The paper extends first-order asymptotics for path-dependent derivatives in multiscale stochastic volatility.
problem Analyzing path-dependent derivatives in a multiscale stochastic volatility environment.
method First-order asymptotics analysis using Dupire's functional Ito calculus.
result Market parameters calibrated to vanilla options can price path-dependent derivatives to the same order.
Study stabilizes second-order systems to first-order dynamics.
problem Stabilizing second-order systems to first-order dynamics.
method Feedback control of second-order systems on manifolds.
result Second-order systems can globally exponentially stabilize first-order dynamics for fully actuated systems.
The purpose of this paper is to provide a sharp analysis on the asymptotic behavior of the Durbin-Watson statistic. We focus our attention on the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We establish the almost sure convergence and the asymptotic n…
Constructs flows with prescribed tangent and asymptotic behavior.
problem Creating mean curvature flows with specific tangent flows and asymptotics.
method General theorem allowing construction of flows with prescribed tangent and asymptotic behavior.
result Constructs mean curvature flows with prescribed tangent and asymptotic behavior.
Unified bounds for iterative algorithms with Gaussian data matrices.
problem Establishing non-asymptotic bounds for iterative algorithms with Gaussian data.
method Explicit coupling between iterates and Gaussian process with deterministic covariance.
result Tight, dimension-free bounds for generalized first-order methods.
Study heat content on RCD(K,N) spaces with specific boundary conditions.
problem Analyzing heat content in RCD(K,N) spaces with irregular boundaries.
method Proved first-order asymptotics using measured interior geodesic condition.
result Established first-order heat content asymptotics on RCD(K,N) spaces.
First-order ANIL learns shared representations even with overparametrization.
problem Lack of theoretical evidence for model-agnostic meta-learning (ANIL) learning shared representations.
method First-order ANIL with a linear two-layer network architecture, showing asymptotically low-rank solutions with overparametrization.
result First-order ANIL learns linear shared representations, even with overparametrization, and performs well in adaptation.
The paper calculates option prices using Mellin transform for stochastic volatility models.
problem Calculating prices for path-dependent options under stochastic volatility.
method Asymptotic approach and Mellin transform for deriving closed-form formulas.
result Derives closed-form formulas for option prices with first-order approximation.
In the context of the multi-dimensional infinite horizon optimal consumption-investment problem with proportional transaction costs, we provide the first order expansion in small transact costs. Similar to the one-dimensional derivation in our accompanying paper [42], the asymptotic expansion is expressed in terms of a…
Paper studies the full asymptotic torsion forms of flat bundles.
problem Analytic torsion forms of flat bundles and their expansions.
method Proves the existence of the full expansion and gives a formula for the sub-leading term.
result Existence and formula for the full asymptotic expansion of torsion forms.
OSGM uses online learning to adapt stepsize for faster convergence.
problem Improving convergence rates of first-order methods.
method OSGM combines online learning and feedback functions to adjust stepsize.
result OSGM achieves convergence rates asymptotically no worse than optimal.
For a 3-manifold M with boundary, we study the Kauffman module with indeterminate equal to −1+ε where ε2=0. We conjecture an explicit relation between this module and the Reidemeister torsion of M which we prove in particular cases. As a maybe useful tool, we then introduce a notion of twisted self-linking and…
We develop a first order expansion for convex penalized estimators in high-dimensional regression.
problem High-dimensional regression problems with random designs.
method Construct a first order expansion η of the penalized estimator β^. result The risk of β^ is asymptotically the same as the risk of η. New methods bound estimation error in high-dimensional statistical problems.
problem Fundamental limits of first order methods in high-dimensional estimation.
method Introduces general first order methods for high-dimensional regression and low-rank matrix estimation.
result Derives optimal lower bounds on estimation error for these methods.
Geometrical properties of holonomic and non holonomic varieties defined by the Pfaff equations connected with a first order systems of differential equations are studied. The Riemann extensions of affine connected spaces for investigation of geodesics and asymptotic lines are used.
