arXiv research

We develop a comprehensive mathematical framework for polynomial jump-diffusions in a semimartingale context, which nest affine jump-diffusions and have broad applications in finance. We show that the polynomial property is preserved under polynomial transformations and Lévy time change. We present a generic method for…

Develops polynomial diffusion models for multi-factor commodity futures dynamics.

The paper examines differential smoothness in specific algebra types.

Infinite dimensional measure-valued processes modeled as polynomial diffusions.

We provide new examples of diffusion operators in dimension 2 and 3 which have orthogonal polynomials as eigenvectors. Their construction rely on the finite subgroups of O(3) and their invariant polynomials.

PolyNSD improves Neural Sheaf Diffusion with polynomial operators and spectral rescaling.

PDSim simulates and estimates commodity futures prices using polynomial diffusion models.

Derives a series expansion for Asian option pricing with polynomial jump-diffusion moments.

The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming--Viot process is a particular example. The defining property of finite dimensional polynomial processes considered by Cuchiero et al. (2012) and Filipovic and Larsson (2016) is transferred to this infinite …

The model uses signatures to accurately calibrate SPX and VIX options without jumps or rough volatility.

New method uses Hermite polynomials for American option valuation.

In this paper we study the pricing and hedging problem of a portfolio of life insurance products under the benchmark approach, where the reference market is modelled as driven by a state variable following a polynomial diffusion on a compact state space. Such a model guarantees not only the positivity of the OIS short …

This work extends diffusion models to handle heavy-tailed targets, improving score estimation and sampling guarantees.

Develops new bounds for deterministic samplers in diffusion models.

Algorithm learns mixtures of Gaussians efficiently using diffusion models.

Combining diffusion models with Langevin dynamics improves posterior sampling efficiency.

New framework uses score-based priors to solve ill-conditioned polynomial equations, improving signal recovery from noisy data.

New sampling method improves efficiency for diffusion models.

Posterior sampling from diffusion models is computationally hard.

Improved sample complexity for training diffusion models.

In this article, we explore a class of tractable interest rate models that have the property that the price of a zero-coupon bond can be expressed as a polynomial of a state diffusion process. Our results include a classification of all such time-homogeneous single-factor models in the spirit of Filipovic's maximal deg…

ML-EM method speeds up diffusion model sampling.

We develop a new model for VIX derivatives with closed-form solutions.

Langevin dynamics fails to produce accurate samples even with small score function errors.

This work sets lower bounds on the number of score queries needed for diffusion sampling.

New method speeds up generative modeling without requiring diffusion steps.

New research shows some distributions hard to sample via diffusions.

Paper optimizes diffusion models for denoising tasks with theoretical guarantees.

This paper presents a novel one-factor stochastic volatility model where the instantaneous volatility of the asset log-return is a diffusion with a quadratic drift and a linear dispersion function. The instantaneous volatility mean reverts around a constant level, with a speed of mean reversion that is affine in the in…

In the setting of polynomial jump-diffusion dynamics, we provide an explicit formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula appears as a linear combination of exponentials of the generator matrix, extending the well-known moment formula for p…

Diffusion models converge linearly to complex data manifolds.

We introduce the {\it diffusion $K$-means} clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion $K$-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as …

Typically flat filling, linear or polynomial interpolation methods to generate missing historical data. We introduce a novel optimal method for recreating data generated by a diffusion process. The results are then applied to recreate historical data for stocks.

Efficiently learns mixtures of Gaussians without separation assumptions.

New bounds show diffusion models converge nearly linearly in data dimension.

New framework analyzes regret in guided diffusion for optimizing structured inputs.

New polynomial convergence guarantees for SGM on general data distributions.

Algorithm learns diffusion processes with high-dimensional state spaces.

We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. We establish parametric conditions which guarantee existen…

We extend the viscosity solution characterization proved in [5] for call/put American option prices to the case of a general payoff function in a multi-dimensional setting: the price satisfies a semilinear re-action/diffusion type equation. Based on this, we propose two new numerical schemes inspired by the branching p…

Study shows neural operators can efficiently solve complex reaction-diffusion systems.

We consider a Markov process $X$, which is the solution of a stochastic differential equation driven by a Lévy process $Z$ and an independent Wiener process $W$. Under some regularity conditions, including non-degeneracy of the diffusive and jump components of the process as well as smoothness of the Lévy density of $Z…

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)…

Unified analysis for deterministic samplers in diffusion models.

This work explores the generalization properties of diffusion models, providing theoretical and empirical insights.

Paper tackles sampling from non-log-concave distributions using denoising diffusion.

Identifies directed graphs from node measurements using polynomial filters.