Walter Farkas

Atomic and Subatomic Decompositions in Anisotropic Function Spaces

This work deals with decompositions in anisotropic function spaces. Defining anisotropic atoms as smooth building blocks which are the counterpart of the atoms from the works of M. Frazier and B. Jawerth, it is shown that the study of anisotropic function spaces can be done with the help of some sequence spaces in a similar way as it is done in the isotropic case. It is also shown that the subatomic decomposition theorem for isotropic function spaces, recently proved by H. Triebel, can be extended to the anisotropic case if the mean smoothness parameter is sufficiently large.

Sobolev Spaces on Non Smooth Domains and Dirichlet Forms Related to Subordinate Reflecting Diffusions

Let Ω be a bounded domain with fractal boundary, for instance von Kochs snowflake domain. First we determine the range and the kernel of the trace on ∂Ω of Sobolev spaces of fractional order defined on Ω. This extends some earlier results of H. Wallin and J. Marschall Secondly we apply these results in studying Dirichlet forms related to subordinate reflecting diffusions in non–smooth domains.

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.

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

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

Local Growth Envelopes of Besov Spaces of Generalized Smoothness

The concept of local growth envelope (ELGA, u) of the quasi-normed function space A is applied to the Besov spaces of generalized smoothness B p,q (Rn).

Beyond Cash-Additive Risk Measures: When Changing the Numeraire 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.

On the Hartree-Fock Equations of the Electron-Positron Field

We study the energy of relativistic electrons and positrons interacting via the second quantized Coulomb potential in the field of a nucleus of charge Z within the Hartree-Fock approximation. We show that the associated functional has a minimizer. In addition, all minimizers are purely electronic states, they are projections, and fulfill the no-pair Dirac-Fock equations.

Capital Requirements with Defaultable Securities

We study capital requirements for bounded financial positions defined as the minimum amount of capital to invest in a chosen eligible asset targeting a pre-specified acceptability test. We allow for general acceptance sets and general eligible assets, including defaultable bonds. Since the payoff of these assets is not necessarily bounded away from zero, the resulting risk measures cannot be transformed into cash-additive risk measures by a change of numeraire. However, extending the range of eligible assets is important because, as exemplified by the recent financial crisis, assuming the existence of default-free bonds may be unrealistic. We focus on finiteness and continuity properties of these general risk measures. As an application, we discuss capital requirements based on Value-at-Risk and Tail-Value-at-Risk acceptability, the two most important acceptability criteria in practice. Finally, we prove that there is no optimal choice of the eligible asset. Our results and our examples show that a theory of capital requirements allowing for general eligible assets is richer than the standard theory of cash-additive risk measures.

Local Volatility of Volatility for the VIX Market

Following a trend of sustained and accelerated growth, the VIX futures and options market has become a closely followed, active and liquid market. The standard stochastic volatility models -- which focus on the modeling of instantaneous variance -- are unable to fit the entire term structure of VIX futures as well as the entire VIX options surface. In contrast, we propose to model directly the VIX index, in a mean-reverting local volatility-of-volatility model, which will provide a global fit to the VIX market. We then show how to construct the local volatility-of-volatility surface by adapting the ideas in Carr (2008) and Andreasen, Huge (2010) to a mean-reverting process.

The Distribution of Eigenfrequencies of Anisotropic Fractal Drums

Let Γ be an anisotropic fractal. The aim of the paper is to investigate the distribution of the eigenvalues of the fractal differential operator formula here acting in the classical Sobolev space W˚ 1 2 (Ω) where Ω is a bounded C ∞ domain in the plane ℝ 2 with Γ⊂Ω. Here −Δ is the Dirichlet Laplacian in Ω and tr Γ is closely related to the trace operator tr Γ .