New method learns SDEs without integrators, speeding up computation.
problem Computational expense in learning SDEs using neural networks.
method Importance-sampling estimator for SDEs, leveraging parallelism.
result Lower-variance gradient estimates and massive computation time reductions.
Study on the smoothness of solutions to a specific type of stochastic differential equation.
problem Regularity of solutions to mean-field G-SDEs. method Analysis of first and second order Fréchet differentiability in the random initial condition.
result Established the Fréchet differentiability of the solution and specified the corresponding equations.
Sig-SDE model integrates signatures with SDEs for financial data.
problem Calibrating models to exotic financial products with non-linear dependencies.
method Integrating signatures from stochastic analysis with neural SDEs.
result Sig-SDE provides theoretical guarantees for convergence.
A new machine learning method solves high-dimensional Kolmogorov PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs and SDEs.
method Stochastic weighted minimization and stochastic gradient descent with Malliavin weights.
result Accurate approximation of high-dimensional Kolmogorov PDEs and SDEs without curse of dimensionality.
Itô maps provide a method for any-step SDE integration.
problem Stochastic dynamics
method Itô map formulation
result Empirical results on synthetic and image-generation benchmarks
Paper extends Poincaré's work to stochastic differential equations.
problem Existence of first integrals in stochastic differential equations.
method Introduce two definitions of local first integrals for SDEs.
result Stochastic version of Poincaré non-integrability theorem.
New schemes for SDEs on manifolds keep solutions close to the manifold.
problem Solving SDEs constrained to manifolds in high accuracy.
method Geometrically invariant numerical schemes that remain close to the manifold.
result The schemes converge under standard assumptions and outperform existing methods.
New estimator for SDEs is shown to be an adjoint state method.
problem Estimating gradients for overparameterized SDEs efficiently.
method Demonstrates generator gradient estimator as an adjoint state method.
result Generator gradient estimator is an adjoint state method for SDEs.
New algorithm optimizes nonlinear SDEs online with convergence guarantees.
problem Optimizing nonlinear stochastic differential equations (SDEs) is computationally challenging.
method Forward propagation algorithm that solves an SDE derived using forward differentiation.
result Convergence theorem for nonlinear dissipative SDEs with bounds on stochastic fluctuations.
The paper develops methods to price and hedge options in path-dependent stock models.
problem Pricing and hedging options under complex stock models.
method Develops a path-dependent PDE for option pricing and differentiability of path-dependent SDE solutions.
result Provides formulas for option Greeks and differentiability of path-dependent SDE solutions.
Improved SDE-BNN model reduces NFEs and accelerates convergence.
problem High computational cost and convergence instability in SDE-BNNs.
method Nesterov's Accelerated Gradient (NAG) method integrated into SDE-BNN framework.
result Significantly reduced number of function evaluations (NFEs) and improved predictive accuracy.
AdjointDEIS simplifies diffusion model optimization.
problem Optimizing diffusion models with respect to a differentiable metric.
method Novel bespoke ODE solvers for continuous adjoint equations.
result Continuous adjoint equations simplify to a simple ODE, improving efficiency.
Extends unbiased simulation method to Asian options.
problem Simulating path-dependent dynamics for Asian options.
method Extension of unbiased simulation method for SDEs to path-dependent dynamics.
result Extension applies to numerical resolution of path-dependent PDEs.
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. Study of stochastic differential equations on non-compact manifolds, solving open problem on strong completeness.
problem Open problem on strong completeness of SDEs on non-compact spaces.
method Systematic study of stochastic differential equations, solving open problem on strong completeness.
result Existence of global smooth solution flow of SDEs on R^n, including substantial growth of coefficients.
SING improves state inference in latent SDE models for better drift function estimation.
problem Intractable posterior inference in latent SDE models.
method Natural gradient variational inference.
result SING provides faster and more reliable inference in latent SDE models.
Rex solves the inverse problem for ODE/SDE solvers, improving precision and stability.
problem Inversion of ODE/SDE solvers is inaccurate and impractical for precision applications.
method Rex uses Lawson methods to convert explicit Runge-Kutta schemes into algebraically reversible ones.
result Rex achieves near-machine-precision reconstruction and improves generative models.
