Study finds range-based volatility estimators are more predictable than the standard close-to-close estimator.
problem Comparing predictability of range-based volatility estimators to the standard close-to-close estimator.
method Used LSTM recurrent neural networks to analyze patterns of volatility changes in the Dow Jones Industrial Average index.
result Changes in range-based volatility estimators are more predictable than the standard close-to-close estimator.
Paper connects CR geometry conditions to closed range of ∂ˉ-operator.
problem Establishing closed range of the ∂ˉ-operator on CR manifolds. method Defined third and fourth order CR invariants and used them to show closed range for ∂ˉ-Laplacian. result Third and fourth order CR invariants provide sufficient conditions for closed range of ∂ˉ-operator. Sharp fractional Sobolev inequalities on closed manifolds identified.
problem Critical fractional Sobolev embedding on closed Riemannian manifolds.
method Intrinsic heat-kernel based framework, determining optimal coefficients, proving sharp inequalities.
result Sharp p-power inequality and almost sharp inequality established. This paper extends homological stability results for configuration spaces of manifolds.
problem Homological stability of configuration spaces of manifolds.
method Analyzing the cohomology of configuration spaces of manifolds, focusing on stability in odd and even degrees.
result The stable range for homology groups of configuration spaces depends on the dimension of the manifold and the number of configuration points.
Simple curve in 3D sums to cube, non-rectifiable.
problem Constructing a simple closed curve in 3D whose convex hull equals half-sum of the curve with itself.
method Constructing a specific curve in R^3 and proving its properties.
result A simple closed curve in R^3 exists whose convex hull equals half-sum of the curve with itself.
New invariant for classifying 4-manifolds up to cobordism.
problem Classifying closed, oriented topological 4-manifolds up to s-cobordism. method Introducing a stable range invariant after stabilization by a fixed number of S2imesS2. result A new invariant for classifying 4-manifolds up to s-cobordism. Extended univariate Range Value-at-Risk to multivariate settings.
problem Inability of traditional risk measures for heavy-tail distributions and infinite tail expectations.
method Multivariate definitions of robust truncated tail expectations, robustness and properties derived, closed-form expressions and special cases discussed.
result Empirical estimators accuracy examined through numerical and graphical examples.
Geometrically studies Moore-Penrose inverse and polar decomposition continuity.
problem Continuity of Moore-Penrose inverse for perturbations by operator ideals.
method Geometric construction using essential codimension and Banach-Lie group action.
result Moore-Penrose inverse is a real analytic map between manifolds.
Method approximates first passage times for birth-death processes.
problem Approximating first passage times for birth-death processes.
method General method using birth-death process properties, Keilson's theorem, and Riemann sums.
result Closed-form expressions for first passage times.
We give a complete description of the closure of the space of one-generator closed subgroups of PSL2(R) for the Chabauty topology, by computing explicitly the matrices associated with elements of Aut(D) = PSL2(R), and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions …
Let M be a quasi-Fuchsian three-manifold that contains a closed incompressible surface with principal curvatures within the range of the unit interval, for a prescribed function H (with mild conditions) on M, we construct a closed incompressible surface with mean curvature H . A direct application is the existe…
Proves triviality of inertia groups in high-dimensional manifolds.
problem Classifying manifolds in the metastable range.
method Understanding the second extended power functor in synthetic spectra.
result Inertia groups of high-dimensional manifolds are trivial.
Proving that next-token prediction makes language models generate coherent long documents.
problem Understanding why language models generate coherent documents despite focusing on next-token prediction.
method Proving the power of next-token prediction in learning longer-range structure using Recurrent Neural Networks (RNN).
result Optimizing next-token prediction in RNNs yields a model that closely approximates the training distribution, even for long-range coherence.
In this note we explain how the computation of the spectrum of the lamplighter group from \cite{Grigorchuk-Zuk(2000)} yields a counterexample to a strong version of the Atiyah conjectures about the range of L2-Betti numbers of closed manifolds.
The paper introduces closed-form expressions for interpreting Tsetlin Machines.
problem Interpreting complex Tsetlin Machines with a large number of clauses.
method Developed closed-form expressions for local and global interpretability of Tsetlin Machines.
result The expressions enable real-time feature importance assessment and data clustering.
