What is an infinitely divisible distribution and how it can be used in financial modeling?

An infinitely divisible distribution is technically defined in terms of terms of its characteristic function.As is well known, the PDF of a sum of independent random variablesis obtained by a convolution of the probabiility densities of the RV's being summed.The CF of the sum is obtained by multiplying characteristic functions. If the densities are furthermore identical, then the sum of n IID RV's has a CF obtained by raising the CF of one of them to the nth power.Working backwards, if we are handed a CF, we may ask if it can be represented as another CF raised to the nth power.If this can be done for all n =1,2,3,, , the CF is one of an infinitely divisible distribution.The link to option pricing occurs through Levy processes.For each infinitely divisible random variable, there corresponds a unique Levy process which has that random variable as the process level at time 1.It's common now to model log prices as levy processes.

Let me add a bit more.Infinitely divisible distribution is a concept in theoretical frameworkof Levy processes. In application you can just forget the concept and onlykeep in mind that Levy process is a representitative of the processes withinfinitely divisible distribution. Those processes may have weird samplepaths, but only Levy process has(almost surely, in probability jargon) nice property on its sample paths: right continuous with left limits.Sato's book has an excellent coverage on this topic.