We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metr…
Paper studies second order tail probabilities in risk models.
problem Analyzing tail probabilities in risk models with constant interest force.
method Asymptotic expansion and weighted Kesten-type inequality for second order subexponential random variables.
result Second order asymptotic formulae for continuous-time renewal risk models are derived.
This work analyzes actor-critic methods for faster convergence.
problem Finite-time analysis and sample complexity of two-time-scale actor-critic methods.
method Non-asymptotic analysis under non-i.i.d. setting, proving convergence to first-order stationary point.
result Actor-critic method finds a first-order stationary point with ildeO(ε−2.5) sample complexity. The paper deals with a formally self-adjoint first order linear differential operator acting on m-columns of complex-valued half-densities over an n-manifold without boundary. We study the distribution of eigenvalues in the elliptic setting and the propagator in the hyperbolic setting, deriving two-term asymptotic form…
Paper improves risk estimation for extreme events.
problem Estimating extreme risks accurately.
method Modified Bayes risk for expectiles, asymptotic expansions, efficient estimators.
result Asymptotic normality of estimators proved.
The paper uses EVT to improve tail risk measures under ambiguity sets.
problem Misspecification of tail risk measures leads to inflated risk estimates.
method Applies Extreme Value Theory to derive worst-case tail risk under ambiguity sets.
result Proposes a tail-calibrated ambiguity design that preserves nominal tail asymptotic scaling.
First-order method solves stochastic bilevel optimization with linear constraints.
problem Stochastic bilevel optimization with linear constraints and noise.
method Developed a novel framework using gradient-based techniques and smoothed penalty functions.
result Achieved finite-time convergence guarantees for (δ,ε)-Goldstein stationary points. The paper establishes preferred coordinates for AE 3-manifolds, improving ADM center of mass convergence.
problem Establishing preferred coordinates for asymptotically Euclidean 3-manifolds.
method Analyzing regularity of conformal compactifications via elliptic theory.
result Improves Sobolev regularity of conformally compactified AE 3-manifolds.
Study on deformations of Spin(7)-structures on manifolds.
problem Analyzing deformations of Spin(7)-structures on asymptotically conical manifolds.
method Examined the moduli space of torsion-free, asymptotically conical Spin(7)-structures, showing it is an orbifold for generic decay rates.
result Found that the classical Bryant-Salamon metric on positive spinors on S4 has no continuous deformations as an AC Spin(7)-metric. In this paper we prove an approximate formula expressed in terms of elementary functions for the implied volatility in the Heston model. The formula consists of the constant and first order terms in the large maturity expansion of the implied volatility function. The proof is based on saddlepoint methods and classical …
We consider the classical Merton problem of lifetime consumption-portfolio optimization problem with small proportional transaction costs. The first order term in the asymptotic expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form in the one-dimensional case. …
Paper examines risk measure expansions under FGM dependence, improving accuracy at extreme levels.
problem Capturing higher-order tail behavior and dependence effects in risk measures.
method Second-order asymptotic expansions using extreme value theory and regular variation theory.
result Second-order approximations reduce approximation errors, especially at extreme confidence levels.
We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kern…
In this paper, we study the portfolio optimization problem with general utility functions and when the return and volatility of underlying asset are slowly varying. An asymptotic optimal strategy is provided within a specific class of admissible controls under this problem setup. Specifically, we first establish a rigo…
A new algorithm solves bilevel optimization with linear constraints.
problem Solving bilevel optimization problems with coupled linear constraints.
method Penalty and augmented Lagrangian methods reformulate the problem; a single-loop, first-order algorithm proposed.
result Improved convergence rates compared to prior methods.
We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of m complex-valued half-densities over a connected compact n-dimensional manifold without boundary. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator …
A new algorithm for decentralized optimization over directed graphs.
problem Decentralized stochastic optimization over directed networks.
method Gradient tracking and S-ADDOPT algorithm with constant and decaying step-sizes.
result S-ADDOPT converges linearly with constant step-size and sublinearly with decaying step-size.
