In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive form…
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48 results for “scalar measures”
Study connects curvature bounds to map existence and flow solutions.
problem Existence of lower scalar curvature bounds and measures.
method Relates curvature bounds to map existence and backward limit of Ricci flow solutions.
result Sufficient condition for existence of limiting scalar curvature measure.
We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space …
In this paper we present results on scalar risk measures in markets with transaction costs. Such risk measures are defined as the minimal capital requirements in the cash asset. First, some results are provided on the dual representation of such risk measures, with particular emphasis given on the space of dual variabl…
Upper bound found for systolic geometry on manifolds with positive scalar curvature.
problem Relationship between systolic geometry and positive scalar curvature.
method Spinorial methods combined with geometric measure theory and curvature estimates.
result Upper bound for the two-dimensional stable systole on certain manifolds.
Synthetic scalar curvature defined via Gaussian integrals, applied to manifolds and flows.
problem Defining scalar curvature for non-smooth spaces and flows.
method Gaussian integral approach to scalar curvature, applied to manifolds and flows.
result Characterizes Ricci flows as minimal super-Ricci flows.
New scalars measure failure of CC metrics to solve singular Yamabe problem.
problem Measuring failure of CC metrics to solve singular Yamabe problem.
method Introducing conformally invariant scalar curvature quantities along conformal infinity.
result CC boundary curvature scalars compute canonical expansion coefficients for singular Yamabe metrics.
The study examines conditions for scalar curvature and Dirac operators on singular spaces.
problem Existence of scalar curvature measures and Dirac operators on singular spaces.
method Investigation of smooth manifolds with singular Riemannian metrics.
result Sufficient conditions for the existence of scalar curvature measures and Dirac operators.
Study bounds on curvature for special Finsler metrics.
problem Curvature and topological properties of -Einstein Finsler metrics.
method Construct special metrics, analyze equivalence, impose curvature bounds.
result Establish bounds for curvature and distortion on -Einstein Finsler manifolds.
The paper concerns primal and dual representations as well as time consistency of set-valued dynamic risk measures. Set-valued risk measures appear naturally when markets with transaction costs are considered and capital requirements can be made in a basket of currencies or assets. Time consistency of scalar risk measu…
Extends Dirac operator results to foliations with invariant measures.
problem Positive scalar curvature on foliations with invariant measures.
method Relative measured index theorem for foliations with invariant transverse measures.
result Space of positive scalar curvature metrics has infinitely many path components.
Paper studies convergence of nonnegative scalar curvature metrics to a specific limit space.
problem Understanding convergence of nonnegative scalar curvature metrics.
method Analyzes a sequence of warped product metrics on $\Sph^2 imes \Sph^1$.
result Sequence converges to an extreme limit space in specific senses.
A new risk measure framework captures multivariate risk in banking.
problem Scalar risk measures fail to capture the multivariate nature of risk in banking.
method A novel multivariate risk measure framework based on the Magnitude-Propensity approach.
result The proposed framework provides a more comprehensive characterization of extreme events.
New method removes scalar curvature assumption in Ricci flow smoothing.
problem Uniform bounds on scalar curvature and other factors for Ricci flow.
method Quantitative short-time existence of Ricci flow without scalar curvature assumption.
result Ricci flow smoothing for measure space limits, Gromov-Hausdorff compactness, and topological rigidity results.
Study on deformation of weighted scalar curvature, proving geometric results and stability.
problem Deformation of weighted scalar curvature and related geometric properties.
method Linearization of weighted scalar curvature, studying kernel of formal adjoint.
result Definition and study of weighted vacuum static spaces, stability results on flat spaces.
Proposes new deviation measures using Minkowski gauges.
problem Lack of suitable acceptance sets for deviation measures.
method Derives deviation measures through Minkowski gauges of acceptable sets.
result Any positive homogeneous deviation measure can be accommodated in the framework.
