The Bass model is calibrated to vanilla options using a fixed-point equation.
problem Calibration of the Bass local volatility model to vanilla options.
method Solving a fixed-point equation to achieve calibration.
result Existence and uniqueness of the solution to the fixed-point equation, and linear convergence of the fixed-point iteration scheme.
Unified framework for solving fixed-point equations in deterministic and stochastic settings.
problem Solving fixed-point equations for seminorm-contractive operators in both deterministic and stochastic contexts.
method Fixed-point theorem and stochastic approximation analysis.
result Unified finite-sample bounds for various reinforcement learning algorithms.
Classifies Toda-type tt*-structures and their fixed points.
problem Classifying Toda-type tt*-structures and their fixed points.
method Fixed point description and reduction of anti-symmetry conditions.
result Reduces possibilities of anti-symmetry condition to two cases.
Study asymptotically almost periodic solutions on real hyperbolic manifolds.
problem Existence and asymptotic behavior of solutions to parabolic equations.
method Dispersion and smoothing estimates, fixed point argument.
result Existence and uniqueness of asymptotically almost periodic solutions.
Study solves optimal portfolio selection using HJB equation.
problem Optimal portfolio selection problem.
method Maximal monotone operator method, Banach fixed-point theorem, Fourier transform, monotone operators technique.
result Existence and uniqueness of solution to HJB equation.
In this paper, we introduce new methods for solving the vacuum Einstein constraints equations: the first one is based on Schaefer's fixed point theorem (known methods use Schauder's fixed point theorem) while the second one uses the concept of half-continuity coupled with the introduction of local supersolutions. These…
Study optimizes solving fixed-point equations using subspace search.
problem Solving linear fixed point equations in Hilbert spaces.
method Linear stochastic approximation scheme with Polyak--Ruppert averaging.
result Established optimal approximation factor for temporal difference learning methods.
Proves existence of solution to Lichnerowicz equation on non-CMC manifolds.
problem Existence of positive solution to Lichnerowicz equation on non-CMC closed manifolds with supercritical terms.
method Employed a fixed-point argument involving sub- and supersolutions, with conditions on coefficients to prevent classical solutions.
result Proves existence of a positive and essentially bounded solution.
Study fixed-point sets of S1-actions on quaternionic manifolds.
problem Characterize fixed-point sets and compatible complex structures on quaternionic manifolds.
method Analyze fixed-point sets and derive equations involving first Chern classes.
result Conditions for the existence of hypercomplex structures on quaternionic manifolds.
The Jacobi-Maupertuis metric allows one to reformulate Newton's equations as geodesic equations for a Riemannian metric which degenerates at the Hill boundary. We prove that a JM geodesic which comes sufficiently close to a regular point of the boundary contains pairs of conjugate points close to the boundary. We prove…
A new Poisson bracket defined on Poisson structures with applications to fixed points and cohomology.
problem Defining a Poisson bracket on the space of Poisson structures.
method Constructing a Poisson bracket on P(M) depending on a volume form, and defining invariant of Poisson structures. result Invariant of Poisson structures detects unimodularity and related Poisson bracket for symplectic structures.
Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find \emph{all} fixed poi…
IGNN captures long-range graph dependencies using fixed-point equations.
problem Limited GNN ability to capture long-range graph dependencies.
method Fixed-point equilibrium equations involving implicitly defined state vectors, leveraging Perron-Frobenius theory and projected gradient descent.
result IGNN consistently captures long-range dependencies and outperforms state-of-the-art GNNs.
New approach to analyze matrix denoising using gradient flow and fixed point equations.
problem Positive semi-definite matrix denoising in extensive-rank and high-dimensional settings.
method Gradient flow and fixed point equations derived from linear pencil techniques of random matrix theory.
result Continuous phase transitions in the extensive-rank and high-dimensional regime.
New method stabilizes DEQ models by regularizing Jacobian of fixed-point equations.
problem Stability and performance of DEQ models.
method Jacobian regularization to stabilize DEQ models.
result Significant stabilization of fixed-point convergence in DEQ models.
FNO-DEQ solves steady-state PDEs as fixed points, outperforming traditional FNOs.
problem Lack of understanding in designing neural network architectures for PDEs.
method Proposes FNO-DEQ, a deep equilibrium architecture that solves steady-state PDEs as fixed points.
result FNO-DEQ outperforms FNO-based architectures in predicting solutions to steady-state PDEs.