In this note we show that for any proper action of a Banach--Lie group G on a Banach manifold M, the corresponding tangent maps $\g \to T_x(M)$ have closed range for each x∈M, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $…
Alternative closed-form formula for spread call option prices under log-normal models.
problem Valuation of spread call options under log-normal models.
method Developed an alternative closed-form formula for spread call option prices.
result Our formula performs better for certain range of model parameters than existing closed-form formula.
Study calculates homotopy groups and derivatives for disc diffeomorphisms.
problem Understanding the homotopy groups of diffeomorphisms of discs.
method Computes rational homotopy groups and uses Weiss' orthogonal calculus.
result Determines optimal rational concordance stable range for high-dimensional discs.
Paper presents new expansions for option pricing with cash dividends.
problem No exact formula for European options with cash dividends.
method Uses Etore and Gobet's technique for piecewise lognormal process with jumps.
result Provides more robust first, second, and third-order expansions.
New model incorporates long-range dependence in mortality rates for better valuation and risk management.
problem Lack of appropriate models for valuing and managing mortality securities with long-range dependence.
method Proposes a novel class of Volterra mortality models that incorporate LRD, derived in closed-form solution.
result Models provide flexibility and tractability for valuing and hedging mortality-related products.
We describe the range of a restricted spherical mean transform, which sends a function supported inside a closed ball in a hyperbolic space to its mean values on the geodesics spheres centered at the boundary of the ball. The description resembles that of the same transform on the Euclidean spaces obtained by Mark Agra…
DLGNet improves dialogue response generation by leveraging transformer architecture.
problem Lack of relevance, diversity, and coherence in dialogue responses.
method Transformer-based model for dialogue response generation, incorporating long-range structures and random paddings.
result Significant improvements over state-of-the-art models on multiple datasets.
New G_2-manifolds created from quotients of twisted connected sums.
problem Creating new G_2-manifolds with specific topological properties.
method Generalizing Kovalev's twisted connected sums by taking quotients of pieces before gluing.
result Realization of a wider range of topological types of G_2-manifolds.
Paper tackles distinguishing discrete distributions with local minimax rates.
problem Distinguishing between two discrete distributions when they are close in L1-norm. method Local minimax approach to adapt rates to distribution shapes.
result First local minimax rate for separation distance up to logarithmic factors.
New method produces coherent forecasts for long-range data.
problem Inaccurate and non-coherent forecasts on long-horizon data.
method Probabilistic forecasting with KL-divergence for coherent aggregates.
result Improves forecast performance across base levels and aggregates.
Reconstructs geodesic ray transforms on simple Riemannian surfaces with connections.
problem Reconstructing geodesic ray transforms on Riemannian surfaces with connections.
method Derives reconstruction formulas and injectivity conditions for geodesic ray transforms with connections.
result Injectivity of geodesic ray transforms in a neighborhood of constant curvature metrics and non-unitary connections.
Uniswap markets perform well and closely track reference prices.
problem Understanding and validating the performance of Uniswap markets.
method Formal analysis and numerical simulation of constant product markets.
result Uniswap markets closely track reference prices under common conditions.
The study classifies homomorphisms from mapping class groups using finite subgroups.
problem Classifying homomorphisms from mapping class groups.
method Using finite subgroups to classify homomorphisms.
result Only finitely many mapping class groups have non-trivial homomorphisms into Homeo(S^n) for any n.
We use the expectation of the range of an arithmetic Brownian motion and the method of moments on the daily high, low, opening and closing prices to estimate the volatility of the stock price. The daily price jump at the opening is considered to be the result of the unobserved evolution of an after-hours virtual tradin…
Paper introduces robust Gaussian process regression without sacrificing computational efficiency.
problem Violation of independent and identically distributed Gaussian observation noise assumption in Gaussian process regression.
method Proves robust and conjugate Gaussian process regression (RCGP) at no additional cost using generalised Bayesian inference.
result RCGP enables exact conjugate closed form updates in all settings where standard GPs admit them.
The paper studies efficient Hessian fitting methods for stochastic optimization.
problem Efficient Hessian fitting for stochastic optimization.
method Preconditioned Stochastic Gradient Descent (PSGD) method and Lie groups.
result Hessian fitting problem is strongly convex in certain Lie groups.
