Progressively measurable process

Key takeaways

Progressively measurable process belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Progressively measurable process to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Progressively measurable process from memory before moving on to harder problems.

Reference excerpt

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}}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Progressively measurable process

Start with the simplest possible case. Write down what Progressively measurable process claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Progressively measurable process before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Progressively measurable process ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Progressively measurable process

In research: Progressively measurable process appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Progressively measurable process in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Progressively measurable process is common in secondary-school and first-year university syllabi. It links to neighbouring topics Measure theory, Stochastic processes, so understanding it makes those chapters shorter.
In everyday life: Look for Progressively measurable process outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Progressively measurable process in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Progressively measurable process means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Progressively measurable process out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Progressively measurable process in simple terms?

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.

Why does Progressively measurable process matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Progressively measurable process?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Progressively measurable process.