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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[edit]

Let

$(\Omega ,{\mathcal {F}},\mathbb {P} )$ be a probability space;
$(\mathbb {X} ,{\mathcal {A}})$ be a measurable space, the state space;
$\{{\mathcal {F}}_{t}\mid t\geq 0\}$ be a filtration of the sigma algebra ${\mathcal {F}}$ ;
$X:[0,\infty )\times \Omega \to \mathbb {X}$ be a stochastic process (the index set could be $[0,T]$ or $\mathbb {N} _{0}$ instead of $[0,\infty )$ );
$\mathrm {Borel} ([0,t])$ 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 $[0,t]\times \Omega \to \mathbb {X}$ defined by $(s,\omega )\mapsto X_{s}(\omega )$ is $\mathrm {Borel} ([0,t])\otimes {\mathcal {F}}_{t}$ -measurable. This implies that $X$ is ${\mathcal {F}}_{t}$ -adapted.^[1]

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

Properties[edit]

It can be shown^[1] that $L^{2}(B)$ , the space of stochastic processes $X:[0,T]\times \Omega \to \mathbb {R} ^{n}$ for which the Itô integral

\int _{0}^{T}X_{t}\,\mathrm {d} B_{t}

with respect to Brownian motion

B

is defined, is the set of equivalence classes of

\mathrm {Prog}

-measurable processes in

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]

References[edit]

^ ^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.

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise-deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Karhunen–Loève theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
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