Lets look at brownian motion now. And yes, its the same as what our high school teachers taught about the particles moving in random motion. Here, we attempt to give it a proper structure and definition to work with.

A Brownian Motion is a random process with parameters if

For , are mutually independent. This is often called the independent increments property.

For

is a continuous function of t.

We say that is a Brownian motion with drift and volatility .

For the special case of and , we have a standard Brownian motion. We can denote it with and assume that

If and then where is a standard brownian motion. Thus,

The next concept is important in finance, that is, Information Filtrations.

For any random process, we will use to denote the information available at time t.

– the set is then the information filtration.

– denotes an expectation conditional on time t information available.

Note: The independent increment property of Brownian Motion implies that any function of is independent of and that .

So let us do a bit of math to obtain for instance.

Using condition expectation identity, we have

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