probability theory notation

[edtost]

trying to understand a section from a book....here's the passage i'm having trouble with:

"To be reasonably precise about asset dynamics, we fix a probability space ([Sigma],F,P) on which there is a standard Brownian motion W. The information set (or [sigma]-algebra) generated by this Brownian motion up to time t is denoted Ft."

what would [Sigma] and F denote, and what is a [sigma]-algebra? I'm somewhat familiar with the model they are trying to describe, it seems like Ft is the history of the Brownian motion and F is the set of all possible histories, and P is obviously some sort of probability measure. But I have no idea what [Sigma] could be, or what a [sigma]-algebra is (though a search brought up things relating to measure theory, with which I am unfamiliar).

thanks,
ed

http://mathworld.wolfram.com/ is a good resource for math stuff.

Roughly speaking, it's something like this:
[Sigma] is all possible results
F is the collection of subsets of [Sigma] that have probability
P is a function from F to [0,1] that gives the probability of F occuring (more or less).

For example:
[Sigma] might be the set of all possible heads up 7 card stud games.
F is a list of all subsets. For example, an element of F might be all games where the big blind has a hidden ace.
P gives the probability for an element of F (for example, the probability of a game where the big blind has a hidden ace).

[Luzion]

[ QUOTE ]
trying to understand a section from a book....here's the passage i'm having trouble with:

"To be reasonably precise about asset dynamics, we fix a probability space ([Sigma],F,P) on which there is a standard Brownian motion W. The information set (or [sigma]-algebra) generated by this Brownian motion up to time t is denoted Ft."

what would [Sigma] and F denote, and what is a [sigma]-algebra? I'm somewhat familiar with the model they are trying to describe, it seems like Ft is the history of the Brownian motion and F is the set of all possible histories, and P is obviously some sort of probability measure. But I have no idea what [Sigma] could be, or what a [sigma]-algebra is (though a search brought up things relating to measure theory, with which I am unfamiliar).

thanks,
ed

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I have no idea what Brownian motion is. But when dealing with discrete probability, the Sigma symbol is used to show the total summation of some variables. A book might show E(x) = Sigma p(x)u(x) or something like that to tell you to sum up all possible probabilities and the values.

[mosdef]

brownian motion refers (imprecisely) to random walks. back in the day, a physicist (or biologist? the distinction was not so clear once upon a time) named brown studied the movement of particles in fluid, modelling them as random walks. hence the name.

[AaronBrown]

Actually it was a Scottish botanist named Robert Brown who first studied the phenomenom in 1827. Jan Ingenhousz had noticed it 62 years earlier, but Ingenhouszian motion took too long to say. Brown was observing pollen grains that had fallen on water, Ingenhousz noticed carbon particles on alcohol. Louis Bachelier gave the first mathematical explanation in 1900 in "The theory of speculation"; however the mathematical model call Brownian motion is not a good model of the physical Brownian motion studied by Brown and Ingenhousz.

The important thing about Brownian motion historically is that it provided the definitive proof of the molecular theory of matter.

[AaronBrown]

The sigma algebra tells you the granularity at which you can distinguish states. In finance, this comes up most often in credit default models. With a standard Merton model of default, defaults are always predictable. You can see the stock price approaching the default barrier, there are no surprise defaults. But suppose you only observed the stock price sampled every second and rounded to the nearest penny. The underlying stock price is still a random walk, but you don't have the full information. In this model, there could be a surprise default.

[edtost]

[ QUOTE ]
The sigma algebra tells you the granularity at which you can distinguish states. In finance, this comes up most often in credit default models. With a standard Merton model of default, defaults are always predictable. You can see the stock price approaching the default barrier, there are no surprise defaults. But suppose you only observed the stock price sampled every second and rounded to the nearest penny. The underlying stock price is still a random walk, but you don't have the full information. In this model, there could be a surprise default.

[/ QUOTE ]

Thanks a lot, that was exactly what I needed.

Bonus points for figuring out that I was reading a derivation of a standard Merton model for a seminar on credit risk.

[edtost]

[ QUOTE ]
we fix a probability space ([Sigma],F,P)

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wow, I suck at life, I can't even copy straight from a text anymore.

that should read ([Omega],F,P); not that it really makes any difference.

[AaronBrown]

Just curious, you're preparing for a finance seminar on credit default swaps, and you look for advice on a Poker forum? Do you think Poker players know everything? Or are credit default swap professionals unfriendly?

[edtost]

[ QUOTE ]
Just curious, you're preparing for a finance seminar on credit default swaps, and you look for advice on a Poker forum? Do you think Poker players know everything? Or are credit default swap professionals unfriendly?

[/ QUOTE ]

although i'm preparing for a finance seminar, the question is more related to probability or analysis than finance. i wouldn't know who to ask for advice about analysis, and there are a lot of smart people who post here, many of whom have a good knowledge of probability.

also, since my goal is to not look like an idiot the first day of class, i was trying to avoid asking the professor, i don't know any others in the department who specialize in credit risk, and the one i knew who taught probability theory is away this semester.

in short, i had no obvious person to ask without going against the reason for asking, and i thought there was a pretty good chance that someone here might know the answer to my question, even if they didn't know anything about credit risk.