The paper analyzes optimal dividend and capital injection strategies under time-inconsistent preferences.
problem Optimal dividend and capital injection strategies under time-inconsistent preferences.
method Diffusion risk model with general discount functions, weak equilibrium definition, HJB equation system.
result Explicit solutions and threshold types of optimal strategies derived under different discount functions.
Nonuniform tubular neighborhoods of curves in Euclidean n-space are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtaine…
Injective and surjective neural operators for function spaces.
problem Tackles injective and surjective neural operators in function spaces.
method Combines prior work in ReLU and operator learning, uses Fredholm theory and Leray-Schauder degree theory.
result Injective and surjective neural operators are universal approximators and maintain their properties in finite-rank implementations.
The Fridman function is bounded by the injectivity radius for certain hyperbolic manifolds.
problem Bounding the Fridman function for hyperbolic manifolds.
method Analyzing the relationship between the Fridman function and the injectivity radius function.
result The Fridman function is bounded above by the injectivity radius function for certain hyperbolic manifolds.
Injective X-ray transform on Heisenberg group for regular functions.
problem Injectivity of X-ray transform on sub-Riemannian manifolds.
method Group Fourier Transform and analysis of taming metrics.
result Sufficiently regular functions on Heisenberg group are determined by their line integrals.
Injectivity of geodesic ray transform on specific Finsler manifolds proven.
problem Injectivity of geodesic ray transform on spherically symmetric reversible Finsler manifolds.
method Reduction to invertibility of generalized Abel transforms using angular Fourier series and Taylor expansions of geodesics.
result Injectivity of geodesic ray transform proven on specified Finsler manifolds.
Non-injectivity proven for trace map on character varieties.
problem Proving non-injectivity of trace map on character varieties.
method Using Amitsur-Levitzki identity and free homotopy classes.
result Explicit construction of nonzero elements in kernel of trace map.
Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar func…
GNIs induce a regulariser that penalizes high-frequency components in neural network activations.
problem Understanding the regularizing effect of Gaussian noise injections on neural network activations.
method Deriving the explicit regularizer by marginalizing out injected noise and analyzing its effect in the Fourier domain.
result GNIs induce a regularizer that produces calibrated classifiers with large margins.
Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on L2 for metrics with finitely differentiable tensor. Let P be a knot in a solid torus, K a knot in 3-space and P(K) the satellite knot of K with pattern P. This defines an operator on the set of knot types and induces a satellite operator P:C--> C on the set of smooth concordance classes of knots. There has been considerable interest in whether certain such functions are…
The function on the Teichmueller space of complete, orientable, finite-area hyperbolic surfaces of a fixed topological type that assigns to a hyperbolic surface its maximal injectivity radius has no local maxima that are not global maxima.
In this paper we propose and solve an optimal dividend problem with capital injections over a finite time horizon. The surplus dynamics obeys a linearly controlled drifted Brownian motion that is reflected at the origin, dividends give rise to time-dependent instantaneous marginal profits, whereas capital injections ar…
In this paper, we study the optimal control problem for a company whose surplus process evolves as an upward jump diffusion with random return on investment. Three types of practical optimization problems faced by a company that can control its liquid reserves by paying dividends and injecting capital. In the first pro…
The paper analyzes insurance risk with Parisian ruin and capital injection.
problem Analyzing insurance risk with Parisian ruin and capital injection.
method Using fluctuation and excursion theory of spectrally negative Levy processes.
result Distributional identities and ruin probabilities are derived.
We study functions whose truncations are convex or quasiconvex.
problem Understanding functions with specific truncation properties.
method Analyzing C2-smooth functions with positive definite Hessians. result Injectivity of restricted gradient in positive definite region.
Optimal control problem for firm cash flow with dividend and capital injection strategies.
problem Maximizing dividends while managing capital injections in a firm's cash flow.
method Proved two optimal strategies: mean-reverting dividends with capital injections or no injections until ruin.
result Optimal strategies are dichotomous: either mean-reverting dividends with injections or no injections.
