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 Itō integrals.

Definition[edit]

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 ).

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[edit]

  • It can be shown[1] that , the space of stochastic processes for which the Ito 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[edit]

  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) PDE and Martingale Methods in Option Pricing. Berlin: Springer[page needed]