Can a right-continuous stochastic process be predictable?

Question

I know that a stochastic process is said to be predictable if it's measurable with respect to the predictable $\sigma$-field $\mathcal P$, namely the $\sigma$-field generated by all left-continuous adapted processes.

I furthermore know that if $X$ is a càdlàg process then $X(t-)$, (the left hand limit) is a predictable process.

Nevertheless I have some difficulties discerning whether a right-continuous process (the process being non-continuous) can be predictable or not.

Could you give me a hand with this? Thanks in advance.

predictability will hold for sure for all continuous processes since a continuous process is by definition left-continuous. I mean some process which is right continuous and whose left limit does not necessarily exists. — Chaos, Feb 11 '20 at 08:02

score 3 · Answer 1 · answered Feb 11 '20 at 08:15

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Yes. In fact, stochastic processes of the form $Z 1_{[\tau,\infty)}$ where $Z \in \mathcal{F}_{\tau-}$ and $\tau$ is a predictable stopping time generate the predictable sigma-algebra (see here), and any such process is right-continuous.

answered Feb 11 '20 at 08:15

user159517

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You got a point, One additional question, assume we are working with generic right-continuous processes, in order to have predictability do we need the left-limit to exits? – Chaos Feb 11 '20 at 08:21
No, just take a deterministic process which is right-continuous but doesn't have left limits – user159517 Feb 19 '20 at 12:04

Can a right-continuous stochastic process be predictable?

1 Answers1