Doob's maximum inequality

The Doob's martingale inequality is one of the main inequalities in the stochastics . In addition to the Burkholder inequality , it is one of the most common calculation methods for the (stochastic) order of magnitude of (continuous) martingales . It is named after Joseph L. Doob and can be found in the literature under different names ( Doob's inequality ${\displaystyle L^{p}}$ , Doob's inequality (s) , Doob's extremal inequality , maximum inequality , Doob's maximum inequality ) as well as in slightly different formulations that are differ by the number of specified inequalities and the conditions. The naming as inequality follows from the use of the norm, the naming as "maximal", since the supremum of the first elements of the process is estimated. There are also differences in the notation, so either the norm or the expected value are used for the formulation. ${\displaystyle L^{p}}$ ${\displaystyle L^{p}}$ ${\displaystyle L^{p}}$

Discrete index set

Be a stochastic process . Define ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$

${\displaystyle X_{n}^{*}:=\sup\{X_{k}\,|\,k\leq n\}}$ and ${\displaystyle |X|_{n}^{*}:=\sup\{|X_{k}|\,|\,k\leq n\}}$

If there is a submartingale , then applies to everyone ${\displaystyle X}$ ${\displaystyle \lambda >0}$

${\displaystyle \lambda P(X_{n}^{*}\geq \lambda )\leq \operatorname {E} (X_{n}\mathbf {1} _{\{X_{n}^{*}\geq \lambda \}})\leq \operatorname {E} (|X_{n}|\mathbf {1} _{\{X_{n}^{*}\geq \lambda \}})}$.

If a martingale or a positive submartingale is and is as well , then applies ${\displaystyle X}$ ${\displaystyle \lambda >0}$ ${\displaystyle p\geq 1}$

${\displaystyle \lambda ^{p}P(|X|_{n}^{*}\geq \lambda )\leq \operatorname {E} (|X_{n}|^{p})}$.

Furthermore, always applies to everyone ${\displaystyle p>1}$

${\displaystyle \operatorname {E} (|X_{n}|^{p})\leq \operatorname {E} \left(\left(|X|_{n}^{*}\right)^{p}\right)\leq \left({\frac {p}{p-1}}\right)^{p}\operatorname {E} (|X_{n}|^{p})}$

There are various differences in the formulation. For example, some authors do not include the first inequality, others only formulate the first and second inequality, and these only for positive submartingales, only show a special case for fixes or call the first inequality Doob's extremal inequality and the second Doob's inequality . ${\displaystyle p}$ ${\displaystyle L^{p}}$

Continuous index set

Let it be a martingale or a nonnegative submartingale and be right-continuous . Then applies to all : ${\displaystyle (M_{t})_{t\geq 0}\;}$ ${\displaystyle p>1}$ ${\displaystyle (M_{t})}$ ${\displaystyle T>0}$

${\displaystyle \|\sup _{t\leq T}|M_{t}|\,\|_{p}\leq {\frac {p}{p-1}}\|M_{T}\|_{p}}$.

The Lp standard denotes . Note that the conjugate real number is to, i. H. it applies . Accordingly, the central proof step is the application of the Hölder inequality . ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle q={\tfrac {p}{p-1}}}$ ${\displaystyle p}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$

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 .
• 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 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .

Individual evidence

1. ↑ Klenke: Probability Theory. 2013, p. 222.
2. ↑ Klenke: Probability Theory. 2013, p. 484.
3. ↑ Kusolitsch: Measure and probability theory. 2014, p. 284.
4. ^ Schmidt: Measure and probability. 2011, p. 430.
5. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 327.
6. ↑ Klenke: Probability Theory. 2013, p. 222.
7. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 327.
8. ^ Schmidt: Measure and probability. 2011, p. 430.
9. ↑ Kusolitsch: Measure and probability theory. 2014, pp. 284–286.
10. ↑ Heinz Bauer : Probability Theory. 5th edition. De Gruyter textbook, Berlin 2002, ISBN 3-11-017236-4 , p. 412f