New method handles unknown task boundaries in continual learning.
problem Catastrophic forgetting in neural networks.
method Fixed-point equations for online variational Bayes optimization.
result Approximates online Bayes update for non-stationary data.
Study on Navier-Stokes equations on non-compact manifolds, proving existence and decay of solutions.
problem Existence and asymptotic behavior of solutions to Navier-Stokes equations on non-compact manifolds.
method Used Lp−Lq-dispersive and smoothing estimates of the Stokes semigroup, fixed point arguments, and Gronwall's inequality. result Established existence and exponential decay of almost periodic and asymptotically almost periodic mild solutions.
Study compares methods for computing hypergradients in machine learning problems.
problem Computing exact hypergradients in machine learning is difficult.
method Investigates reverse mode iterative differentiation and approximate implicit differentiation methods.
result Unified analysis provides iteration complexity bounds and hierarchy of methods.
VR-GHAL method solves stochastic fixed-point equations with high probability.
problem Solving stochastic fixed-point equations in normed spaces with nonexpansive or contractive operators.
method VR-GHAL, a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces, using clipped stochastic differences.
result The method achieves a high-probability residual bound, reducing the residual nearly geometrically across epochs.
Paper proves new method for constructing initial data in general relativity.
problem Proving the existence of solutions for initial data in general relativity.
method Using the Banach fixed point theorem to prove existence, with guarantees of uniqueness and explicit construction.
result Guaranteed uniqueness and explicit construction of solutions to the conformal method equations.
The paper solves a complex financial optimization problem using a novel mathematical technique.
problem Optimizing portfolio selection in financial markets.
method Maximal monotone operator method and Riccati transformation.
result Existence and uniqueness of a solution to the transformed parabolic equation in a Sobolev space.
We obtain general formulae expressing Hirzebruch genera of a manifold with Z/p-action in terms of invariants of this action (the sets of weights of fixed points). As an illustration, we consider numerous particular cases of well-known genera, in particular, the elliptic genus. We also describe the connection with the s…
We consider vector fixed point (FP) equations in large dimensional spaces involving random variables, and study their realization-wise solutions. We have an underlying directed random graph, that defines the connections between various components of the FP equations. Existence of an edge between nodes i, j implies the …
We examine the fixed points to first-order RG flow of a non-linear sigma model with background metric, dilaton and tachyon fields. We show that on compact target spaces, the existence of fixed points with non-zero tachyon is linked to the sign of the second derivative of the tachyon potential V′′(T) (this is the anal…
Paper tackles time inconsistency in portfolio management with stochastic volatility and power utility.
problem Time inconsistency in portfolio management with stochastic volatility and power utility.
method Extended Hamilton Jacobi Bellman (HJB) equation, fixed point iteration, and linear parabolic PDE.
result Subgame perfect strategies are characterized and solved through numerical experiments.
We consider a stochastic control problem with the assumption that the system is controlled until the state process breaks the fixed barrier. Assuming some general conditions, it is proved that the resulting Hamilton Jacobi Bellman equations has smooth solution. The aforementioned result is used to solve the optimal div…
The principle of convergence stability for geometric flows is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points. It implies that if the flow from an initial state g0 exists for all time and converges to a stable fixed point, then the flows of solutions…
Let P be a principal U(1)-bundle over a closed manifold M. On P, one can define a modified version of the Ricci flow called the Ricci Yang-Mills flow, due to these equations being a coupling of Ricci flow and the Yang-Mills heat flow. We use maximal regularity theory and ideas of Simonett concerning the asymptoti…
Core-Halo solves large-scale fixed-point problems by decentralizing updates.
problem Large-scale fixed-point equations with block dependencies.
method Core-Halo decomposition separates write ownership from read-only context, aligning with block-dependence structure.
result Core-Halo achieves near-centralized performance while retaining parallelism.
We use the energy gap result of pure Yang-Mills equation [Feehan P.M.N., Adv. Math. 312 (2017), 547-587, arXiv:1502.00668] to prove another energy gap result of complex Yang-Mills equations [Gagliardo M., Uhlenbeck K., J. Fixed Point Theory Appl. 11 (2012), 185-198, arXiv:1401.7366], when Riemannian manifold X of dim…
Paper defines when early exercise of American options is optimal under negative rates.
problem Determining optimal exercise times for American options with negative interest rates.
method Developed a new integral equation to price options and find exercise boundaries under negative rates, using modified fixed point method.
result Successfully developed and validated a new algorithm for pricing American options under negative rates.