The paper proves bounds on curvature and injectivity radius for convex sums of Riemannian metrics.
problem Understanding the geometry of convex sums of Riemannian metrics.
method Quantitative inverse function theorem and Riemannian geometry techniques.
result Injectivity radii of convex sums have uniform lower bounds.
The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.
problem Uniform Lipschitz bounds on geometric functions in Teichmüller space.
method Injectivity radius analysis and Lipschitz bounds on systole function.
result Uniform Lipschitz constant for the square root of the systole function on Teichmüller space.
X-ray transform on H-type groups solved, revealing function injectivity.
problem Injectivity in sub-Riemannian geometry.
method Fourier Slice Theorem adapted to H-type groups.
result Integrable functions on H-type groups are uniquely determined by their integrals over geodesics.
Study optimal dividend and capital injection in insurance portfolios with self-exciting claim arrivals.
problem Optimal dividend and capital injection in insurance portfolios with Hawkes process claim arrivals.
method Analytical properties, explicit threshold, HJB variational inequality, finite-difference scheme, policy-gradient, actor-critic methods.
result Learned strategies closely match the PDE benchmark and remain stable across initial conditions.
If G is a finite group, is a function f:G→C determined by its sums over all cosets of cyclic subgroups of G? In other words, is the Radon transform on G injective? This inverse problem is a discrete analogue of asking whether a function on a compact Lie group is determined by its integrals over all ge…
The paper constructs Morse homology for functionals involving the p-Laplacian in Banach spaces.
problem Constructing Morse homology for functionals involving the p-Laplacian in Banach spaces.
method The approach involves constructing critical points and ensuring injectivity of the second differential.
result Provides a positive answer to Smale's suggestion for injectivity of the second differential.
The paper optimizes insurer's dividend, reinsurance, and capital injection strategies for two collaborating business lines.
problem Maximizing expected total dividend payments while managing risk and preventing ruin.
method Solving the problem using a closed-form value function for optimal strategies.
result Optimal strategies include threshold dividend payout, decreasing reinsured risk, and capital injection to prevent ruin.
The paper studies harmonic maps between surfaces homotopic to a covering map, proving uniqueness and injectivity of Hopf differential.
problem Analyzing harmonic maps between surfaces in the homotopy class of a covering map.
method Proving the uniqueness of critical points and injectivity of Hopf differential for harmonic maps.
result The uniqueness of critical points of energy function and injectivity of Hopf differential are proven under specific conditions.
In the present paper, we investigate the optimal capital injection behaviour of an insurance company if the interest rate is allowed to become negative. The surplus process of the considered insurance entity is assumed to follow a Brownian motion with drift. The changes in the interest rate are described via a Markov-s…
In this paper, we study a risk process modeled by a Brownian motion with drift (the diffusion approximation model). The insurance entity can purchase reinsurance to lower its risk and receive cash injections at discrete times to avoid ruin. Proportional reinsurance and excess-of-loss reinsurance are considered. The obj…
Paper proves injectivity of non-abelian X-ray transform on certain spaces.
problem Injectivity of non-abelian X-ray transform on asymptotically hyperbolic spaces.
method Gauge equivalence for unitary connections and skew-Hermitian Higgs fields.
result Injectivity result for non-abelian X-ray transform over skew-Hermitian Higgs fields.
Study the discontinuity of functions not embeddable in Euclidean space.
problem Understanding discontinuity of non-embeddable functions.
method Define a modulus of discontinuity and establish lower bounds.
result Quantified nonembeddability results and topological Tverberg theorem.
Injectivity of geodesic X-ray transform on low-regularity manifolds.
problem Injectivity of geodesic X-ray transform on manifolds with low regularity.
method Calculus of differential and curvature operators on non-smooth structures.
result Injectivity of geodesic X-ray transform on simple Riemannian manifolds with C1,1-regularity. We study the optimal financing and dividend distribution problem with restricted dividend rates in a diffusion type surplus model where the drift and volatility coefficients are general functions of the level of surplus and the external environment regime. The environment regime is modeled by a Markov process. Both cap…
Spatial processes with nonstationary and anisotropic covariance structure are often used when modelling, analysing and predicting complex environmental phenomena. Such processes may often be expressed as ones that have stationary and isotropic covariance structure on a warped spatial domain. However, the warping functi…
We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author's result to the with-boundary setti…
Injectivity of ReLU networks is characterized for generative models and inverse problems.
problem Injectivity in ReLU networks for generative models and inverse problems.
method Layerwise analysis, worst-case Lipschitz constants, differential topology, random projections.
result Global injectivity of ReLU networks requires expansivity between 3.4 and 10.5 for Gaussian matrices.
