Elementary predictable stochastic process

The elementary (predictable) stochastic processes , usually simply called elementary processes , are a class of stochastic processes in probability theory . They are characterized by their simplicity and correspond to a stochastic generalization of the staircase functions . The Ito integral can be obtained from the elementary processes through completion, similar to the construction of the Lebesgue integral from the simple functions . Thus the elementary processes belong to the basis of stochastic analysis .

definition

Be given

• a probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$
• the index set ${\displaystyle T=[0,\infty )}$
• a filtration . ${\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}$

Then a real-valued stochastic process is called an elementary process, if there is one such that numbers ${\displaystyle H=(H_{t})_{t\in T}}$ ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle 0=t_{0}<t_{1}<\dots <t_{n-1}<t_{n}<\infty }$

exist and for - measurable random variables exist such that ${\displaystyle i=1,\dots ,n}$ ${\displaystyle {\mathcal {F}}_{t_{i-1}}}$ ${\displaystyle h_{i-1}}$

${\displaystyle H_{t}(\omega )=\sum _{i=1}^{n}h_{i-1}(\omega )\mathbf {1} _{(t_{i-1},t_{i}]}(t)}$

is. The characteristic function denotes on the set ${\displaystyle \mathbf {1} _{A}}$ ${\displaystyle A}$

Variants in the definition

The definitions differ in the literature partly in that the definition requires the restriction of the random variables . If this does not happen in the definition, the elementary processes are subsequently restricted to the set of restricted elementary processes. ${\displaystyle h_{i}}$

Furthermore, it is assumed for the entire construction of the Ito integral that the usual conditions apply, but this additional assumption has no influence on the properties discussed in this article.

Explanation

If one interprets the course of the process over time, the elementary process consists in the period ${\displaystyle T}$

${\displaystyle (t_{i-1},t_{i}]}$unchanged from the random variable , ${\displaystyle h_{i-1}}$

and then move on to the next random variable at the point in time . Thus the process can be viewed as "piecewise constant". This becomes even clearer when you select one and the function ${\displaystyle t_{i}}$ ${\displaystyle \omega _{0}\in \Omega }$

${\displaystyle t\mapsto H_{t}(\omega _{0})}$

considered. It is a step function and takes on the value on the interval . ${\displaystyle (t_{i-1},t_{i}]}$ ${\displaystyle h_{i-1}(\omega _{0})}$

properties

An elementary process is always a left-steady process . This follows from the fact that the intervals on the right are closed. Therefore, the step function (see above) ${\displaystyle (t_{i-1},t_{i}]}$ ${\displaystyle \omega _{0}\in \Omega }$

${\displaystyle t\mapsto H_{t}(\omega _{0})}$

a left continuous function and thus also the process left continuous.

Furthermore, elementary processes are always adapted because they can be measured . ${\displaystyle {\mathcal {F}}_{t_{i-1}}}$ ${\displaystyle h_{i-1}}$

Due to the definition of the predictable σ-algebra , the predictability of the process follows from the continuity and adaptation of a process to the left . As a result, elementary processes are always predictable.

In addition, the set of elementary processes forms a vector space , which is a subspace of the real-valued functions on . If one denotes the set of restricted elementary processes, a mapping can be made on it ${\displaystyle \Omega \times T}$ ${\displaystyle {\mathcal {E}}_{b}}$

${\displaystyle \|\cdot \|_{{\mathcal {E}}_{b}}\colon {\mathcal {E}}_{b}\to \mathbb {R} }$

by

${\displaystyle \|H\|_{{\mathcal {E}}_{b}}:=\operatorname {E} \left(\int _{T}H_{s}^{2}\mathrm {d} \lambda (s)\right)^{\frac {1}{2}}=\left(\sum _{i=1}^{n}\operatorname {E} (h_{i-1}^{2})(t_{i}-t_{i-1})\right)^{\frac {1}{2}}}$

define. This is a semi-standard .

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 3-540-21676-6 , doi : 10.1007 / b137972 .

Individual evidence

1. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 413.