In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, progressive measurability is a property of stochastic processes. A progressively measurable process is one for which events defined in terms of values of the process across a range of times can be assigned probabilities (that is, are measurable). Being progressively measurable is a strictly stronger property than the notion of being an adapted process

Adapted process

In the study of stochastic processes, an adapted process is one that cannot "see into the future". An informal interpretation is that X is adapted if and only if, for every realisation and every n, Xn is known at time n...

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

Definition

Let

• be a probability space
  Probability space
  In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
  
  ;
• be a measurable space, the state space;
• be a filtration
  Filtration (abstract algebra)
  In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj...
  
   of the sigma algebra ;
• be a stochastic process
  Stochastic process
  In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
  
   (the index set could be or instead of ).

The process is said to be progressively measurable (or simply progressive) if, for every time , the map defined by is -measurable

Measurable function

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...

. This implies that is -adapted.

A subset is said to be progressively measurable if the process is progressively measurable in the sense defined above. The set of all such subsets form a sigma algebra on , denoted by , and a process is progressively measurable in the sense of the previous paragraph if, and only if, it is -measurable.

Properties

• It can be shown that , the space of stochastic processes for which the Ito integral

with respect to Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

  is defined, is the set of equivalence classes of -measurable processes in .

• Every adapted process with left- or right-continuous paths is progressively measurable. Consequently, every adapted process with càdlàg
  Càdlàg
  In mathematics, a càdlàg , RCLL , or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere...
  
  paths is progressively measurable.
• Every measurable and adapted process has a progressively measurable modification.