Any model in finance is based on random variables and stochastic processes of any kind: continuous or discrete, one-dimensional or multi-dimensional, etc. The answer to any question arising is given via variety of properties of the involved probability distributions.
The following topics will be discussed: role of the moments of positive integer orders, infinite divisibility, nonlinear (Box-Cox) transformations, limit theorems. One of the goals is to explain why some distributions are M-determinate and others M-indeterminate. It is more than useful to see a complete picture of existing uncheckable and checkable conditions for either M-determinacy or M-indeterminacy. Here is an exciting question: How to “fight” with the M-indeterminacy in the most popular models such as Black-Scholes and others? The main ideas, techniques and results can and will be demonstrated by analysing in detail two of the most important distributions, the normal and the lognormal, and related classes of SDEs. Some facts are not so-well known and even look a little surprising. Along with classical results for M-(in)determinacy (Cramer, Hardy, Carleman, Krein, Heyde, Berg, Pakes), a few recent results will be reported. Challenging open questions will be outlined.