Stochastic process, compatible

adapted stochastic process

A family of random variables $X=(X_t(\omega))_{t\geq0}$ defined on a measurable space $(\Omega,\mathcal F)$, with an increasing family $\mathbf F=(\mathcal F_t)_{t\geq0}$ of sub-$\sigma$-fields $\mathcal F_t\subseteq\mathcal F$, such that the $X_t$ are $\mathcal F_t$-measurable for every $t\geq0$. In order to stress this property for such processes, one often uses the notation

$$X=(X_t,\mathcal F_t)_{t\geq0}$$

or

$$X=(X_t,\mathcal F_t),$$

and says that $X$ is $\mathbf F$-adapted, or adapted to the family $\mathbf F=(\mathcal F_t)_{t\geq0}$, or that $X$ is an adapted process. Corresponding definitions can also be given in the case of discrete time, and then "adapted process" is sometimes replaced by "adapted sequence" .

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For additional references, see Stochastic process.