Adapted process

A stochastic process $X$ is adapted to a filtration $({\mathcal{F}}_t)$ when $X_t$ is ${\mathcal{F}}_t$-measurable at every time $t$. Its present value therefore depends only on information already available.

For an adapted process with right-continuous paths and a stopping time $T$, $X_t^T=X_{t\wedge T}$ is adapted. At fixed $t$, round $t\wedge T$ upward on finite partitions of $[0,t]$. Every approximating value is ${\mathcal{F}}_t$-measurable, and right continuity gives the limit. Completeness handles a null exceptional set if right continuity holds only almost surely. A martingale assumption and boundedness of $T$ are unnecessary.

A process $H$ is progressively measurable when, for every $T$, its restriction to $[0,T]\times \Omega$ is measurable for $\mathcal{B}([0,T])\otimes {\mathcal{F}}_T$. Continuous adapted processes are progressively measurable, and a Borel function of such a process remains so. A progressively measurable square-integrable Brownian integrand has an equivalent predictable integrand for the Itô integral, up to equality for $dtd\mathbb{P}$.

Let a probability space have two sample points, one of probability zero and one of probability one, and give it the complete power-set sigma-algebra at every time. Let $A\subseteq [0,1]$ be non-Borel. Assign the stochastic process the path $1_A(t)$ at the null sample point and the zero path at the other point. Each fixed-time variable is measurable and the paths are càdlàg almost surely, but the stochastic process is not progressively measurable: a section in the time variable of a product-measurable function must be Borel measurable at every sample point. The zero stochastic process is an indistinguishable progressively measurable version. Completing the filtration does not complete the time-product sigma-algebra or remove this distinction.

For an adapted process with pathwise right continuity, fix $T$ and approximate its time argument by the next endpoint of a deterministic partition of $[0,T]$. Every approximation is jointly measurable for the terminal product sigma-algebra because all sampled values are ${\mathcal{F}}_T$-measurable. Right continuity makes them converge pointwise, proving progressive measurability. Sampling from the right need not preserve adaptedness at intermediate times, but that is not required in this proof.