Geometric Brownian Motion (GBM)
For GBM, I would suggest any reputable textbook on quantitative finance, such as Joshi or Shreve Vol. II. In fact, finance is where this concept was originally developed mathematically.
Let's say we wish to model the price of a stock by Geometric Brownian Motion (GBM), also known as a "random walk". This model can be represented by the following Stochastic Differential Equation (SDE):
$$ dS_t = \mu S_t dt + \sigma S_t dW_t $$
Some key points are as follows:
- This is a stochastic process. Although the drift/growth term $\mu S_t dt$ is deterministic, the diffusion/volatility term $\sigma S_t dW_t$ is random.
- The increments $dW_t$ are stationary, independent & normally distributed
- It is relatively simple to see that GBM is both a Markov process and a Martingale.
- We can gain insight into where the "Geometric" in GBM comes from by rewriting the SDE as $ \frac{dS_t}{S_t} = \mu dt + \sigma dW_t $. By interpreting $\frac{dS_t}{S_t}$ as the percentage change in price (i.e. the return) we can see that movements in the stock price are proportional to the current stock price.
- The famous Black-Scholes pricing model, among others, is based on asset price movements following GBM.
Fractional Brownian Motion (fBm)
fBm is a more advanced topic. Shevchenko's lecture notes and the Wikipedia page are a good place to start.
The original Mandelbrot (1968) paper summarises the idea best. With a slight change of notation:
We propose to designate a family of Gaussian random functions defined
as follows: $B_t$ being ordinary Brownian motion, and $H$ a parameter
satisfying $0 < H < 1$, fBm of exponent $H$ is a moving average of
$dB_t$, in which past increments of $B_t$ are weighted by the kernel
$(t - s)^{H-1/2}$.
So we are no longer assuming that increments are independent of one another across time; rather, they are related by a covariance function parameterised by the Hurst exponent $H$:
$$ E[B_t^H B_s^H] = \frac{1}{2} \big( t^{2H} + s^{2H} - |t-s|^{2H} \big) $$
The Hurst exponent $H$ is the key parameter of this stochastic process:
- Setting $H>\frac{1}{2}$ is appropriate for modelling "slow" processes like interest rates.
- Setting $H=\frac{1}{2}$ recovers GBM,
- Setting $H<\frac{1}{2}$ is appropriate for modelling "fast" processes like mean-reverting financial spreads.
(image from Shevchenko)
I think it's difficult to "get" fBm and the purpose of the covariance function without understanding Gaussian Processes, of which it is an example. Luckily, many machine learning textbooks now cover this topic, including MacKay and Rasmussen.
