Is a predictable process adapted?

Question

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 $(\mathcal{F}_{t-})_t$?

Answer 1

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.

Answer 2

I wanted to note another method for proving the first part. I shall only sketch the argument.

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.
2. Now use the Monotone-Class theorem (see David Williams, Page 37) to show that every bounded predictable process is progressively measurable. (Again very straight-forward)
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.
4. Now use the fact that a progressively measurable process is measurable and adapted. (This too is easy to show).

Answer 3

Denote

$ \mathcal{P} = \sigma(\mathcal{I}) = \sigma(\{ (s,t]\times E, 0 \leq s < t, E \in \mathcal{F}_s \} )$

Consider the map

$ f: (\Omega, \mathcal{F}_{t-}) \to ((0,\infty) \times \Omega , \sigma(\mathcal{I})) $

$ \omega \mapsto (t,\omega) $

This map $f$ is $\mathcal{F}_{t-}-\mathcal{P} $ measurable. For this, notice that $f^{-1}(\mathcal{I}) \subset \mathcal{F}_{t-}$, since for any element in $\mathcal{I}$:

$ f^{-1}( (s,r] \times E ) = \begin{cases} E \quad \text{if } s<t\leq r \\ \emptyset \quad \text{if } t \leq s \text{ or } t > r \end{cases} \in \mathcal{F}_s \subset \mathcal{F}_{t-} $

Hence $f^{-1}(\sigma(\mathcal{I})) \subset \mathcal{F}_{t-}$ (see Proving measurability of a function only by checking generating sets). Finally as by the assumption $ X : ((0,\infty) \times \Omega, \mathcal{P}) \to (\mathbb{R}, \mathscr{B}(\mathbb{R})) $ a measurable map, the composition of the two measurable maps $ X_t = X \circ f : (\Omega, \mathcal{F}_{t-}) \to (\mathbb{R}, \mathscr{B}(\mathbb{R}))$ is $\mathcal{F}_{t-}$ measurable.