predictable processPredictableProcess2008-12-13 15:07:072008-12-20 22:09:42Definitionmeasurability of stochastic processespredictableprevisible$\sigma$-algebrafiltration% almost certainly you want these
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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 \emph{previsible}.
\section{predictable processes in discrete time}
Suppose we have a \PMlinkname{filtration}{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 $\mathcal{F}_{n-1}$-\PMlinkname{measurable}{MeasurableFunctions} for every $n\ge 1$ 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.
\section{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
\begin{align*}
&X\colon\mathbb{R}_+\times\Omega\rightarrow\mathbb{R}\\
&(t,\omega)\mapsto X_t(\omega)
\end{align*}
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 $\mathbb{R}_+\times\Omega$
\begin{align*}
\wp &=\sigma\left(\left\{(t,\infty)\times A:t\ge 0,A\in\mathcal{F}_t\right\}\cup\{\{0\}\times A:A\in\mathcal{F}_0\}\right)\\
&=\sigma\left(\left\{(T,\infty):T\textrm{ is a stopping time}\right\}\cup\{\{0\}\times A:A\in\mathcal{F}_0\}\right)\\
&=\sigma\left(\left\{[T,\infty):T\textrm{ is a predictable stopping time}\right\}\right)
\end{align*}
Note that in these definitions, the sets $(T,\infty)$ and $[T,\infty)$ are stochastic intervals, and subsets of $\mathbb{R}_+\times\Omega$.
\section{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{equation*}\begin{split}
\wp &=\sigma\left(\left\{(t,\infty]\times A:t\in\mathbb{T},A\in\mathcal{F}_t\right\}\cup S\right)\\
&= \sigma\left(\left\{(T,\infty]:T\colon\Omega\rightarrow\mathbb{T}\textrm{ is a stopping time}\right\}\cup S\right).
\end{split}\end{equation*}
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.