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Due to the intermittent and unstable nature of PV systems, maximum power point tracking (MPPT) of PV systems is necessary in practice. At the same time, PV cells are also shaded by objects such as trees and houses in the environment causing local shadows, so the Lévy flight particle swarm optimisation (LFPSO) is proposed in this paper. The random walk process of Lévy flight is added to the particle swarm optimisation (PSO) to increase the diversity of the search, which can avoid falling into local optimal solutions. The experimental simulation results show that the algorithm proposed in this paper can still accurately track the global maximum power in the case of partial shading, which well avoids falling into the local maximum power. The performance is much higher than the traditional maximum power point tracking algorithm, which improves the efficiency of the PV system. [ABSTRACT FROM AUTHOR]

This paper introduces a novel metaheuristic algorithm named the opposition-based cooperation search algorithm with Nelder–Mead (OCSANM). This enhanced algorithm builds upon the cooperation search algorithm (CSA) by incorporating opposition-based learning (OBL) and the Nelder–Mead simplex search method. The primary application of this algorithm is the design of a fractional-order proportional–integral–derivative (FOPID) controller for a buck converter system. A comprehensive evaluation is conducted using statistical boxplot analysis, nonparametric statistical tests and convergence response comparisons to assess the algorithm's performance and confirm its superiority over CSA. Furthermore, the FOPID-controlled buck converter system based on OCSANM is compared with two top-performing algorithms: one using a hybridized approach of Lévy flight distribution with simulated annealing (LFDSA) and the other employing the improved hunger games search (IHGS) algorithm. This comparison encompasses transient and frequency responses, performance indices and robustness analysis. The results reveal the notable advantages of the proposed OCSANM-based system, including 25.8% and 8.7% faster rise times, 26% and 8.8% faster settling times compared with the best-performing approaches, namely LFDSA and IHGS, respectively. In addition, the OCSANM-based system exhibits a 34.7% and 9.6% wider bandwidth than the existing approaches-based systems. Incorporating voltage and current responses of the buck converter's switched circuit with the OCSANM-based FOPID controller further underscores the algorithm's effectiveness. To provide a comprehensive assessment, the paper also compares the proposed approach's time and frequency domain responses with those of 17 other state-of-the-art approaches attempting to control buck converter systems similarly. These findings affirm the effectiveness of the OCSANM in designing FOPID controllers for buck converter systems. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new method to obtain the accessibility of stochastic partial differential equations driven by additive pure jump noise. An important novelty of this paper is to allow the driving noises to be degenerate. As an application, for the first time, we obtain the accessibility of a class of stochastic equations driven by pure jump (possibly degenerate) noise, including stochastic 2D Navier-Stokes equations, stochastic Burgers equations, stochastic singular p-Laplace equations, and stochastic fast diffusion equations. As a further application, we establish the ergodicity of stochastic singular p-Laplace equations and stochastic fast diffusion equations driven by additive pure jump noise, and we remark that the driving noises could be Compound Poisson processes or Lévy processes with heavy tails. [ABSTRACT FROM AUTHOR]

The classical Liouville property says that all bounded harmonic functions in Rn$\mathbb {R}^n$, that is, all bounded functions satisfying Δf=0$\Delta f = 0$, are constant. In this paper, we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator m(D)$m(D)$, such that the solutions f$f$ to m(D)f=0$m(D)f=0$ are Lebesgue a.e. constant (if f$f$ is bounded) or coincide Lebesgue a.e. with a polynomial (if f$f$ is polynomially bounded). The class of Fourier multipliers includes the (in general non‐local) generators of Lévy processes. For generators of Lévy processes, we obtain necessary and sufficient conditions for a strong Liouville theorem where f$f$ is positive and grows at most exponentially fast. As an application of our results above, we prove a coupling result for space‐time Lévy processes. [ABSTRACT FROM AUTHOR]

