What is Stochastic Differential Equation?

Consider the continuous compound interest formula


Where A is the total amount after time t when the amount P was invested at time 0 at an interest rate r . So, A is the rate of change of P , i.e.


Thus, the differential equation for this change is given as




Where dP represents change in the amount P and dt represents change in time t . And, e^{rt} is called drift coefficient (or drift rate) that is the growth rate of the amount P .

In general, S is used to represent an asset price, and \mu is used to represent drift coefficient. So, by replacing P with S and e^{rt} with \mu , the above equation is given by

dS=\mu Sdt

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 \sigma . Like drift coefficient \mu that is in relation with the asset price S , i.e. \mu S , the uncertainty \sigma is also given in relation with the asset price S , i.e. \sigma S . Besides, this uncertainty is assumed to follow Brownian motion z . The above differential equation is now

dS=\mu Sdt+\sigma Sdz

This differential equation has a stochastic variable z 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.