Study short maturity Asian options in jump-diffusion models with local volatility.
problem Analyzing Asian options pricing in models with jumps and local volatility.
method Asymptotic analysis for short maturity, considering fixed and floating strike options.
result Explicit results for Asian option prices in several models, including Merton, double-exponential, and Variance Gamma models.
Study on spectral asymptotics of Toeplitz operators on CR manifolds.
problem Analyzing spectral properties of Toeplitz operators on CR manifolds.
method Full asymptotic expansion of functional calculus of Toeplitz operators.
result Established several CR analogues of complex geometry results.
We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditio…
We consider the problem of optimizing the expected logarithmic utility of the value of a portfolio in a binomial model with proportional transaction costs with a long time horizon. By duality methods, we can find expressions for the boundaries of the no-trade-region and the asymptotic optimal growth rate, which can be …
This is the first in a series of papers in which we study an efficient approximation scheme for solving the Hamilton-Jacobi-Bellman equation for multi-dimensional problems in stochastic control theory. The method is a combination of a WKB style asymptotic expansion of the value function, which reduces the second order …
In high dimensions, the mean and geometric median are nearly identical.
problem Understanding the relationship between mean and geometric median in high-dimensional spaces.
method Analytical derivation and simulation of the distance between mean and geometric median.
result The distance between mean and geometric median vanishes with dimensionality in high dimensions.
New method evaluates LLMs fairness in universal prediction.
problem Evaluating fairness of large language models in universal prediction.
method Introducing batch regret as a modification of average regret for LLMs.
result Asymptotical value of batch regret for add-constant predictors on memoryless and first-order Markov sources.
ROOT-SGD solves convex optimization problems with optimal nonasymptotic and near-optimal asymptotic performance.
problem Solving strongly convex and smooth unconstrained optimization problems using stochastic first-order algorithms.
method ROOT-SGD: Recursive One-Over-T SGD, averaging past stochastic gradients.
result Achieves state-of-the-art performance in both nonasymptotic and asymptotic senses.
We analyze training dynamics in Gaussian mixture models using a comparison theorem.
problem Analyzing training algorithms with Gaussian mixture data.
method Applying a Gaussian comparison theorem to a specific family of training algorithms.
result Validated dynamic mean-field expressions and provided iterative refinement schemes.
BMM algorithm improves convergence for nonconvex optimization problems.
problem Constrained nonsmooth nonconvex optimization problems.
method Block majorization-minimization with diminishing radius.
result Improved convergence rate for nonconvex optimization problems.
The paper studies the asymptotic expansion of Gaussian integral operators on Riemannian submanifolds.
problem Analyzing the asymptotic behavior of Gaussian integral operators on Riemannian submanifolds.
method Deriving a full asymptotic expansion of the Gaussian integral operator and computing the first-order correction term.
result Explicit computation of the first-order correction term in terms of mean curvature vector and scalar curvature.
Information geometry applies concepts in differential geometry to probability and statistics and is especially useful for parameter estimation in exponential families where parameters are known to lie on a Riemannian manifold. Connections between the geometric properties of the induced manifold and statistical properti…
We consider an elliptic self-adjoint first order differential operator L acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of the operator L is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Gi…
The paper refines classical covariance asymptotics using geometric information geometry.
problem Deviation of finite-sample behavior from classical predictions in curved models.
method Develops a curvature-aware refinement by viewing parametric families as Riemannian manifolds with Fisher-Rao metric.
result Derives an \(n^{-2}\) correction to the leading \(n^{-1}I(θ)^{-1}\) covariance term for score-root estimators.
In this paper, we develop a general study of contributions at infinity of Bochner-Weitzenböck-type formulas on asymptotically flat manifolds, inspired by Witten's proof of the positive mass theorem. As an application, we show that similar proofs can be obtained in a much more general setting as any choice of an irreduc…
Localized Kasner-like singularities constructed in spacetime.
problem Constructing localized singular solutions to Einstein vacuum equations.
method First order symmetric hyperbolic formulation, adapted orthonormal frame.
result Localized Kasner-like singularities with refined uniqueness and general asymptotic data.