Constructing Sublinear Expectations on Path Space

We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G-expectation, and an optional sampling theorem that holds without exceptional set. Our results also shed light on the inherent limitations to constructing sublinear expectations through aggregation.Comment: 28 pages; forthcoming in 'Stochastic Processes and their Applications

An introduction to quantum filtering

This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum It\^o calculus and its use in the modelling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.Comment: 41 pages, 1 figur

Cartan subalgebras in C*-algebras of Hausdorff etale groupoids

The reduced $C^*$-algebra of the interior of the isotropy in any Hausdorff \'etale groupoid $G$ embeds as a $C^*$-subalgebra $M$ of the reduced $C^*$-algebra of $G$. We prove that the set of pure states of $M$ with unique extension is dense, and deduce that any representation of the reduced $C^*$-algebra of $G$ that is injective on $M$ is faithful. We prove that there is a conditional expectation from the reduced $C^*$-algebra of $G$ onto $M$ if and only if the interior of the isotropy in $G$ is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, $M$ is a Cartan subalgebra. We prove that for a large class of groupoids $G$ with abelian isotropy---including all Deaconu--Renault groupoids associated to discrete abelian groups---$M$ is a maximal abelian subalgebra. In the specific case of $k$-graph groupoids, we deduce that $M$ is always maximal abelian, but show by example that it is not always Cartan.Comment: 14 pages. v2: Theorem 3.1 in v1 incorrect (thanks to A. Kumjain for pointing out the error); v2 shows there is a conditional expectation onto $M$ iff the interior of the isotropy is closed. v3: Material (including some theorem statements) rearranged and shortened. Lemma~3.5 of v2 removed. This version published in Integral Equations and Operator Theor