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
Let
- [math]\displaystyle{ (\Omega, \mathcal{F}, \mathbb{P}) }[/math] be a probability space;
- [math]\displaystyle{ (\mathbb{X}, \mathcal{A}) }[/math] be a measurable space, the state space;
- [math]\displaystyle{ \{ \mathcal{F}_{t} \mid t \geq 0 \} }[/math] be a filtration of the sigma algebra [math]\displaystyle{ \mathcal{F} }[/math];
- [math]\displaystyle{ X : [0, \infty) \times \Omega \to \mathbb{X} }[/math] be a stochastic process (the index set could be [math]\displaystyle{ [0, T] }[/math] or [math]\displaystyle{ \mathbb{N}_{0} }[/math] instead of [math]\displaystyle{ [0, \infty) }[/math]);
- [math]\displaystyle{ \mathrm{Borel}([0, t]) }[/math] be the Borel sigma algebra on [math]\displaystyle{ [0,t] }[/math].
The process [math]\displaystyle{ X }[/math] is said to be progressively measurable[2] (or simply progressive) if, for every time [math]\displaystyle{ t }[/math], the map [math]\displaystyle{ [0, t] \times \Omega \to \mathbb{X} }[/math] defined by [math]\displaystyle{ (s, \omega) \mapsto X_{s} (\omega) }[/math] is [math]\displaystyle{ \mathrm{Borel}([0, t]) \otimes \mathcal{F}_{t} }[/math]-measurable. This implies that [math]\displaystyle{ X }[/math] is [math]\displaystyle{ \mathcal{F}_{t} }[/math]-adapted.[1]
A subset [math]\displaystyle{ P \subseteq [0, \infty) \times \Omega }[/math] is said to be progressively measurable if the process [math]\displaystyle{ X_{s} (\omega) := \chi_{P} (s, \omega) }[/math] is progressively measurable in the sense defined above, where [math]\displaystyle{ \chi_{P} }[/math] is the indicator function of [math]\displaystyle{ P }[/math]. The set of all such subsets [math]\displaystyle{ P }[/math] form a sigma algebra on [math]\displaystyle{ [0, \infty) \times \Omega }[/math], denoted by [math]\displaystyle{ \mathrm{Prog} }[/math], and a process [math]\displaystyle{ X }[/math] is progressively measurable in the sense of the previous paragraph if, and only if, it is [math]\displaystyle{ \mathrm{Prog} }[/math]-measurable.
Properties
- It can be shown[1] that [math]\displaystyle{ L^2 (B) }[/math], the space of stochastic processes [math]\displaystyle{ X : [0, T] \times \Omega \to \mathbb{R}^n }[/math] for which the Itô integral
- [math]\displaystyle{ \int_0^T X_t \, \mathrm{d} B_t }[/math]
- with respect to Brownian motion [math]\displaystyle{ B }[/math] is defined, is the set of equivalence classes of [math]\displaystyle{ \mathrm{Prog} }[/math]-measurable processes in [math]\displaystyle{ L^2 ([0, T] \times \Omega; \mathbb{R}^n) }[/math].
- 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
- ↑ 1.0 1.1 1.2 1.3 1.4 Karatzas, Ioannis; Shreve, Steven (1991). Brownian Motion and Stochastic Calculus (2nd ed.). Springer. pp. 4–5. ISBN 0-387-97655-8.
- ↑ 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.
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