Study finds financial market data follows power-law exponents typical of stochastic processes.
problem Testing long-range memory in financial markets.
method Analyzed empirical return and trading activity time series from Forex.
result Power-law exponents of burst and inter-burst duration probability density functions are close to 3/2.
Increasingly, a huge amount of statistics have been gathered which clearly indicates that income and wealth distributions in various countries or societies follow a robust pattern, close to the Gibbs distribution of energy in an ideal gas in equilibrium. However, it also deviates in the low income and more significantl…
Constructs operations on stable moduli spaces to compare manifold cohomology.
problem Comparing cohomology of moduli spaces of closed manifolds.
method Constructs operations on stable moduli spaces and uses them to compare cohomology.
result Obtains isomorphisms in a stable range for all primes not invertible in coefficients.
Doodles on surfaces expanded from 2-sphere to any closed orientable surface.
problem Extending doodles from 2-sphere to any closed orientable surface.
method Introduced and proved uniqueness of minimal representatives, gave examples, and introduced virtual doodles.
result Natural one-to-one correspondence between doodles on surfaces and virtual doodles on the plane.
An almost Fuchsian manifold is a quasi-Fuchsian hyperbolic three-manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1,1). In such a hyperbolic three-manifold, the minimal surface is unique and embedded, hence one can parametrize these three-manifolds by…
We develop closed-form approximations for European put options under stochastic volatility models.
problem Tackling the pricing of European put options under stochastic volatility models with time-dependent parameters.
method Using a second-order Taylor expansion around the mean of the argument, we write the option price as an expectation of a Black-Scholes formula. We then simplify the resulting expectations and derive closed-form pricing formulas under the assumption of piecewise-constant parameters.
result We derive closed-form pricing formulas and bounds on the remainder term generated by the Taylor expansion, showing that the errors are well within acceptable ranges for practical applications.
A well known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface Mn in the unit sphere Sn+1(1) is just its dimension n. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that t…
In this paper we present an algorithm for pricing barrier options in one-dimensional Markov models. The approach rests on the construction of an approximating continuous-time Markov chain that closely follows the dynamics of the given Markov model. We illustrate the method by implementing it for a range of models, incl…
The local Hurst exponent, a measure employed to detect the presence of dependence in a time series, may also be used to investigate the source of intraday variation observed in the returns in foreign exchange markets. Given that changes in the local Hurst exponent may be due to either a time-varying range, or standard …
Paper optimizes liquidity provision in decentralized finance markets.
problem Strategic LPs face predictable losses and concentration risk in CL pools.
method Derive optimal liquidity provision strategy based on fees, PL, and concentration risk.
result Optimal strategy increases fee revenue and profit from marginal rate changes.
Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. Th…
We model a closed economic system with interactions that generates the features of empirical wealth distribution across all wealth brackets, namely a Gibbsian trend in the lower and middle wealth range and a Pareto trend in the higher range, by simply limiting the an agents' interaction to only agents with nearly the s…
Hedging in the presence of transaction costs leads to complex optimization problems. These problems typically lack closed-form solutions, and their implementation relies on numerical methods that provide hedging strategies for specific parameter values. In this paper we use a genetic programming algorithm to derive exp…
We report an empirical study of the Ibovespa index of the Sao Paulo Stock Exchange in which we detect the existence of long-range correlations. To analyze our data we introduce a rescaled variant of the usual Detrended Fluctuation Analysis that allows us to obtain the Hurst exponent through a one-parameter fitting. We …
We discuss various compatibility criteria for overdetermined systems of PDEs generalizing the approach to formal integrability via brackets of differential operators. Then we give sufficient conditions that guarantee that a PDE possessing a Lie algebra of symmetries has invariant solutions with respect to this Lie alge…
Study on autoencoder denoising in high dimensions.
problem Denoising data from Gaussian mixtures.
method Two-layer non-linear autoencoder with skip connection in high-dimensional limit.
result Closed-form expressions for denoising mean-squared test error.
Unified treatment of stability problems in geometry and analysis.
problem Spherical closeness of hypersurfaces under geometric constraints.
method Estimate relating distance to geodesic spheres with norms of traceless Hessian operator.
result Unified treatment of stability problems in geometry and analysis.