σ-algebra of the τ past

The σ-algebra of the τ-past , also called the past of τ , is a special set system in probability theory , more precisely a σ-algebra . It is created by combining a filtration with a stopping time and is mostly used for statements about stopped processes , i.e. stochastic processes that are stopped at a random point in time. These statements include, for example, the Optional Stopping Theorem , the Optional Sampling Theorem and the definition of the strong Markov property .

definition

Given is a probability space as well as a filter with regard to the upper σ-algebra and a stop time with regard to . Then is called ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \tau }$ ${\displaystyle \mathbb {F} }$

${\displaystyle {\mathcal {F}}_{\tau }=\{A\in {\mathcal {A}}\,|\,A\cap \{\tau \leq t\}\in {\mathcal {F}}_{t}{\text{ für alle }}t\in T\}}$

the σ-algebra of the τ-past .

properties

Are stop times and is , so is . ${\displaystyle \sigma ,\tau }$ ${\displaystyle \sigma \leq \tau }$ ${\displaystyle {\mathcal {F}}_{\sigma }\subset {\mathcal {F}}_{\tau }}$

Furthermore, it is always - measurable . ${\displaystyle \tau }$ ${\displaystyle {\mathcal {F}}_{\tau }}$

It can be turned into a stochastic process ${\displaystyle \tau <\infty }$

${\displaystyle X=(X_{t})_{t\in T}}$

a "sampled" random variable

${\displaystyle X_{\tau }\colon \omega \mapsto X_{\tau (\omega )}(\omega )}$

define. If, in addition, at most it can be counted and the stochastic process is adapted , then it is always measurable. The random variable should not be confused with the stopped process , especially since the notation in the literature is not uniform. ${\displaystyle T}$ ${\displaystyle X_{\tau }}$ ${\displaystyle {\mathcal {F}}_{\tau }}$ ${\displaystyle X_{\tau }}$ ${\displaystyle X^{\tau }}$

Clearly, there is a random variable in the case of index set on the amount of the random variable , on the amount of , etc. This results in this case, the alternative definition ${\displaystyle X_{\tau }}$ ${\displaystyle T=\mathbb {N} }$ ${\displaystyle \{\tau =0\}}$ ${\displaystyle X_{0}}$ ${\displaystyle \{\tau =1\}}$ ${\displaystyle X_{1}}$

${\displaystyle X_{\tau }=\sum _{n=0}^{\infty }\mathbf {1} _{\{\tau =n\}}X_{n}}$.

literature

• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .
• 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 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .

Individual evidence

1. ↑ Klenke: Probability Theory. 2013, p. 197.
2. ↑ Kusolitsch: Measure and probability theory. 2009, p. 278.