Itô maps for any-step SDEsstat.ML
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
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48 results for “Ito SDE”
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…
The paper develops a method for stochastic differential equations on manifolds using Schwartz morphisms and diffusion generators.
problem Representing stochastic differential equations on smooth manifolds.
method Using Schwartz morphisms and diffusion generators to construct SDEs on manifolds.
result An extended Ito formula for SDEs on manifolds.
We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allows one to systematically develop low dimensional approximations to high dimensional SDEs using differential geometric techniques. The approach generalizes the notion o…
The paper clarifies the approximation of SGD with Ito SDEs for finite learning rates.
problem Theoretical justification and experimental verification of the Ito SDE approximation for finite learning rates in SGD.
method An efficient simulation algorithm SVAG and a necessary condition test for the SDE approximation.
result The Ito SDE approximation can meaningfully capture training and generalization properties of deep nets with finite learning rates.
NANSDE-Net models time series with memory using neural ARMA-type noise.
problem Modeling time series with long- or short-memory characteristics.
method Developed NANSDE-Net, a generative model that incorporates Neural Network-kernel ARMA-type noise.
result NANSDE-Net matches or outperforms existing models in reproducing long- and short-memory features of data.
Neural Options Pricingq-fin.MF
Neural models price financial options without assuming underlying price forms.
problem Pricing financial options under flexible price processes.
method Apply neural SDEs as universal approximators, use Wasserstein distance for training.
result Error in option prices bounded by Wasserstein distance used for training.
This article provides a new representation for pricing adjustments in derivatives.
problem Derivative pricing adjustments and XVA (Expected Value of All Risk) models.
method An Ito SDE/parabolic PDE framework to encapsulate pricing adjustments.
result A new representation that encompasses various past adjustments.
Study on stochastic mean curvature flow on networks using Ito calculus.
problem Understanding the dynamics of network structures under random influences.
method Application of Ito calculus to derive a stochastic differential equation (SDE) for network edges.
result New insights into the stability, long-term behavior, and pattern formation of complex networks under stochastic influences.
Optimal Projection Filtersmath.PR
Develops optimal low-dimensional approximations to high-dimensional SDEs.
problem Approximating solutions to high-dimensional SDEs in a low-dimensional space.
method Introduces Ito-vector and Ito-jet projections for optimal approximation.
result Optimal projection filters yield better approximations than Stratonovich projection.
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.
Inverting the Markovian projection, with an application to local stochastic volatility modelsmath.PR
We study two-dimensional stochastic differential equations (SDEs) of McKean--Vlasov type in which the conditional distribution of the second component of the solution given the first enters the equation for the first component of the solution. Such SDEs arise when one tries to invert the Markovian projection developed …
Method learns latent SDEs from high-dimensional time series.
problem Learning latent stochastic differential equations from time series data.
method Self-supervised learning with variational autoencoders and Euler-Maruyama approximation.
result Can recover SDE coefficients and latent variables up to isometry with infinite data.
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.
Paper proposes a new Taylor moment expansion for non-linear Gaussian filtering and smoothing.
problem Non-linear Gaussian filtering and smoothing in continuous-discrete state-space models.
method Taylor moment expansion (TME) for moment functions directly and in time variable.
result Significantly outperforms state-of-the-art methods in terms of estimation accuracy and numerical stability.
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…
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.
GANs can approximate SDEs for large time steps.
problem Approximating SDEs for large time steps using GANs.
method Proposed a conditional GAN architecture to enable strong approximation of SDEs.
result Supervised GAN outperformed standard GAN and other schemes in strong error.
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.
In deep latent Gaussian models, the latent variable is generated by a time-inhomogeneous Markov chain, where at each time step we pass the current state through a parametric nonlinear map, such as a feedforward neural net, and add a small independent Gaussian perturbation. This work considers the diffusion limit of suc…
This paper proposes a governing equation for stock market indexes that accounts for non-stationary effects. This is a linear Fokker-Planck equation (FPE) that describes the time evolution of the probability distribution function (PDF) of the price return. By applying Ito's lemma, this FPE is associated with a stochasti…
G-framework is presented by Peng [41] for measure risk under uncertainty. In this paper, we define fractional G-Brownian motion (fGBm). Fractional G-Brownian motion is a centered G-Gaussian process with zero mean and stationary increments in the sense of sub-linearity with Hurst index . This process has sta…
A novel approach models rating transitions using Lie groups and Deep Learning.
problem Modeling rating transitions with geometric properties and stochastic processes.
method Introducing Itô-SDEs on Lie groups, using TimeGAN for calibration, and examining rating matrix properties.
result The geometric approach using Lie groups and Deep Learning generates a good fit for rating transitions.
We derive asymptotic expansions for option data to detect infinite variation volatility.
problem Detecting infinite variation volatility in high-frequency option data.
method Nonparametric higher-order asymptotic expansions for small-time changes of characteristic functions of Itô semimartingales.
result Evidence of infinite variation volatility in high-frequency option data.
DLPM replaces Gaussian noise with α-stable noise in DDPM, improving data distribution coverage and robustness.
problem Handling mode collapse and class imbalance in datasets with heavy-tailed noise.
method Extending DDPM to use α-stable noise, simplifying the process with elementary proof techniques.
result DLPM yields better coverage of data distribution tails, improved robustness to unbalanced datasets, and faster computation times.
