Process with independent increments

The process with independent increments , also called process with independent increments , is a term from the theory of stochastic processes , a sub-area of probability theory . A process with independent increments is a process in which the course of the future of the process is independent of the past. Many important classes of processes such as the Lévy process and thus also the Wiener process and the Poisson process are processes with independent increments.

definition

A real-valued stochastic process is given . The process is called a process with independent increments , if for anyone with ${\displaystyle (X_{t})_{t\in T}}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle t_{0},\ldots ,t_{N}\in T}$

${\displaystyle t_{0}<t_{1}<\dots <t_{N-1}<t_{N}}$

holds that the random variables ${\displaystyle N}$

${\displaystyle Z_{i}=X_{t_{i}}-X_{t_{i-1}}{\text{ für }}i=1,\dots ,N}$

are stochastically independent . This is called, in an obvious way, growth . ${\displaystyle Z_{i}}$

example

We consider the discrete-time symmetric random walk on as an example . To do this, be independent and identically Rademacher distributed for everyone , that is . The random walk is then defined as ${\displaystyle \mathbb {Z} }$ ${\displaystyle Y_{n}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle P(Y_{n}=-1)=P(Y_{n}=1)={\tfrac {1}{2}}}$

${\displaystyle X_{0}=0{\text{ und }}X_{n}=\sum _{i=1}^{n}Y_{i}{\text{ für }}n\geq 1}$.

Accordingly, the difference is at any two points in time and with always ${\displaystyle t_{i}}$ ${\displaystyle t_{i-1}}$ ${\displaystyle t_{i-1}<t_{i}}$

${\displaystyle Z_{i}=\sum _{t_{i-1}+1}^{t_{i}}Y_{i}}$.

But since they are all independent of each other, each subfamily formed from them without overlapping is also independent. Accordingly, they are also independent of each other and the process is a process with independent increments. ${\displaystyle Y_{n}}$ ${\displaystyle Z_{i}}$ ${\displaystyle (X_{n})}$

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 .