Improved numerical solution for BSDEs with reduced boundary errors.
problem Boundary errors in numerical solution of BSDEs.
method Modified damping and shifting schemes to transform target function into a bounded periodic function, applying Fourier transforms.
result Significant reduction in boundary errors with improved accuracy and convergence.
Method detects errors in numerical data using regression models.
problem Noise and errors in numerical datasets.
method Introduced veracity scores and a filtering procedure for error detection.
result Method outperforms other approaches in identifying incorrect values.
Deep learning method improves numerical approximation of FBSDEs with jumps.
problem Improving numerical solutions for FBSDEs with jumps.
method Deep learning-based approach for decoupled FBSDEs with jumps.
result A priori and a posteriori error estimates for finite and infinite activity cases.
In this paper we propose a new kind of high order numerical scheme for backward stochastic differential equations(BSDEs). Unlike the traditional θ-scheme, we reduce truncation errors by taking θ carefully for every subinterval according to the characteristics of integrands. We give error estimates of this nonlinear…
The study analyzes numerical stability in large language models using mixed-precision arithmetic.
problem Numerical stability of large language models using low-precision arithmetic.
method Developed a mixed-precision analysis of transformer inference, deriving bounds for condition numbers and forward error.
result Established that numerical stability is determined by the interplay between weight magnitude and the growth of the residual stream.
Estimates domain truncation error for option pricing PDEs.
problem Estimating error in option pricing models with domain truncation.
method Derives an estimate of domain truncation error for a multidimensional PDE system.
result Proposes a sharper error estimate for option pricing models.
A fast Monte Carlo method for additive processes and option pricing.
problem Efficiently pricing path-dependent options with additive processes.
method Developed a fast Monte Carlo scheme for additive processes, analyzing and reducing numerical error sources.
result Shows significant reduction in error (1 bp or below) for pricing path-dependent options.
A research frontier has emerged in scientific computation, wherein numerical error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly dete…
New findings show score matching's accuracy doesn't ensure numerical stability in diffusion sampling.
problem Numerical stability issues in diffusion sampling despite small forward-marginal error.
method Constructing a smooth score field with arbitrarily small forward-marginal L2 error, showing nonexplosive behavior and moments of every order. result Euler--Maruyama discretizations can converge in probability even when moments diverge, demonstrating failure of weak convergence.
In this paper, we treat the problem of evaluating the asymptotic error in a numerical integration scheme as one with inherent uncertainty. Adding to the growing field of probabilistic numerics, we show that Gaussian process regression (GPR) can be embedded into a numerical integration scheme to allow for (i) robust sel…
The literature on optimal reinsurance does not deal with how much the effectiveness of such solutions is degraded by errors in parameters and models. The issue is investigated through both asymptotics and numerical studies. It is shown that the rate of degradation is often O(1/n) as the sample size n of historical …
We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast F…
Study on error rates for approximating rough volatility models.
problem Simulation of rough volatility models with fractional Brownian motion.
method Analysis of weak error rates for numerical schemes, focusing on fBm and cubic test functions.
result Convergence rates for approximations are (3H+21)∧1 for exact left-point discretization and H+21 for hybrid schemes. Two-layer networks struggle with high frequencies due to numerical and computational limitations.
problem High frequency approximation and learning in shallow networks.
method Mathematical and computational analysis focusing on numerical error, computational cost, and stability.
result Explicit answers to fundamental computational issues in shallow networks' high frequency handling.
We provide a bound for the error committed when using a Fourier method to price European options when the underlying follows an exponential \levy dynamic. The price of the option is described by a partial integro-differential equation (PIDE). Applying a Fourier transformation to the PIDE yields an ordinary differential…
L-HNNs improve Bayesian inference by reducing gradient requirements and improving ESS.
problem Efficient Bayesian inference with complex target densities.
method Latent Hamiltonian Neural Networks (L-HNNs) with NUTS, incorporating online error monitoring.
result L-HNNs in NUTS with online error monitoring required 1--2 orders of magnitude fewer numerical gradients and improved ESS by an order of magnitude.
The paper analyzes numerical instability in variational flows and proposes a diagnostic method.
problem Numerical instability in variational flows affects sampling, density evaluation, and ELBO estimation.
method Treated variational flows as dynamical systems, used shadowing theory for theoretical guarantees, and developed a diagnostic procedure.
result Despite numerical instability, results from variational flows can be accurate enough for practical applications.
