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

→ Main article : Stopped process

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 ${\displaystyle X=(X_{t})_{t\in T}}$ ${\displaystyle (\Omega ,{\mathcal {A}},({\mathcal {F}})_{t\in T},P)}$ ${\displaystyle T=\mathbb {N} _{0}}$ ${\displaystyle T=\mathbb {R} _{0}^{+}}$ ${\displaystyle \tau }$

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

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 .