Enlargements, semiabundancy and unipotent monoids1

Enlargements, Semiabundancy and Unipotent Monoids1

The relation R̃ on a monoid S provides a natural generalisation of Green’s relation R. If every R̃-class of S contains an idempotent, S is left semiabundant; if R̃ is a left congruence then S satisfies (CL). Regular monoids, indeed left abundant monoids, are left semiabundant and satisfy (CL). However, the class of left semiabundant monoids is much larger, as we illustrate with a number of example...

Enlargements of Filtrations

Enlargements of Filtrations and Applications

In this paper we review some old and new results about the enlargement of filtrations problem, as well as their applications to credit risk and insider trading problems. The enlargement of filtrations problem consists in the study of conditions under which a semimartingale remains a semimartingale when the filtration is enlarged, and, in such a case, how to find the Doob-Meyer decomposition. Fi...

Monotone Operators without Enlargements

Enlargements have proven to be useful tools for studying maximally monotone mappings. It is therefore natural to ask in which cases the enlargement does not change the original mapping. Svaiter has recently characterized non-enlargeable operators in reflexive Banach spaces and has also given some partial results in the nonreflexive case. In the present paper, we provide another characterization...

Enlargements of Categories

In order to apply nonstandard methods to modern algebraic geometry, as a first step in this paper we study the applications of nonstandard constructions to category theory. It turns out that many categorical properties are well behaved under enlargements.