MEDIDA discovers model errors in chaotic systems using sparse regression and data assimilation.
problem Model errors in chaotic systems lead to significant discrepancies between model predictions and real-world states.
method MEDIDA combines Bayesian sparse regression and data assimilation to estimate and interpret model errors from noisy observations.
result MEDIDA successfully identifies different types of model errors in the chaotic Kuramoto-Sivashinsky system.
Improved method for numerical conformal mappings on complex domains.
problem Accurate and efficient computation of conformal mappings on multiply connected domains.
method Generalization and refinement of the conjugate function method using high-order finite element methods.
result Achieved accurate and efficient construction of boundary values for multiply connected domains.
We present a numerical scheme to calculate fluctuation identities for exponential Lévy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-L…
Corrects local error estimates for UBU integrator in SDEs, improving complexity guarantees.
problem Improper local error estimates in UBU integrator for SDEs.
method Reconciles theory with practice by correcting local error estimates.
result Stronger assumptions needed for O(d1/4ε−1/2) steps in Wasserstein-2 distance. CNN improves medium-range temperature forecasts with limited resources.
problem Limited computational resources for high-resolution temperature forecasts.
method CNN post-processing with ensemble NWP models for bias correction and spatial downscaling.
result High-resolution (5-km) surface temperature forecasts with lead times up to 5.5 days.
FBSJNN solves PIDEs and FBSDEJs with deep learning, offering theoretical and numerical efficiency.
problem Solving Partial Integro-Differential Equations and Forward-Backward Stochastic Differential Equations with Jumps.
method FBSJNN framework using a single neural network for both solution approximation and non-local integral.
result FBSJNN achieves numerical solutions with a relative error of 10−3, demonstrating efficiency. A new adaptive splitting method improves accuracy for Cox-Ingersoll-Ross model.
problem Improving numerical solution accuracy for Cox-Ingersoll-Ross model.
method Adaptive splitting method over deterministic and random meshes, with uniform moment bound and strong error results.
result Uniform moment bound and strong error results of order 1/4 in L1 and L2 for κθ>σ^2, and order 1 for large noise.
Integration of the form ∫a∞f(x)w(x)dx, where w(x) is either sin(ωx) or cos(ωx), is widely encountered in many engineering and scientific applications, such as those involving Fourier or Laplace transforms. Often such integrals are approximated by a numerical integration…
New methods reduce extrapolation errors in feature importance.
problem Flawed feature importance methods using unrestricted permutations lead to extrapolation errors.
method Three new approaches: conditional model reliance, Knockoffs with Gaussian transformation, and restricted ALE plot designs.
result Theoretical and numerical results show our strategies reduce/eliminate extrapolation.
Calibrated probabilistic solvers improve accuracy of ODE estimates.
problem Uncertainty in probabilistic ODE solutions is not well-calibrated for adaptive step sizes.
method Introduce and assess several calibration methods for probabilistic ODE solvers.
result Calibration methods interact efficiently with adaptive step-size selection, improving posteriors.
The sigma-point filters, such as the UKF, which exploit numerical quadrature to obtain an additional order of accuracy in the moment transformation step, are popular alternatives to the ubiquitous EKF. The classical quadrature rules used in the sigma-point filters are motivated via polynomial approximation of the integ…
Unified kernel framework extends to stochastic systems, improving numerical stability.
problem Extending kernel methods to stochastic dynamical systems with diffusion.
method Unified kernel framework, Feynman-Kac path-integral representations, collocation-based computational framework.
result Kernel equivalence under uniform ellipticity assumptions and improved numerical stability with moderate diffusion.
