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Robust Long-Term Growth Rate of Expected Utility for Leveraged ETFs

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arXiv 2023年

作者： Leung, Tim Park, Hyungbin Yeo, Heejun Department of Applied Mathematics Computational Finance & Risk Management Program University of Washington SeattleWA98195 United States Department of Mathematical Sciences Research Institute of Mathematics Seoul National University 1 Gwanak-ro Gwanak-gu Seoul08826 Korea Republic of Department of Mathematical Sciences Seoul National University 1 Gwanak-ro Gwanak-gu Seoul08826 Korea Republic of

This paper analyzes the robust long-term growth rate of expected utility and expected return from holding a leveraged exchange-traded fund (LETF). When the Markovian model parameters in the reference asset are uncertain, the robust long-term growth rate is derived by analyzing the worst-case parameters among an uncertainty set. We compute the growth rate and describe the optimal leverage ratio maximizing the robust long-term growth rate. To achieve this, the worst-case parameters are analyzed by the comparison principle, and the growth rate of the worst-case is computed using the martingale extraction method. The robust long-term growth rates are obtained explicitly under a number of models for the reference asset, including the geometric Brownian motion (GBM), Cox–Ingersoll–Ross (CIR), 3/2, and Heston and 3/2 stochastic volatility models. Additionally, we demonstrate the impact of stochastic interest rates, such as the Vasicek and inverse GARCH short rate models. This paper is an extended work of Leung and Park [2017]. Copyright © 2023, The Authors. All rights reserved.

关键词： Growth rate

SEMISIMPLE ALGEBRAS OF VECTOR FIELDS ON C3

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arXiv 2024年

作者： Ali, Sajid Azad, Hassan Biswas, Indranil Mahomed, Fazal M. Shah, Said Waqas College of Interdisciplinary Studies Zayed University Dubai19282 United Arab Emirates Abdus Salam School of Mathematical Sciences GCU Lahore54600 Pakistan Department of Mathematics Shiv Nadar University NH91 Tehsil Dadri Uttar Pradesh Greater Noida201314 India DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences School of Computer Science and Applied Mathematics University of the Witwatersrand Johannesburg Wits2050 South Africa

关键词： C (programming language)

Iterative Methods for Navier–Stokes Inverse Problems

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arXiv 2024年

作者： O’Connor, Liam Lecoanet, Daniel Anders, Evan H. Augustson, Kyle C. Burns, Keaton J. Vasil, Geoffrey M. Oishi, Jeffrey S. Brown, Benjamin P. Department of Engineering Sciences and Applied Mathematics Northwestern University EvanstonIL60208 United States Center for Interdisciplinary Exploration and Research in Astrophysics Northwestern University EvanstonIL60201 United States Kavli Institute for Theoretical Physics University of California Santa Barbara Santa BarbaraCA93106 United States Department of Mathematics Massachusetts Institute of Technology CambridgeMA02142 United States School of Mathematics Edinburgh University EH9 3FD United Kingdom Department of Physics and Astronomy Bates College LewistonME04240 United States Department of Astrophysical and Planetary Sciences University of Colorado Boulder BoulderCO80309 United States

Even when the partial differential equation underlying a physical process can be evolved forward in time, the retrospective (backward in time) inverse problem often has its own challenges and applications. Direct Adjoint Looping (DAL) is the defacto approach for solving retrospective inverse problems, but it has not been applied to deterministic retrospective Navier–Stokes inverse problems in 2D or 3D. In this paper, we demonstrate that DAL is ill-suited for solving retrospective 2D Navier–Stokes inverse problems. Alongside DAL, we study two other iterative methods: Simple Backward Integration (SBI) and the Quasi-Reversible Method (QRM). Our iterative SBI approach is novel while iterative QRM has previously been used. Using these three iterative methods, we solve two retrospective inverse problems: 1D Korteweg–de Vries–Burgers (decaying nonlinear wave) and 2D Navier–Stokes (unstratified Kelvin–Helmholtz vortex). In both cases, SBI and QRM reproduce the target final states more accurately and in fewer iterations than DAL. We attribute this performance gap to additional terms present in SBI and QRM’s respective backward integrations which are absent in DAL. © 2024, CC0.

