stochastic differential equations

This work addresses the challenge of modeling a single nonstationary, non-ergodic stochastic differential equation (SDE) trajectory—a setting where conventional SDE identification methods fail due to their reliance on ergodicity or stationarity assumptions. Method: We propose Stochastic Sparse Identification of SDEs (SSISDE), the first data-driven algorithm capable of jointly estimating drift and diffusion functions while reconstructing Brownian increments from a single trajectory. SSISDE integrates stochastic Taylor expansions with the Girsanov transformation, constructing solvable estimators initialized from drift function approximations—bypassing the need for stationary or ergodic data. Contribution/Results: SSISDE establishes the first SDE modeling paradigm tailored to single-trajectory, nonstationary, non-ergodic regimes. It achieves high-fidelity model discovery on benchmark nonstationary linear and quadratic systems—including Black–Scholes dynamics—significantly improving estimation accuracy over existing approaches. This framework enables real-time, interpretable modeling of complex dynamical systems in domains such as finance and biophysics, where ergodic assumptions are fundamentally violated.

This work addresses the nonparametric learning of drift functions in stochastic differential equations (SDEs) from sparse and noisy observations. We propose EM-SMC-RKHS—a unified framework integrating sequential Monte Carlo (SMC) for efficient Bayesian inference of latent state trajectories, penalized likelihood estimation within a reproducing kernel Hilbert space (RKHS), and the generalized representer theorem to enable structure-free function learning; we further introduce a Bayesian shrinkage prior to automatically control model complexity. Unlike conventional parametric approaches, our method imposes no assumptions on the functional form of the drift, substantially reducing reliance on domain-specific prior knowledge. Experiments demonstrate high accuracy and robustness in drift estimation even under extremely low sampling rates and high observational noise. The framework provides an interpretable, data-efficient, nonparametric inference paradigm for modeling dynamic systems with complex or partially unknown mechanisms.

This work addresses the structural identifiability of parameters in stochastic differential equation (SDE) models under multiple interventions—i.e., whether SDE parameters can be uniquely recovered from samples of post-intervention stationary distributions. Theoretically, we establish the first uniqueness guarantee for SDE parameter recovery under multi-intervention settings; for linear SDEs, we derive a tight lower bound on the minimum number of required interventions; for weak-noise nonlinear SDEs, we obtain an upper bound on identifiability. Methodologically, we propose a parametric framework featuring learnable activation functions, integrating intervention modeling, stationary distribution analysis, and weak-noise asymptotic theory. Experiments on synthetic data demonstrate that our approach accurately recovers ground-truth parameters, and the theory-guided learnable architecture significantly improves both estimation accuracy and robustness.

Multidimensional stochastic differential equations (SDEs) generally lack closed-form solutions, and existing numerical methods are often constrained in strong convergence order and computational efficiency, particularly when handling multiple stochastic integrals where accuracy and complexity are difficult to balance. This work proposes an improved Milstein scheme that incorporates two novel algorithms for efficiently computing multiple stochastic integrals and establishes a theoretical framework enabling verifiable strong and weak convergence orders. The method accurately assesses convergence performance even in the absence of analytical solutions, significantly enhancing both accuracy and efficiency for high-dimensional SDEs. Numerical experiments and applications to financial models demonstrate that the proposed approach outperforms current techniques in convergence rate and computational cost, offering a highly accurate and scalable numerical tool for simulating high-dimensional stochastic systems.

Existing generalization bounds for heavy-tailed stochastic optimization either rely on intractable information-theoretic quantities or yield only expectation-based guarantees. To address this, this paper establishes the first *computable, dimension-friendly, high-probability generalization bound* for heavy-tailed stochastic differential equation (SDE) optimizers. Methodologically, we introduce a novel entropy flow analysis framework grounded in the fractional-order Fokker–Planck equation, unifying heavy-tailed SDE theory with fractional PDE techniques. Our analysis reveals a structural-phase transition phenomenon: the impact of heavy tails on generalization—beneficial or detrimental—is governed by the underlying problem geometry. The resulting bound is fully computable, contains no unmeasurable terms, and exhibits improved dimension dependence compared to prior work. Extensive experiments across multiple models and datasets empirically validate the theoretical insights.

