# progressively measurable process

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

 $X\colon\mathbb{T}\times\Omega\rightarrow\mathbb{R},\ (t,\omega)\mapsto X_{t}(\omega)$

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

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

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

 $S\cap\left((-\infty,s]\times\Omega\right)\in\mathcal{B}(\mathbb{T})\otimes% \mathcal{F}_{s}$

for every $s\in\mathbb{T}$. 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.

Title progressively measurable process ProgressivelyMeasurableProcess 2013-03-22 18:37:31 2013-03-22 18:37:31 gel (22282) gel (22282) 4 gel (22282) Definition msc 60G05 progressive process PredictableProcess OptionalProcess progressive progressively measurable