We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance to a fixed point in M. We are, in particular, interested in finding optimal (or…
We characterize the price of an Asian option, a financial contract, as a fixed-point of a non-linear operator. In recent years, there has been interest in incorporating changes of regime into the parameters describing the evolution of the underlying asset price, namely the interest rate and the volatility, to model sud…
L-CNNs approximate gauge actions, revealing fixed points with no lattice artifacts.
problem Approximating gauge actions with lattice artifacts.
method Lattice gauge-equivariant convolutional neural networks (L-CNNs).
result L-CNNs provide fixed point actions with no lattice artifacts.
Model financial network dynamics to avoid systemic risk.
problem Avoid systemic risk in financial networks.
method Model financial network as random liability graph, agents adapt strategies based on learning, analyze using ODE.
result Emerging strategies converge to evolutionary stable strategies (all risky or all less risky agents).
New TD algorithms stabilize RL tasks by reformulating updates into fixed point equations.
problem TD learning's sensitivity to step size specification.
method Implicit TD algorithms reformulate TD updates into fixed point equations.
result Implicit TD algorithms are more stable and less sensitive to step size.
Local Neural Operators enable efficient system-level analysis of complex PDEs.
problem System-level analysis of large-scale dynamical systems using neural operators.
method Integrating local Neural Operators with Krylov subspace iterative methods for stability and bifurcation analysis.
result Demonstrated effectiveness of local Neural Operators in fixed-point, stability, and bifurcation analysis of nonlinear PDEs.
Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projecti…
We complement a recent work on the stability of fixed points of the CMC-Einstein-Λ flow. In particular, we modify the utilized gauge for the Einstein equations and remove a restriction on the fixed points whose stability we are able to prove by this method, and thereby generalize the stability result. In addition, we…
Constructs solutions of Einstein equations for black holes gluing along timelike geodesics.
problem Constructing solutions of Einstein equations for black holes gluing along timelike geodesics.
method Constructs solutions gε of the Einstein equations describing a mass ε Kerr black hole traveling along a timelike geodesic C. result Constructs true solutions gε of the Einstein equations for black holes gluing along timelike geodesics. Study curve shortening flow on Riemann surfaces with conical singularities.
problem Evolution of curves on Riemann surfaces with singular points.
method Curve shortening flow governed by a degenerate quasilinear parabolic equation.
result Evolving curves stay fixed at singular points and show collapsing and convergence results.
New method estimates Schrödinger bridge potentials via empirical risk minimization.
problem Estimating Schrödinger bridge potentials from samples.
method Rewriting Schrödinger system as a fixed-point equation and estimating the potential via empirical risk minimization.
result Uniform concentration of empirical risk around population counterpart under sub-Gaussian assumptions.
In this paper we prove a lower bound for the least number of one-periodic solutions of nondegenerate locally Hamiltonian equations on compact symplectic manifolds in terms of the Betti numbers of the Novikov homology associated to the Calabi invariant of the locally Hamiltonian equations. Our result improves lower boun…
New findings on Helmholtz equation solutions show exponential growth in constant for three ball inequality.
problem Analyzing solutions of Helmholtz equation on different manifolds.
method Examining the three ball inequality for solutions of Helmholtz equation on Rn, Sn, or Hn. result The constant in the three ball inequality grows exponentially with the wave number.
The paper offers a framework to analyze machine learning problems using concentration of measure.
problem Analyzing machine learning algorithms defined by implicit equations.
method Develops a concentration of measure framework to solve convex problems and implicit formulations.
result Provides precise estimations for the first moments of the solution, describing the behavior and performance of machine learning classifiers.
We classify SIC-POVMs of rank one in CP^2, or equivalently sets of nine equally-spaced points in CP^2, without the assumption of group covariance. If two points are fixed, the remaining seven must lie on a pinched torus that a standard moment mapping projects to a circle in R^3. We use this approach to prove that any S…
Consider the following variational problem: among all curves in Rn of fixed length with prescribed end points and prescribed tangents at the end points, minimise the L∞-norm of the curvature. We show that the solutions of this problem, and of a generalised version, are characterised by a system of d…