Stopping Time - Localization

Localization

Stopping times are frequently used to generalize certain properties of stochastic processes to situations in which the required property is satisfied in only a local sense. First, if X is a process and is a stopping time, then X is used to denote the process X stopped at time .

Then, X is said to locally satisfy some property P if there exists a sequence of stopping times _n, which increases to infinity and for which the processes satisfy property P. Common examples, with time index set I = [0,∞), are as follows;

• (Local martingale) A process X is a local martingale if it is càdlàg and there exists a sequence of stopping times _n increasing to infinity, such that is a martingale for each n.
• (Locally integrable) A non-negative and increasing process X is locally integrable if there exists a sequence of stopping times _n increasing to infinity, such that for each n.