diffusion process with geometric Brownian motion

SDE : stochastic differential equation
dAt = rAt dt

The price process we select must satisfy three requirements
1. The price should always be greater than or equal to zero.
2. If the stock price hits zero, corporate bankruptcy takes place. -> absorbing (cemetery) boundary.
3. The expected percentage return required by investors from a stock should be independent of the stock’s price. -> We will assume initially that this excess return is constant over time.(risk premium)

dSt = mSt dt + b(St , t)dzt

m : drift coefficient ; constant expected rate of return on the stock
b(St , t) : some diffusion coefficient
zt : Wiener process ; zt ~ N(0,1)

If b = 0, then it is the SDE for the risk-free asset.
For any risky asset, b cannot be zero.

b(S, t) = σ(S, t)S ; volatility of the stock price
σ is any positive function of S and t, or possibly some other stochastic variables influencing the stock.
a measure of variability or uncertainty in stock price movements

constant volatility process:
dSt = mSt dt + σSt dzt
or:
dS = mSdt + σSdz

m and σ are the constant instantaneous expected rate of return on the stock (drift rate) and volatility of the stock price

=> although not entirely realistic, as we will see, is very robust and leads to a tractable model.

σ2dt is the variance of the proportional change in the stock price in time dt
σ2S2dt is the variance of the actual change in the stock price, S, during dt.