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 . ${\displaystyle (Y_{n})_{n\in \mathbb {N} }}$

Be and ${\displaystyle X_{0}:=0}$

${\displaystyle X_{n}:=\sum _{i=1}^{n}Y_{i}}$

Then applies to everyone and everyone ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle r>0}$

${\displaystyle P\left(\sup _{m\leq n}X_{m}\geq r\right)\leq 2P(X_{n}\geq r)-P(X_{n}=r)}$

If they almost certainly assume values ​​from , then equality holds for all in the above inequality. ${\displaystyle Y_{i}}$ ${\displaystyle \{-1,0,1\}}$ ${\displaystyle r\in \mathbb {N} }$

Principle of reflection for the Wiener process

Be a Wiener process as well as and . Then applies ${\displaystyle (W_{t})_{t\in \mathbb {R} ^{+}}}$ ${\displaystyle T>0}$ ${\displaystyle r>0}$

${\displaystyle P(\sup\{B_{t}\,|\,t\in [0,T]\}>r)=2P(B_{T}>r)=P(|B_{T}|>r)}$.

The further estimate is obtained from the density of the normal distribution

${\displaystyle P(|B_{T}|>r)\leq {\sqrt {\frac {4T}{2\pi r^{2}}}}\exp \left({\frac {-r^{2}}{2T}}\right)}$.

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 .

Individual evidence

1. ↑ Klenke: Probability Theory. 2013, p. 363.
2. ↑ Klenke: Probability Theory. 2013, p. 520.
3. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 364.
4. ↑ Klenke: Probability Theory. 2013, p. 363.
5. ↑ Klenke: Probability Theory. 2013, p. 480.
6. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 366.