How to properly define a stochastic process as a random variable valued in the Skorokhod space

Question

There are different ways to define a (cadlag) stochastic process (I add cadlag because I consider this a minimal requirement for all applications that are relevant to my personal interests). Starting from a probability space $(\Omega,\mathcal{F},\mathsf{P})$ one of the most classical is to consider the product space $\Omega\times[0,\infty)$ equipped with the product sigma-algebra $\mathcal{F}\otimes\mathcal{B}([0,\infty))$ and to define a stochastic process as an application $$ X:\Omega\times[0,\infty)\rightarrow \mathbb{R} $$ measurable with respect to $\mathcal{B}(\mathbb{R})$ in arrival and $\mathcal{F}\otimes\mathcal{B}([0,\infty))$ in departure. Another possibility (useful for results on functional convergence) is to take the space $\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ of cadlag functions defined form $\mathbb{R}_{+}=[0,\infty)$ with values in $\mathbb{R}$ equipped with the Skorokhod metric. In this case we associate to each $\omega\in\Omega$ the "entire trajectory" and so a stochastic process is an application $$ X:\Omega\rightarrow\mathbb{D}(\mathbb{R}_{+},\mathbb{R}) $$ which is also a random variable once some measurability constraint is imposed. In order to do so: which sigma-algebra is implicitly considered on $\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ ? I assume that it is the Borel sigma algebra associated with the topology, but I might be mistaken since I did not find an explicit mention of it in the textbooks that I am reading, so any confirmation/disproof is greatly appreciated.

Answer 1

I may add some further details to the other answers by mentioning what Jacod and Shiryaev say.

The authors denote with $\mathscr{D}_t^0$ in VI.1.1.(c) the $\sigma$-algebras generated by the evaluation maps $x\mapsto x_s,\,s\leq t$, where of course the input $x \in \mathcal{D}_{\mathbb{R}^d}[0,\infty)$, that is our space of $\mathbb{R}^d$-valued càdlàg functions over $[0,\infty)$. Then, they define the $\sigma$-algebra $\mathscr{D}:=\vee_{t \geq 0}\mathscr{D}_t^0$. As the authors state, one of the purposes of the whole Skorokhod machinery is to make $\mathscr{D}$ the Borel $\sigma$-algebra of $\mathcal{D}_{\mathbb{R}^d}[0,\infty)$: this result appears in VI.1.14.(c). These play a role in making familiar functions measurable such as (strict) stopping times: we define a filtration $(\mathscr{D}_t)_{t \geq 0}$ where $\mathscr{D}_t:=\cap_{s>t}\mathscr{D}_s^0$, then a function $\tau:\mathcal{D}_{\mathbb{R}^d}[0,\infty)\to \mathbb{R}^+$ is a strict stopping time for such filtration if $\{x:\tau(x)\leq t\}\in \mathscr{D}_t^0,\,\forall t$ (see e.g. VI.2.10).

Answer 2

From what I understood, Skorokhod metric induces a topological space which is a Polish space. Polish spaces are pretty much standard probability spaces, and they are canonically assumed to be endowed with the Borel $\sigma$-algebra.

Speaking of disproof etc. Suppose you see someone introducing a topological space in a probabilistic setting, but let's say you are not sure which $\sigma$-algebra is used there. Just see which kind of sets are presumed to be measurable there, or which functions are expectations taken over. If those would be arbitrary open or closed balls, or continuous function, then you need to have a $\sigma$-algebra which is at least as rich as the Borel one.

Answer 3

There are indeed many different ways how to consider a càdlàg stochastic process. In my opinion, two perspectives are predominent: the canonical representation and the representation through an abstract probability space.

Let us start with the canonical representation. To define a random càdlàg function, the simplest way is to define a probability measure $\mathbb P$ on the Skorokhod space $\mathbb D(\mathbb R_+, \mathbb R)$. If you want to describe it in terms of random variables, you would look at the canonical process $X:\omega \mapsto \omega$ which is simply the identity on $\mathbb D(\mathbb R_+, \mathbb R)$. The distribution of $X$ is then exactly $\mathbb P$.

The other option is the one you cite: I take some abstract probability space $(\Omega, \mathcal F, \mathbb P)$ which I do not want to characterize further and I consider a random variable $X:\Omega \rightarrow \mathbb D(\mathbb R_+, \mathbb R)$. More precisely, one usually prescribes a law of $X$ and uses the fact that there exists some abstract space such that I find such an $X$.

As you mentioned, this does not completely answer the question as a probability measure is only defined on a $\sigma$-algebra. As one usually wants to have a $\sigma$-algebra compatible with a suitable topology, one usually takes the Borel $\sigma$-algebra induced by some topology. A quest for a suitable $\sigma$-algebra, then, is simply a quest for a good topology. Motivated by similar questions as yours, I wrote a short introduction to the major topologies that one defines on the Skorokhod space. If you want to take a look, you can find it here. Here is the short version of it: Skorokhod defined four different topologies that are known as the $J_1$, $J_2$, $M_1$ and $M_2$ topologies. His motivation was to find a topologies that would extend the topology of uniform convergence from the space of continuous functions to the space of càdlàg functions. (The problem being that the topology of uniform convergence does not behave well on the $\mathbb D(\mathbb R_+,\mathbb R)$ as it is not separable!) All four of his propositions are different interpretations of the thought "Let's take uniform convergence, but let's allow for a finite number of jumps by extending the $\epsilon$-tube". As far as I know, most people use the $J_1$ topology which is usually referred to as the Skorokhod topology. However, I have seen papers which use the $M_1$ or $M_2$ topologies as they are weaker (and thus allow more sequences to converge). I have never seen anybody use the $J_2$ topology.

This leaves the question whether the $J_1$ topology is suitable for doing probability theory. As @Ilya notes in her answer, it is very nice to have a Polish space (i.e. a topologically that is separable and completely metrisable). This is important, because this ensures that Prokhorov's Theorem holds, and thus allow to prove convergence of measures via tightness and identification of the limit. Fortunately, $J_1$ is indeed Polish. The same holds true for $M_1$. The topologies $J_2$ and $M_2$ are not Polish, but they do still imply a version of Prokhorov's Theorem.

To be complete, note that there have been other proposals of topologies, known as the $N$ and the $S$ topologies. They are both weaker than the $J_1$ topologies, are not Polish, but still imply some version of Prokhorov's Theorem.

To conclude, it will depend on your particular case. Probably, you will be using the $J_1$ topology which is standard throughout the literature. However, if this topology is too strong for you, you might want to look at one of the other options.

Also, I am not completely sure about this, but I think that the induced $\sigma$-algebras might coincide for some of these topologies...