Communications in Mathematical Research CMR Volume 40, Number 1

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Volume 40, Number 1

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Removing Splitting/Modeling Error in Projection/Penalty Methods for Navier-Stokes Simulations with Continuous Data Assimilation

Elizabeth Hawkins, Leo G. Rebholz & Duygu Vargun

10.4208/cmr.2023-0008

Commun. Math. Res., 40 (2024), pp. 1-29.

Preview Full PDF 2061 17983 Abstract

Abstract

We study continuous data assimilation (CDA) applied to projection and penalty methods for the Navier-Stokes (NS) equations. Penalty and projection methods are more efficient than consistent NS discretizations, however are less accurate due to modeling error (penalty) and splitting error (projection). We show analytically and numerically that with measurement data and properly chosen parameters, CDA can effectively remove these splitting and modeling errors and provide long time optimally accurate solutions.

Omni-Representations of Leibniz Algebras

Zhangju Liu & Yunhe Sheng

10.4208/cmr.2023-0019

Commun. Math. Res., 40 (2024), pp. 30-42.

Preview Full PDF 2093 19157 Abstract

Abstract

In this paper, first we introduce the notion of an omni-representation of a Leibniz algebra $g$ on a vector space $V$ as a Leibniz algebra homomorphism from $g$ to the omni-Lie algebra $gl(V)⊕V.$ Then we introduce the omni-cohomology theory associated to omni-representations and establish the relation between omni-cohomology groups and Loday-Pirashvili cohomology groups.

Convergence Towards the Population Cross-Diffusion System from Stochastic Many-Particle System

Yue Li, Li Chen & Zhipeng Zhang

10.4208/cmr.2023-0002

Commun. Math. Res., 40 (2024), pp. 43-63.

Preview Full PDF 1852 18961 Abstract

Abstract

In this paper, we derive rigorously a non-local cross-diffusion system from an interacting stochastic many-particle system in the whole space. The convergence is proved in the sense of probability by introducing an intermediate particle system with a mollified interaction potential, where the mollification is of algebraic scaling. The main idea of the proof is to study the time evolution of a stopped process and obtain a Grönwall type estimate by using Taylor’s expansion around the limiting stochastic process.

On a Partially Non-Stationary Vector AR Model with Vector GARCH Noises: Estimation and Testing

Chor-yiu Sin, Zichuan Mi & Shiqing Ling

10.4208/cmr.2023-0005

Commun. Math. Res., 40 (2024), pp. 64-101.

Preview Full PDF 2185 19128 Abstract

Abstract

This paper studies a partially nonstationary vector autoregressive (VAR) model with vector GARCH noises. We study the full rank and the reduced rank quasi-maximum likelihood estimators (QMLE) of parameters in the model. It is shown that both QMLE of long-run parameters asymptotically converge to a functional of two correlated vector Brownian motions. Based these, the likelihood ratio (LR) test statistic for cointegration rank is shown to be a functional of the standard Brownian motion and normal vector, asymptotically. As far as we know, our test is new in the literature. The critical values of the LR test are simulated via the Monte Carlo method. The performance of this test in ﬁnite samples is examined through Monte Carlo experiments. We apply our approach to an empirical example of three interest rates.

Uncertainty Comparison Between Value-at-Risk and Expected Shortfall

Qing Liu, Weimin Liu, Liang Peng & Gengsheng Qin

10.4208/cmr.2022-0071

Commun. Math. Res., 40 (2024), pp. 102-124.

Preview Full PDF 2194 20472 Abstract

Abstract

Value-at-Risk (VaR) and expected shortfall (ES) are two key risk measures in financial risk management. Comparing these two measures has been a hot debate, and most discussions focus on risk measure properties. This paper uses independent data and autoregressive models with normal or $t$-distribution to examine the effect of the heavy tail and dependence on comparing the nonparametric inference uncertainty of these two risk measures. Theoretical and numerical analyses suggest that VaR at 99% level is better than ES at 97.5% level for distributions with heavier tails.

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