progressively measurable process

A stochastic process ${(X_t)}_{t\in {\mathbb{Z}}_{+}}$ is said to be adapted to a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) $({ℱ}_t)$ on the measurable space $(\mathrm{\Omega },ℱ)$ if $X_t$ is an ${ℱ}_t$-measurable random variable for each $t=0,1,\mathrm{\dots }$. However, for continuous-time processes, where the time $t$ ranges over an arbitrary index set $𝕋\subseteq ℝ$, the property of being adapted is too weak to be helpful in many situations. Instead, considering the process as a map

$$X:𝕋\times \mathrm{\Omega }\to ℝ,(t,\omega )\mapsto X_t(\omega )$$

it is useful to consider the measurability of $X$.

The process $X$ is progressive or progressively measurable if, for every $s\in 𝕋$, the stopped process $X_t^s\equiv X_{\min (s,t)}$ is $ℬ(𝕋)\otimes {ℱ}_s$-measurable. In particular, every progressively measurable process will be adapted and jointly measurable. In discrete time, when $𝕋$ is countable, the converse holds and every adapted process is progressive.

A set $S\subseteq 𝕋\times \mathrm{\Omega }$ is said to be progressive if its characteristic function $1_S$ is progressive. Equivalently,

$$S\cap \left((-\mathrm{\infty },s]\times \mathrm{\Omega }\right)\in ℬ(𝕋)\otimes {ℱ}_s$$

for every $s\in 𝕋$. The progressively measurable sets form a $\sigma$-algebra, and a stochastic process is progressive if and only if it is measurable with respect to this $\sigma$-algebra.