Hello. This post will be about Brownian Motion. This topic is a probability and stochastic calculus topic.

**Prerequisites and Audience**

This topic appeals to the probability and financial mathematics audience. One should be familiar with the normal distribution. It would also be helpful to read another previous article on the random walk.

**What Is Brownian Motion?**

Brownian Motion is a stochastic (random) process which is a normal distribution with a mean of 0 and a variance of time t. The variance of Brownian motion varies over time.

Brownian Motion is a normal random variable and is denoted by . Sometimes is used instead of and sometimes you see or instead of or .

**From Random Walks to Brownian Motion**

The derivation of Brownian Motion comes from the random walk. In fact, it comes from the scaled symmetric random walk. The scaled symmetric random walk is defined by:

__Central Limit Theorem__

Letting be fixed. As n approaches infinity, then at time t the distribution of the scaled random walk converges to a normal distribution with mean 0 and variance t.

The limit of these scaled random walks as is Brownian motion.

**Properties**

Some basic properties of Brownian Motion include:

__Covariance of Brownian Motion__

If we have two times and with the covariance of and is the minimum of the times and which is time . In notation, we have:

__The Moment Generating Function of Brownian Motion__

The moment generating function (mgf) of a normal random variable X is given by

Since Brownian Motion is , the moment generating function of Brownian motion would be

__Martingale Property of BM__

Brownian Motion is a martingale. That is with times and and . (Source: http://quicklatex.com/cache3/d0/ql_dece83635bb784804e805c32766983d0_l3.png)

This means that the expectation of Brownian motion at time t given the filtration or history of the random process up to time s is just Brownian Motion at time s. For example, the expectation of Brownian motion at a later time given the present time is just Brownian motion at the present.

__Independent Brownian Increments__

For , the increments:

Their variances are

__Covariance-Variance Matrix of Brownian Motion__

Since each Brownian increment from above are independent and normally distributed, the random variables are jointly normally distributed.

The mean vector of the -dimensional vector is the zero vector.

From before we had the covariance of Brownian Motion as assuming that . In the covariance-variance matrix for Brownian motion, each entry is

For example if we have the first row and second column entry in the covariance matrix, we have as .

Assume that we have the ordered times . The m by m covariance-variance matrix of the -dimensional vector is as follows:

Note that for .

**Notes**

The covariance-variance matrix and covariance matrix refer to the same matrix.

Topics such as filtration and the Brownian Motion reflection principle are not discussed in depth for simplicity.

Multidimensional Brownian Motion is a more difficult case. Here we are mostly in one dimension.

Brownian Motion is sometimes called the Wiener process. Don’t ask me why.

**References**

Certain parts of this material was taken from the book Stochastic Calculus for Finance II – Continuous Time Models by Steven E. Shreve.

The image is from http://www.maplesoft.com/SizedImages/image.ashx?file=53b9a179b4602655dc2b26b5a9fbbebb_364_280.jpg.

Update on June 21, 2017: QuickLaTeX was used to fix a few math parts.