Crossover inequality

The crossing inequality , sometimes also called the upcrossing inequality or Doob's crossing theorem (after Joseph L. Doob ), is an inequality about the temporal behavior of submartingales in discrete time. Thus, the statement is to be assigned to the theory of stochastic processes and thus to the probability theory . The crossing inequality is an important tool for deriving the martingale convergence theorems and analogous statements for backward martingales.

idea

The basic idea is to think of the submartingale as a stock price. If the price falls below the value , one buys shares; if the value rises above it , one sells. If you now know how often the interval has been crossed (i.e. how often the interval has been crossed from bottom to top), you can estimate the total profit based on the number of crossings. The crossing inequality makes exactly this estimate. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle [a,b]}$

Formalization

The formulation of falling below and exceeding works using stop times . You bet for the submartingale ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle X=(X_{n})_{n\in \mathbb {N} }}$

${\displaystyle \sigma _{0}=0}$

as a start,

${\displaystyle \tau _{k}:=\inf\{n\geq \sigma _{k-1}\,|\,X_{n}\leq a\}{\text{ für }}k=1,2,\dots }$

as the point in time of the k-th falling below and ${\displaystyle a}$

${\displaystyle \sigma _{k}:=\inf\{n\geq \tau _{k}\,|\,X_{n}\geq b\}{\text{ für }}k=1,2,\dots }$

as the point in time of the k-th exceeding of . The number of crossings from to is then given by ${\displaystyle b}$ ${\displaystyle [a,b]}$ ${\displaystyle n}$

${\displaystyle U_{n}^{a,b}=\sup\{k\in \mathbb {N} \,|\,\sigma _{k}\leq n\}}$.

The crossover inequality is now

${\displaystyle \operatorname {E} (U_{n}^{a,b})\leq {\frac {\operatorname {E} ((X_{n}-a)^{+})-\operatorname {E} ((X_{0}-a)^{+})}{b-a}}}$

Derivation of convergence statements

The derivation of convergence statements mostly follows the scheme that one

${\displaystyle U^{a,b}:=\lim _{n\to \infty }U_{n}^{a,b}}$

considered. Can one now show, under suitable additional conditions and the intersection inequality, that

${\displaystyle \operatorname {E} (U^{a,b})<\infty {\text{ für alle }}a<b}$

applies and the process is unrestricted upwards or downwards, the process must be in the interval in the long term , since it can neither cross the interval indefinitely nor leave the range of the interval. But since this is true for everyone , it can be shown that the process converges. ${\displaystyle [a,b]}$ ${\displaystyle a<b}$

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 .
• Christian Hesse: Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .
• 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. ↑ Kusolitsch: Measure and probability theory. 2014, p. 269.