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Stochastic multi-scale modeling and simulation for nonlinear thermo-mechanical problems of composite materials with complicated random microstructures remains a challenging issue. In this paper, we develop a novel statistical higher-order multi-scale (SHOMS) method for nonlinear thermo-mechanical simulation of random composite materials, which is designed to overcome limitations of prohibitive computation involving the macro-scale and micro-scale. By virtue of statistical multi-scale asymptotic analysis and Taylor series method, the SHOMS computational model is rigorously derived for accurately analyzing nonlinear thermo-mechanical responses of random composite materials both in the macro-scale and micro-scale. Moreover, the local error analysis of SHOMS solutions in the point-wise sense clearly illustrates the crucial indispensability of establishing the higher-order asymptotic corrected terms in SHOMS computational model for keeping the conservation of local energy and momentum. Then, the corresponding space-time multi-scale numerical algorithm with off-line and on-line stages is designed to efficiently simulate nonlinear thermo-mechanical behaviors of random composite materials. Finally, extensive numerical experiments are presented to gauge the efficiency and accuracy of the proposed SHOMS approach.

This paper examines the effectiveness of several forecasting methods for predicting inflation, focusing on aggregating disaggregated forecasts - also known in the literature as the bottom-up approach. Taking the Brazilian case as an application, we consider different disaggregation levels for inflation and employ a range of traditional time series techniques as well as linear and nonlinear machine learning (ML) models to deal with a larger number of predictors. For many forecast horizons, the aggregation of disaggregated forecasts performs just as well survey-based expectations and models that generate forecasts using the aggregate directly. Overall, ML methods outperform traditional time series models in predictive accuracy, with outstanding performance in forecasting disaggregates. Our results reinforce the benefits of using models in a data-rich environment for inflation forecasting, including aggregating disaggregated forecasts from ML techniques, mainly during volatile periods. Starting from the COVID-19 pandemic, the random forest model based on both aggregate and disaggregated inflation achieves remarkable predictive performance at intermediate and longer horizons.

We combine Kronecker products, and quantitative information flow, to give a novel formal analysis for the fine-grained verification of utility in complex privacy pipelines. The combination explains a surprising anomaly in the behaviour of utility of privacy-preserving pipelines -- that sometimes a reduction in privacy results also in a decrease in utility. We use the standard measure of utility for Bayesian analysis, introduced by Ghosh at al., to produce tractable and rigorous proofs of the fine-grained statistical behaviour leading to the anomaly. More generally, we offer the prospect of formal-analysis tools for utility that complement extant formal analyses of privacy. We demonstrate our results on a number of common privacy-preserving designs.

Ghost, or fictitious points allow to capture boundary conditions that are not located on the finite difference grid discretization. We explore in this paper the impact of ghost points on the stability of the explicit Euler finite difference scheme in the context of the diffusion equation. In particular, we consider the case of a one-touch option under the Black-Scholes model. The observations and results are however valid for a much wider range of financial contracts and models.

In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients of the Zakai equation, the corresponding random PDE admits a solution random field which, for almost all realizations of the random coefficients, can be written as a classical solution of a linear parabolic PDE. This makes it possible to apply the Feynman--Kac formula to obtain an efficient Monte Carlo scheme for computing approximate solutions of Zakai equations. The approach achieves good results in up to 25 dimensions with fast run times.

This paper develops power series expansions of a general class of moment functions, including transition densities and option prices, of continuous-time Markov processes, including jump--diffusions. The proposed expansions extend the ones in Kristensen and Mele (2011) to cover general Markov processes. We demonstrate that the class of expansions nests the transition density and option price expansions developed in Yang, Chen, and Wan (2019) and Wan and Yang (2021) as special cases, thereby connecting seemingly different ideas in a unified framework. We show how the general expansion can be implemented for fully general jump--diffusion models. We provide a new theory for the validity of the expansions which shows that series expansions are not guaranteed to converge as more terms are added in general. Thus, these methods should be used with caution. At the same time, the numerical studies in this paper demonstrate good performance of the proposed implementation in practice when a small number of terms are included.

This study presents a novel high-order numerical method designed for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The Caputo definition is employed to characterize the time-fractional derivative. A weak singularity at the initial time ($t=0$) is encountered in the considered problem, which is effectively managed by adopting a discretization approach for the time-fractional derivative, where Alikhanov's high-order L2-1$_\sigma$ formula is applied on a non-uniform fitted mesh, resulting in successful tackling of the singularity. A high-order two-dimensional compact operator is implemented to approximate the spatial variables. The alternating direction implicit (ADI) approach is then employed to solve the resulting system of equations by decomposing the two-dimensional problem into two separate one-dimensional problems. The theoretical analysis, encompassing both stability and convergence aspects, has been conducted comprehensively, and it has shown that method is convergent with an order $\mathcal O\left(N_t^{-\min\{3-\alpha,\theta\alpha,1+2\alpha,2+\alpha\}}+h_x^4+h_y^4\right)$, where $\alpha\in(0,1)$ represents the order of the fractional derivative, $N_t$ is the temporal discretization parameter and $h_x$ and $h_y$ represent spatial mesh widths. Moreover, the parameter $\theta$ is utilized in the construction of the fitted mesh.

This paper focuses on investigating the density convergence of a fully discrete finite difference method when applied to numerically solve the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noises. The main difficulty lies in the control of the drift coefficient that is neither globally Lipschitz nor one-sided Lipschitz. To handle this difficulty, we propose a novel localization argument and derive the strong convergence rate of the numerical solution to estimate the total variation distance between the exact and numerical solutions. This along with the existence of the density of the numerical solution finally yields the convergence of density in $L^1(\mathbb{R})$ of the numerical solution. Our results partially answer positively to the open problem emerged in [J. Cui and J. Hong, J. Differential Equations (2020)] on computing the density of the exact solution numerically.

This paper studies model checking for general parametric regression models having no dimension reduction structures on the predictor vector. Using any U-statistic type test as an initial test, this paper combines the sample-splitting and conditional studentization approaches to construct a COnditionally Studentized Test (COST). Whether the initial test is global or local smoothing-based; the dimension of the predictor vector and the number of parameters are fixed or diverge at a certain rate, the proposed test always has a normal weak limit under the null hypothesis. When the dimension of the predictor vector diverges to infinity at faster rate than the number of parameters, even the sample size, these results are still available under some conditions. This shows the potential of our method to handle higher dimensional problems. Further, the test can detect the local alternatives distinct from the null hypothesis at the fastest possible rate of convergence in hypothesis testing. We also discuss the optimal sample splitting in power performance. The numerical studies offer information on its merits and limitations in finite sample cases including the setting where the dimension of predictor vector equals the sample size. As a generic methodology, it could be applied to other testing problems.