So I have just started to learn about stochastic processes, and I got to learn about progressively measurable processes.
Definition
Let $(\Omega, \mathcal{F},P)$ be a probability space and $\{\mathcal{F_t}\}_{t\in[0,\infty)}$ be a filtration. A stochastic process $X:[0,\infty)\times \Omega \to \mathbb{R}$ is said to be $\mathcal{F}$-progressively measurable if, for any $T$, $X|_{[0,T]}$ is $\mathcal{B}([0,T])\otimes\mathcal{F}_T$-measurable.
So that is the definition, and I am failing to see how it is progressive. I would like to know why it was named in such a way.
Best Answer
The word "progressive" refers to the progression (in time) of measurability conditions that a progressively measurable process satisfies: for each $t\ge 0$, the restriction of $X$ to $[0,t]\times \Omega$ is $\mathcal B([0,t])\otimes\mathcal F_t$-measurable. A progressive process is a little better than just adapted. One consequence is that if $X$ is progressive then $t\mapsto \int_0^t X_s ds$ is well defined and progressive (even predictable) provided $\int_0^t|X_s| ds<\infty$ for $t>0$, by Fubini's theorem.