Localization (stochastics)

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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

than when the process is stopped . The process thus agrees with the process up to this point , but then remains at its current value and no longer changes its state.

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:

  • It is almost certainly true for , ie for almost all the (deterministic) sequence diverges towards plus infinity .
  • For everyone , the stopped process is in .

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 .


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 .

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


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