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

The process X is said to be progressively measurable[2] (or simply progressive) if, for every time t, the map [0, t] \times \Omega \to \mathbb{X} defined by (s, \omega) \mapsto X_{s} (\omega) is \mathrm{Borel}([0, t]) \otimes \mathcal{F}_{t}-measurable. This implies that X is  \mathcal{F}_{t} -adapted.[1]

A subset P \subseteq [0, \infty) \times \Omega is said to be progressively measurable if the process X_{s} (\omega) := \chi_{P} (s, \omega) is progressively measurable in the sense defined above, where \chi_{P} is the indicator function of P. The set of all such subsets P form a sigma algebra on [0, \infty) \times \Omega, denoted by \mathrm{Prog}, and a process X is progressively measurable in the sense of the previous paragraph if, and only if, it is \mathrm{Prog}-measurable.

Properties[edit]

  • It can be shown[1] that L^2 (B), the space of stochastic processes X : [0, T] \times \Omega \to \mathbb{R}^n for which the Ito integral
\int_0^T X_t \, \mathrm{d} B_t
with respect to Brownian motion B is defined, is the set of equivalence classes of \mathrm{Prog}-measurable processes in L^2 ([0, T] \times \Omega; \mathbb{R}^n)\,.
  • 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]