For the numerical solution of the American option valuation problem, we provide a script written in MATLAB implementing an explicit finite difference scheme. Our main contribute is the definition of a posteriori error estimator for the American options pricing which is based on Richardson's extrapolation theory. This e…
The aim of this article is to design a moment transformation for Student- t distributed random variables, which is able to account for the error in the numerically computed mean. We employ Student-t process quadrature, an instance of Bayesian quadrature, which allows us to treat the integral itself as a random variable…
We consider assets for which price Xt and squared volatility Yt are jointly driven by Heston joint stochastic differential equations (SDEs). When the parameters of these SDEs are estimated from N sub-sampled data (XnT,YnT), estimation errors do impact the classical option pricing PDEs. We estimate thes…
Neural ODEs' performance varies with numerical method, requiring adaptive step size control.
problem Neural ODEs' performance depends on the numerical method used during training.
method Proposes an adaptive step size control algorithm to ensure a valid ODE without increasing computational cost.
result Valid Neural ODEs require careful numerical method selection and step size adaptation.
New method reduces Monte Carlo error in option pricing and Greeks estimation.
problem Reducing Monte Carlo error in option pricing and Greeks estimation.
method Denoised Monte Carlo technique for LSV models.
result Reduces Monte Carlo error by an order of magnitude.
In this report, we derive a non-negative series expansion for the Jensen-Shannon divergence (JSD) between two probability distributions. This series expansion is shown to be useful for numerical calculations of the JSD, when the probability distributions are nearly equal, and for which, consequently, small numerical er…
Prior information can be incorporated in matrix completion to improve estimation accuracy and extrapolate the missing entries. Reproducing kernel Hilbert spaces provide tools to leverage the said prior information, and derive more reliable algorithms. This paper analyzes the generalization error of such approaches, and…
Improved pricing of vanilla options using modified Adams method and sinh-acceleration.
problem Calibration of rough Heston model leads to incorrect implied volatility surfaces.
method Modified Adams method and sinh-acceleration for Fourier inversion.
result Corrected implied volatility surface is significantly flatter and fits data poorly.
In this paper, we explore various statistical techniques for anomaly detection in conjunction with the popular Long Short-Term Memory (LSTM) deep learning model for transportation networks. We obtain the prediction errors from an LSTM model, and then apply three statistical models based on (i) the Gaussian distribution…
Analysts use vague language in reports to convey useful information about future payoffs.
problem Lack of precise numerical forecasts in analyst reports.
method Empirical analysis of analyst reports to assess the predictive power of linguistic tone.
result The textual tone of analyst reports has predictive power for forecast errors and subsequent revisions, especially when language is vague and uncertainty is high.
The accuracy of deep learning, i.e., deep neural networks, can be characterized by dividing the total error into three main types: approximation error, optimization error, and generalization error. Whereas there are some satisfactory answers to the problems of approximation and optimization, much less is known about th…
New method uses extreme value theory to estimate neural network errors.
problem Quantifying the error of neural networks, especially for large values.
method Applying extreme value theory to approximate the distribution of error.
result Developed a new estimator for the shape parameter of the Pareto distribution.
Study on CVA in volatility models, including rough volatility.
problem Calculating CVA in fractional and rough volatility models.
method General representation formula, specialized for volatility models, numerical and theoretical error analysis.
result Roughness influences the claim's price, and provides accurate approximations.
We find approximate solutions of partial integro-differential equations, which arise in financial models when defaultable assets are described by general scalar Lévy-type stochastic processes. We derive rigorous error bounds for the approximate solutions. We also provide numerical examples illustrating the usefulness a…
Differentially private (DP) machine learning has recently become popular. The privacy loss of DP algorithms is commonly reported using (ε,δ)-DP. In this paper, we propose a numerical accountant for evaluating the privacy loss for algorithms with continuous one dimensional output. This accountant can be appl…
This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the valu…
Deep learning improves probabilistic PPDE solution accuracy.
problem Approximating solutions to path-dependent PDEs with limited basis selection.
method Deep learning for conditional expectation estimation with error bounds.
result Deep learning yields more accurate PPDE solutions, especially in high dimensions.
This paper examines error bounds for deep learning classifiers with noisy labels.
problem Understanding the performance of classifiers trained on noisy data.
method Derives error bounds for excess risk, decomposing it into statistical and approximation errors. Uses independent block construction for statistical dependencies and vector-valued setting for approximation error.
result Established theoretical results for error bounds in deep learning with noisy labels, mitigating the impact of high-dimensional input spaces.
Unified framework for blending ML and mechanistic models in dynamical systems.
problem Learning dynamical systems from noisy, partially observed data.
method A unifying framework that combines mechanistic and machine learning approaches.
result Proves that hybrid models can learn memory-dependent model error.