The financial crisis showed the importance of measuring, allocating and regulating systemic risk. Recently, the systemic risk measures that can be decomposed into an aggregation function and a scalar measure of risk, received a lot of attention. In this framework, capital allocations are added after aggregation and can…
The weighted Yamabe flow converges on smooth metric measure spaces.
problem Analyzing convergence of the weighted Yamabe flow on metric measure spaces.
method Introduced the weighted Yamabe flow and proved its long-time existence and convergence under certain conditions.
result Long-time existence and convergence of the weighted Yamabe flow on smooth metric measure spaces.
Study shows tori metrics converging to flat under specific conditions.
problem Understanding convergence of metrics on tori with non-negative scalar curvature.
method Uniformly conformal metrics and controlled geometry sequences.
result Sequence of metrics converges to flat metric in multiple senses.
Most previous contributions to BSDEs, and the related theories of nonlinear expectation and dynamic risk measures, have been in the framework of continuous time diffusions or jump diffusions. Using solutions of BSDEs on spaces related to finite state, continuous time Markov chains, we develop a theory of nonlinear expe…
Study systemic risk measures and capital allocation rules, showing commonalities.
problem Systemic risk measures and capital allocation in financial systems.
method Developed a general framework to embed axiomatic and injective capital approaches, introduced Aumann-Shapley CAR.
result Aumann-Shapley CAR provides a universal method for capital allocation regardless of risk measurement.
This work extends set-valued risk measures to discrete time, using difference inclusions and equations.
problem Defining set-valued dynamic risk measures in discrete time.
method Investigates discrete time setting with difference inclusions and difference equations.
result Provides insights for continuous time representations of set-valued dynamic risk measures.
A new flow method solves the weighted Yamabe problem with boundary.
problem Solving the weighted Yamabe problem on metric measure spaces with boundary.
method Introduced a Yamabe-type flow with a specific geometric setup.
result Long-time existence and convergence of the flow proved.
Proves a conjecture about manifolds and scalar curvature.
problem Determining when a manifold admits a positive scalar curvature metric.
method Uses a geometric bound to measure discrepancies between vector fields.
result Proves the conjecture in codimension two.
Paper proves Gromov's conjecture on manifolds with certain group properties.
problem Gromov's conjecture on positive scalar curvature and simplicial volume.
method Proves conjecture under a fundamental group decay property.
result Proves Gromov's conjecture for manifolds with a weakened rapid decay property.
Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P(M) to the space of positive scalar curvature metrics on this manifold. The group of all diffeomorphisms of the manifold naturally acts on P(M). The moduli group of positive scalar curvature metrics is…
Yamabe Classification and Prescribed Scalar Curvature in the Asymptotically Euclidean Settingmath.DG
We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yam…
Estimates scalar curvature of point clouds without embedding.
problem Estimating scalar curvature of data sets without embedding.
method Intrinsic estimator based on metric structure, consistent and stable.
result Estimator converges to scalar curvature as sample size increases.
We introduce a measure for estimating the best risk-return relation of power production in wind farms within a given time-lag, conditioned to the velocity field. The velocity field is represented by a scalar that weighs the influence of the velocity at each wind turbine at present and previous time-steps for the presen…
The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we d…
Prove that collapsing CSC metrics can be perturbed to invariant collapsing CSC metrics.
problem Prove that collapsing constant scalar curvature metrics can be perturbed to invariant collapsing constant scalar curvature metrics.
method Prove that a sequence of constant scalar curvature metrics which is collapsing with bounded curvature to a manifold can be perturbed to a sequence of invariant collapsing constant scalar curvature metrics.
result Prove that a sequence of constant scalar curvature metrics which is collapsing with bounded curvature to a manifold can be perturbed to a sequence of invariant collapsing constant scalar curvature metrics.
This paper introduces a new scalarization method for multi-objective optimization.
problem Efficiently optimizing multiple conflicting objectives in black box settings.
method Introduces a novel hypervolume scalarization function and uses it to approximate the hypervolume indicator metric.
result Provable convergence to the entire Pareto frontier using random scalarizations and Bayesian optimization.