In this paper, we study one-dimensional hyperbolic Anderson models (HAM) driven by space-time pure-jump Lévy white noise in a finite-variance setting. Motivated by recent active research on limit theorems for stochastic partial differential equations driven by Gaussian noises, we present the first study in this Lévy setting. In particular, we first establish the spatial ergodicity of the solution and then a quantitative central limit theorem (CLT) for the spatial averages of the solution to HAM in both Wasserstein distance and Kolmogorov distance, with the same rate of convergence. To achieve the first goal (i.e. spatial ergodicity), we exploit some basic properties of the solution and apply a Poincaré inequality in the Poisson setting, which requires delicate moment estimates on the Malliavin derivatives of the solution. Such moment estimates are obtained in a soft manner by observing a natural connection between the Malliavin derivatives of HAM and a HAM with Dirac delta velocity. To achieve the second goal (i.e. CLT), we need two key ingredients: (i) a univariate second-order Poincaré inequality in the Poisson setting that goes back to Last, Peccati, and Schulte (Probab. Theory Related Fields, 2016) and has been recently improved by Trauthwein (arXiv:2212.03782); (ii) aforementioned moment estimates of Malliavin derivatives up to second order. We also establish a corresponding functional CLT by (a) showing the convergence in finite-dimensional distributions and (b) verifying Kolmogorov's tightness criterion. Part (a) is made possible by a linearization trick and the univariate second-order Poincaré inequality, while part (b) follows from a standard moment estimate with an application of Rosenthal's inequality. [ABSTRACT FROM AUTHOR]

In this paper, we study McKean–Vlasov stochastic differential equations driven by Lévy processes. Firstly, under the non-Lipschitz condition which include classical Lipschitz conditions as special cases, we establish the existence and uniqueness for solutions of McKean–Vlasov stochastic differential equations using Carathéodory approximation. Then under certain averaging conditions, we establish a stochastic averaging principle for McKean–Vlasov stochastic differential equations driven by Lévy processes. We find that the solutions to stochastic systems concerned with Lévy noise can be approximated by solutions to averaged McKean–Vlasov stochastic differential equations driven by Lévy processes in the sense of convergence in p th moment. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of stochastic Riemann-Liouville type fractional differential equations driven by Lévy noise. By using Itô formula for the considered equation, we attempt to explore the non-confluence property of solution for the considered equation under some appropriate conditions. Our approach is to construct some suitable Lyapunov functions which is novel in exploring the non-confluence property of differential equations. [ABSTRACT FROM AUTHOR]

First hitting times (FHTs) describe the time it takes a random "searcher" to find a "target" and are used to study timescales in many applications. FHTs have been well-studied for diffusive search, especially for small targets, which is called the narrow capture or narrow escape problem. In this paper, we study the FHT to small targets for a one-dimensional superdiffusive search described by a Lévy flight. By applying the method of matched asymptotic expansions to a fractional differential equation we obtain an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order s ε (0, 1) (describing a (2s)-stable Lévy flight whose squared displacement scales as t1/s in time t) and targets of radius \varepsilon \ll 1, we show that the MFHT is order one for s ε (1/2, 1) and diverges as log(1/\varepsilon) for s = 1/2 and \varepsilon 2s 1 for s ε (0, 1/2). We then use our asymptotic results to identify the value of s ε (0, 1] which minimizes the average MFHT and find that (a) this optimal value of s vanishes for sparse targets and (b) the value s = 1/2 (corresponding to an inverse square Lévy search) is optimal in only very specific circumstances. We confirm our results by comparison to both deterministic numerical solutions of the associated fractional differential equation and stochastic simulations. [ABSTRACT FROM AUTHOR]

We aimed to improve the detection accuracy of laser methane sensors in expansive temperature application environments. In this paper, a large-scale dataset of the measured concentration of the sensor at different temperatures is established, and a temperature compensation model based on the ISSA-BP neural network is proposed. On the data side, a large-scale dataset of 15,810 sets of laser methane sensors with different temperatures and concentrations was established, and an Improved Isolation Forest algorithm was used to clean the large-scale data and remove the outliers in the dataset. On the modeling framework, a temperature compensation model based on the ISSA-BP neural network is proposed. The quasi-reflective learning, chameleon swarm algorithm, Lévy flight, and artificial rabbits optimization are utilized to improve the initialization of the sparrow population, explorer position, anti-predator position, and position of individual sparrows in each generation, respectively, to improve the global optimization seeking ability of the standard sparrow search algorithm. The ISSA-BP temperature compensation model far outperforms the four models, SVM, RF, BP, and PSO-BP, in model evaluation metrics such as MAE, MAPE, RMSE, and R-square for both the training and test sets. The results show that the algorithm in this paper can significantly improve the detection accuracy of the laser methane sensor under the wide temperature application environment. [ABSTRACT FROM AUTHOR]