We consider models of the population or opinion dynamics which result in the non-linear stochastic differential equations (SDEs) exhibiting the spurious long-range memory. In this context, the correspondence between the description of the birth-death processes as the continuous-time Markov chains and the continuous SDE…
Identifies most probable flows for Kunita SDEs in fluid dynamics.
problem Modeling stochastic processes with Eulerian noise and deterministic drifts.
method Equipping the domain with a Riemannian metric from the noise, solving the resulting PDEs.
result Most probable flows differ from deterministic flows, especially under noise.
We prove the existence and uniqueness of solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free martingales. The main tool in our reasoning is Picard's iterative procedure and a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous martingale…
Novel method for SDE calibration from sparse data using neural flows.
problem Calibrating SDEs from sparse, noisy observations.
method Characterization of posterior SDE using neural networks trained to solve a PDE with multiplicative updates.
result Significant improvement in scalability and accuracy compared to classical methods.
We consider the approximation of stochastic differential equations (SDEs) with non-Lipschitz drift or diffusion coefficients. We present a modified explicit Euler-Maruyama discretisation scheme that allows us to prove strong convergence, with a rate. Under some regularity and integrability conditions, we obtain the opt…
Proposes PI-VAE for solving SDEs with limited measurements.
problem Solving SDEs with limited measurements of system parameters.
method Physics-informed Variational Autoencoder (PI-VAE) integrating VAE and governing equations.
result Satisfactory accuracy and efficiency compared to PI-WGAN.
Generative model for Lévy area improves SDE simulation accuracy.
problem Simulating Lévy areas for high-order SDEs is challenging due to non-Gaussian nature and lack of fast sampling algorithms.
method LévyGAN, a deep-learning model with a GNN-inspired architecture, generates approximate samples of Lévy area.
result LévyGAN matches all joint and conditional odd moments exactly and achieves state-of-the-art performance in 4D Brownian motion.
SigMA uses signatures and attention to estimate parameters in fBm-driven SDEs.
problem Estimating parameters in SDEs driven by fBm is challenging due to non-Markovian and semimartingale issues.
method SigMA integrates path signatures with multi-head self-attention, using convolutional and MLP layers.
result SigMA outperforms other methods in accuracy, robustness, and model compactness.
We propose and analyze numerical methods for the Heath-Jarrow-Morton (HJM) model. To construct the methods, we first discretize the infinite dimensional HJM equation in maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite dimensional system of stochastic dif…
A new method simulates implied volatility surfaces for multiple assets.
problem Generating consistent market scenarios for multiple asset implied volatilities.
method Combining functional data analysis and neural SDEs with a penalty for model misspecification.
result Simulated market scenarios are consistent with historical features and lie within the sub-manifold of essentially free static arbitrage.
A new framework models uncertainty in structured temporal data using SDEs and neural networks.
problem Uncertainty quantification in machine learning applications involving structured and temporal data.
method Integrates stochastic differential equations (SDEs) with deep generative models in a variational autoencoder framework.
result Improves uncertainty quantification in machine learning applications involving structured and temporal data.
New SDEs from affine and polynomial perspectives for path-dependent processes.
problem Characterizing path-dependent stochastic processes.
method Affine and polynomial processes, signature SDEs, Fourier-Laplace transform, Riccati and linear ODEs.
result Explicit formulas for the Fourier-Laplace transform and expected values of entire functions of signature processes.
New integration method improves BSDE-based PDE solvers.
problem Discretization bias in standard BSDE-based solvers.
method Proposed Stratonovich-based BSDE formulation with stochastic Heun integration.
result Eliminates bias issues and outperforms EM-based variants.
SDE Matching eliminates simulation for training Latent SDEs, achieving similar performance.
problem Training Latent SDEs with adjoint sensitivity methods is computationally expensive and limited.
method SDE Matching, inspired by Score- and Flow Matching, eliminates simulation for training Latent SDEs.
result SDE Matching achieves performance comparable to adjoint sensitivity methods while reducing computational complexity.
