Definition of Stochastic Process

Question

My question is about the definition of a stochastic process.

From the definition of the majority of textbooks, we know a stochastic process is defined as a collection of random variables defined on a common probability space $(\Omega, \mathcal{F}, \mathcal{P})$, indexed by some set T, and all take values in the same measurable space, e.g. $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, so the stochastic process can be written as {$X_t$: t $\in$ T}.

My question is since a random variable is essentially a measurable function e.g. $f$ and here mapping from $\Omega$ to $\mathbb{R}$, do the random variables in the definition of stochastic process take the same functional form, or can they have different measurable functional form individually as long as they are defined on the same probability space?

One situation is each of the random variable $X_t$ in the collection has different functional form $f_t$ and defined on their own probability space $(\Omega_t, \mathcal{F}_t, \mathcal{P}_t)$, and now I define a common probability space which is a product measurable space ($\times_t \Omega_t$, $\times_t \mathcal{F}_t)$, where each $(\Omega_t, \mathcal{F}_t)$ is the measurable space on which each random variable $X_t$ is defined. I want to know does such a collection of random variables that each of them has different functional form yet now have all been defined on a common probability space consist a stochastic process?

Thanks in advance!

Answer 1

There are different ways to define a Stochastic process, there are also different types of stochastic processes depending on measurability properties. Here is one way to do it:

Suppose $(\Omega\mathscr{F},\mathbb{P})$ be a probability space, $(S,\mathscr{S})$ a measurable space, and $T$ an index set (typically $\mathbb{Z}_+$, or $[0,\infty)$). Consider the collection of functions from $T$ to $S$, $S^T$, equipped with product $\sigma$-algebra $\mathscr{S}^{\otimes T}$ (the later is the $\sigma$-algebra generated by the projections $\pi_t:S^T\rightarrow S$ given by $\mathbf{s}\mapsto \mathbf{s}(t):=\mathbf{s}_t$, which is to say the $\sigma$-algebra generated by the collection of sets of the form $\{\mathbf{s}:s_t\in A\}$ with for all $t\in T$ and $A\in \mathscr{S}$).

Definition G: A measurable function $X: (\Omega,\mathscr{F})\rightarrow(S^T,,\mathscr{S}^{\otimes})$ is called a stochastic process on $\Omega$ with values on $S$. The law (or distribution) of $X$ is the probability measure on $(S^T,\mathscr{S}^{\otimes})$ induced by $X$, that is $\mathbb{P}(X^{-1}(\cdot))$.

Notice that for each $\omega\in\Omega$, $X(\omega)$ is a function defines on $T$. It is customary to denote the value of $X(\omega)$ at $t$ as $X(\omega,t):=X_t(\omega)$. In other words, one can view the a stochastic process $X$ in the sense of definition A as a function $X:\Omega\times T\rightarrow S$ such that for each $t\in T$ fixed, $X_t:\Omega\rightarrow S$ is $\mathscr{F}/\mathscr{S}$-measurable. (This is a little of an abuse of notation but there is no need to be too rigid here).

In many applications, $(S,\mathscr{S})$ is a nice space (Polish spaces for example, in particular $S=\mathbb{R}$ and $\mathscr{S}=\mathscr{B}(\mathbb{R})$), and the index $T$ denotes time ($\mathbb{Z}_+$ or $T=[0,\infty)$ for example). Henceforth let us assume for simplicity that $(S,\mathscr{S})$ is the real line with the Borel $\sigma$-algebra.

It is often the case that one does not know the whole of the information $\mathscr{F}$ but just what is observed up to time $t$, that is, we have a collection of $\sigma$-algebras $\{\mathscr{F}_t:t\in T\}$ such that $\mathscr{F}_t\subset \mathscr{F}_u\subset \mathscr{F}$ for $t,u\in T$ and $t<u$. This is know as a (measurable) filtration.

Definition A: If for each $t\in T$, $X_t:\Omega\rightarrow S$ is $\mathscr{F}_t/\mathscr{S}$-measurable, then $X$ is a stochastic process adapted to the filtration $\{\mathscr{F}_t:t\in T\}$. All this can be is denoted as $X:(\Omega,\{\mathcal{F}_t:t\in T\})\rightarrow(S,\mathscr{S})$.

One can go on and define stronger types of stochastic processes based on their measurable properties. Suppose that $T=[0,\infty)$, and that $\{\mathscr{F}_t:t\in T\}$ is a filtration

Definition P: If $X$ is a stochastic process (in the sense of definition G) and for each $t\in T$, $X^t:\Omega\times[0,t]\rightarrow S$ is $\mathscr{F}_t\otimes\mathscr{B}([0,t])$ given by $X^t(\omega,u)=X(\omega,\min(t,u))$, then $X$ is a progressively measurable process.

Here $\mathscr{F}\otimes\mathscr{B}([0,t])$ is the product $\sigma$-algebra of $\mathscr{F}$ and the Borel $\sigma$-algebra of the interval $[0,t]$.

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Final comments:

1. As with random variables, given two processes $X$ and $Y$, possibly defined on different probability spaces (filtered or not) $(\Omega_1,\mathscr{F}^1,\mu_1)$ and $(\Omega_2,\mathscr{F}^2,\mu_2)$, if $\mu_1(X^{-1}(\cdot))=\mu_2(Y^{-1}(\cdot))$, then $X=Y$ in law.

One can also talk about a canonical stochastic process with value on $S$: Define $\Omega=S^T$, $\mathscr{F}=\mathscr{S}^{\otimes}$ and $X_t(\omega)=\omega(t)$.

2. Given a stochastic process (we omit the underlaying domain $(\Omega,\mathscr{F})$) $X$, the natural filtration of $X$ is the collection of $\sigma$-algebras defined as $\mathscr{F}^X_t:=\sigma(X_u: u\in T,\, u\leq t)$. It is clear that every stochastic process is adapted to its natural filtration.