Progressively measurable process

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In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process.[1] Progressively measurable processes are important in the theory of Itô integrals.

Definition

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Let

  •   be a probability space;
  •   be a measurable space, the state space;
  •   be a filtration of the sigma algebra  ;
  •   be a stochastic process (the index set could be   or   instead of  );
  •   be the Borel sigma algebra on  .

The process   is said to be progressively measurable[2] (or simply progressive) if, for every time  , the map   defined by   is  -measurable. This implies that   is  -adapted.[1]

A subset   is said to be progressively measurable if the process   is progressively measurable in the sense defined above, where   is the indicator function of  . The set of all such subsets   form a sigma algebra on  , denoted by  , and a process   is progressively measurable in the sense of the previous paragraph if, and only if, it is  -measurable.

Properties

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  • It can be shown[1] that  , the space of stochastic processes   for which the Itô integral
 
with respect to Brownian motion   is defined, is the set of equivalence classes of  -measurable processes in  .
  • Every adapted process with left- or right-continuous paths is progressively measurable. Consequently, every adapted process with càdlàg paths is progressively measurable.[1]
  • Every measurable and adapted process has a progressively measurable modification.[1]

References

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  1. ^ a b c d e Karatzas, Ioannis; Shreve, Steven (1991). Brownian Motion and Stochastic Calculus (2nd ed.). Springer. pp. 4–5. ISBN 0-387-97655-8.
  2. ^ Pascucci, Andrea (2011). "Continuous-time stochastic processes". PDE and Martingale Methods in Option Pricing. Bocconi & Springer Series. Springer. p. 110. doi:10.1007/978-88-470-1781-8. ISBN 978-88-470-1780-1. S2CID 118113178.