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48 pages, 1081 KB

Open AccessArticle

Survival Probabilities for Correlated Drifted Brownian Motions via Exit from Simplicial Cones

by Tristan Guillaume

AppliedMath 2026, 6(3), 45; https://doi.org/10.3390/appliedmath6030045 - 10 Mar 2026

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This paper investigates the finite-horizon survival probability for a system of correlated arithmetic Brownian motions with heterogeneous drifts and volatilities, focusing on the event in which one component remains strictly below all others. Using a whitening transformation of the covariance structure, we reduce the problem to the survival of a standard Brownian motion in a simplicial cone, characterized by its spherical cross-section. While explicit solutions are available in low dimensions, we address the computationally challenging tetrahedral angular case. We derive a semi-analytic formula for the survival probability via an eigenfunction expansion of the Dirichlet Laplace–Beltrami operator on this curved domain. For efficient implementation, we construct a diffeomorphism from the spherical tetrahedron to a fixed Euclidean tetrahedron, enabling the computation of angular eigenpairs through a stable finite-element scheme. For higher-dimensional regimes, we also introduce a covariance-based difficulty index and geometric bounds based on an inscribed spherical cap to assess spectral convergence and estimate long-time decay rates. Numerical experiments show that this offline–online approach achieves high accuracy and substantial speedups relative to Monte Carlo benchmarks. Full article

(This article belongs to the Section Probabilistic & Statistical Mathematics)

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10 pages, 1848 KB

Open AccessArticle

Local Stochastic Correlation Models for Derivative Pricing

by Marcos Escobar-Anel

Stats 2025, 8(3), 65; https://doi.org/10.3390/stats8030065 - 18 Jul 2025

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Abstract

This paper reveals a simple methodology to create local-correlation models suitable for the closed-form pricing of two-asset financial derivatives. The multivariate models are built to ensure two conditions. First, marginals follow desirable processes, e.g., we choose the Geometric Brownian Motion (GBM), popular for stock prices. Second, the payoff of the derivative should follow a desired one-dimensional process. These conditions lead to a specific choice of the dependence structure in the form of a local-correlation model. Two popular multi-asset options are entertained: a spread option and a basket option. Full article

(This article belongs to the Section Applied Stochastic Models)

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16 pages, 6235 KB

Open AccessArticle

Multivariate Accelerated Degradation Modeling and Reliability Assessment for Ball Screw Grease Based on Fractional Brownian Motion Process Model

by Chuanhai Chen, Chaoyi Wang, Zhifeng Liu, Jinyan Guo, Peijuan Cui and Jigui Zheng

Fractal Fract. 2024, 8(10), 556; https://doi.org/10.3390/fractalfract8100556 - 26 Sep 2024

Cited by 6 | Viewed by 1755

Abstract

Considering that the degradation of ball screw grease involves fractal characteristics, which exhibit long-term dependency and autocorrelation, a multivariate accelerated degradation modeling and reliability assessment method based on the fractional Brownian motion process model is proposed in this paper. Firstly, a nonlinear accelerated degradation model of grease is established using fractional Brownian motion, considering the heterogeneity of samples as well as the memory effect and long-term dependence in the deterioration process, and realizing parameter estimation. Secondly, a multivariate reliability evaluation model is established by considering multivariate performance indicators in combination with the Frank copula function. Finally, the effectiveness and potential engineering application value of this method are verified through actual degradation data of the grease. Full article

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20 pages, 942 KB

Open AccessArticle

Assessing Asymmetrical Rates in Multivariate Phylogenetic Trait Evolution: An Extension of Statistical Models for Heterogeneous Rate Estimation

by Dwueng-Chwuan Jhwueng

Symmetry 2023, 15(7), 1445; https://doi.org/10.3390/sym15071445 - 19 Jul 2023

Cited by 2 | Viewed by 2587

Abstract

Understanding the rate of evolution provides insight into how rapidly species have historically evolved. We investigate the often-overlooked concept of asymmetry in evolutionary rates. We observe the variation in the rates at which different traits within the same organism, or the same traits across different organisms, evolve. Influenced by factors such as environmental pressures and genetic constraints, this asymmetry might lead to inconsistent rates of biological changes. To capture these diverse rates, we propose three advanced statistical models, transcending the traditionally employed Brownian motion model. These models—the phylogenetic multivariate Ornstein–Uhlenbeck model, the early burst model, and the mixed model—were applied to body length, forelimbs, and head length in salamanders. The results from our substantial dataset show these models’ effectiveness in highlighting the asymmetrical patterns of trait evolution, enhancing our understanding of the complex dynamics in species evolution. Therefore, our study underscores the importance of considering asymmetry when studying evolutionary rates. Full article

(This article belongs to the Special Issue Fluctuating Asymmetry in Evolutionary Biology)

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19 pages, 30444 KB

Open AccessArticle

A Multi-Spectral Fractal Image Model and Its Associated Fractal Dimension Estimator

by Mihai Ivanovici

Fractal Fract. 2023, 7(3), 238; https://doi.org/10.3390/fractalfract7030238 - 7 Mar 2023

Cited by 3 | Viewed by 3353

Abstract

We propose both a probabilistic fractal model and fractal dimension estimator for multi-spectral images. The model is based on the widely known fractional Brownian motion fractal model, which is extended to the case of images with multiple spectral bands. The model is validated mathematically under the assumption of statistical independence of the spectral components. Using this model, we generate several synthetic multi-spectral fractal images of varying complexity, with seven statistically independent spectral bands at specific wavelengths in the visible domain. The fractal dimension estimator is based on the widely used probabilistic box-counting classical approach extended to the multivariate domain of multi-spectral images. We validate the estimator on the previously generated synthetic multi-spectral images having fractal properties. Furthermore, we deploy the proposed multi-spectral fractal image estimator for the complexity assessment of real remotely sensed data sets and show the usefulness of the proposed approach. Full article

(This article belongs to the Special Issue Fractal Analysis for Remote Sensing Data)

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12 pages, 414 KB

Open AccessArticle

Variance and Entropy Assignment for Continuous-Time Stochastic Nonlinear Systems

by Xiafei Tang, Yuyang Zhou, Yiqun Zou and Qichun Zhang

Entropy 2022, 24(1), 25; https://doi.org/10.3390/e24010025 - 24 Dec 2021

Cited by 4 | Viewed by 3379

Abstract

This paper investigates the randomness assignment problem for a class of continuous-time stochastic nonlinear systems, where variance and entropy are employed to describe the investigated systems. In particular, the system model is formulated by a stochastic differential equation. Due to the nonlinearities of the systems, the probability density functions of the system state and system output cannot be characterised as Gaussian even if the system is subjected to Brownian motion. To deal with the non-Gaussian randomness, we present a novel backstepping-based design approach to convert the stochastic nonlinear system to a linear stochastic process, thus the variance and entropy of the system variables can be formulated analytically by the solving Fokker–Planck–Kolmogorov equation. In this way, the design parameter of the backstepping procedure can be then obtained to achieve the variance and entropy assignment. In addition, the stability of the proposed design scheme can be guaranteed and the multi-variate case is also discussed. In order to validate the design approach, the simulation results are provided to show the effectiveness of the proposed algorithm. Full article

(This article belongs to the Special Issue Entropy and Stochastic Distribution Optimization for Large-Scale Dynamical Systems)

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