant.
Thus, the last version, seemingly more realisitic, dLn(S)=(v-a*s)dt+b*S^(1/2)dW. It not only includes the third version's
merits, but also, gives the volatility a vivd exlaination, the higher the price, the higher the volatility, or the fluctuation.