Arithmetic Brownian Motion Process and SDEs

We discuss various things related to the Arithmetic Brownian Motion Process- these include solution of the SDEs, derivation of its Characteristic Function and Moment Generating Function, derivation of the mean, variance, and covariance, and explanation of the calibration and simulation of the process.

Summary

The SDE of the Arithmetic Brownian Motion is as follows,

$$dX_t=\mu \,dt+\sigma \,dB_t$$

What it says is that in a small period of time, or more formally an infinitesimal period of time, the process changes by a constant amount, which depends on the length of the period, and a random component. The first term can be interpreted as trend, whilst the second term is the random fluctuations around the trend. More formally, the first term is called the drift, whilst the second term is called the diffusion term. Beware: The physicist’s definition of diffusion takes a different form as we shall see when we discuss the heat/diffusion equation.

We assume $\mu \,{\rm and}\, \sigma $ are constant, but they can be deterministic functions. These parameters need to satisfy certain conditions, which essentially boil down to the requirement that they do not grow too fast, or vary too much. Here we have constant parameters,so these conditions don't matter much, and we will discuss these technical conditions in a separate section to avoid distractions

We study this simple process to introduce some of the basic concepts as they apply to stochastic processes, though it is worth pointing out that this simple process has applications in the sense that it usually leads to a very tractable model that can be used to gain useful insights into some hard problems. An application of this SDE in finance is the Normal model for the price of European options, which is a useful tool. The model also has applications in signal processing, and many other disciplines. So it is both simple and useful! Also a large variety of processes can be obtained by changing the drift from a constant to a deterministic function, of say time, or of the stochastic process.

In these notes, we solve the SDE, then characterise the process using 1) mean/variance/covariance,2) characteristic function, and 3) moment generating function, and then discuss how the model can be calibrated using observed data of a financial series. We then also show how to simulate the process.