Reflection principle (stochastics)
The reflection principle , also known as the reflection principle or reflection principle , is a statement about random walks from the theory of stochastic processes and thus to the probability theory . The reflection principle is a consequence of the strong Markov quality and is formulated in different versions, including for the Wiener trial . The reflection principle clearly provides an estimate of the probability that a stochastic process has already exceeded a predetermined threshold value before a certain point in time.
Reflection principle for the symmetrical random walk
A sequence of independently identically distributed as well as symmetrical and real-valued random variables is given .
Be and
Then applies to everyone and everyone
If they almost certainly assume values from , then equality holds for all in the above inequality.
Principle of reflection for the Wiener process
Be a Wiener process as well as and . Then applies
- .
The further estimate is obtained from the density of the normal distribution
- .
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 978-3-540-21676-6 , doi : 10.1007 / b137972 .