# Is a predictable process adapted?

Let us consider a measurable space $(\Omega, \mathcal{F})$, with a filtration $(\mathcal{F}_t)_t$ of sub $\sigma$-algebras of $\mathcal{F}$. The predictable $\sigma$-algebra $\mathcal{P}$ is the $\sigma$-algebra generated by all processes of the form $$X : [0,\infty) \times \Omega \rightarrow \mathbb{R}$$ such that

• $X(t,\cdot)$ is $\mathcal{F}_t$-measurable for all $t>0$ (i.e. $X$ is adapted to the filtration $(\mathcal{F}_t)_t$),

• $X(\cdot, \omega)$ is left-continuous for all $\omega \in \Omega$.

My question is:

Is every $\mathcal{P}$-measurable process necessarily adapted? If so, is it necessarily adapted to the coarser filtration $(\mathcal{F}_{t-})_t$?

• The answers are yes and yes. The question is not adapted to MO. // Upvoters: why the upvote?
– Did
Jan 17, 2014 at 12:56
• Thanks! Even though I saw more basic questions on probability, I agree that my question is not adapted to MO and I do apologize. Would you be so kind to give me a hint on how to prove it? Thanks again.
– Tom
Jan 17, 2014 at 18:05
– Did
Jan 17, 2014 at 19:28
• Sorry to bother again.. does anybody have by chance a hint on this one?
– Tom
Aug 16, 2015 at 18:39
• No idea; but it's a good question though! May 22, 2016 at 3:13

There is also the concept of an optional process:

Definition 1: The optional $\sigma$-algebra $O$ is generated by all adapted càdlàg processes (continue à droite et limit à gauche/continuous from the right and limit from the left). A stochastic process $X=(X_t)_{t\geq 0}$ is called optional, if $X$ is measurable w.r.t. $O\subseteq \mathcal{P}(\Omega\times \mathbb{R}_{\geq 0})$.

It is possible to show that $P\subseteq O$ ($P$ denotes the predictable $\sigma$-algebra). Thus, every predictable process is optional. For optional process, we have the following result:

Theorem 2: If $X$ is optional, then $X$ is $\mathcal{F}\otimes\mathcal{B}(\mathbb{R}_{\geq 0})-\mathcal{B}(\mathbb{R})$ measurable and the random variable $$X_T 1_{T<\infty}:\Omega\rightarrow \mathbb{R}$$ $$\omega\mapsto X_{T(\omega)}1_{T(\omega)<\infty}$$ is $\mathcal{F}_T-\mathcal{B}(\mathbb{R})$-measurable for any stopping time $T$.

Applying this to the stopping time $T\equiv t$ for $t\geq 0$ gives $X_t=X_T 1_{T<\infty}$ is $\mathcal{F}_T$-measurable. Since $\mathcal{F}_T=\mathcal{F}_t$, we see that a predictable process is adapted to the filtration $(\mathcal{F}_t)_{t\geq 0}$

Regarding your second question, we have the following statement:

Theorem 3: Let $T$ be a stopping time and $X$ a predictable process. Then $X_T 1_{\{T<\infty\}}$ is $\mathcal{F}_{T -}$ measurable.

Applying this to the stopping time $T\equiv t$ yields $X_t=X_t 1_{t<\infty}$ is $\mathcal{F}_{t-}$ measurable for all $t\geq 0$ and a predictable process $X$.

The proof for theorem 3 can be found in the first edition of "Foundations of modern probability theory" by Olav Kallenberg in lemma 22.3.

• Hi peer - thx! Two comments: 1. in the first definition you meant "The optional σ-algebra O", not "The predictable σ-algebra O", right? 2. This would answer yes to my first question, not to the second, right?
– Tom
Nov 18, 2016 at 17:02
• @Tom Yes, sorry for that. I fixed it. I will think about your second question.
– peer
Nov 18, 2016 at 22:05
1. Show that the indicator function of every predictable rectangle is progressively measurable. This is very easy to show. Note that the the predictable rectangles form a $\pi$-system which generate the predictable $\sigma$-algebra.
3. Now, if $X$ is predictable $X \wedge n$ is bounded predictable and therefore progressively measurable from the previous step. Now just take limit $n \to \infty$ to conclude that $X$ is progressively measurable.