Study finds conditions for metrics on curved spaces.
problem Finding metrics with specific curvature properties.
method Analyzes a conformal class with prescribed scalar and boundary mean curvatures.
result Establishes a necessary and sufficient condition involving a conformal invariant.
Set-valued risk measures on with for conical market models are defined, primal and dual representation results are given. The collection of initial endowments which allow to super-hedge a multivariate claim are shown to form the values of a set-valued sublinear (coherent) risk measure. Sc…
Non-property (t) for SO(n,1)math.GT
Concrete proof that SO(n,1) is not a T-group.
problem Proving SO(n,1) is not a T-group.
method Constructed a concrete model for hyperbolic space and measure, proving hyperbolic distance equals measure up to a constant.
result Concrete proof that SO(n,1) is not a T-group.
Controlling systemic risk - network structures that minimize it and node properties to calculate itq-fin.RM
Evaluation of systemic risk in networks of financial institutions in general requires information of inter-institution financial exposures. In the framework of Debt Rank algorithm, we introduce an approximate method of systemic risk evaluation which requires only node properties, such as total assets and liabilities, a…
Formula calculates mass using cube faces and edges.
problem Measuring mass of 3-manifolds.
method Cube faces and edges, mean curvature, dihedral angle, geodesic curvature, angle defect.
result Mass formula connects to Gromov's theory and Gauss-Bonnet theorem.
The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk m…
Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior does not extend to the boundary (because for example the interior is complete) w…
On evaluation of joint risk for non-negative multivariate risks under dependence uncertaintyq-fin.RM
Paper proposes a new method to evaluate joint risk under uncertainty.
problem Evaluating joint risk of multiple insurance risks under dependence uncertainty.
method Axiomatic approach to scalar and vector-valued distortion joint risk measures.
result Established a new scalar distortion joint risk measure with positive homogeneity.
On the range of the relative higher index and the higher rho-invariant for positive scalar curvaturemath.KT
Let be a closed spin manifold which supports a positive scalar curvature metric. The set of concordance classes of positive scalar curvature metrics on forms an abelian group after fixing a positive scalar curvature metric. The group measures the size of the space of positive scalar curvature metr…
Conditions for scalar curvature on compact manifolds under conformal deformation.
problem Finding conditions for scalar curvature functions on compact manifolds.
method Analyzing sufficient and necessary conditions for scalar curvature problems within conformal classes.
result Conditions for scalar curvature functions on various compact manifolds.
New set-valued star-shaped risk measures introduced for better risk assessment.
problem Improving risk assessment in financial contexts.
method Developed new set-valued star-shaped risk measures and proved their representation theorems.
result Set-valued star-shaped risk measures can be represented as unions of set-valued convex risk measures.
The report explores conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds.
problem Conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds on non-smooth spaces.
method Description of results in dimension 3, exploration of weak forms of Ricci curvature, use of volume entropy and Bishop-Gromov inequality.
result Recent results on weak Ricci curvature bounds and conditions for positive or non-negative scalar curvature in 3-manifolds.
Choosing a portfolio of risky assets over time that maximizes the expected return at the same time as it minimizes portfolio risk is a classical problem in Mathematical Finance and is referred to as the dynamic Markowitz problem (when the risk is measured by variance) or more generally, the dynamic mean-risk problem. I…
Extends spectral torus band inequalities for compact manifolds with scalar curvature bounds.
problem Proving upper bounds for the width of compact manifolds with boundary.
method Utilizes spacetime harmonic functions, μ-bubbles, and spinorial Callias operators.
result Generalizes Schoen-Yau black hole existence theorem to higher dimensions.
The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R^3, delta_{ij}). In this paper, we quantify this statement using spinors and prove that if a complete, asymptotically flat manifold with non-negative…
Set risk measuresq-fin.MF
Set risk measures extend traditional risk measures to handle sets of positions.
problem Handling sets of positions with a single capital requirement.
method Developed an axiomatic framework for set risk measures, dual representation through topology and measures.
result Characterized worst-case set risk measures and provided examples.