Progressively measurable process

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.

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 $[0, \infty)$ );
$B o r e l ([0, t])$ be the Borel sigma algebra on $[0, t]$ .

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 Itô 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

↑ ^1.0 ^1.1 ^1.2 ^1.3 ^1.4 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.

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Progressively measurable process

Definition

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References

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