# Is every filtration the natural filtration of some stochastic process?

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.

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.

• Very clear example! Thanks a ton! Mar 18, 2019 at 5:26
• @Will. Could you please elaborate on this expression: $Y_t(w) = w$. Jan 30, 2021 at 18:10
• It's just the definition of $Y_t$, whose purpose was to illustrate that any filtration can be seen as the natural filtration of some process $Y$. But here $Y$ takes values in $\Omega$, which might seem a bit strange. If we look for a real-valued random process $Y$, then the answer is negative in general.
– Will
Jan 30, 2021 at 22:29