关键词： Inverse problems

Understanding degeneracy of two-point correlation functions via Debye random media

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Physical Review E 2021年第4期104卷 045306-045306页

作者： Murray Skolnick Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

It is well known that the degeneracy of two-phase microstructures with the same volume fraction and two-point correlation function S2(r) is generally infinite. To elucidate the degeneracy problem explicitly, we examine Debye random media, which are entirely defined by a purely exponentially decaying two-point correlation function S2(r). In this work, we consider three different classes of Debye random media. First, we generate the “most probable” class using the Yeong-Torquato construction algorithm [Yeong and Torquato, Phys. Rev. E 57, 495 (1998)]. A second class of Debye random media is obtained by demonstrating that the corresponding two-point correlation functions are effectively realized in the first three space dimensions by certain models of overlapping, polydisperse spheres. A third class is obtained by using the Yeong-Torquato algorithm to construct Debye random media that are constrained to have an unusual prescribed pore-size probability density function. We structurally discriminate these three classes of Debye random media from one another by ascertaining their other statistical descriptors, including the pore-size, surface correlation, chord-length probability density, and lineal-path functions. We also compare and contrast the percolation thresholds as well as the diffusion and fluid transport properties of these degenerate Debye random media. We find that these three classes of Debye random media are generally distinguished by the aforementioned descriptors, and their microstructures are also visually distinct from one another. Our work further confirms the well-known fact that scattering information is insufficient to determine the effective physical properties of two-phase media. Additionally, our findings demonstrate the importance of the other two-point descriptors considered here in the design of materials with a spectrum of physical properties.

关键词： Complex systems

Structural characterization of many-particle systems on approach to hyperuniform states

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Physical Review E 2021年第5期103卷 052126-052126页

作者： Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The study of hyperuniform states of matter is an emerging multidisciplinary field, impinging on topics in the physical sciences, mathematics, and biology. The focus of this work is the exploration of quantitative descriptors that herald when a many-particle system in d-dimensional Euclidean space Rd approaches a hyperuniform state as a function of the relevant control parameter. We establish quantitative criteria to ascertain the extent of hyperuniform and nonhyperuniform distance-scaling regimes as well as the crossover point between them in terms of the “volume” coefficient A and “surface-area” coefficient B associated with the local number variance σ2(R) for a spherical window of radius R. The larger the ratio B/A, the larger the hyperuniform scaling regime, which becomes of infinite extent in the limit B/A→∞. To complement the known direct-space representation of the coefficient B in terms of the total correlation function h(r), we derive its corresponding Fourier representation in terms of the structure factor S(k), which is especially useful when scattering information is available experimentally or theoretically. We also demonstrate that the free-volume theory of the pressure of equilibrium packings of identical hard spheres that approach a strictly jammed state either along the stable crystal or metastable disordered branch dictates that such end states be exactly hyperuniform. Using the ratio B/A, as well as other diagnostic measures of hyperuniformity, including the hyperuniformity index H and the direct-correlation function length scale ξc, we study three different exactly solvable models as a function of the relevant control parameter, either density or temperature, with end states that are perfectly hyperuniform. Specifically, we analyze equilibrium systems of hard rods and “sticky” hard-sphere systems in arbitrary space dimension d as a function of density. We also examine low-temperature excited states of many-particle systems interacting with “stealt

关键词： Fluctuations & noise

An Assessment of the Supremizer and Aggregation Methods of Stabilization for Reduced-Order Models

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arXiv 2023年

作者： Davie, Kayla D. Elman, Howard C. Applied Mathematics & Statistics Scientific Computation Program University of Maryland College ParkMD20742 United States Department of Computer Science Institute for Advanced Computer Studies University of Maryland College ParkMD20742 United States

MSC Codes 65N22, 65K10, 65F45We explore the features of two methods of stabilization, aggregation and supremizer methods, for reduced-order modeling of parametrized optimal control problems. In both methods, the reduced basis spaces are augmented to guarantee stability. For the aggregation method, the reduced basis approximation spaces for the state and adjoint variables are augmented in such a way that the spaces are identical. For the supremizer method, the reduced basis approximation space for the state-control product space is augmented with the solutions of a supremizer equation. We implement both of these methods for solving several parametrized control problems and assess their performance. Results indicate that the number of reduced basis vectors needed to approximate the solution space to some tolerance with the supremizer method is much larger, possibly double, that for aggregation. There are also some cases where the supremizer method fails to produce a converged solution. We present results to compare the accuracy, efficiency, and computational costs associated with both methods of stabilization which suggest that stabilization by aggregation is a superior stabilization method for control problems. © 2023, CC BY.

