series expansion methods

This paper addresses the limitation of classical Taylor expansion in modeling stochasticity. We propose a stochastic Taylor theorem grounded in Poisson point processes, establishing a novel nonlinear regression framework. Our approach generalizes deterministic polynomial approximation to a random-process-driven functional expansion, unifying univariate and multivariate settings while enabling statistical inference for model parameters. Theoretical analysis establishes that the proposed estimator converges almost surely to the true regression function. Extensive simulations and empirical analysis on stock market data demonstrate superior fitting accuracy and robustness compared to conventional methods. To our knowledge, this is the first work to rigorously integrate Poisson point processes into the Taylor theoretical framework, yielding a new nonparametric regression paradigm that simultaneously offers probabilistic interpretability and inferential validity.

This study addresses estimation bias arising from model misspecification in statistical modeling by proposing a local polynomial regression framework that incorporates first-order differential equation constraints—specifically, exponential growth structures. The method innovatively integrates differential equation priors into local kernel estimation, constructing estimators via Taylor expansions of varying orders to substantially enhance the robustness of local regression under model misspecification. A rigorous theoretical analysis characterizes the asymptotic bias and variance properties of the proposed estimator. Extensive simulations demonstrate its superior performance over conventional approaches across multiple misspecification scenarios, achieving both higher accuracy and improved robustness.

This paper addresses the estimation of dynamic parameters in nonlinear time-varying time series models—including VAR, ARCH/GARCH, and Poisson autoregression—by developing asymptotic theory for local polynomial (quasi-)maximum likelihood estimation. Under substantially weakened smoothness and moment conditions—dispensing with higher-order differentiability or strict stationarity—we establish consistency and asymptotic normality of the estimator and precisely characterize the leading bias induced by smoothing. Key contributions include: (i) the first feasible framework for consistent and asymptotically normal time-varying parameter estimation in Poisson autoregressive models; (ii) a significant relaxation of theoretical regularity conditions required for existing VAR and ARCH-type models; and (iii) a simplified, closed-form bias expression directly applicable to bias correction. The unified theory accommodates a broad class of nonstationary nonlinear models, thereby extending the scope of valid dynamic parameter inference.

Pricing high-dimensional American options and computing their Greeks remain computationally challenging due to the curse of dimensionality and the need for accurate gradient estimation. Method: This paper proposes a gradient-enhanced sparse Hermite polynomial expansion method, the first to incorporate pathwise gradient information into the Least-Squares Monte Carlo (LSM) framework. It formulates a weighted $H^1$-norm constrained least-squares problem to simultaneously approximate both the continuation value function and its spatial derivatives. Contribution/Results: Theoretically, we establish a weighted $H^1$ error bound, ensuring accuracy and interpretability. Algorithmically, the method preserves sparsity and scales efficiently to high dimensions. Numerical experiments demonstrate that, in 100-dimensional settings, it achieves significantly higher accuracy in both option prices and Greeks than classical LSM—matching or surpassing state-of-the-art neural network approaches—while maintaining comparable computational cost and yielding more robust optimal stopping policies.

To address the insufficient local accuracy of global models in modeling complex, unknown processes, this paper proposes the PCEGP method: Polynomial Chaos Expansion (PCE) is employed—novelly and for the first time—to dynamically generate input-dependent hyperparameters for Gaussian Processes (GPs), thereby constructing a nonstationary covariance function and an heteroscedastic noise estimation mechanism. The approach ensures mathematical interpretability, modeling transparency, and prediction traceability, eliminating the need for explicit spatial partitioning and multiple model training inherent in conventional local modeling strategies. Evaluated across multiple regression benchmarks, PCEGP consistently achieves significantly lower prediction errors, with accuracy and generalization performance matching or surpassing state-of-the-art methods. It establishes a new paradigm for high-confidence surrogate modeling in complex system identification and uncertainty quantification.

This study addresses the limitations of standard first-order semiparametric estimators in causal inference and missing data problems, which often fail to achieve asymptotic efficiency due to slow convergence of the nuisance functions and exhibit poor finite-sample performance. The authors systematically compare three classes of higher-order efficient estimators—Higher-Order Influence Functions (HOIF), kernel-based HOTMLE, and HAL-HOTMLE—evaluating, for the first time within a unified simulation framework, how their higher-order expansion constructions and regularization strategies affect estimation accuracy. Results demonstrate that higher-order debiasing substantially reduces bias, with HAL-HOTMLE showing robust performance, whereas HOIF proves sensitive to basis truncation and tuning parameters. The work clarifies the conditions under which higher-order corrections are effective in both theory and practice, while highlighting their limitations and key trade-offs for method selection.

This study addresses the lack of efficient computational methods for capturing the dependence of the fractional Riccati equation on the Hurst parameter $H$ in the rough Heston model. The authors establish, for the first time, the analyticity of the solution with respect to $H$ and develop a unified Taylor expansion framework valid around any $H_0 \in (-1/2, 1/2]$. Expansion coefficients are obtained recursively by solving linear Volterra equations with fractional-logarithmic kernels. Combined with an affine transformation and Fourier-based techniques, this approach enables accurate approximation of the characteristic function and European call option prices. Numerical experiments demonstrate that low-order expansions around $H_0 = 1/2$ (classical Heston) or $H_0 = 0$ yield highly precise implied volatility surfaces across a broad range of $H$, including the super-rough regime.

This study addresses the high sensitivity to noise inherent in differential-algebraic parameter estimation, which arises from its reliance on exact derivatives and severely limits practical applicability. To overcome this limitation, this work integrates Gaussian process regression (GPR) into the differential-algebraic framework, proposing a robust parameter estimation method for ordinary differential equations that synergizes GPR with algebraic elimination. Furthermore, a first-order error analysis theoretical framework is established to characterize noise propagation and parameter sensitivity. Benchmark evaluations demonstrate that the proposed approach achieves state-of-the-art performance, successfully recovering parameters within a 10% relative error in 88.5% of experimental runs. These results confirm that the method effectively resolves the challenge of parameter identification from noisy observational data.

This study addresses the O(h²) bias bottleneck inherent in local linear derivative estimation by proposing an iterative data sharpening method. By constructing sharpened estimators based on residual operators, this approach reduces the bias order to O(h^{2l+2}) while preserving the simplicity of local fitting. Furthermore, a closed-form single-bandwidth expression is derived specifically for Gaussian kernels. Simulation results demonstrate that the proposed algorithm significantly mitigates estimation bias and clearly elucidates the bias-variance trade-off mechanism. Consequently, this work establishes a novel paradigm for nonparametric derivative estimation that effectively balances high precision with computational efficiency, offering a robust solution to longstanding limitations in local polynomial regression techniques.

This work addresses the suboptimal convergence rates of classical influence functions when estimating complex, implicitly defined causal parameters such as quantile treatment effects. While existing higher-order methods are limited to explicitly defined parameters, this paper extends the higher-order influence function framework to implicit M- and Z-estimation problems for the first time. By integrating U-process theory with nonparametric estimation, the authors construct a debiased estimator that substantially relaxes the stringent Hölder smoothness assumptions typically imposed on nuisance parameters. The proposed approach achieves improved convergence rates in settings like quantile treatment effect estimation and reduces requirements on model complexity, thereby broadening the applicability of higher-order influence function methodology to a wider class of semiparametric problems.