X is a predictable proccess iff X is $\{ F_{t^-}\}$-adapted?

Question

I have read in a remark that:

A stochastic process X is a predictable process iff X is $\{F_{t^-}\}$-adapted. $\quad$ (*)

1. Does the filtration need to satisfy any requirements for (*) to be true?

2. Can someone prove (*) or tell me a book where I can I find a proof?

Answer 1

More precisely, the $\sigma$-field $\mathcal F_t$ is generated by events of the form $$ A=\left[\cap_{k=1}^n\{X_{s_k}\in I_k\}\right]\cap\{X_t\in I\}, $$ where $0\le s_1<s_1<\cdots<s_n<t$, and $I$ and each $I_k$ is an open interval in $\Bbb R$. If you replace $\{X_t\in I\}$ in this display with $\{X_{t-}\in I\}$, you get a new event (let's call it $B$) that lies in $\mathcal F_{t-}$ and is such that $$ \Bbb P[A\Delta B]\le\Bbb P[X_t\not=X_{t-}]=0. $$ That is, each of the generators of $\mathcal F_t$ is an element of $\mathcal G_{t-}$. It follows that $\mathcal F_t\subset\mathcal G_{t-}$, whence $\mathcal G_t\subset\mathcal G_{t-}$.

Answer 2

(If you read ($*$) in a book, it is probably best to throw that book out the window or start using it as a doorstop.)

The "prototype" predictable process (in the context of some filtered probability space $(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\Bbb P)$) is a process that is left-continuous and adapted to $(\mathcal F_t)$. The $\sigma$-algebra on $[0,\infty)\times\Omega$ generated by such processes, call it $\mathcal P$, is the predictable $\sigma$-algebra. Finally, a process $X: (\omega,t)\to X_t(\omega)$ is predictable provided it is $\mathcal P$ measurable. A convenient place to read about such things is the blog Almost Sure of Geo. Lowther: https://almostsure.wordpress.com/.