# predictable process

## 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 PredictableProcess 2013-03-22 18:36:30 2013-03-22 18:36:30 gel (22282) gel (22282) 12 gel (22282) Definition msc 60G07 PredictableStoppingTime ProgressivelyMeasurableProcess OptionalProcess predictable previsible