# simple predictable process

Simple predictable processes are a particularly simple class of stochastic processes^{}, for which the Ito integral can be defined directly. They are often used as the starting point for defining stochastic integrals of more general predictable integrands.

Suppose we are given a filtration^{} (http://planetmath.org/FiltrationOfSigmaAlgebras) $({\mathcal{F}}_{t})$ on the measurable space^{} $(\mathrm{\Omega},\mathcal{F})$ with time index $t$ ranging over the nonnegative real numbers.
A simple predictable process $\xi $ is a left-continuous and adapted process which can be written as

$$ |

for some $n\ge 0$, stopping times $$, ${\mathcal{F}}_{0}$-measurable and bounded random variable^{} ${A}_{0}$ and ${\mathcal{F}}_{{S}_{k}}$-measurable and bounded random variables ${A}_{k}$.
In the case where ${S}_{k}$ and ${T}_{k}$ are deterministic times, then $\xi $ is called an *elementary predictable* process.

The stochastic integral of the simple predictable process $\xi $ with respect to a stochastic process ${X}_{t}$ can then be written as

$${\int}_{0}^{t}\xi \mathit{d}X=\sum _{k=1}^{n}{1}_{\{t>{S}_{k}\}}{A}_{k}({X}_{\mathrm{min}(t,{T}_{k})}-{X}_{{S}_{k}}).$$ |

Title | simple predictable process |
---|---|

Canonical name | SimplePredictableProcess |

Date of creation | 2013-03-22 18:36:33 |

Last modified on | 2013-03-22 18:36:33 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 7 |

Author | gel (22282) |

Entry type | Definition |

Classification | msc 60G07 |

Defines | simple predictable |

Defines | elementary predictable |