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The idea behind the question arises from my intuition about the concepts of 'adapted to filtration' and 'previsbility'.

If a process is adapted, it essentially means that the evolution of the universe upto time t reveals the history of our process upto time t also.

On the other hand, if a process is previsible, it means that the evolution of the universe upto time t reveals information about the process beyond time t.

If we are thinking in these terms, it becomes natural to ask if we can construct processes where the evolution of the universe upto time t reveals information about such processes only upto a time prior to t, say t-1.

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    $\begingroup$ I take it that by "evolution of the universe up to time $t$" you mean the $\sigma$-algebra $\mathcal{F}_t$ of the underlying filtration $(\mathcal{F}_s)_{s \geq 0}$. Consider e.g. a Brownian motion $(B_t)_{t \geq 0}$ with canonical filtration $\mathcal{F}_t := \sigma(B_s; s \leq t)$. If we consider the process $X_{t} := B_{t+1}$, then the evolution up to time $t$ (i.e. $\mathcal{F}_t$) reveals information about $X$ only up to time $t-1$. $\endgroup$ – saz Apr 23 at 17:00
  • $\begingroup$ Yes, that is exactly what I mean by 'evolution of the universe upto time t'. So is it correct to say that the process X you have constructed is neither adapted nor previsible? If yes, then is there a special name for such type of processes? $\endgroup$ – Dhruv Gupta Apr 23 at 19:11
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    $\begingroup$ Yes, that's correct. And no, I'm not aware of a particular name for such kind of processes. $\endgroup$ – saz Apr 23 at 20:25
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    $\begingroup$ In fact, most processes are of this type (for any given fixed filtration). It just so happens that we end up working with adapted or previsible ones because they tend to be more amenable to analysis. $\endgroup$ – Rhys Steele Apr 23 at 21:01

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