Twofish's Blog

August 19, 2006

Fiber bundles and probability spaces

Filed under: academia, quantitative finance, quantlib — twofish @ 2:29 pm

Cool.

I’ve managed to convince myself that a probability space is almost a fiber bundle.  A probability space consists of a triplet consisting of an event space, a sigma algebra, and a probability measure.  A fiber bundle consists of a fiber space, a transformation function \pi, and a base space.  If you set the base space to be the real number line 0 to 1,  and if you relax the condition that the fiber space be a manifold, and you set the transformation function to be the probability measure, you turn the probability space into something that is almost a fiber bundle.  If the event space has the structure of a manifold, then what you end up with *is* a fiber bundle.

Once you do this then you can bring in all of the machinery of differential geometry into quantitative finance.  Let me think some more about how you can do Mallavin calculus in this and if you can calculate deltas.

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