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.

Contents

Definition

Let

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

with respect to Brownian motion is defined, is the set of equivalence classes of -measurable processes in .

References

  1. 1 2 3 4 5 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.