Meaning of the "existence" proof

Question

After doing some reading on stochastic processes for work I've found that a proof that the specific process exists is often one of the first things presented.

Could someone please explain, in layman's terms, the purpose/necessity of this proof?

Answer 1

Existence proofs are notoriously difficult to justify and appreciate. One reason is that they don't seem to have any consequences. Whether or not you read and understand an existence proof does not change your subsequent work with a given stochastic process because that work relies on properties of the process and not the fact that it exists.

The purpose of existence proofs is to justify the mathematical foundation of working with a given process. Without these proofs it is like in Star Trek. They can do all sorts of cool things, but in reality the materials and technology do not exist. Hence it is all fiction. If your stochastic process does not exists, that is, if there is no mathematical object with a given set of properties, your subsequent work is all fiction. So it is a fundamental step in science to justify that we are doing science and not science fiction.

Edit: In response to @Nick's comment. When it comes to stochastic processes it is, indeed, of central importance to be able to produce an infinite dimensional measure from a consistent family of finite dimensional distributions. This is what Kolmogorov's consistency theorem is about. This theorem, or a variation of it, often lurks in the background even if it is not used explicitly. For instance, a stochastic process could be a solution to a stochastic differential equation (SDE) involving Brownian motion. Then the existence of the process is a question about whether the SDE has a solution - based on the existence of Brownian motion. The existence of Brownian motion can be proved using Kolmogorov's consistency theorem.

There are, however, alternative ways to obtain existence of processes. My favorite alternative is via compactness arguments.

I kept my reply above non-specific because it goes for all other mathematical topics as well. The existence of, for instance, the uniform distribution on $[0,1]$, which is absolutely non-trivial. And the existence of the real numbers for that matter. I wonder how many on this site have proved that the real numbers exist, or do we just take for granted that we can "fill out the gaps" in the rational numbers to get completeness?

N.B. Just to clarify, by the first line in my original reply it was not my intention to imply that an existence proof can not be justified or appreciated, but to a newcomer it may look like a soccer match where the players are given an object and the referee says it's a ball, but then the players use the entire first half to clarify that, indeed, it is a ball before they start playing.