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'.

  • $\begingroup$ This is a common beginners question. Some answers here quant.stackexchange.com/questions/25481/… $\endgroup$ – noob2 Jan 16 '18 at 16:56
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    $\begingroup$ You will also notice that the book never mentions "unadapted" processes, because those are ill defined and not worth studying. So as a beginner you can just assume that the processes you will learn about are adapted, that they (and the filtration) have been setup properly so they are adapted. As you read, replace the words "adapted process" with "properly defined process". $\endgroup$ – noob2 Jan 16 '18 at 17:21
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    $\begingroup$ Do you have a good understanding of a probability space: a set of outcomes $\Omega$, a sigma algebra $\mathcal{F}$, and probability measure $P$? Do you have a good understanding of what's a measurable function? $\endgroup$ – Matthew Gunn Jan 16 '18 at 23:13

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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