Measuring risk with multiple eligible assets

The risk of financial positions is measured by the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate nondegeneracy, finiteness and continuity properties of these risk measures with respect to multiple eligible assets. Our finiteness and continuity results highlight the interplay between the acceptance set and the class of eligible portfolios. We present a simple, alternative approach to the dual representation of convex risk measures by directly applying to the acceptance set the external characterization of closed, convex sets. We prove that risk measures are nondegenerate if and only if the pricing functional admits a positive extension which is a supporting functional for the underlying acceptance set, and provide a characterization of when such extensions exist. Finally, we discuss applications to set-valued risk measures, superhedging with shortfall risk, and optimal risk sharing

Beyond cash-additive risk measures: when changing the num\'{e}raire fails

We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified eligible asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a risk-free bond, on the grounds that the general case can be reduced to the cash-additive case by a change of numeraire. However, discounting does not work in all financially relevant situations, typically when the eligible asset is a defaultable bond. In this paper we fill this gap allowing for general eligible assets. We provide a variety of finiteness and continuity results for the corresponding risk measures and apply them to risk measures based on Value-at-Risk and Tail Value-at-Risk on $L^p$ spaces, as well as to shortfall risk measures on Orlicz spaces. We pay special attention to the property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. For important examples, we provide characterizations of cash subadditivity and show that, when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasiconvex and show that cash subadditivity is only compatible with continuous pricing rules

Feller semigroups, Lp-sub-Markovian semigroups, and applications to pseudo-differential operators with negative definite symbols

The question of extending L-p-sub-Markovian semigroups to the spaces L-q, q > P, and the interpolation of LP-sub-Markovian semigroups with Feller semigroups is investigated. The structure of generators of L-p-sub-Markovian semigroups is studied. Subordination in the sense of Bochner is used to discuss the construction of refinements of L-p-sub-Markovian semigroups. The role played by some function spaces which are domains of definition for L-p-generators is pointed out. The problem of regularising powers of generators as well as some perturbation results are discussed

The Emma Family Backyard Deck – Thousand Oaks, CA

This paper outlines the planning, design, and construction of an 80 square foot deck/outdoor patio. The main purpose of this project was to assist an elder couple fund and construct a new deck in their backyard. Due to the current world pandemic, COVID-19 the Emma Family was practicing social distancing and adhering to the State and County wide stay at home order. The project plan was to design a deck for Emma family in the design, construction it and receive funding. The deck was a rectangular 8’ x 10’ and raised 6” off the ground. In addition to the deck the project also consisted of a removable housing unit for the main sprinkler and water line. This paper will focus on the preconstruction phase, construction phase, lessons learned, and how this knowledge will be applied to the construction industry

Characterisations of function spaces of generalised smoothness

We investigate function spaces of generalised smoothness of Besov and Triebel-Lizorkin type. Equivalent quasi-norms in terms of maximal functions and local means are given. An atomic decomposition theorem for this type of spaces is prove

Anlegen mit KI - Herausforderung und Chance

Künstliche Intelligenz hat die Effizienz der Datenanalyse revolutioniert und bietet die Möglichkeit, das Investment Research und Portfoliomanagement zu automatisieren. Ihr Einsatz ist aber kein uneingeschränkter Erfolgsgarant