## Diffusion Processes

By Lachlan Bridges, The University of Adelaide

Diffusion processes are a type of random process. What makes these processes interesting, aside from their mathematical properties, is how diverse their applications are. Diffusion processes have been used to estimate the trajectory of spacecraft, to price financial instruments, and even to model the paths that animals take when searching for food.

The most fundamental and simple type of diffusion processes is the Brownian motion, with which Albert Einstein famously described the movement of water molecules in a paper in 1905.

In one dimension, a Brownian motion is just a process that starts at zero and randomly fluctuates over time. A diffusion process extends this idea, by allowing a few extra flexibilities:

• Firstly, the process doesn’t need to start at zero.
• The process is allowed to have a deterministic “drift” term, which allows the process to trend either upwards or downwards over time.
• Finally, the magnitude of the fluctuations also known as the “diffusion” term can be adjusted, to make the processes’ fluctuations either larger or smaller.

One of the most famous uses of a diffusion process is in the Black-Scholes model in finance. This model is used to estimate the price of a European option at some time in the future. The model makes the assumption that the price of the underlying stock can be estimated by a diffusion process. In this process, the drift term represents the interest rate, which always makes the stock price tend upward over time. The diffusion term represents the random fluctuations in the stock price, with the volatility of the stock determining how large these fluctuations are.

My research extends these processes to a new and much less studied processes known as a Markov-modulated diffusion processes, or MMDP for short. The advantage of MMDP are that the drift and diffusion terms of the process are allowed to change at random times.

For example, consider modelling the price of a stock over time, such as in the Black-Scholes model. Certain random events can cause the volatility of a stock to change. Using a MMDP allows the model of the stock to switch from low to high volatility, and vice versa, at random times.

The figure shows an example MMDP representing the price of a stock. The stock is assumed to have a positive interest rate, meaning the process should drift upwards slightly over time. The stock also has two modes of volatility, low volatility (in red), and high volatility (in blue).

Lachlan Bridges was one of the recipients of a 2016/17 AMSI Vacation Research Scholarship.