Re: GBM and wiener process

Hey noblewhale.

Geometric Brownian Motion is a specific kind of process that is a function of a Wiener Process. A Wiener process has specific properties and is basically a normal distribution with the properties of additive variance, independence for independent intervals, the first value is 0 and there's another one I can't recall currently.

For the first one (and the second but you have a different variable), you have a random variable that is a function of t where for every value of t you get a random variable in itself and this is a stochastic process.

You need to solve for X(t) (i.e. the distribution or definition of the random variable) and then find the expectation given particular values of t (think about each t being an individual random variable).

There are two main ways in probability to do this: the first is through stochastic calculus and the 2nd one is through the transformation process to get the distribution of a random variable if it's some kind of function of another (you should get some kind of log-normal distribution if you do it this way).

What have you covered with respect to the above concepts?

Re: GBM and wiener process

Hi Chiro:

First of all thank you for your reply.

I am scratching my head when I read your response. Basically I understand what a wiener process is. It is some process with stochastic additive for change in t. The change is normally distributed.

Geometric Brownian Motion is a case that somehow involves the logarithm (I am not sure how...)

We have only covered the basic definition of the two processes. I tried to read through wikipidia to understand them & I am still clueless about the relationship btw GBM and Wiener. I guess my confusion comes from not able to find an example.

I can only say that I can technically name the properties of both processes, but unable to apply them.

:S

NW

Re: GBM and wiener process

The idea of these particular processes is that you have a Stochastic process that has a time component (deterministic) and a random component (typically a Weiner process) and that the process evolves over time taking into account the deterministic and non-deterministic components.

For example if the dWt was multiplied by 0, then you would have a deterministic first order differential equation with dX = adt which gives dX/dt = a so X(t) = at + C but if you have a random component, then this will affect the evolution of the process by adding in "random changes" at each time-step which will make it "jagged" as a result.

So what happened was that mathematicians like Ito developed a way to calculate derivatives of stochastic processes in the above form (dX = a(X,t)dt + b(X,t)dt) where one could find the derivatives of a particular variable with respect to both the time and the Weiner process variables of which one is a random variable and the other is deterministic.

With this lemma, you then use this to help get to a solution for X(t).

In a geometric brownian motion, and in this kind of situation, your distribution will be a log-normal one where X(t) = S_0*e((mu - [1/2]*sigma^2)*t + sigma*Wt).

The proof for this involves using the various forms of Ito's lemma and knowledge of integrating deterministic and non-deterministic parts (on top of differentiating).

If you are studying this in class and you haven't gone through this, then that is a problem you need to address.

Re: GBM and wiener process

after days of confusion, thank you