Adapted Processes and the Information Available in a Filtration

Summary

The document explains an adapted stochastic process in financial mathematics through the information represented by a filtration. At each time, the process value must be measurable with respect to the sigma algebra available at that time. Intuitively, this condition prevents the process from revealing information that is not yet observable under the chosen information structure.

The example uses a process's natural filtration, which records its history up to each point in time. The process is adapted to that filtration because its current value is part of the information contained in its own past and present history. This gives a concise conceptual explanation relevant to stochastic models and arbitrage theory, where information timing matters. The document does not develop a trading strategy or provide a worked market example; its scope is the definition and intuition, and the meaning of adaptation depends on which filtration is being used.

Key ideas

• A process is adapted when its value at each time is measurable using the information available by that time.
• A filtration represents the accumulation of information over time.
• A process is adapted to its natural filtration because that filtration contains its history.
• Adaptation rules out a model process that uses information unavailable at the stated time.

Tags

# What is an adapted process


# What is an adapted process












I am reading Björk, Arbitrage theory in Continous Time and I have noticed that he uses the term adapted proces a lot. I can't seem to understand what an 'adapted proces' is by the wikipedia article. So, in terms of financial mathematics, please provide an example of an 'adapted proces' and why it is called an 'adapted proces'.

## Answer by Matthew Gunn (score 11, accepted)

https://quant.stackexchange.com/a/37764

Let $\{X_t\}$ be a stochastic process and $\mathcal{F}$ be a filtration.

The intuitive idea is that for $\{X_t\}$ to be adapted, it can't reveal what's unknowable (according to the filtration). By requiring random variable $X_t$ be measurable with respect to sigma algebra $\mathcal{F}_t$, the random variable $X_t$ can't reveal more information than sigma algebra $\mathcal{F}_t$ allows.

As an example, the natural filtration of a stochastic process contains information on all the past history of the process. A stochastic process is adapted with respect to it's natural filtration.

Perhaps this example can help build some intuition how technically a filtration works.

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