Backward martingale

A backward martingale , also called an inverse martingale or backward martingale , is a stochastic process that arises from a martingale by reversing the index set. So clearly it is a martingale that is "played backwards". As for martingales, there are also convergence theorems for backward martingales. These are used, for example, in the proof of de Finetti's representation theorem on the structure of interchangeable families of random variables .

definition

Consider a filtration and a martingale. Then the process is called ${\displaystyle \mathbb {F} =({\mathcal {F}}_{n})_{n\in -\mathbb {N} }}$ ${\displaystyle X=(X_{n})_{n\in -\mathbb {N} }}$ ${\displaystyle \mathbb {F} }$

${\displaystyle X:=(X_{-n})_{n\in \mathbb {N} }}$

a reverse martingale .

properties

Note that with continues to apply to filtration . thus contains all relevant information of the process. ${\displaystyle {\mathcal {F}}_{s}\subseteq {\mathcal {F}}_{t}}$ ${\displaystyle s,t\in -\mathbb {N} }$ ${\displaystyle s<t}$ ${\displaystyle {\mathcal {F}}_{0}}$

Backward martingales can always be integrated to the same degree because, due to the martingale property, they are always the representation

${\displaystyle X_{-n}=\operatorname {E} (X_{0}|{\mathcal {F}}_{-n})}$

have.

Convergence theorem for backward martingales

statement

If a martingale is related , it exists ${\displaystyle X=(X_{n})_{n\in -\mathbb {N} }}$ ${\displaystyle \mathbb {F} =({\mathcal {F}}_{n})_{n\in -\mathbb {N} }}$

${\displaystyle \lim _{n\to \infty }X_{-n}=X_{-\infty }}$

on average and almost certainly . With

${\displaystyle {\mathcal {F}}_{-\infty }:=\bigcap _{n=1}^{\infty }{\mathcal {F}}_{-n}}$

then applies

${\displaystyle X_{-\infty }=\operatorname {E} (X_{0}|{\mathcal {F}}_{-\infty })}$.

Analogous to the Martingale Convergence Theorem , the proof follows by means of the intersection inequality by considering the intersections between and above . ${\displaystyle -n}$ ${\displaystyle 0}$ ${\displaystyle [a,b]}$

Inference

An important conclusion from the above statement for the derivation of de Finetti's theorem is the following: Is

${\displaystyle \varphi \colon E^{k}\to \mathbb {R} {\text{ messbar, }}\operatorname {E} (|\varphi (X_{1},\dots ,X_{k})|)<\infty }$

and an interchangeable family of random variables with values ​​in and the permutation of the random variables under and ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle E}$ ${\displaystyle (X_{1},\dots ,X_{n})^{\sigma }}$ ${\displaystyle \sigma }$

${\displaystyle A_{n}(\varphi )={\tfrac {1}{n!}}\sum _{\sigma \in S(n)}\varphi ((X_{1},\dots ,X_{n})^{\sigma })}$

the symmetrized mean . Then on average and almost certainly applies

${\displaystyle \operatorname {E} (\varphi (X)|{\mathcal {E}})=\operatorname {E} (\varphi (X)|{\mathcal {T}})=\lim _{n\to \infty }A_{n}(\varphi )}$.

The term denotes the terminal σ-algebra and the interchangeable σ-algebra . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {E}}}$

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 .
• 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. ^ Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 84 , doi : 10.1007 / 978-3-642-41997-3 .
2. ↑ Kusolitsch: Measure and probability theory. 2014, p. 267.