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Adapted, Predictable, and Neither: Brownian Motion Examples

Article Quant Q&A · Author: Dhruv Gupta
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Summary

The document distinguishes adapted and predictable processes using a Brownian motion and its natural filtration. It compares the Brownian path itself, a version delayed by one time unit, and a version shifted forward by one time unit. The first is adapted; the delayed process is both adapted and predictable because each value is measurable using earlier information; the forward-shifted process is neither adapted nor predictable under the stated filtration.

These examples illustrate how a process’s time shift affects what is knowable from the filtration. The explanation is conceptual rather than a trading method, and it assumes the stated Brownian-motion setup and time indexing. It does not address more general filtration choices, edge cases at the start of the time domain, or applications to financial models.

Key ideas

  • Adaptedness means each process value is measurable with respect to information available by that time.
  • Predictability requires process values to be measurable using information available strictly before the relevant time.
  • A Brownian motion is adapted to its natural filtration.
  • A delayed Brownian path is predictable in the example because its value at each time is known earlier.
  • A forward-shifted Brownian path is neither adapted nor predictable under the stated filtration.

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


# 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.

## Answer by Kevin (score 2, accepted)

https://quant.stackexchange.com/a/46664

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

- $X_t = B_t$,

- $Y_t = B_{t-1}$ and

- $Z_t = B_{t+1}$.

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 $Z_t$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.