STS clarifies chaos and stochastic dynamics, linking algebraic topology and physics.
problem Chaos and stochastic dynamics in arbitrary form SDEs.
method Supersymmetric theory of stochastic dynamics (STS) using generalized transfer operator (GTO) and topological field theories (TFT).
result Positive 'pressure' in GTOs corresponds to spontaneous breakdown of topological supersymmetry, explaining 1/f noise.
Motivated by marginals-mimicking results for Itô processes via SDEs and by their applications to volatility modeling in finance, we discuss the weak convergence of the law of a hypoelliptic diffusions conditioned to belong to a target affine subspace at final time, namely if $X_{\cdot}=(Y_\cd…
Counterexample shows Ito integrand needn't be locally square integrable.
problem Ito integrand's square integrability condition is not always met.
method Provided a counterexample to Ito's Lemma's integrability condition.
result Ito integrand needn't be locally square integrable.
Projects Markovian processes from Itô semimartingales with jumps.
problem Modeling Itô semimartingales with jumps using Markovian projections.
method Construct Markovian projections for Itô semimartingales with jumps using non-local FPKEs.
result Markovian projections match the marginal laws of the original process.
Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-Itô, applies to one dimensional semimartingales and convex functions. There are also satisfactory generalizations of Itô's formula for diffusion processes whe…
Developed a machine-checked Itô calculus for Brownian motion.
problem Formal verification of Itô calculus for Brownian motion.
method Machine-checked formalization in Lean over Mathlib.
result First machine-checked constructions of the Itô integral and Itô's formula.
This paper extends Markovian projections to semimartingales with jumps.
problem Extending Markovian projections to semimartingales with jumps.
method Using Markovian projections to match marginal laws of Itô semimartingales with jumps.
result Existence of Markovian projections for Itô semimartingales with jumps.
SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equationsstat.ML
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.
A machine-checked Itô calculus for Brownian motion on
problem Developing an Itô calculus for Brownian motion
method Formalized in Lean 4 on top of Mathlib and the BrownianMotion package
result First machine-checked proof of Itô's formula and construction of Itô integral as martingale-valued process
A new approach to continuous-time universal portfolios using pathwise Itô calculus.
problem Continuous-time version of Cover's universal portfolio strategies.
method Pathwise Itô calculus approach to establish existence and properties of universal portfolio strategies.
result The universal portfolio strategy's portfolio value process is the average of all values of constant rebalanced strategies.
New time series generation models improve accuracy and correlation identification.
problem Generating accurate and correlated time series from limited data.
method Conditional Euler Generator (CEGEN) using Euler discretization of SDEs and Wasserstein metrics.
result CEGEN outperforms state-of-the-art models on various metrics and real-world datasets.
New SDEs use -Brownian motion, extending mean-field models.
problem Extending mean-field models to new types of stochastic processes.
method Introduced -SDEs with coefficients dependent on current state and solution as random variable.
result Validated new SDE framework for complex stochastic systems.
Extends Itô's formula for path-dependent functions in finance.
problem Modeling and hedging of path-dependent financial options.
method Functional extension of Itô's formula for C^{0,1}-functions of continuous weak Dirichlet processes.
result Validates the hedging or superhedging problems for path-dependent options.
Study on the smoothness of solutions to a specific type of stochastic differential equation.
problem Regularity of solutions to mean-field -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.
Neural Jump ODEs model Itô processes without adversarial training.
problem Generating samples from Itô processes with irregular data.
method Neural Jump ODEs framework for drift and diffusion approximation.
result NJODEs can recover true parameters of Itô processes in the limit.
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.
The paper introduces a new volatility model for state heterogeneous financial markets using high-frequency data.
problem State heterogeneity in financial volatility processes.
method Developed a state heterogeneous GARCH-Ito (SG-Ito) model based on continuous Ito diffusion process.
result Empirical studies reveal various state heterogeneities in S&P 500 index volatility.
We analyze a new Markov chain model for better sampling and optimization.
problem Developing a new Markov chain model for improved sampling and optimization.
method We introduce a new class of Ito chains with arbitrary noise and inexact drift/diffusion coefficients, proving a bound in -distance.
result Our analysis provides improved or first results for various applications like SGLD, sampling, and boosting.
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.
-player and Mean-field Games in Itô-diffusion Markets with Competitive or Homophilous Interactionq-fin.MF
Study -player and mean-field games in Itô-diffusion markets with competitive or homophilous interactions.
problem Optimal portfolio choice in a common market with interacting players.
method Analyzes -player and mean-field games in incomplete and complete markets with CARA utilities and random risk tolerances.
result Derives explicit or closed-form solutions for equilibrium processes and game values.
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.
Implicit Transfer Operator Learning: Multiple Time-Resolution Surrogates for Molecular Dynamicsphysics.chem-ph
Framework learns surrogates for molecular dynamics across multiple time-scales.
problem Stable molecular dynamics simulations require small time-steps, but long-time-scale moments need repeated simulations.
method Implicit Transfer Operator Learning with denoising diffusion probabilistic models and SE(3) equivariant architecture.
result Models can generate self-consistent stochastic dynamics across multiple time-scales.
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