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The Gaussian distribution is central in continuous-time models for different reasons: https://almostsure.wordpress.com/2010/04/13/levys-characterization-of-bro
by thanatropism 10y ago
The Gaussian distribution is central in continuous-time models for different reasons:
https://almostsure.wordpress.com/2010/04/13/levys-characterization-of-brownian-motion/ https://almostsure.wordpress.com/2010/04/13/levys-characteri...
Basically any reasonable* stochastic continuous process is driven by a brownian motion. Also: discontinuous processes are more or less* the sum of a brownian motion and a poisson-type process.
https://en.wikipedia.org/wiki/L%C3%A9vy_process#L.C3.A9vy.E2.80.93Khintchine_representation https://en.wikipedia.org/wiki/L%C3%A9vy_process#L.C3.A9vy.E2...
(* Much details about filtrations, Banach spaces yadda yadda omitted)
- srean 10y agoYes indeed, but the Brownian motion story is weaker than the CLT story. Lot more conditions required, you have to look at it the right scale, in the right way ... then stochastic processes look very much like a Brownian motion. Well, technically the bog standard CLT is a special case of this, hence has a simpler story.