analyze stochastic differential equations

This work addresses the modeling challenge of non-Markovian stochastic differential equations (SDEs), where temporal correlations induced by colored noise invalidate conventional Markovian SDE frameworks. We propose a generalized motion coordinate theory that unifies treatment of both Markovian and non-Markovian systems—driven by white or colored noise—via pathwise analysis and an extended state-space formulation. Innovatively, we construct the first generalized coordinate framework enabling exact short-time solutions and global long-time analytical characterization—including flow and perturbation analysis—for non-Markovian SDEs, circumventing the approximation limitations of traditional Markovian embedding approaches. Our methodology integrates rough path theory, analytical flow analysis, generalized Bayesian filtering derivation, and efficient numerical simulation. Key contributions include: (i) exact solutions for linear SDEs with analytically characterized perturbations; (ii) reconstruction of generalized Bayesian filtering; (iii) novel high-accuracy algorithms for simulation, filtering, and control; and (iv) smooth path approximations for rough SDEs.

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 addresses parameter identification of stochastic continuous-time linear systems from a single finite-length state trajectory, particularly when the open-loop system matrix may be unstable. We propose an estimation method incorporating randomized control inputs and derive a non-asymptotic upper bound on the estimation error—establishing, for the first time, finite-sample theoretical guarantees for unstable stochastic continuous-time systems. Our analysis integrates tools from non-stationary martingale theory, the generalized iterated logarithm law, and continuous-time system identification frameworks, rigorously proving that the estimation error converges at a $1/sqrt{T}$ rate with respect to trajectory length $T$. Numerical experiments corroborate the theoretical convergence rate and uncover intrinsic dynamic learning behavior. The developed stochastic analytical techniques possess independent theoretical significance and are extendable to broader classes of stochastic dynamical system learning problems.

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.

This paper addresses the stability of stochastic approximation (SA) algorithms under Markovian noise—a long-standing challenge in reinforcement learning. We extend the Borkar–Meyn theorem for the first time to non-i.i.d., state-dependent Markov noise settings by introducing a novel asymptotic step-size decay condition. Our analysis integrates ergodicity theory for Markov chains, the strong law of large numbers, and the law of the iterated logarithm, yielding a tight ODE-based convergence framework. Theoretically, we establish—rigorously and for the first time—that the parameter iterates of off-policy temporal-difference algorithms with linear function approximation and eligibility traces (e.g., GTD, TDC) remain almost surely bounded. This provides the first unified, mathematically rigorous stability guarantee for such algorithms, thereby filling a fundamental theoretical gap in off-policy reinforcement learning under non-i.i.d. Markovian noise.

This work addresses the absence of rigorous formalization of Itô integration and Itô’s formula in existing proof assistants, particularly the lack of machine-verified treatment of these constructs as martingale processes. Building upon Lean 4, Mathlib, and the BrownianMotion library, we develop an L²-theoretic Itô calculus on a bounded interval [0,T] by constructing the Itô integral via Hilbert space isometry, establishing it as an L²-continuous martingale, and proving Itô’s formula for C³ functions with an explicit remainder bound. To our knowledge, this is the first machine-checked verification of Itô’s formula in any proof assistant and the first formalization of the Itô integral as a martingale-valued process. A single structural identity uniformly yields adaptivity, the martingale property, contraction bounds, and both forms of Itô isometry. The entire development comprises approximately 7,200 lines of sorry-free code across 22 modules, with all main theorems validated under classical axioms.

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.

This work addresses the finite-time convergence of stochastic iterative algorithms for fixed-point equations accessible only through a noisy oracle. The authors propose a norm-independent, unified Lyapunov function framework constructed via a generalized Moreau envelope, which integrates Lyapunov stability theory with stochastic approximation analysis. This framework accommodates complex settings such as Markovian noise, seminorm contractive operators, and dissipative operators, yielding sharp non-asymptotic convergence bounds in both high-probability and mean-square senses. As a result, it provides a unified and refined finite-time convergence guarantee for a broad class of algorithms, including stochastic gradient descent, linear stochastic approximation, Q-learning, and temporal difference learning.

This work addresses the challenge of effectively extending kernel-based methods—originally developed for deterministic dynamical systems—to stochastic differential equations (SDEs) for approximating eigenfunctions of the Koopman operator. By leveraging the Feynman–Kac path integral representation, the study unifies three distinct kernel constructions—variational principles, Green’s function convolutions, and resolvent operators—into a coherent framework for stochastic systems with diffusion, establishing a corresponding reproducing kernel Hilbert space (RKHS) approximation scheme. Theoretically, under uniform ellipticity, these three approaches are shown to be equivalent, revealing that diffusion enhances numerical conditioning through elliptic regularization. The analysis further provides error bounds that separate RKHS approximation error from Monte Carlo sampling error. Numerical experiments on the Ornstein–Uhlenbeck process, nonlinear SDEs, and high-dimensional systems demonstrate the method’s efficacy, showing that moderate diffusion significantly improves numerical stability.

This study investigates the stationary distribution and stability of stochastic differential equation systems with multimodal uncertain parameters exhibiting superposition effects, using the nonlinear Rosenzweig–MacArthur predator–prey model as a case study. For the first time, multimodal mixture-distributed parameters are incorporated into the stationary analysis of stochastic dynamical systems. System stability is quantified through the eigenvalue distribution of the Jacobian matrix, and posterior stationary density estimates are obtained via the Monte Carlo method proposed by Hoegele (2026). The results reveal that under multimodal parameter uncertainty, the system exhibits a multimodal stationary distribution, accurately delineating regions of stability. This demonstrates the effectiveness and novelty of the proposed framework for uncertainty quantification in complex ecological dynamics.