Progressively measurable process (original) (raw)

From Wikipedia, the free encyclopedia

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.

Let

( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )} $(\Omega ,{\mathcal {F}},\mathbb {P} )$ be a probability space;
( X , A ) {\displaystyle (\mathbb {X} ,{\mathcal {A}})} $(\mathbb {X} ,{\mathcal {A}})$ be a measurable space, the state space;
{ F t ∣ t ≥ 0 } {\displaystyle \{{\mathcal {F}}_{t}\mid t\geq 0\}} $\{{\mathcal {F}}_{t}\mid t\geq 0\}$ be a filtration of the sigma algebra F {\displaystyle {\mathcal {F}}} ${\mathcal {F}}$ ;
X : [ 0 , ∞ ) × Ω → X {\displaystyle X:[0,\infty )\times \Omega \to \mathbb {X} } ![{\displaystyle X:0,\infty )\times \Omega \to \mathbb {X} } be a stochastic process (the index set could be [ 0 , T ] {\displaystyle [0,T]} $[0,T]$ or N 0 {\displaystyle \mathbb {N} _{0}} $\mathbb {N} _{0}$ instead of [ 0 , ∞ ) {\displaystyle [0,\infty )} ![{\displaystyle 0,\infty )});
B o r e l ( [ 0 , t ] ) {\displaystyle \mathrm {Borel} ([0,t])} $\mathrm {Borel} ([0,t])$ be the Borel sigma algebra on [ 0 , t ] {\displaystyle [0,t]} $[0,t]$ .

The process X {\displaystyle X} $X$ is said to be progressively measurable[2] (or simply progressive) if, for every time t {\displaystyle t} $t$ , the map [ 0 , t ] × Ω → X {\displaystyle [0,t]\times \Omega \to \mathbb {X} } $[0,t]\times \Omega \to \mathbb {X}$ defined by ( s , ω ) ↦ X s ( ω ) {\displaystyle (s,\omega )\mapsto X_{s}(\omega )} $(s,\omega )\mapsto X_{s}(\omega )$ is B o r e l ( [ 0 , t ] ) ⊗ F t {\displaystyle \mathrm {Borel} ([0,t])\otimes {\mathcal {F}}_{t}} $\mathrm {Borel} ([0,t])\otimes {\mathcal {F}}_{t}$ -measurable. This implies that X {\displaystyle X} $X$ is F t {\displaystyle {\mathcal {F}}_{t}} ${\mathcal {F}}_{t}$ -adapted.[1]

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

∫ 0 T X t d B t {\displaystyle \int _{0}^{T}X_{t}\,\mathrm {d} B_{t}} $\int _{0}^{T}X_{t}\,\mathrm {d} B_{t}$

with respect to Brownian motion B {\displaystyle B} $B$ is defined, is the set of equivalence classes of P r o g {\displaystyle \mathrm {Prog} } $\mathrm {Prog}$ -measurable processes in L 2 ( [ 0 , T ] × Ω ; R n ) {\displaystyle L^{2}([0,T]\times \Omega ;\mathbb {R} ^{n})} $L^{2}([0,T]\times \Omega ;\mathbb {R} ^{n})$ .

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]

^ 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.
^ 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.