Chapter 64: The fundamental theorm of martingales and the S-integral
The algebras Aτ-; the previsible version u- of a near-simple process; jumps and residual oscillations; integrating u-; previsible processes; the previsible σ-algebra.
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.
Extracting jumps from a non-decreasing process; ∥∫udv∥2 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)dv〉n∈N.
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.
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.
Changing the filtration
Simultaneously expanding every algebra in a filtration by a single element; controlling [[S-∫u dv≠0]].
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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