Study shows how heat leaks from material sets in low diffusivity scenarios.
problem Understanding heat leakage from material sets in low diffusivity limits.
method Generalized leading-order asymptotics for time-dependent diffusion processes.
result Diffusive transport out of a material set is proportional to the surface area of the set boundary.
Paper develops semi-analytic method for American options in time-dependent jump-diffusion models.
problem Pricing American options in models with time-dependent and exponential jumps.
method Generalizes existing methods for barrier and American options to handle arbitrary time dependencies and solves the problem through algebraic and Fredholm-Volterra equations.
result Presents a semi-analytic solution for American options in time-dependent jump-diffusion models with exponential jumps.
We propose the time-dependent generalization of an `ordinary' autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds derived from the extended musculo-skeletal configuration manifold. The …
The paper justifies time-dependent loss reweighting schemes for flow matching and diffusion models.
problem Theoretical justification for time-dependent loss reweighting schemes in flow matching and diffusion models.
method Clarifies that the loss can depend on both time and state, and shows theoretical justification for time-dependent loss weighting schemes.
result Time-dependent loss weighting schemes are theoretically justified for Generator Matching and Edit Flows.
The paper proposes a time-dependent Markov model for a limit order book.
problem Understanding the convergence of a limit order book to a more complex diffusion.
method A simple time-dependent Markov model is proposed, describing the arrival of different orders.
result Empirical studies verify the validity of the modeling assumptions for certain stocks.
We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection-diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or h…
We study the problem of non-explosion of diffusion processes on a manifold with time-dependent Riemannian metric. In particular we obtain that Brownian motion cannot explode in finite time if the metric evolves under backwards Ricci flow. Our result makes it possible to remove the assumption of non-explosion in the pat…
The latter author, together with collaborators, proposed a numerical scheme to calculate the price of barrier options. The scheme is based on a symmetrization of diffusion process. The present paper aims to give a mathematical credit to the use of the numerical scheme for Heston or SABR type stochastic volatility model…
New reweighted losses improve diffusion model training and image quality.
problem Training and improving diffusion models for image generation.
method Constructing a cascade of time-dependent variational lower bounds.
result Significant improvements in pixel-space image modeling quality.
CDM models counterfactual outcomes in longitudinal data with improved accuracy.
problem Predicting counterfactual outcomes in longitudinal data with complex time-dependent confounding.
method Causal Diffusion Model (CDM) using denoising diffusion architecture with relational self-attention.
result CDM outperforms state-of-the-art methods in generating full probabilistic distributions of counterfactual outcomes.
We present a novel approximate inference method for diffusion processes, based on the Wasserstein gradient flow formulation of the diffusion. In this formulation, the time-dependent density of the diffusion is derived as the limit of implicit Euler steps that follow the gradients of a particular free energy functional.…
Efficient semi-analytic methods for pricing double barrier options with time-dependent parameters.
problem Pricing and calibration of double barrier options with time-dependent parameters.
method Two approaches: General Integral transform method and Heat Potential method.
result Semi-analytic techniques are more efficient for pricing double barrier options than traditional numerical methods.
This paper gives a brief overview on the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inferences of instantaneous returns and volatility functions of time-homogeneous and time-dependent diffusion processes, and estimation of transition densities and st…
Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Emery's curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distribut…
In this paper the unconditional stability of four well-known ADI schemes is analyzed in the application to time-dependent multidimensional diffusion equations with mixed derivative terms. Necessary and sufficient conditions on the parameter theta of each scheme are obtained that take into account the actual size of the…
A time-dependent double-barrier option is a derivative security that delivers the terminal value φ(ST) at expiry T if neither of the continuous time-dependent barriers $b_\pm:[0,T]\to \RR_+$ have been hit during the time interval [0,T]. Using a probabilistic approach we obtain a decomposition of the barrier opti…
We consider Feller mean-reverting square-root diffusion, which has been applied to model a wide variety of processes with linearly state-dependent diffusion, such as stochastic volatility and interest rates in finance, and neuronal and populations dynamics in natural sciences. We focus on the statistical mixing (or sup…
The financial market is nonpredictable, as according to the Bachelier, the mathematical expectation of the speculator is zero. Nevertheless, we observe in the price fluctuations the two distinct scales, short and long time. Behaviour of a market in long terms, such as year intervals, is different from that in short ter…
A new diffusion model tackles brightness issues with a probabilistic approach.
problem Brightness-related limitations in diffusion models.
method Introduces a novel diffusion model with a probabilistic framework, modifying both forward and reverse diffusion processes.
result The model mitigates brightness-related limitations and improves performance in high-dimensional settings.
