# Reference Request for Fractional Brownian motion

This question has been asked several times on this website. But the problem is that all the references suggested are mathematics oriented and difficult to understand. I am looking for a reference, which explains it in easy language but without over-simplifying. I want to understand the basic ideas behind Brownian and fractional Brownian motion, with some mathematical understanding.

Unfortunately, I have not been able to find any text which contains a good exposition of the concepts without making the mathematics impenetrable. Any suggestions would be useful for both Brownian and fractional Brownian motion. I will prefer to read two or three chapters over a whole book on the subject.

I have a fairly good background in probability theory and engineering calculus.

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.

• I have a good understanding of Gaussian processes including the mathematics of it. However, fBM's mathematics is somewhat out of my reach. Thanks a lot for the suggestions. I will go through the references you have suggested. Also, would understanding GBM help understanding fBM? – Abhinav Gupta Feb 9 '18 at 1:24
• Absolutely, understanding GBM (and stochastic calculus in general) would help you understand fBm, since it is just a special case of it. Those 2 finance textbooks should help you with that. – A. G. Feb 9 '18 at 5:23