Recent progress and open questions on the numerical index of Banach spaces

The aim of this paper is to review the state-of-the-art of recent research concerning the numerical index of Banach spaces, by presenting some of the results found in the last years and proposing a number of related open problems.Comment: 27 pages, 4 figures, to appear in RACSA

Default times, no-arbitrage conditions and changes of probability measures

In this paper, we give a financial justification, based on no-arbitrage conditions, of the (H)-hypothesis in default time modeling. We also show how the (H)-hypothesis is affected by an equivalent change of probability measure. The main technique used here is the theory of progressive enlargements of filtration

Single Jump Processes and Strict Local Martingales

Many results in stochastic analysis and mathematical finance involve local martingales. However, specific examples of strict local martingales are rare and analytically often rather unhandy. We study local martingales that follow a given deterministic function up to a random time $\gamma$ at which they jump and stay constant afterwards. The (local) martingale properties of these single jump local martingales are characterised in terms of conditions on the input parameters. This classification allows an easy construction of strict local martingales, uniformly integrable martingales that are not in $H^1$, etc. As an application, we provide a construction of a (uniformly integrable) martingale $M$ and a bounded (deterministic) integrand $H$ such that the stochastic integral $H\bullet M$ is a strict local martingale.Comment: 21 pages; forthcoming in 'Stochastic Processes and their Applications

Postquantum Br\`{e}gman relative entropies and nonlinear resource theories

We introduce the family of postquantum Br\`{e}gman relative entropies, based on nonlinear embeddings into reflexive Banach spaces (with examples given by reflexive noncommutative Orlicz spaces over semi-finite W*-algebras, nonassociative L$_p$ spaces over semi-finite JBW-algebras, and noncommutative L$_p$ spaces over arbitrary W*-algebras). This allows us to define a class of geometric categories for nonlinear postquantum inference theory (providing an extension of Chencov's approach to foundations of statistical inference), with constrained maximisations of Br\`{e}gman relative entropies as morphisms and nonlinear images of closed convex sets as objects. Further generalisation to a framework for nonlinear convex operational theories is developed using a larger class of morphisms, determined by Br\`{e}gman nonexpansive operations (which provide a well-behaved family of Mielnik's nonlinear transmitters). As an application, we derive a range of nonlinear postquantum resource theories determined in terms of this class of operations.Comment: v2: several corrections and improvements, including an extension to the postquantum (generally) and JBW-algebraic (specifically) cases, a section on nonlinear resource theories, and more informative paper's titl

Diametral notions for elements of the unit ball of a Banach space

The first and third authors were supported by grant PID2021-122126NB-C31 funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU, by Junta de Andalucía I+D+i grants P20_00255 and FQM-185, and by “Maria de Maeztu” Excellence Unit IMAG (CEX2020-001105-M) funded by MICIU/AEI/10.13039/501100011033. The second named author was supported by the Estonian Research Council grant SJD58.We introduce extensions of Δ-points and Daugavet points in which slices are replaced by relatively weakly open subsets (super Δ-points and super Daugavet points) or by convex combinations of slices (ccs Δ-points and ccs Daugavet points). These notions represent the extreme opposite to denting points, points of continuity, and strongly regular points. We first give a general overview of these new concepts and provide some isometric consequences on the spaces. As examples: (1) If a Banach space contains a super Δ-point, then it does not admit an unconditional FDD (in particular, unconditional basis) with suppression constant smaller than 2. (2) If a real Banach space contains a ccs Δ-point, then it does not admit a one-unconditional basis. (3) If a Banach space contains a ccs Daugavet point, then every convex combination of slices of its unit ball has diameter 2. We next characterize the notions in some classes of Banach spaces, showing, for instance, that all the notions coincide in L1-predual spaces and that all the notions but ccs Daugavet points coincide in L1-spaces. We next comment on some examples which have previously appeared in the literature, and we provide some new intriguing examples: examples of super Δ-points which are as close as desired to strongly exposed points (hence failing to be Daugavet points in an extreme way); an example of a super Δ-point which is strongly regular (hence failing to be a ccs Δ-point in the strongest way); a super Daugavet point which fails to be a ccs Δ-point. The extensions of the diametral notions to points in the open unit ball and consequences on the spaces are also studied. Lastly, we investigate the Kuratowski measure of relatively weakly open subsets and of convex combinations of slices in the presence of super Δ-points or ccs Δ-points, as well as for spaces enjoying diameter-two properties. We conclude the paper with some open problems.MICIU/AEI/10.13039/501100011033 PID2021-122126NB-C31ERDF/EUJunta de Andalucía I+D+i P20_00255, FQM-185MICIU/AEI/10.13039/501100011033 “Maria de Maeztu” (CEX2020-001105-M

Representation formulas for pairings between divergence-measure fields and BV functions

The purpose of this paper is to find pointwise representation formulas for the density of the pairing between divergence-measure fields and BV functions, in this way continuing the research started in [17, 20]. In particular, we extend a representation formula from an unpublished paper of Anzellotti [7] involving the limit of cylindrical averages for normal traces, and we exploit a result of [35] in order to derive another representation in terms of limits of averages in half balls

Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints