Adapted process

Given a growing family of information $(\mathcal{F}_t)$, a stochastic process $X$ is adapted to $(\mathcal{F}_t)$ if $X_t$ is $\mathcal{F}_t$-measurable for every $t$. For Brownian motion, Shigekawa's lecture notes say that the information is often taken to be the one generated by the Brownian motion up to time $t$¹. In the wording of the MIT notes, the value at each time $t$ depends only on $X_s$ for $s \le t$².

For example, the price $X_t$ at each time and $\min\{X_t, c\}$ are adapted. The maximum over the whole period, $\max_{0 \le t \le T} X_t$, is not adapted, because at time $t$ it includes future prices that are not yet known. A strategy that decides trade sizes must also be adapted, because a strategy that looks at future prices cannot be carried out².

The Ito integral of an adapted integrand is a martingale under conditions such as square integrability². In the LIBOR market model as well, the drift and volatility of the forward rates are specified as adapted processes³.

Notes

1. Ichiro Shigekawa, "数理ファイナンス入門" (Introduction to Mathematical Finance), Kyoto University, intensive course at Ehime University, 2009 (in Japanese), page 42 (the definition of a stochastic process in Chapter 6, "Brownian motion"; "adapted to $(\mathcal{F}_t)$"). The end of the same section says that the information generated by the Brownian motion is often taken. ↩

2. Massachusetts Institute of Technology, "Lecture 18: Itō Calculus", pages 4 to 5, Definition 2.1 (adapted process), Example 2.2, the explanation of trading strategies that follows, and Theorem 2.3. Example 2.2 (iii) shows that the maximum over the period is not adapted. ↩ ↩² ↩³

3. Martin Haugh, "Market Models" (Columbia University, Term Structure Models: IEOR E4710, 2010), page 6, right after equation (14) ($\mu_n(t)$ and $\sigma_n(t)$ are adapted processes). ↩