Definition of optional sigma algebra require the left hand limit or just right continuous adapted processes?

Question

While reading the following post : https://almostsuremath.com/2009/11/08/filtrations-and-adapted-processes/

I have come across a question on the definition of optional and predictable processes. In most books I have read before, optional sigma algebra is defined as the sigma algebra generated by cadlag adapted processes, and predictable sigma algebra is defined as the sigma algebra generated by caglad adapted processes or in some books just left continuous adapted processes.

In George's post he defines both as just right-continuous or left-continuous adapted processes. In the predictable case, I can see that both are in fact the same as it is generated by continuous adapted processes. However, in the case of optional sigma algebra I am not sure if they are the same. Does the omission of left-limit here matter?

Answer 1

The books quoted from in the following links were the ultimate bibles of stochastic processes when I was a student:

• The optional sigma algebra ${\cal O}(\boldsymbol F)$ is generated by the càdlàg adapted processes.

• The predictable sigma algebra ${\cal P}(\boldsymbol F)$ is by the adapted processes that are continuous from the left (not necessarily with limits from the right).

Interestingly, ${\cal P}(\boldsymbol F)\subset {\cal O}(\boldsymbol F)\,.$

Answer 2

Generally, on a filtered probability space $(\Omega, \mathscr{F}, \mathbf{F}=(\mathscr{F}_t)_{t\in \mathbb{R}_+},\mathsf{P})$, no matter the completeness of $\mathbf{F}$, the optional sigma algebra $\mathscr{O}(\mathbf{F})$ is generated by the càdlàg adapted processes.

Also, the following theorem is important: (cf. C. Dellacherie & P. Meyer, Probabilities and Potential, volume 29 of North-Holland Mathematics Studies. North-Holland, Amsterdam, (1978), p.124, Th.VI.65)

Theorem: Under the usual conditions, the optional $\sigma$-field is also generated by the right-continuous processes adapted to the $\mathbf{F}$.

Due to this theorem, the definition in above mentioned post, is also reasonable.