# Can a stochastic process be neither adapted to filtration nor previsible?

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.

Let $$(\Omega,\mathcal{F},\mathbb{P})$$ be a probability space carrying a Brownian motion $$(B_t)$$ whose natural filtration I denote by $$(\mathcal{F}_t)$$. By definition, for every $$t\geq0$$, $$B_t$$ is $$\mathcal{F}_t$$-measurable, i.e. $$(B_t)$$ is adapted to $$(\mathcal{F}_t)$$.

As I understand, you wonder whether we may construct a process which is neither adapted nor previsible (aka predictable)? Consider the three processes

1. $$X_t = B_t$$,
2. $$Y_t = B_{t-1}$$ and
3. $$Z_t = B_{t+1}$$.

Then, $$(X_t)$$ is adapted to $$(\mathcal{F}_t)$$ by definition. So is $$(Y_t)$$ which is also known from the information contained in $$\mathcal{F}_t$$. Indeed, $$(Y_t)$$ is previsible as every $$Y_t$$ is $$\mathcal{F}_{t-1}$$-measurable. The process $$(Z_t)$$ however is neither adapted nor previsible. Knowledge about the evolution of the universe up to time $$t$$ merely reveals information about $$Z_{t-1}$$ yet you have no clue about the value of $$Z_t$$.

• This is a very clear example! Thank your for writing this out. One quick question: do these processes have a particular name? If we have assigned a name to the other two classes - adapted and previsible - it seems we should also have a name for this class of processes. – Dhruv Gupta Jul 18 '19 at 11:51
• Sorry but as far as I am aware these processes do not have an extra name. But I suppose not everything has a name: How do you call a stochastic process which is not a martingale? I dont believe we have an extra term for that. Neither am I aware of a name for a non-adapted process other than non-adapted''. Note that being previsible implies adapted. Things only get a name if they're just often. A non-adapted process is not used that often (in finance) as far as I know... – KeSchn Jul 18 '19 at 12:11
• Yes, even I thought that non-adapted processes won't have a special name because they don't pop up that often in quantitative finance; just wanted to confirm from someone who knows more! Thanks a lot for your effort. :) – Dhruv Gupta Jul 18 '19 at 12:19