# Localization (stochastics)

In stochastics , localization is understood to mean the expansion of a class of stochastic processes by those that can be made belonging to the class by deliberately stopping . The term local martingales , which play an important role in stochastic analysis , is particularly important here .

## Stopped processes

Let be a stochastic process on a filtered probability space , where or is. If there is any stopping time in relation to the filtration, then the process is called ${\ displaystyle X = (X_ {t}) _ {t \ in T}}$ ${\ displaystyle (\ Omega, {\ mathcal {A}}, ({\ mathcal {F}}) _ {t \ in T}, P)}$${\ displaystyle T = \ mathbb {N} _ {0}}$${\ displaystyle T = \ mathbb {R} _ {0} ^ {+}}$${\ displaystyle \ tau}$

${\ displaystyle X ^ {\ tau} = (X_ {t} ^ {\ tau}) _ {t \ in T} \ quad {\ text {with}} \ quad X_ {t} ^ {\ tau} (\ omega): = X _ {\ min (\ tau (\ omega), t)} (\ omega), \ quad \ omega \ in \ Omega}$

than when the process is stopped${\ displaystyle \ tau}$ . The process thus agrees with the process up to this point , but then remains at its current value and no longer changes its state. ${\ displaystyle X ^ {\ tau}}$${\ displaystyle \ tau}$${\ displaystyle X}$

## Localization of process classes

Let us now be a set of processes with the same index set , say the set of all Martingales or all Lévy processes . A process is said to be local of the class if there is a sequence of stop times that meets the following two properties: ${\ displaystyle {\ mathcal {C}}}$${\ displaystyle T}$${\ displaystyle X}$${\ displaystyle {\ mathcal {C}}}$ ${\ displaystyle (\ tau _ {i}), \; i \ in \ mathbb {N}}$

• It is almost certainly true for , ie for almost all the (deterministic) sequence diverges towards plus infinity .${\ displaystyle \ tau _ {i} \ to \ infty}$ ${\ displaystyle i \ to \ infty}$${\ displaystyle \ omega \ in \ Omega}$${\ displaystyle \ tau _ {1} (\ omega), \; \ tau _ {2} (\ omega) \ ldots}$
• For everyone , the stopped process is in .${\ displaystyle i \ in \ mathbb {N}}$${\ displaystyle X ^ {\ tau _ {i}}}$${\ displaystyle {\ mathcal {C}}}$

The localization of the set is now defined as the class of all processes that are local to the class . A sequence of stop times belonging to a local process (but not unique!) With the above properties is also referred to as a localizing sequence of . ${\ displaystyle {\ mathcal {C}} _ ​​{\ mathrm {loc}}}$${\ displaystyle {\ mathcal {C}}}$${\ displaystyle {\ mathcal {C}}}$${\ displaystyle X}$${\ displaystyle X}$

## properties

The mapping is not an envelope operator : it always applies (for every process the constant sequence fs can be chosen as the localizing sequence ), and the condition also applies, but generally does not apply , so the mapping is not idempotent . ${\ displaystyle {\ mathcal {C}} \ mapsto {\ mathcal {C}} _ ​​{\ mathrm {loc}}}$${\ displaystyle {\ mathcal {C}} \ subseteq {\ mathcal {C}} _ ​​{\ mathrm {loc}}}$${\ displaystyle X \ in {\ mathcal {C}}}$${\ displaystyle \ tau _ {n}: = \ infty}$ ${\ displaystyle {\ mathcal {C}} \ subseteq {\ mathcal {D}} \ Rightarrow {\ mathcal {C}} _ ​​{\ mathrm {loc}} \ subseteq {\ mathcal {D}} _ {\ mathrm { loc}}}$${\ displaystyle ({\ mathcal {C}} _ ​​{\ mathrm {loc}}) _ {\ mathrm {loc}} = {\ mathcal {C}} _ ​​{\ mathrm {loc}}}$

To a closure operator, the figure is only when one is limited to amounts of processes that are stable under Stop: A lot of stochastic processes is stable under Stop if for all and all stop times applies: . Then the above idempotency applies as well as the property${\ displaystyle {\ mathcal {C}}}$${\ displaystyle X \ in {\ mathcal {C}}}$${\ displaystyle \ tau}$${\ displaystyle X ^ {\ tau} \ in {\ mathcal {C}}}$${\ displaystyle ({\ mathcal {C}} \ cap {\ mathcal {D}}) {\ mathrm {loc}} = {\ mathcal {C}} _ ​​{\ mathrm {loc}} \ cap {\ mathcal {D}} _ {\ mathrm {loc}}}$

## literature

• Daniel Revuz, Marc Yor: Continuous Martingales and Brownian motion . Springer-Verlag, New York 1999, ISBN 978-3540643258 .