In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Itô's formula, we get rid, in the asset price dynamics, of the stochastic i…
New SDEs use G-Brownian motion, extending mean-field models.
problem Extending mean-field models to new types of stochastic processes.
method Introduced G-SDEs with coefficients dependent on current state and solution as random variable. result Validated new SDE framework for complex stochastic systems.
ProGen improves spatiotemporal forecasting with SDEs and diffusion models.
problem Complex spatial and temporal dependencies in spatiotemporal data.
method ProGen uses Stochastic Differential Equations and diffusion-based generative models.
result ProGen outperforms state-of-the-art models on traffic datasets.
Unified ML approach for SDEs in bounded domains.
problem Challenges in simulating SDEs with particle exit phenomena.
method Hybrid approach combining diffusion model and exit prediction network.
result Accurate modeling of interior dynamics and boundary interactions.
New SDE model for continuous-time reinforcement learning.
problem Modeling exploration in continuous-time reinforcement learning.
method Introduced grid-sampling SDE as a proxy model.
result Wellposedness of the SDE in the presence of jumps.
We use a path integral approach for solving the stochastic equations underlying the financial markets, and we show the equivalence between the path integral and the usual SDE and PDE methods. We analyze both the one-dimensional and the multi-dimensional cases, with point dependent drift and volatility, and describe a c…
Faster training of neural ODEs using Gauß-Legendre quadrature.
problem Training neural ODEs is slow due to solving ODEs numerically.
method Use Gauß-Legendre quadrature to solve integrals faster than ODE-based methods.
result Faster training of neural ODEs, especially for large models.
This paper uses SDEs to analyze GANs training and long-run behavior.
problem Understanding the training process and long-run behavior of GANs.
method Established SDE approximations for GANs training and analyzed long-run behavior via invariant measures.
result The long-run behavior of GANs training can be studied via the invariant measures of its SDE approximations.
Proposes a method for approximating transition densities of SDEs driven by gamma processes.
problem Calculating transition densities for SDEs driven by gamma processes.
method Taylor-type approximation and conditional expectation of multiple stochastic integrals.
result Efficiency of the proposed method demonstrated through numerical tests.
The paper identifies generators of linear SDEs with noise types.
problem Identifying the generator of linear SDEs from their solution distribution.
method Deriving sufficient and necessary conditions for additive noise, and sufficient conditions for multiplicative noise.
result Generic conditions for identifying the generator of linear SDEs with both types of noise.
Neural networks estimate SDEs with jump noise using a Tamed-Milstein scheme.
problem Estimating drift and diffusion functions in SDEs with jump noise.
method Tamed-Milstein scheme with neural networks as non-parametric approximators.
result Flexible estimation of complex nonlinear dynamics in systems with state-dependent noise.
Simulation-free VI closes the approximation gap in latent SDEs
problem Recovering dynamical systems from noisy observations
method Helmholtz-SDE
result Recovers dynamics more faithfully than prior methods
Proposes methods to include distributional information in MV-SDEs for better modeling of interacting particle systems.
problem Modeling the behavior of an infinite number of interacting particles with distributional information.
method Semi-parametric methods and estimators for MV-SDEs.
result Explicitly including distributional dependence improves performance in modeling temporal data with interaction.
A new method estimates SDEs using occupation kernels.
problem Learning multivariate stochastic differential equations (SDEs).
method Two-step procedure: estimate drift, then diffusion. Occupation kernels used in RKHS.
result Validated on simulated and real-world data.
We explain how Itô Stochastic Differential Equations (SDEs) on manifolds may be defined using 2-jets of smooth functions. We show how this relationship can be interpreted in terms of a convergent numerical scheme. We show how jets can be used to derive graphical representations of Itô SDEs. We show how jets can be used…
New method speeds up SDE inference by matching moments to FPK equation.
problem Efficiency of sampling schemes in high-dimensional SDEs.
method Direct approximation of Fokker-Planck-Kolmogorov equation by matching moments.
result Fast, scalable inference in high-dimensional latent spaces.
Proposes neural SDEs with change points for better time series modeling.
problem Restrictions in modeling time series with distributional shift.
method Generative adversarial networks (GANs) for SDEs and change point detection.
result Jointly learns change points and SDE model parameters.