关键词： Stabilization

Diffusion spreadability as a probe of the microstructure of complex media across length scales

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Physical Review E 2021年第5期104卷 054102-054102页

Understanding time-dependent diffusion processes in multiphase media is of great importance in physics, chemistry, materials science, petroleum engineering, and biology. Consider the time-dependent problem of mass transfer of a solute between two phases and assume that the solute is initially distributed in one phase (phase 2) and absent from the other (phase 1). We desire the fraction of total solute present in phase 1 as a function of time, S(t), which we call the spreadability, since it is a measure of the spreadability of diffusion information as a function of time. We derive exact direct-space formulas for S(t) in any Euclidean space dimension d in terms of the autocovariance function as well as corresponding Fourier representations of S(t) in terms of the spectral density, which are especially useful when scattering information is available experimentally or theoretically. These are singular results because they are rare examples of mass transport problems where exact solutions are possible. We derive closed-form general formulas for the short- and long-time behaviors of the spreadability in terms of crucial small- and large-scale microstructural information, respectively. The long-time behavior of S(t) enables one to distinguish the entire spectrum of microstructures that span from hyperuniform to nonhyperuniform media. For hyperuniform media, disordered or not, we show that the “excess” spreadability, S(∞)−S(t), decays to its long-time behavior exponentially faster than that of any nonhyperuniform two-phase medium, the “slowest” being antihyperuniform media. The stealthy hyperuniform class is characterized by an excess spreadability with the fastest decay rate among all translationally invariant microstructures. We obtain exact results for S(t) for a variety of specific ordered and disordered model microstructures across dimensions that span from hyperuniform to antihyperuniform media. Moreover, we establish a remarkable connection between the spreadability

关键词： Complex systems

Emulating complex synapses using interlinked proton conductors

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Physical Review applied 2025年第1期23卷 014027-014027页

作者： Lifu Zhang Ji-An Li Yang Hu Jie Jiang Rongjie Lai Marcus K. Benna Jian Shi Department of Materials Science and Engineering Rensselaer Polytechnic Institute Troy New York 12180 USA Neurosciences Graduate Program University of California San Diego La Jolla California 92093 USA Department of Mathematics Purdue University West Lafayette Indiana 47907 USA Department of Neurobiology School of Biological Sciences University of California at San Diego La Jolla California 92093 USA Department of Physics Applied Physics and Astronomy Rensselaer Polytechnic Institute Troy New York 12180 USA

In terms of energy efficiency and computational speed, neuromorphic electronics based on nonvolatile memory devices are expected to be one of most promising hardware candidates for future artificial intelligence (AI). However, catastrophic forgetting, networks rapidly overwriting previously learned weights when learning new tasks, remains a pivotal obstacle in either digital or analog AI chips for unleashing the true power of brainlike computing. To address catastrophic forgetting in the context of online memory storage, a complex synapse model (the Benna-Fusi model) was proposed recently [M. K. Benna and S. Fusi, Nat. Neurosci. 19, 1697 (2016)], the synaptic weight and internal variables of which evolve following diffusion dynamics. In this work, by designing a proton transistor with a series of charge-diffusion-controlled storage components, we have experimentally realized the Benna-Fusi artificial complex synapse. Memory consolidation from coupled storage components is revealed by both numerical simulations and experimental observations. Different memory timescales for the complex synapse are engineered by the diffusion length of charge carriers and the capacity and number of coupled storage components. The advantages of the demonstrated complex synapse for both memory capacity and memory consolidation are revealed by neural network simulations of face-familiarity detection. Our experimental realization of the complex synapse suggests a promising approach to enhance memory capacity and to enable continual learning.

关键词： Nonvolatile storage

Fractional Integrable and Related Discrete Nonlinear Schrödinger Equations

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arXiv 2022年

作者： Ablowitz, Mark J. Been, Joel B. Carr, Lincoln D. Department of Applied Mathematics University of Colorado BoulderCO80309 United States Department of Applied Mathematics and Statistics Colorado School of Mines GoldenCO80401 United States Department of Physics Colorado School of Mines GoldenCO80401 United States Quantum Engineering Program Colorado School of Mines GoldenCO80401 United States

Integrable fractional equations such as the fractional Korteweg-deVries and nonlinear Schrödinger equations are key to the intersection of nonlinear dynamics and fractional calculus. In this manuscript, the first discrete/differential difference equation of this type is found, the fractional integrable discrete nonlinear Schrödinger equation. This equation is linearized;special soliton solutions are found whose peak velocities exhibit more complicated behavior than other previously obtained fractional integrable equations. This equation is compared with the closely related fractional averaged discrete nonlinear Schrödinger equation which has simpler structure than the integrable case. For positive fractional parameter and small amplitude waves, the soliton solutions of the integrable and averaged equations have similar behavior. © 2022, CC BY.

关键词： Nonlinear equations