Chapter 64: The fundamental theorm of martingales and the S-integral
641
Previsible versions
The algebras Aτ-; the previsible version u<
of a moderately oscillatory process; jumps and residual
oscillations; the previsible version of an indefinite integral;
quadratic variations; the previsible version of a previsible version;
integrating a previsible version.
642
Previsible processes
Previsible processes; and previsible versions; previsible
σ-algebras; previsibly measurable processes; and
order*-convergence.
643
The fundamental theorem of martingales
uτ- and the conditional expectation Pτ-; 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
A kind of sequential smoothness for the Riemann-sum integral; a weak
topology on Mn-s; sufficient conditions to ensure that
order*-convergence of ⟨un<⟩n∈N implies convergence
of ⟨∫undv⟩n∈N.
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 u-;
S-integration is uniformly continuous on uniformly previsibly bounded
sets; a dominated convergence theorem; previsible, previsibly
order-bounded processes are S-integrable.
646
Basic properties of the S-integral
Splitting a lattice; indefinite S-integrals; martingales and
uniformly integrable capped-stake variation sets; change of
variable; Itô's formula again.
647
Changing the filtration II
Simultaneously expanding every algebra in a filtration by a single
element; controlling
[[S-∫u dv≠0]].
649
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.
648
Changing the algebra II
Subalgebras, induced stochastic integration structures and
S-integrals.
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