Study fails C-property in Orlicz spaces and proves w∗-representation for risk measures.
problem Identifying Orlicz spaces where the C-property fails and proving w∗-representation for risk measures. method Identifying Orlicz spaces, proving C-property variants, and using Delbaen's Fatou property.
result Established a variant of the C-property and proved w∗-representation for risk measures. Extends risk measure theory to general Orlicz spaces.
problem Applying risk measure theory to non-standard spaces.
method Generalizes results from bounded random variables to general Orlicz spaces, proving new characterizations and extensions.
result Characterizations and extensions of the Fatou property and Kusuoka representation in Orlicz spaces.
The strong Fatou property is crucial for risk measures' dual representations.
problem Ensuring nice dual representations of risk measures.
method Exploring Fatou-type properties and inf-convolutions of law-invariant or surplus-invariant risk measures.
result Every quasiconvex law-invariant functional on a rearrangement invariant space with the strong Fatou property is σ(X, L∞)-lower semicontinuous.
The paper characterizes risk measures with the Fatou property in function spaces.
problem Investigating the Fatou property of law-invariant risk measures in function spaces.
method Characterization of the Fatou property using the AOCEA property and dual representations.
result Risk measures with the Fatou property exist under the AOCEA property in most classical model spaces.
Investigates stability properties of Haezendonck-Goovaerts premium principles in Orlicz spaces.
problem Stability properties of Haezendonck-Goovaerts premium principles in various Orlicz spaces.
method Analysis of stability properties including Fatou and Lebesgue properties, and continuity with respect to Φ-weak convergence. result Haezendonck-Goovaerts principles satisfy the Fatou property and Lebesgue property under certain conditions.
Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms f:CPk→CPk, when k>1. Classical theory describes U(f) as the complement in CPk of the support of a dynamically-defined closed positive (1,1) current. Given any closed positive $(…
Extends Fatou theorem to bounded harmonic maps.
problem Classical Fatou theorem for bounded harmonic functions.
method Extending theorem to bounded harmonic maps.
result Identifies bounded harmonic maps on unit disk with bounded measurable functions on boundary.
Characterizes Fatou closedness for robust asset pricing.
problem Robust asset pricing under model uncertainty.
method Characterization in terms of Fatou closedness for weakly closed monotone convex sets.
result Characterization of Fatou closedness for robust asset pricing.
In this paper we study a class of functions that appear naturally in some equidistribution problems and that we call F-harmonic. These are functions of the universal cover of a closed and negatively curved which possess an integral representation analogous to the Poisson representation of harmonic functions, where th…
Proves theorem for Riemannian manifolds, extending previous work.
problem Proving Quantitative Fatou Theorem on Riemannian manifolds.
method Extending ε-approximation lemma to manifold setting.
result Proves Quantitative Fatou Theorem for Lipschitz domains on Riemannian manifolds.
The present paper deals with the characterization of no-arbitrage properties of a continuous semimartingale. The first main result, Theorem \refMainTheoremCharNA, extends the no-arbitrage criterion by Levental and Skorohod [Ann. Appl. Probab. 5 (1995) 906-925] from diffusion processes to arbitrary continuous semimartin…
New risk measures for incomplete markets without lattice structures.
problem Risk measures on incomplete markets without lattice structures.
method Study of risk measures without lattice structures, focusing on tractable dual representations and solid superspaces.
result Existence of a tractable dual representation equivalent to a Fatou-like property, and extension theorems under certain conditions.
The paper explores properties of functions on Teichmüller space, proving theorems about limits and non-ergodicity.
problem Properties of bounded pluriharmonic and holomorphic functions on Teichmüller space.
method Analyzes the boundary behavior of functions and proves theorems about limits and non-ergodicity.
result Proves the existence of radial limits for bounded pluriharmonic functions and non-constant bounded holomorphic functions.
We present simple new examples of pure-jump strict local martingales. The examples are constructed as exponentials of self-exciting affine Markov processes. We characterize the strict local martingale property of these processes by an integral criterion and by non-uniqueness of an associated ordinary differential equat…
The study extends classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.
problem Extending classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.
method Investigated the restricted mean-value property on Riemannian manifolds, focusing on non-tangential boundary behavior.
result Extended a classical result of Fenton to non-positively curved Harmonic manifolds of purely exponential volume growth.
New technique explains convergence in ML models with data modifications.
problem Understanding convergence of ML models under data changes.
method Analogue of Fatou's lemma and gamma-convergence.
result Relevance and applications in general ML tasks and domain adaptation.
