Doob martingale

A doob martingale is a special stochastic process in stochastics . According to the name Doob martingales belong to the class of martingales . Doob martingales are characterized by their simple presentation. They are also closely related to the martingale convergence theorems . Doob martingales themselves already converge due to their properties, which follow from the definition. The martingale convergence theorems then answer the question which martingales can be represented as doob martingales.

The Doob Martingale are named after Joseph L. Doob .

definition

A probability space , an index set as well as a filtering into and an integrable random variable , that is, is given . ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle T\subset \mathbb {N} _{0}}$ ${\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle X}$ ${\displaystyle \operatorname {E} (|X|)<\infty }$

Then the stochastic process is called the by

${\displaystyle X_{n}:=\operatorname {E} (X|{\mathcal {F}}_{n})}$

is defined, a doob martingale.

It denotes the conditional expected value of the random variable , given the σ-algebra . ${\displaystyle \operatorname {E} (Y|{\mathcal {A}})}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {A}}}$

Proof of martingale property

The integrability of the Doob martingale follows from

${\displaystyle \operatorname {E} (|X_{n}|)=\operatorname {E} (\left|\operatorname {E} (X|{\mathcal {F}}_{n})\right|)\leq \operatorname {E} (\operatorname {E} (\left|X\right||{\mathcal {F}}_{n}))=\operatorname {E} (|X|)}$

according to the definition, the triangle inequality for the conditional expected value and the rule of forming the expected value using the conditional expected value.

The adaptation of the Doob martingale follows from this, which by definition is always - measurable . ${\displaystyle X_{n}=\operatorname {E} (X|{\mathcal {F}}_{n})}$ ${\displaystyle {\mathcal {F}}_{n}}$

The proof of the defining property for Martingales follows from the tower property of the conditional expectation value:

${\displaystyle X_{n}=\operatorname {E} (X|{\mathcal {F}}_{n})=\operatorname {E} (\operatorname {E} (X|{\mathcal {F}}_{n})|{\mathcal {F}}_{n+1})=\operatorname {E} (\operatorname {E} (X|{\mathcal {F}}_{n+1})|{\mathcal {F}}_{n})=\operatorname {E} (X_{n+1}|{\mathcal {F}}_{n})}$.

properties

Equal integrability

Every Doob martingale can always be integrated equally . This can be shown by inferring from the random variable , which is uniformly integrable, via a criterion for uniform integrability, which uses convex functions, by means of Jensen's inequality for the conditional expectation value, to the uniform integrability. ${\displaystyle X}$

As a closed martingale

Every closed martingale can be represented as a Doob martingale: If the last element of the closed martingale is ${\displaystyle X=(X_{n})_{n\in T}}$ ${\displaystyle X_{u}}$

${\displaystyle X_{n}=\operatorname {E} (X_{u}|{\mathcal {F}}_{n})}$for everyone . ${\displaystyle n\leq u}$

Conversely, every Doob martingale can be completed. To do this, one sets as the last element ${\displaystyle T'=T\cup u}$

${\displaystyle X_{u}:=X}$and , ${\displaystyle {\mathcal {F}}_{u}:={\mathcal {A}}}$

the σ-algebra of the underlying probability space.

convergence

If you set

${\displaystyle {\mathcal {F}}_{\infty }:=\sigma \left(\bigcup _{n\in \mathbb {N} }{\mathcal {F}}_{n}\right)}$

the following statement can be derived from the martingale convergence theorem :

If a martingale is related , then it can be represented as a Doob martingale with respect to a random variable if and only if one of the following two equivalent conditions are met: ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mathbb {F} =({\mathcal {F}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$

1. ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ is equally integrable
2. ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$Converges on the first mean and almost certainly

If then the limit value is , then applies ${\displaystyle X_{\infty }}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$

${\displaystyle X_{\infty }=\operatorname {E} (X|{\mathcal {F}}_{\infty })}$

Levy's theorem

Sometimes a martingale convergence theorem for Doob Martingales or for the conditional expected value is also formulated as an independent statement and then referred to as the Lévy theorem (after Paul Lévy ). It reads:

If it is an integrable random variable, then converges almost certainly and in the first mean against . ${\displaystyle X}$ ${\displaystyle \operatorname {E} (X|{\mathcal {F}}_{n})}$ ${\displaystyle \operatorname {E} (X|{\mathcal {F}}_{\infty })}$

Depending on the source, it is also required that the random variable can be square-integrated. The convergence is then the root mean square.

literature

• 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. 273.
2. ↑ Klaus D. Schmidt: Measure and probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 431 , doi : 10.1007 / 978-3-642-21026-6 .
3. ↑ TO Shiryaev: Martingales . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).