What is meant by existence of a (discrete-time) stochastic process?

Question

What is meant by existence of a (discrete-time) stochastic process?
How do I know whether a process exists or not?
Could anyone offer a simple example of an existent and another of a nonexistent process?

These questions arose when discussing in the thread "Need for existence of stochastic processes behind models of conditional variance".

Answer 1

In this answer I collect and summarize some bits of insight I have received through comments and gathered myself. Considerable credit goes to @IsabellaGhement and @whuber

What is meant by existence of a (discrete-time) stochastic process?

A stochastic process exists if the relevant mathematical objects demanded of the axioms of probability exist:

1. a set of outcomes (sample paths),
2. a collection of measurable events (which generate a sigma field), and
3. a valid probability function on those events.

How do I know whether a process exists or not?

To show existence of a stochastic process, the objects listed above have to be exhibited.

How come some processes do not exist?

A stochastic process is commonly defined by designating a set of properties that its distribution and sample paths must satisfy. If these properties are contradictory, no triplet of the form (outcomes, sigma field, probability function) can satisfy them, hence the process does not exist.

Could anyone offer a simple example of an existent and another of a nonexistent process?

I am still looking for such simple examples of processes that do exist and these that do not. (I suppose it is not too difficult to construct a nonexisting process and show why it is such. The interesting part is to find an example where the contradiction leading to nonexistence is not immediately obvious, so that the example is pedagogically useful.)

Answer 2

Could anyone offer a simple example of an existent process?

A well-known theorem which guarantees the existence of a stochastic process ${(X_n)}_{n \geqslant 1}$, say with $\mathbb{R}$-valued random variables $X_n$, is the Daniell-Kolmogorov extension theorem. A typical application of this theorem gives the existence of a sequence of independent random variables ${(X_n)}_{n \geqslant 1}$, with $X_n$ following any probability law (possibly depending on $n$).

Here is the statement of this theorem. Denote by $\mathcal{B}_n$ the Borel $\sigma$-field on $\mathbb{R}^n$. Suppose that for every $n \geqslant 1$ we have a probability measure $\mu_n$ on $(\mathbb{R}^n, \mathcal{B}_n)$. Suppose that the sequence of probability measures ${(\mu_n)}_{n \geqslant 1}$ is consistent, in the sense that $\mu_{n+1}(A \times \mathbb{R}) = \mu_n(A)$ for every $n \geqslant 1$ and every $A \in \mathcal{B}_n$. Then the theorem asserts that there exists a probability measure $\mu$ on $(\mathbb{R}^\mathbb{N}, \mathcal{B}_\infty)$ which extends all the $\mu_n$, in the sense that $\mu(A \times \mathbb{R}^\mathbb{N}) = \mu_n(A)$ for every $n \geqslant 1$ and every $A \in \mathcal{B}_n$.

Let's see how to apply this theorem to show the existence of a sequence of independent random variables ${(X_n)}_{n \geqslant 1}$ with $X_n \sim \nu_n$, where ${(\nu_n)}_{n \geqslant 1}$ is a given sequence of probability measures on $\mathbb{R}$. One takes the product measure $\mu_n = \nu_1 \otimes \cdots \otimes \nu_n$ for every $n \geqslant 1$. Then the consistency condition of ${(\mu_n)}_{n \geqslant 1}$ is easy to check. Then the Daniell-Kolmogorov extension theorem provides a probability measure $\mu$ on $\mathbb{R}^\mathbb{N}$ which extends the $\mu_n$. Take the probability space $$ (\Omega, \mathcal{A}, \mathbb{P}) = (\mathbb{R}^\mathbb{N}, \mathcal{B}_\infty, \mu). $$ An element $\omega$ of $\Omega$ is a sequence of real numbers $(\omega_1, \omega_2, \ldots)$. Then it suffices to define for each $n \geqslant 1$ the random variable $X_n$ on $(\Omega, \mathcal{A}, \mathbb{P})$ by $X_n(\omega) = \omega_n$. In other words the random sequence ${(X_n)}_{n \geqslant 1}$ is a $\mathbb{R}^\mathbb{N}$-valued random variable whose probability distribution is $\mu = \nu_1 \otimes \nu_2 \otimes \cdots$. The theorem guarantees the existence of this infinite product measure.

Could anyone offer a simple example of a nonexistent process?

I like this example: there does not exist a "non-trivial" martingale ${(M_n)}_{n \geqslant 1}$ such that $M_n$ takes its values in $\{0,1\}$ for every $n \geqslant 1$. Indeed that would mean that $M_n = \mathbf{1}_{A_n}$ for a certain event $A_n$, for every $n \geqslant 1$. The martingale condition is $$ \mathbb{E}[M_{n+1} \mid M_1, \ldots, M_n] = \mathbb{E}[M_{n+1} \mid M_n] = M_n. $$

We have $$ \begin{align} \mathbb{E}\bigl[(\mathbf{1}_{A_{n+1}}-\mathbf{1}_{A_{n}})^2\bigr] & =\mathbb{E}[\mathbf{1}_{A_{n+1}}^2]+\mathbb{E}[\mathbf{1}_{A_{n}}^2]- 2\mathbb{E}[\mathbf{1}_{A_{n+1}}\mathbf{1}_{A_{n}}] \\ & =\mathbb{E}[\mathbf{1}_{A_{n+1}}]+\mathbb{E}[\mathbf{1}_{A_{n}}]- 2\mathbb{E}[\mathbf{1}_{A_{n+1}}\mathbf{1}_{A_{n}}]. \end{align} $$ But $$ \mathbb{E}[\mathbf{1}_{A_{n+1}}] = \mathbb{E}\bigl[\mathbb{E}[\mathbf{1}_{A_{n+1}} \mid \mathbf{1}_{A_{n}}]\bigr] = \mathbb{E}[\mathbf{1}_{A_{n}}] $$ and $$ \mathbb{E}[\mathbf{1}_{A_{n+1}}\mathbf{1}_{A_{n}}] = \mathbb{E}\bigl[\mathbb{E}[\mathbf{1}_{A_{n+1}}\mathbf{1}_{A_{n}} \mid \mathbf{1}_{A_{n}}]\bigr] = \mathbb{E}\bigl[\mathbf{1}_{A_{n}}\mathbb{E}[\mathbf{1}_{A_{n+1}} \mid \mathbf{1}_{A_{n}}]\bigr] = \mathbb{E}[\mathbf{1}_{A_{n}}^2] = \mathbb{E}[\mathbf{1}_{A_{n}}]. $$ Finally, $\mathbb{E}\bigl[(\mathbf{1}_{A_{n+1}}-\mathbf{1}_{A_{n}})^2\bigr] = 0$, which means that $A_{n+1} = A_n$ (almost surely): our martingale is "trivial".