Stochastic Differential Equations (SDEs)

The previous article on Brownian motion and the Wiener process introduced standard Brownian motion as a method for simulating asset price paths. However, standard Brownian motion has a non-zero probability of being negative, which is clearly not a characteristic of real-world assets—stock prices cannot fall below zero. Therefore, while the stochastic nature of Brownian motion for our model should be retained, it is necessary to adjust the random distribution method correctly. Specifically, the concept of geometric Brownian motion (GBM) will be presented now, which addresses the issue of negative stock prices.

However, before considering geometric Brownian motion, it is necessary to discuss the concept of stochastic differential equations (SDE). This will allow us to construct GBM and solve it to obtain a function for the asset price path.

Stochastic differential equation

Now that we have defined Brownian motion, we can use it as a building block to start creating stochastic differential equations (SDE). We need a stochastic differential equation (SDE) to discuss the behavior of the function f = f(S) and its derivative with respect to S, where S is the stock price determined by Brownian motion.

Some rules of regular calculus do not work as expected in a stochastic world. We need to modify them to account for both the random behavior of Brownian motion and its non-differentiable nature. We will start by discussing stochastic integrals, which will naturally lead us to the concept of SDE.

Definition (Stochastic Integral)

The stochastic integral of the function f = f(t)

is the function W = W(t),$t\in [0,T]$ defined by:$$\begin{array}{r}W(t)={\int }_0^{t}f(s)dB(s)=\underset{n\to \mathrm{\infty }}{lim}\sum _{k=1}^{N}f(t_{k-1})(B(t_k)-B(t_{k-1}))\end{array}$$

where $t_k=\frac{kt}{N}$.

Note that the function $f$ is non-anticipatory, in the sense that it is evaluated within the summation sign at time $t_{k-1}$. This means that it has no information as to what the random variable at $X(t_k)$ is. Supposing that $f$ represented some portfolio allocation based on $B$, then if it were not evaluated at $t_{k-1}$, but rather at $t_k$, we would be able to anticipate the future and modify the portfolio accordingly.

It can be seen that $\mu$ and $\sigma$ are both functions of $t$ and $W$. $\mu$ has the interpretation of a non-stochastic drift coefficient, while $\sigma$ represents the coefficient of volatility - it is multiplied by the stochastic $dB$ term. Hence, stochastic differential equations have both a non-stochastic and stochastic component.

In the following section on geometric Brownian motion, a stochastic differential equation will be utilised to model asset price movements.

Reference: Stochastic Differential Equations

From https://www.quantstart.com/articles/Stochastic-Differential-Equations/