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

stochastic-analysisstochastic-calculusstochastic-processes

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?

Best Answer

(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/.

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