Study dynamics of automorphisms on cubic surfaces and their connection to Painlevé 6.
problem Dynamics of holomorphic automorphisms on cubic surfaces and their relation to Painlevé 6.
method Defined Julia and Fatou sets, studied locally discrete and non-discrete dynamics, and proved existence of non-empty Fatou and Julia sets.
result Existence of non-empty Fatou and Julia sets for the group action.
In Karatzas and Kardaras's paper on semimartingale financial models, it is proved that the NUPBR condition is a property of the local characteristic of the asset process alone. In Takaoka's paper on NUPBR, it is proved that the NUPBR condition is equivalent to the existence of a simga-martingale deflator. However, Taka…
Study of algebraic dynamics on Markov cubics in tropical geometry.
problem Understanding the dynamics of Markov cubics over non-archimedean fields.
method Tropicalization and (∞,∞,∞)-triangle reflection group on hyperbolic plane. result Existence of Fatou domain and finitude of orbits with rational points over prime power denominators.
Paper explores closedness properties of convex sets in rearrangement invariant spaces.
problem Closedness properties of law-invariant convex sets in rearrangement invariant spaces.
method Analyzes equivalence of different closedness types in rearrangement invariant spaces.
result Order closedness, σ(X,Xn∼)-closedness and σ(X,L∞)-closedness of a law-invariant convex set are equivalent. New risk measure extensions preserve key properties.
problem Extending risk measures to larger spaces while preserving properties.
method Unique extension of dilatation monotone risk measures to L1. result Risk measures extend uniquely and preserve monotonicity, convexity, and cash-additivity.
We show that \emph{No unbounded profit with bounded risk} (NUPBR) implies \emph{predictable uniform tightness} (P-UT), a boundedness property in the Emery topology which has been introduced by C. Stricker \cite{S:85}. Combining this insight with well known results from J. Mémin and L. Słominski \cite{MS:91} leads to a …
We consider the problem of decomposing monetary risk in the presence of a fully traded market in {\it some} risks. We show that a mark-to-market approach to pricing leads to such a decomposition if the risk measure is time-consistent in the sense of Delbaen.
Characterizes measures preserving compound mixed renewal process properties.
problem Preserving compound mixed renewal process properties under different probability measures.
method Characterization of progressively equivalent probability measures.
result Any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures.
Investigates the effects of nondominated sets of probability measures in robust models of finance.
problem Uncertainty in financial models due to multiple possible probability measures.
method Analyzes various results from mathematical finance literature under the assumption of nondominated sets of probability measures.
result Many classical results in robust models do not hold when the set of measures is nondominated.
In this paper we extend a recent result of Collin-Rosenberg ({\it a solution to the minimal surface equation in the Euclidean disc has radial limits almost everywhere}) to a large class of differential operators in Divergence form. Moreover, we construct an example (in the spirit of \cite{CR2}) of a minimal graph in $\…
An infinite family of generalized pseudo-Anosov homeomorphisms of the sphere S is constructed, and their invariant foliations and singular orbits are described explicitly by means of generalized train tracks. The complex strucure induced by the invariant foliations is described, and is shown to make S into a complex sp…
We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito, Delbaen, and Kupper (2006). These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably p…
We study strict local martingales via h-transforms, a method which first appeared in Delbaen-Schachermayer. We show that strict local martingales arise whenever there is a consistent family of change of measures where the two measures are not equivalent to one another. Several old and new strict local martingales are i…
This work aims at a deeper understanding of the mathematical implications of the economically-sound condition of absence of arbitrages of the first kind in a financial market. In the spirit of the Fundamental Theorem of Asset Pricing (FTAP), it is shown here that absence of arbitrages of the first kind in the market is…
We extend risk measure stability conditions for coherent risk management.
problem Reserving and hedging claims under dynamic coherent risk measures.
method Dual characterisation of cones in L∞ adapted to a discrete time filtration, proving stability conditions. result Equivalence of V-m-stability and time-consistency in risk management. The paper examines dynamic reserving for multiple currencies under coherent risk measures.
problem Dynamic reserving for risk in multiple currencies under a general coherent risk measure.
method Shows time-consistency of reserving portfolios in multiple currencies when a generalized m-stability condition holds, equivalent to dynamic trading across baskets of currencies with proportional transaction costs.
result A version of the Fundamental Theorem of Asset Pricing holds in this context, proving time-consistency of reserving portfolios.
