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. Progressively measurable processes are important in the theory of Itô integrals.

Definition

Let

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

The process X {\displaystyle X} is said to be progressively measurable (or simply progressive) if, for every time t {\displaystyle t}, the map [ 0 , t ] × Ω → X {\displaystyle [0,t]\times \Omega \to \mathbb {X} } defined by ( s , ω ) ↦ X s ( ω ) {\displaystyle (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}}-measurable. This implies that X {\displaystyle X} is F t {\displaystyle {\mathcal {F}}_{t}}-adapted.

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

Properties

  • It can be shown that L 2 ( B ) {\displaystyle L^{2}(B)}, the space of stochastic processes X : [ 0 , T ] × Ω → R n {\displaystyle X:[0,T]\times \Omega \to \mathbb {R} ^{n}} for which the Itô integral

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

with respect to Brownian motion B {\displaystyle B} is defined, is the set of equivalence classes of P r o g {\displaystyle \mathrm {Prog} }-measurable processes in L 2 ( [ 0 , T ] × Ω ; R n ) {\displaystyle 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.
  • Every measurable and adapted process has a progressively measurable modification.