This work addresses the computationally expensive inverse problem of parameter estimation for stochastic differential equations (SDEs) by proposing an efficient solution framework that, for the first time, integrates Wiener chaos expansion (WCE) with stochastic gradient descent (SGD). By projecting the stochastic solution onto a deterministic system of propagators via an orthogonal Hermite polynomial basis, the method constructs a regularized discrepancy functional amenable to SGD optimization. This transformation effectively converts the original stochastic inverse problem into a deterministic optimization task, substantially reducing computational complexity and data requirements. Numerical experiments on several nonlinear SDE models—including a biological individual growth model—demonstrate that the approach accurately and robustly recovers parameters even from sparse and noisy observational data, exhibiting strong scalability and practical promise.

This work proposes a unified mathematical framework grounded in dynamic information flow for constructing structurally rigorous models of future prediction. By integrating filtering theory, regular conditional probabilities, Markov semigroups, infinitesimal generators, and multiple information geometries—including Hilbert, Fisher–Rao, and Wasserstein—the approach conceptualizes prediction as the construction of conditional distributions governed by informational, geometric, and modeling constraints. The framework elucidates deep connections among classical results such as the tower property and semigroup laws, as well as Itô’s formula and backward equations. Explicit transition laws, spectral decompositions, term structures, and asymptotic behaviors are derived within canonical models like Ornstein–Uhlenbeck and Cox–Ingersoll–Ross, thereby establishing a compact mathematical mapping from idealized theoretical constructs to empirical forecasting.

This study addresses the challenge of estimating diffusion parameters in stochastic differential equation (SDE) models when data and model are compatible only at specific scales. The authors propose an adaptive subsampling method based on the statistics of monotonic runs. By demonstrating that, for a broad class of additive-noise SDEs, the length of monotonic runs at infinitesimal scales approximately follows a geometric distribution with success probability 1/2, they establish a general criterion for selecting the subsampling rate without relying on multiscale diffusion asymptotics. The optimal sampling scale matching the SDE’s infinitesimal behavior is automatically determined solely from the statistical properties of monotonic increasing or decreasing segments in the observed time series. Validation on surrogate modeling of fiber lay-down trajectories in nonwoven fabric production demonstrates that the method yields highly accurate and model-consistent diffusion parameter estimates, proving effective in real-world industrial applications.

This work proposes a weak-form sparse regression framework based on spatial Gaussian test functions to address structural bias introduced by conventional time-domain test functions in the identification of stochastic differential equations (SDEs). By unifying the estimation of drift and diffusion terms into two sparse linear systems sharing a common design matrix, the method uniquely integrates the weak formulation of Weak SINDy with the objective of stochastic SINDy. The use of spatial Gaussian kernels ensures zero conditional mean under noise, thereby eliminating regression bias at its source. Combined with ℓ¹ regularization, grouped cross-validation, and a two-step bias correction scheme, the approach effectively handles state-dependent diffusion. Validated on Ornstein–Uhlenbeck, double-well Langevin, and multiplicative noise systems, it accurately recovers all active generators (coefficient errors < 4%), achieves total variation distances below 0.01 for stationary densities, and precisely reproduces true relaxation timescales in autocorrelation functions.

This work addresses the challenge of simulating sample paths for stochastic differential equations (SDEs) with gradient drift and unit diffusion coefficients under noisy observations, where existing methods often suffer from discretization bias or high sampling complexity. The authors propose an exact Gibbs sampling framework that enables unbiased path simulation without temporal discretization and naturally integrates Gaussian process tools to facilitate parameter inference. This approach achieves, for the first time, discretization-free MCMC sampling for a broad class of SDE models, handling both univariate and multivariate cases within a unified framework—without requiring rejection sampling or debiasing techniques. Empirical evaluations on synthetic and real-world data demonstrate clear advantages over particle MCMC methods, offering superior accuracy and computational efficiency.