# 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^{} $(\mathrm{\Omega},\mathcal{F})$. Then a stochastic process ${X}_{n}$ is predictable if ${X}_{n}$ is ${\mathcal{F}}_{n-1}$-measurable (http://planetmath.org/MeasurableFunctions) for every $n\ge 1$ 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.

## 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

$X:{\mathbb{R}}_{+}\times \mathrm{\Omega}\to \mathbb{R}$ | ||

$(t,\omega )\mapsto {X}_{t}(\omega )$ |

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

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

$\mathrm{\wp}$ | $=\sigma \left(\{(t,\mathrm{\infty})\times A:t\ge 0,A\in {\mathcal{F}}_{t}\}\cup \{\{0\}\times A:A\in {\mathcal{F}}_{0}\}\right)$ | ||

$=\sigma \left(\{(T,\mathrm{\infty}):T\text{is a stopping time}\}\cup \{\{0\}\times A:A\in {\mathcal{F}}_{0}\}\right)$ | |||

$=\sigma \left(\{[T,\mathrm{\infty}):T\text{is a predictable stopping time}\}\right)$ |

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

## 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 $\mathbb{T}\times \mathrm{\Omega}$. 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

$$\begin{array}{cc}\hfill \mathrm{\wp}& =\sigma \left(\{(t,\mathrm{\infty}]\times A:t\in \mathbb{T},A\in {\mathcal{F}}_{t}\}\cup S\right)\hfill \\ & =\sigma \left(\{(T,\mathrm{\infty}]:T:\mathrm{\Omega}\to \mathbb{T}\text{is a stopping time}\}\cup S\right).\hfill \end{array}$$ |

Here, $(t,\mathrm{\infty}]$ and $(T,\mathrm{\infty}]$ 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 $\mathrm{\wp}$ 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 $X:\mathbb{T}\times \mathrm{\Omega}\to \mathbb{R}$ is predictable if it is $\mathrm{\wp}$-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 |