What is Stochastic Differential Equation?

Consider the continuous compound interest formula

$A=Pe^{rt}$

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.

$A=\frac{dP}{dt}$

Thus, the differential equation for this change is given as

$\frac{dP}{dt}=Pe^{rt}$

$dP=Pe^{rt}dt$

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.