Chapter 64: The fundamental theorm of martingales and the S-integral

641
Previsible processes

The algebras **A**_{τ-}; the previsible version *u*_{-}
of a near-simple process; jumps and residual oscillations;
integrating *u*_{-}; previsible processes; the previsible
σ-algebra.

643
The fundamental theorem of martingales

*u*_{τ-} and the conditional expectation *P*_{τ-}; the
region of accessibility of a stopping time; previsible variations;
assembling processes from components on stopping-time intervals;
sublattices with countable cofinality are adequate; the fundamental
theorem.

644
Pointwise convergence

Extracting jumps from a non-decreasing process;
∥∫* udv*∥

645
Construction of the S-integral

Previsibly order-bounded processes; the S-integration topology;
S-integrable processes; definition of the S-integral; the
Riemann-sum integral of ** u** is the S-integral of

646
Basic properties of the S-integral

Splitting a lattice; martingales and uniformly integrable
capped-stake variation sets; indefinite S-integrals; change of
variable; Itô's formula again.

647
Changing the filtration

Simultaneously expanding every algebra in a filtration by a single
element; controlling
[[S-∫*u**d*** v**≠0]].

648
Pathwise integration

Calculating a Riemann-sum integral one path at a time (i) by
Bichteler's construction (ii) with a measure-converging filter; the
S-integral for non-decreasing integrators.

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Revised 25.12.14