Is every filtration the natural filtration of some stochastic process?

Question

We have a notion of natural filtrations, which intuitively represents the history of the process as the process evolves over time.

We also have a notion of filtrations in general, which are increasing sequence of sub-sigma algebras.

Naturally, the latter concept is more abstract than the former, and I am having trouble getting a concrete grip on the latter.

In particular, if we have a stochastic process X, and a filter F, I tend to look at F as a natural filtration (although we only know it's a filtration in general, and not necessarily a natural one) of some other process Y. Can we do that?

As to why I am doing what I am doing, in many practical scenarios (like quantitative finance, which I am studying), we would be directly observing the process Y (say Y is the share price process) and hence our information would be the natural filtration of Y, but we might be interested in a slightly different process X (which might be the log of the share price or some other functional transformation say). In this scenario, the natural filtration of Y is simply a filtration from the perspective of X, and not a natural one.

Thanks a lot in advance!

Answer 1

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space equipped with a filtration $(\mathcal F_t)_{t\in\mathbb R_+}$.

For all $t\in\mathbb R_+$, let $Y_t:(\Omega,\mathcal F)\to(\Omega,\mathcal F_t)$ be defined for all $\omega\in\Omega$ by $Y_t(\omega)=\omega$. Then we clearly have $\sigma(Y_s,s\le t)=\mathcal F_t$, hence $(\mathcal F_t)_{t\in\mathbb R_+}$ is the natural filtration of the stochastic process $Y:(\Omega,\mathcal F)\to(\Omega^{\mathbb R_+},\bigotimes_{t\in\mathbb R_+}\mathcal F_t),\omega\mapsto(\omega)_{t\in\mathbb R_+}$.

So for any filtration $(\mathcal F_t)_{\mathbb R_+}$, there exists a measurable space $(E,\mathcal E)$ and a stochastic process $Y:\Omega\to E$ such that for all $t\in\mathbb R_+$, $\mathcal F_t=\sigma(Y_s,s\le t)$.

A much more interesting question would be to know whether this is true when $(E,\mathcal E)$ is fixed. For instance, for any filtration $(\mathcal F_t)_{t\in\mathbb R_+}$, does there exists a real-valued process $(Y_t:\Omega\to(\mathbb R,\mathcal B(\mathbb R))_{t\in\mathbb R_+}$ such that $(\mathcal F_t)_{t\in\mathbb R_+}$ is the natural filtration of $Y$? The answer is no, as shown is this post Is every sigma-algebra generated by some random variable?.

In many pratical scenarios like quantitative finance, as you said, you are very likely to deal with the natural filtration of an observable process $Y$ and you are interested in a different process $X$. But when $X$ is a one-to-one and onto function of $Y$, then the natural filtrations of $X$ and $Y$ are the same.