Scorochod embedding theorem

The Skorochod embedding theorem (also found in the spellings Skorokhod or Skorohod ) is a mathematical theorem from the theory of stochastic processes , a branch of probability theory . It is named after Anatoly Skorochod . It clearly states that every random variable can (under certain circumstances) be embedded in the mathematical modeling of Brownian molecular motion , the Wiener process .

statement

A Wiener process and the corresponding generated filtration are given . ${\displaystyle (W_{t})_{t\geq 0}}$ ${\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\geq 0}}$

Let be a real-valued random variable with and ${\displaystyle X}$ ${\displaystyle \operatorname {E} (X)=0}$ ${\displaystyle \operatorname {Var} (X)<\infty }$

Then there is a stop time regarding such that ${\displaystyle \tau }$ ${\displaystyle \mathbb {F} }$

${\displaystyle \operatorname {E} (\tau )=\operatorname {Var} (X)}$

and has the same distribution as . ${\displaystyle W_{\tau }}$ ${\displaystyle X}$

application

With the embedding theorem, the law of the iterated logarithm can be more easily derived in the general form. To do this, you first show the law of the iterated logarithm for the Wiener process and then use the embedding theorem to extend this result to the general case.

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 .