predictable process

A predictable process is a real-valued stochastic process whose values are known, in a sense, just in advance of time. Predictable processes are also called previsible.

1 predictable processes in discrete time

Suppose we have a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) $(\mathcal{F}_{n})_{n\in\mathbb{Z}_{+}}$ on a measurable space $(\Omega,\mathcal{F})$ . Then a stochastic process $X_{n}$ is predictable if $X_{n}$ is $\mathcal{F}_{n-1}$ -measurable (http://planetmath.org/MeasurableFunctions) for every $n\geq 1$ and $X_{0}$ is $\mathcal{F}_{0}$ -measurable. So, the value of $X_{n}$ is known at the previous time step. Compare with the definition of adapted processes for which $X_{n}$ is $\mathcal{F}_{n}$ -measurable.

2 predictable processes in continuous time

In continuous time, the definition of predictable processes is a little more subtle. Given a filtration $(\mathcal{F}_{t})$ with time index $t$ ranging over the non-negative real numbers, the class of predictable processes forms the smallest set of real valued stochastic processes containing all left-continuous $\mathcal{F}_{t}$ -adapted processes and which is closed under taking limits of a sequence of processes.

Equivalently, a real-valued stochastic process

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

is predictable if it is measurable with respect to the predictable sigma algebra $\wp$ . This is defined as the smallest $\sigma$ -algebra on $\mathbb{R}_{+}\times\Omega$ making all left-continuous and adapted processes measurable.

Alternatively, $\wp$ is generated by either of the following collections of subsets of $\mathbb{R}_{+}\times\Omega$

	$\displaystyle\wp$	$\displaystyle=\sigma\left(\left\{(t,\infty)\times A:t\geq 0,A\in\mathcal{F}_{t% }\right\}\cup\{\{0\}\times A:A\in\mathcal{F}_{0}\}\right)$
		$\displaystyle=\sigma\left(\left\{(T,\infty):T\textrm{ is a stopping time}% \right\}\cup\{\{0\}\times A:A\in\mathcal{F}_{0}\}\right)$
		$\displaystyle=\sigma\left(\left\{[T,\infty):T\textrm{ is a predictable % stopping time}\right\}\right)$

Note that in these definitions, the sets $(T,\infty)$ and $[T,\infty)$ are stochastic intervals, and subsets of $\mathbb{R}_{+}\times\Omega$ .

3 general predictable processes

The definition of predictable process given above can be extended to a filtration $(\mathcal{F}_{t})$ with time index $t$ lying in an arbitrary subset $\mathbb{T}$ of the extended real numbers. In this case, the predictable sets form a $\sigma$ -algebra on $\mathbb{T}\times\Omega$ . If $\mathbb{T}$ has a minimum element $t_{0}$ then let $S$ be the collection of sets of the form $\{t_{0}\}\times A$ for $A\in\mathcal{F}_{t_{0}}$ , otherwise let $S$ be the empty set.Then, the predictable $\sigma$ -algebra is defined by

$\begin{split}\displaystyle\wp&\displaystyle=\sigma\left(\left\{(t,\infty]% \times A:t\in\mathbb{T},A\in\mathcal{F}_{t}\right\}\cup S\right)\\ &\displaystyle=\sigma\left(\left\{(T,\infty]:T\colon\Omega\rightarrow\mathbb{T% }\textrm{ is a stopping time}\right\}\cup S\right).\end{split}$

Here, $(t,\infty]$ and $(T,\infty]$ are understood to be intervals containing only times in the index set $\mathbb{T}$ . If $\mathbb{T}$ is an interval of the real numbers then $\wp$ can be equivalently defined as the $\sigma$ -algebra generated by the class of left-continuous and adapted processes with time index ranging over $\mathbb{T}$ .

A stochastic process $X\colon\mathbb{T}\times\Omega\rightarrow\mathbb{R}$ is predictable if it is $\wp$ -measurable. It can be verified that in the cases where $\mathbb{T}=\mathbb{Z}_{+}$ or $\mathbb{T}=\mathbb{R}_{+}$ then this definition agrees with the ones given above.

Title	predictable process
Canonical name	PredictableProcess
Date of creation	2013-03-22 18:36:30
Last modified on	2013-03-22 18:36:30
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	12
Author	gel (22282)
Entry type	Definition
Classification	msc 60G07
Related topic	PredictableStoppingTime
Related topic	ProgressivelyMeasurableProcess
Related topic	OptionalProcess
Defines	predictable
Defines	previsible