Extends diffusion-based Schrödinger bridge models to handle time-dependent potentials.
problem Approximating optimal transport dynamics between two boundary distributions with a twisted Brownian motion reference.
method Introduces Twisted Schrödinger Bridge Matching (TSBM) using the Iterative Markovian Fitting (IMF) paradigm, incorporating a gradient-dependent bridge-matching loss.
result Improves trajectory inference across high-dimensional settings, including crowd navigation and single-cell data.
Researchers find the optimal exercise time for American options using a specific type of diffusion process.
problem Finding the optimal time to exercise American options with a time-dependent Ornstein-Uhlenbeck process.
method Optimal stopping problem, probabilistic arguments, non-linear Volterra-type integral equation, Picard iteration algorithm.
result They derive a non-linear Volterra-type integral equation and prove the exercise boundary's Lipschitz continuity and differentiability almost everywhere.
Simulates financial market orders using anomalous diffusion models.
problem Anomalous diffusion in financial market order dynamics.
method Discrete Time Random Walk with Sibuya waiting times, non-uniform sampling, and cubic spline interpolation.
result Demonstrates price impact for different forcing functions and model parameters.
We propose a simple stochastic model for the dynamics of a limit order book, extending the recent work of Cont and de Larrard (2013), where the price dynamics are endogenous, resulting from market transactions. We also show that the conditional diffusion limit of the price process is the so-called Brownian meander.
NDMs enable non-linear transformations in diffusion models for better generative tasks.
problem Limited to linear transformations, diffusion models struggle with generative tasks.
method Presented NDMs that allow time-dependent non-linear transformations.
result NDMs outperform conventional diffusion models in likelihood and sample quality.
We present an improved analysis of the Euler-Maruyama discretization of the Langevin diffusion. Our analysis does not require global contractivity, and yields polynomial dependence on the time horizon. Compared to existing approaches, we make an additional smoothness assumption, and improve the existing rate from $O(η)…
Improved sampling efficiency for inverse problems using variance-reduced diffusion methods.
problem Efficiently estimating noisy scores in inverse problems.
method Developed a nonparametric self-normalized importance sampling estimator and a state-dependent blending rule.
result Improved sample quality for fixed simulation budgets in synthetic targets and PDE-governed inverse problems.
A new method approximates the exact posterior score for diffusion models.
problem Training-free guidance of diffusion models for image restoration and inverse problems.
method Presented a novel expression for the exact posterior score, leveraging it to compute step sizes on the fly.
result Demonstrated competitive performance with fewer time steps compared to state-of-the-art techniques.
Study shows neural operators can efficiently solve complex reaction-diffusion systems.
problem Efficiently solving nonlinear reaction-diffusion systems using neural operators.
method Laplacian-based neural operators applied to a generalized Gierer-Meinhardt system.
result Explicit approximation error bounds established for neural operators in terms of network parameters.
New method calibrates DPMs to improve likelihood bounds.
problem Improving the likelihood bounds of DPMs.
method Deriving concentration bounds and using the optional stopping theorem for data scores to calibrate DPMs.
result Calibrated DPMs can increase likelihood bounds and improve sampling quality.
In this paper we apply Markovian approximation of the fractional Brownian motion (BM), known as the Dobric-Ojeda (DO) process, to the fractional stochastic volatility model where the instantaneous variance is modelled by a lognormal process with drift and fractional diffusion. Since the DO process is a semi-martingale,…
Paper improves American option valuation in complex models.
problem Valuation of American options in time-dependent jump-diffusion models.
method Integral equations and characteristic functions for explicit exercise boundary determination.
result Efficient and accurate pricing method for American options in various models.
The study characterizes diffusion model generalization using data-dependent ridge manifolds.
problem Understanding where diffusion model-generated samples lie when not memorizing the training set.
method Introduced a time-dependent family of log-density ridge manifolds to characterize reverse-time inference.
result Generated samples evolve by a reach-align-slide mechanism, controlled by normal and tangential components of training error.
Theory explains creativity in diffusion models generating novel images.
problem Diffusion models generate highly original images far from training data.
method Identified locality and equivariance as inductive biases to prevent optimal score-matching.
result Analytic models predict diffusion model outputs with high accuracy.
