Level set (data structures): Difference between revisions

Revision as of 10:35, 18 December 2012

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

Let

$(Ω, ℱ, ℙ)$ be a probability space;
$(𝕏, 𝒜)$ be a measurable space, the state space;
${ℱ_{t} | t \geq 0}$ be a filtration of the sigma algebra $ℱ$ ;
$X : [0, \infty) \times Ω \to 𝕏$ be a stochastic process (the index set could be $[0, T]$ or $ℕ_{0}$ instead of $[0, \infty)$ ).

The process $X$ is said to be progressively measurable^[2] (or simply progressive) if, for every time $t$ , the map $[0, t] \times Ω \to 𝕏$ defined by $(s, ω) \mapsto X_{s} (ω)$ is $B o r e l ([0, t]) \otimes ℱ_{t}$ -measurable. This implies that $X$ is $ℱ_{t}$ -adapted.^[1]

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

Properties

It can be shown^[1] that $L^{2} (B)$ , the space of stochastic processes $X : [0, T] \times Ω \to ℝ^{n}$ for which the Ito integral

\int_{0}^{T} X_{t} d B_{t}

with respect to Brownian motion

B

is defined, is the set of equivalence classes of

P r o g

-measurable processes in

L^{2} ([0, T] \times Ω; 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

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Template:Probability-stub Template:Mathanalysis-stub

↑ ^1.0 ^1.1 ^1.2 ^1.3 ^1.4 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ Pascucci, Andrea (2011) PDE and Martingale Methods in Option Pricing. Berlin: Springer Template:Page needed

[Karatzas-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[Pasc-2] Pascucci, Andrea (2011) PDE and Martingale Methods in Option Pricing. Berlin: Springer Template:Page needed

[1]

[2]

@@ Line 1: / Line 1: @@
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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]].<ref name="Karatzas">{{cite book|last=Karatzas|first=Ioannis|last2=Shreve|first2=Steven|year=1991|title=Brownian Motion and Stochastic Calculus|publisher=Springer|edition=2nd|isbn=0-387-97655-8|pages=4–5}}</ref> Progressively measurable processes are important in the theory of [[Itō integral]]s.
+==Definition==
+Let
+* <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a [[probability space]];
+* <math>(\mathbb{X}, \mathcal{A})</math> be a [[measurable space]], the ''state space'';
+* <math>\{ \mathcal{F}_{t} | t \geq 0 \}</math> be a [[Filtration_(abstract_algebra)|filtration]] of the [[sigma algebra]] <math>\mathcal{F}</math>;
+* <math>X : [0, \infty) \times \Omega \to \mathbb{X}</math> be a [[stochastic process]] (the index set could be <math>[0, T]</math> or <math>\mathbb{N}_{0}</math> instead of <math>[0, \infty)</math>).
+The process <math>X</math> is said to be '''progressively measurable'''<ref name=Pasc>Pascucci, Andrea (2011) ''PDE and Martingale Methods in Option Pricing''. Berlin: Springer {{Page needed|date=August 2011}}</ref>  (or simply '''progressive''') if, for every time <math>t</math>, the map <math>[0, t] \times \Omega \to \mathbb{X}</math> defined by <math>(s, \omega) \mapsto X_{s} (\omega)</math> is <math>\mathrm{Borel}([0, t]) \otimes \mathcal{F}_{t}</math>-[[Measurable_function|measurable]]. This implies that <math>X</math> is <math> \mathcal{F}_{t} </math>-adapted.<ref name="Karatzas" />
+A subset <math>P \subseteq [0, \infty) \times \Omega</math> is said to be '''progressively measurable''' if the process <math>X_{s} (\omega) := \chi_{P} (s, \omega)</math> is progressively measurable in the sense defined above, where <math>\chi_{P}</math> is the [[indicator function]] of <math>P</math>. The set of all such subsets <math>P</math> form a sigma algebra on <math>[0, \infty) \times \Omega</math>, denoted by <math>\mathrm{Prog}</math>, and a process <math>X</math> is progressively measurable in the sense of the previous paragraph if, and only if, it is <math>\mathrm{Prog}</math>-measurable.
+==Properties==
+* It can be shown<ref name="Karatzas" /> that <math>L^2 (B)</math>, the space of stochastic processes <math>X : [0, T] \times \Omega \to \mathbb{R}^n</math> for which the [[Ito integral]]
+:: <math>\int_0^T X_t \, \mathrm{d} B_t </math>
+: with respect to [[Brownian motion]] <math>B</math> is defined, is the set of [[equivalence class]]es of <math>\mathrm{Prog}</math>-measurable processes in <math>L^2 ([0, T] \times \Omega; \mathbb{R}^n)\,</math>.
+* Every adapted process with left- or [[Continuous function#Directional continuity|right-continuous]] paths is progressively measurable. Consequently, every adapted process with [[càdlàg]] paths is progressively measurable.<ref name="Karatzas" />
+* Every measurable and adapted process has a progressively measurable modification.<ref name="Karatzas" />
+==References==
+{{reflist}}
+[[Category:Stochastic processes]]
+[[Category:Measure theory]]
+{{probability-stub}}
+{{mathanalysis-stub}}

Level set (data structures): Difference between revisions

Revision as of 10:35, 18 December 2012

Definition

Properties

References

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