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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 filtration 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, 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!

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  • $\begingroup$ I believe the natural filtration of a process is the same as the natural filtration of a bijective transformation of that process. The example that comes to mind is a stock price modelled as a geometric Brownian motion $S_t$ governed by a standard Brownian motion $W_t$. Now, the knowledge of $W$ implies the knowledge of $X$, i.e. $\sigma ( W_t ) = \sigma (X_t)\ \forall t$, where $\sigma(\cdot)$ represents the sigma-algebra generated by the process at all times. As to whether all filtrations are natural filtrations of some process, I am not sure. $\endgroup$
    – AdB
    Mar 18, 2019 at 9:10

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Essentially, yes. And you will even by able to choose your process to be a martingale. Indeed, assume $\lbrace\mathcal{F}_{n}\rbrace_{n\ge 0}$ is some (for simplicity) countable filtration on $(\Omega,\mathcal{F},\mathbb{P})$. Let $X$ be some integrable rv and set \begin{equation} X_{n}:=\mathbb{E}[X|\mathcal{F}_{n}]. \end{equation} Then, $(X_{n})$ is a martingale, also called 'Doob's martingale', and a model for information accumulation. Clearly, by construction $\lbrace\mathcal{F}_{n}\rbrace$ is its natural filtration.

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