Chapter 61: The Riemann-sum integral
611
Stopping times
Filtrations; the lattice T of stopping times; the regions
[[σ<τ]]; suprema and infima in T; the algebra
defined by a stopping time; stopping time intervals; enumerating the
cells of a finite sublattice of T; covered envelopes and
covering ideals.
612
Fully adapted processes
L0 spaces; f-algebras of fully adapted processes; the identity
process; progressively measurable classical processes; actions of
Borel measurable real functions; simple processes; order-bounded
processes; extension of processes to covered envelopes; Brownian
motion; the Poisson process.
613
Definition of the integral
Convergence in measure; interval functions;
Δe(u,dψ), Riemann sums SI(u,dψ),
integrals ∫Su dψ and
∫Su dv; invariance under change of law;
integrating a simple process; indefinite integrals; integration and
covered envelopes.
614
Simple and order-processes and bounded variation
Integration of simple processes; order-bounded processes; processes
of bounded variation; cumulative variations; sample paths of
bounded variation.
615
Moderately oscillatory processes
The ucp topology on Mob; simple processes; moderately oscillatory
processes; the structure of moderately oscillatory processes;
càdlàg sample paths.
616
Integrating interval functions
Capped-stake variation sets; integrating interval functions and
integrators; integration and the ucp topology; integrators are
moderately oscillatory; indefinite integrals; if u is
moderately oscillatory and ψ is an integrating interval
function, v=iiψ(1) is an integrator and
∫u dψ=∫u dv is defined; processes of
bounded variation are integrators.
617
Integral identities and quadratic variations
Approximating moderately oscillatory processes by simple processes;
continuity of indefinite integration; integrating with respect to an
indefinite integral; integrating with respect to a cumulative
variation; covariation and quadratic variation; integrating with
respect to a covariation; the quadratic variations of the identity
process and the Poisson process; change of variable in an integral
∫u dvdv'; the quadratic variation of an
indefinite integral; the quadratic variation of a cumulative
variation.
618
Jump-free processes
Oscillations; jump-free processes; classical processes with
continous sample paths; moderately oscillatory processes;
indefinite integrals, covariations, cumulative variations.
619
Itô's formula
Itô's formula in one dimension; k-tuples of processes; Itô's
formula in k dimensions.
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