Classifies π1-injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All π1-injective proper maps are classified. Electronic power inverters are capable of quickly delivering reactive power to maintain customer voltages within operating tolerances and to reduce system losses in distribution grids. This paper proposes a systematic and data-driven approach to determine reactive power inverter output as a function of local measuremen…
Positive injectivity radius for manifolds with Lie structure at infinity.
problem Injectivity radius positivity for manifolds with specific boundary conditions.
method Lie groupoids to prove injectivity radius positivity.
result Injectivity radius is positive for manifolds with Lie structure at infinity.
Geometric analysis improves noise injection in GANs.
problem Unclear mechanism of noise injection in GANs.
method Geometric framework based on Riemannian geometry.
result A new strategy for noise injection is devised.
Regularity results for geodesic X-ray transform on nonsmooth manifolds
problem Geodesic X-ray transform on nonsmooth simple manifolds
method Symbol smoothing arguments and pseudodifferential operators with low regularity symbols
result Improved injectivity results for Lp functions Study group actions on hyperbolic spaces to find algebraic and geometric properties.
problem Understanding algebraic properties of group actions on hyperbolic spaces.
method Developed an algorithm in the group algebra to show free ideals generated by few elements.
result Derived lower bounds on Morse complexity of closed hyperbolic manifolds.
The paper optimizes insurer's decisions on dividends, reinsurance, and capital injection under model uncertainty.
problem Maximizing insurer's expected discounted dividends while managing model uncertainty and risk.
method Modeling reserve levels as diffusion processes, solving for optimal strategies in closed form.
result Optimal strategies include barrier dividend and capital injection policies.
A quasi-geodesic is Morse if and only if it is strongly contracting in injective spaces.
problem Characterizing Morse quasi-geodesics in injective spaces.
method Proving equivalence between Morse and strongly contracting quasi-geodesics.
result Injective metric spaces have the Morse local-to-global property and acylindrically hyperbolic groups with Morse elements.
This paper connects noise injection to Bayesian inference for neural networks, improving model uncertainty.
problem Improving the reliability and confidence of neural network predictions through uncertainty quantification.
method Introducing noise into neural network parameters during training and inference to estimate prediction uncertainty.
result The MCNI method outperforms baseline models in regression and classification tasks.
Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.
problem Establishing uniform bounds for eigenvalues of the Hodge Laplacian on manifolds with specific geometric constraints.
method Using uniform bounds on Ricci curvature, injectivity radius, and diameter, we derive eigenvalue estimates for the Hodge Laplacian.
result Uniform upper bounds for eigenvalues of the Hodge Laplacian on differential forms on manifolds with given geometric constraints.
Compact theorem for minimal surfaces with lower injectivity radius.
problem Proving compactness of minimal surfaces with lower injectivity radius.
method Variant of Choi--Schoen compactness theorem, focusing on injectivity radius.
result Proved compactness theorem for minimal surfaces.
Lecture notes on group actions on injective spaces and Helly graphs.
problem Understanding group actions on specific metric spaces.
method Review of injective metric spaces and Helly graphs, elementary properties, constructions, and exercises.
result Presentation of various constructions of injective metric spaces and Helly graphs with interesting group actions.
Proves polynomial injectivity of Fubini-Study map for ample line bundles.
problem Injectivity of Fubini-Study map for ample line bundles.
method Polynomial injectivity proof with polynomial dependence on ample line bundle exponent.
result Quantitative version of injectivity proved, polynomial in ample line bundle exponent.
Lower bound on boundary injectivity radius for specific tubes.
problem Estimating the boundary injectivity radius of Margulis tubes.
method Using curvature bounds to derive a lower bound.
result A lower bound on the boundary injectivity radius is provided.