Stochastic process
Mathematical construction

In mathematics, constructions of mathematical objects are needed, which is also the case for stochastic processes, to prove that they exist mathematically. There are two main approaches for constructing a stochastic process. One approach involves considering a measurable space of functions, defining a suitable measurable mapping from a probability space to this measurable space of functions, and then deriving the corresponding finite-dimensional distributions.

Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using Kolmogorov's existence theorem to prove a corresponding stochastic process exists. This theorem, which is an existence theorem for measures on infinite product spaces, says that if any finite-dimensional distributions satisfy two conditions, known as consistency conditions, then there exists a stochastic process with those finite-dimensional distributions.
punctuation:no_comma, length_constraints:number_words[408]