Chapter 63: Back to work
|dv| and dv↑; near-simple and jump-free processes of bounded variation; indefinite integrals; quadratic variation; the jump-free part of a process of bounded variation.
Changing the measure
Radon-Nikodým derivatives and conditional expectations; semimartingales stay semimartingales under change of law.
Martingales and submartingales
Martingales, local martingales and uniform integrability; interval functions PΔv; submartingales; the Doob-Meyer theorem and previsible variations.
Integrators and semimartingales
Supermartingales, quasi-martingales and strong integrators; decomposition of quasi-martingales into differences of supermartingales; convex sets in L1μ which are topologically bounded in L0; changing law to make an integrator a strong integrator (the Bichteler-Dellacherie theorem).
The ucp topology
The ucp topology on the space Mob of order-bounded processes; continuity of basic operations; completeness; the spaces of near-simple and jump-free processes; indefinite integration with respect to a near-simple integrator is a continuous linear operator; if both integrator and integrand are near-simple, the indefinite integral is near-simple; if the integrand is near-simple and the integrator is jump-free, the indefinite integral is jump-free; change of variable in an integral ∫z dvdv'.
Coverings and full lattices
Covering sets and sublattices; extension of processes defined on covering sublattices; properties of such extensions; the ucp topology for full lattices; quadratic variations of jump-free processes.
Separating sublattices; cases when ∫Su dv=∫S′u dv; using a measure-converging filter for path-by-path calculation of a stochastic integral; connections between properties of a process and its restriction to a separating sublattice; a general extension method for near-simple processes.
Changing the algebra
Subalgebras and restricted filtrations and stopping time lattices; fully adapted processes and absolute properties; càdlàg processes and their extensions; relative independence and subalgebras coordinated with filtrations; martingales; stochastic independence and free products.
Changing the filtration
The filtration defined from a totally ordered family of stopping times; the corresponding transformation of fully adapted processes; integration by substitution.
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