New method proves utility maximization without dual problem, simplifying existing results.
problem Maximizing utility from terminal wealth in a continuous-time financial market.
method Utilizes recent Orlicz space theory to prove existence of optimal investment without dual problem.
result Existence of optimal investment strategy for non-smooth utilities and strict concavity.
Dynamic risk measures follow law invariance principles over time.
problem Tackles dynamic risk measurement principles.
method Shows equivalence between adapted law invariance and recursive one-step conditional-law representation for time-consistent risk measures.
result Identifies adapted law invariance as the dynamic counterpart of ordinary law invariance.
Extends utility maximization theory for infinite horizons without strong no-arbitrage assumptions.
problem Maximizing lifetime utility from wealth over an infinite horizon.
method Develops a duality theory using deflators and supermartingale properties, extending previous work.
result Establishes a strong duality theorem for infinite horizon utility maximization under minimal no-arbitrage assumptions.
Paper shows equivalence between NA and ACLMM in diffusion models.
problem No arbitrage condition and existence of ACLMM in general diffusion models.
method Investigates equivalence between NA and ACLMM in single asset diffusion market models.
result NA is equivalent to ACLMM plus mild conditions on scale function and absence of reflecting boundaries.
We propose a new procedure for the risk measurement of large portfolios. It employs the following objects as the building blocks: - coherent risk measures introduced by Artzner, Delbaen, Eber, and Heath; - factor risk measures introduced in this paper, which assess the risks driven by particular factors like the price …
The paper introduces a new risk measure for financial models with jumps.
problem The limitations of point-in-time risk measures in models with jumps.
method Proposes an intra-horizon expected shortfall for profit and loss processes.
result The intra-horizon expected shortfall is a coherent risk measure for various Lévy processes.
New mappings connect 3-sphere geometry to complex analysis problems.
problem Locally homeomorphic quasiregular mappings in 3-space and their properties.
method Constructing non-trivial cobordisms and equivariant mappings.
result Found new mappings related to complex analysis and geometry.
Study of dynamics on cubic surfaces and their connection to Painlevé 6 Equation.
problem Understanding the dynamics of automorphism groups on cubic surfaces.
method Analyzing holomorphic automorphisms and character varieties.
result Several open questions about the dynamics of automorphism groups.
The two main approaches in credit risk are the structural approach pioneered in Merton (1974) and the reduced-form framework proposed in Jarrow & Turnbull (1995) and in Artzner & Delbaen (1995). The goal of this article is to provide a unified view on both approaches. This is achieved by studying reduced-form approache…
We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if A and B are two connected compo…
The paper shows equivalence between order closedness and σ(L^Φ,L^Ψ)-closedness in Orlicz spaces.
problem Characterizing closedness in Orlicz spaces with respect to the σ-topology.
method Analyzing the equivalence of order closedness and σ(L^Φ,L^Ψ)-closedness for convex sets in Orlicz spaces.
result Order closedness and σ(L^Φ,L^Ψ)-closedness are equivalent if and only if either Φ or Ψ satisfies the Δ_2-condition.
The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.
problem Understanding exceptional sets for radial limits of superharmonic functions on curved manifolds.
method Analysis of radial geodesic rays, Poisson integrals, Green potentials, and Riesz decomposition.
result Sharp bounds on Hausdorff dimensions of exceptional sets for superharmonic functions.
The paper reviews historical and modern approaches to asset pricing probability measures.
problem Constructing or selecting probability measures for asset pricing.
method Historical review of various approaches including state price theory, martingale measures, and modern data-driven methods.
result Modern asset pricing involves constructing, transforming, or selecting probability measures to represent market prices.
Paper introduces second-order Esscher densities for continuous-time models.
problem Modeling continuous-time market models with second-order Esscher densities.
method Introduced linear and exponential classes of second-order Esscher densities, characterized using semimartingale characteristics and pointwise equations.
result Characterized the relationship between linear and exponential classes for one-dimensional case and showed their connection in compound Poisson and jump-diffusion models.
The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.
problem Metric inequivalence and characterization of proper holomorphic maps.
method Explicit characterization of proper holomorphic maps from a finitely-connected planar domain onto the unit disk.
result Characterization of proper holomorphic maps from a finitely-connected planar domain onto the unit disk.
Groups with Property (T) have fiber products with Property (T).
problem When does the fiber product of groups with Property (T) have Property (T)?
method Analyzing fiber products of groups with Property (T).
result Fiber products of groups with Property (T) also have Property (T).