Consider the continuous compound interest formula
Where is the total amount after time when the amount was invested at time 0 at an interest rate . So, is the rate of change of , i.e.
Thus, the differential equation for this change is given as
Where represents change in the amount and represents change in time . And, is called drift coefficient (or drift rate) that is the growth rate of the amount .
In general, is used to represent an asset price, and is used to represent drift coefficient. So, by replacing with and with , the above equation is given by
This is the case when we do not have any uncertainty, for example, we invested the amount or asset in a bank that pay fixed interest rate. For an asset price or a stock price, of course, there is uncertainty. The price may go up or down any time in the future. This uncertainty (or standard deviation) is represent by . Like drift coefficient that is in relation with the asset price , i.e. , the uncertainty is also given in relation with the asset price , i.e. . Besides, this uncertainty is assumed to follow Brownian motion . The above differential equation is now
This differential equation has a stochastic variable that changes randomly. Hence, it is called stochastic differential equation. Or any differential equation that has at least one stochastic variable is called stochastic differential equation.