Contents of Measure Theory, by D.H.Fremlin

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

641 Previsible processes
The algebras Aτ-; 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
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
Extracting jumps from a non-decreasing process; ∥∫udv2 and ∥∫u2dv*1 when v is a martingale; a weak topology on Mn-s; order*-convergence of ⟨u(n)-n∈N implies convergence of ⟨∫u(n)dvn∈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; 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 dv≠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.

TeX, PDF, ro-PDF (results-only version).
Return to contents page.

Revised 25.12.14