Study methods to recover unknown processes in PDEs from data.
problem Identifying unknown processes in time-dependent PDEs using observational data.
method Theoretical analysis and numerical approaches including Galerkin and collocation algorithms.
result The Galerkin algorithm is more suitable for practical situations with noisy data.
Parallel score matching accelerates DPM training and improves density estimation.
problem Extended training periods and limited modeling flexibility in DPMs.
method Partitioning the learning task into independent time sub-intervals and modeling the score at each time point separately.
result Significant acceleration of training process and improved density estimation performance.
We characterise the value function of the optimal dividend problem with a finite time horizon as the unique classical solution of a suitable Hamilton-Jacobi-Bellman equation. The optimal dividend strategy is realised by a Skorokhod reflection of the fund's value at a time-dependent optimal boundary. Our results are obt…
The paper studies large deviation principles for stochastic volatility models with reflection, focusing on binary barrier options and call prices.
problem Large deviation principles for stochastic volatility models with reflection.
method Sample path and small-noise large deviation principles for the log-price process.
result Asymptotic behavior of binary barrier options and call prices in the small-noise regime.
G-FuNK learns solutions for nonlinear PDEs on multiple domains and parameters.
problem Predicting time-dependent dynamics of complex systems governed by nonlinear PDEs with varying parameters and domains.
method Graph Fourier Neural Kernels combining domain-adapted and transferable components for non-diffusive and diffusive terms.
result G-FuNK achieves low relative errors on unseen domains and fiber fields, significantly accelerating predictions.
Probabilistic proof of smooth boundaries in optimal stopping problems.
problem Continuous differentiability of time-dependent optimal boundaries in optimal stopping problems.
method Local probabilistic arguments for a wider range of conditions.
result First probabilistic proof of continuous differentiability under general conditions.
Analyzed a generalized voter model with power-law herding intensity, revealing anomalous diffusion and long-range memory.
problem Anomalous diffusion and long-range memory in a generalized voter model.
method Derived analytical expressions for moments and first passage time distribution, confirmed numerically.
result The model exhibits long-range memory indicators despite being a Markov model.
The volatility characterizes the amplitude of price return fluctuations. It is a central magnitude in finance closely related to the risk of holding a certain asset. Despite its popularity on trading floors, the volatility is unobservable and only the price is known. Diffusion theory has many common points with the res…
The paper analyzes the probabilistic structure of DDPMs and bounds their sampling error.
problem Understanding and controlling errors in discrete-time DDPMs.
method Structural analysis of score functions, Schrödinger's problem, and FBSDEs.
result Explicit upper bound for total variation distance between sampling and target distributions.
This work develops robust diffusion recursive least squares algorithms to mitigate the performance degradation often experienced in networks of agents in the presence of impulsive noise. The first algorithm minimizes an exponentially weighted least-squares cost function subject to a time-dependent constraint on the squ…
This paper introduces a linear state-space model with time-varying dynamics. The time dependency is obtained by forming the state dynamics matrix as a time-varying linear combination of a set of matrices. The time dependency of the weights in the linear combination is modelled by another linear Gaussian dynamical model…
Study on uniquely determining thermal properties from boundary temperature and heat flux measurements.
problem Determine thermal conductivity and volumetric heat capacity from boundary measurements.
method Uniqueness proof for isotropic and anisotropic media under thermal diffusivity assumption.
result Uniqueness of thermal properties in all dimensions and up to a gauge in two dimensions.
New boundary treatment improves accuracy for complex PDEs.
problem Order reduction in high-order IMEX schemes for multidimensional PDEs.
method Novel boundary treatment algorithms for Cartesian meshes, treating implicit-explicit stages similarly to interior points.
result Recovery of designed order of convergence by numerical verification.
Develops a new method for pricing GMWBs with jumps and stochastic interest rates.
problem Pricing guaranteed minimum withdrawal benefits (GMWBs) with jumps and stochastic interest rates.
method Combines semi-Lagrangian method with Fourier pricing and Green's function.
result Mathematically demonstrates convergence to the viscosity solution of the HJB-QVI.
Study of time-dependent metrics and connections in geometry.
problem Understanding geodesics and connections in time-dependent Riemannian manifolds.
method Examine connections on product manifolds, explore parallel transport, geodesics, and torsion.
result Define the derivative of a one-parameter family of connections.