This paper presents an enhanced whale optimization algorithm based on virtual force to optimize coverage and address the problem of uneven coverage during the deployment of underwater sensor networks. This method is guided by the Lévy flight and virtual force algorithms and adopts a differential mutation strategy based on random walks. This improves the fitness of the initial population and population richness of the algorithm. The effectiveness of this method was verified through simulation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the optimal portfolio problem for a company that can invest in two risky assets, where a novel Lévy-process-driven model is constructed to describe the dynamics of the wealth process by using a constant elasticity of variance model and a jump-diffusion process. A delicately designed value function is proposed under the mean–variance criterion to reflect the optimal portfolio for the stochastic volatility model. By using the verification theorem, the desired optimal portfolio strategy is proposed by the solution to certain Hamilton–Jacobi–Bellman equations. Furthermore, the corresponding expressions are achieved by using the stochastic analysis theory. Finally, a numerical simulation example is provided to verify the effectiveness of the proposed optimal portfolio strategy. [ABSTRACT FROM AUTHOR]

The paper develops a finite-time quantum state reduction framework via the use of Lévy random bridges (LRBs) that can be understood as Doob h-transformations on Lévy processes. Building upon the non-anticipative semimartingale representation of LRBs, we propose a family of energy-driven stochastic Schrödinger equations that go beyond the purely-continuous Brownian motion setup, and enter the scope of quantum systems involving discontinuities and heavy-tails. In doing so, we allow collapse dynamics to be governed intrinsically by the Markovian statistics of LRBs. The framework can host progressively convoluted stochastic state reduction dynamics in a tractable way and encapsulates jump-diffusion Schrödinger evolutions by use of the Lévy-Itô decomposition. Our construct motivates the usage of more general Markov processes and Doob h-transformations in producing consistent wave function collapse dynamics in finite-time. [ABSTRACT FROM AUTHOR]

The Artificial Electric Field Algorithm (AEFA) stands out as a physics-inspired metaheuristic, drawing inspiration from Coulomb's law and electrostatic force; however, while AEFA has demonstrated efficacy, it can face challenges such as convergence issues and suboptimal solutions, especially in high-dimensional problems. To overcome these challenges, this paper introduces a modified version of AEFA, named mAEFA, which leverages the capabilities of Lévy flights, simulated annealing, and the Adaptive s-best Mutation and Natural Survivor Method (NSM) mechanisms. While Lévy flights enhance exploration potential and simulated annealing improves search exploitation, the Adaptive s-best Mutation and Natural Survivor Method (NSM) mechanisms are employed to add more diversity. The integration of these mechanisms in AEFA aims to expand its search space, enhance exploration potential, avoid local optima, and achieve improved performance, robustness, and a more equitable equilibrium between local intensification and global diversification. In this study, a comprehensive assessment of mAEFA is carried out, employing a combination of quantitative and qualitative measures, on a diverse range of 29 intricate CEC'17 constraint benchmarks that exhibit different characteristics. The practical compatibility of the proposed mAEFA is evaluated on five engineering benchmark problems derived from the civil, mechanical, and industrial engineering domains. Results from the mAEFA algorithm are compared with those from seven recently introduced metaheuristic algorithms using widely adopted statistical metrics. The mAEFA algorithm outperforms the LCA algorithm in all 29 CEC'17 test functions with 100% superiority and shows better results than SAO, GOA, CHIO, PSO, GSA, and AEFA in 96.6%, 96.6%, 93.1%, 86.2%, 82.8%, and 58.6% of test cases, respectively. In three out of five engineering design problems, mAEFA outperforms all the compared algorithms, securing second place in the remaining two problems. Results across all optimization problems highlight the effectiveness and robustness of mAEFA compared to baseline metaheuristics. The suggested enhancements in AEFA have proven effective, establishing competitiveness in diverse optimization problems. [ABSTRACT FROM AUTHOR]

The estimation of absorption time distributions of Markov jump processes is an important task in various branches of statistics and applied probability. While the time‐homogeneous case is classic, the time‐inhomogeneous case has recently received increased attention due to its added flexibility and advances in computational power. However, commuting sub‐intensity matrices are assumed, which in various cases limits the parsimonious properties of the resulting representation. This paper develops the theory required to solve the general case through maximum likelihood estimation, and in particular, using the expectation‐maximization algorithm. A reduction to a piecewise constant intensity matrix function is proposed in order to provide succinct representations, where a parametric linear model binds the intensities together. Practical aspects are discussed and illustrated through the estimation of notoriously demanding theoretical distributions and real data, from the perspective of matrix analytic methods. [ABSTRACT FROM AUTHOR]

• Feature selection is studied by a role-oriented binary grey wolf optimizer. • Foraging-following is combined with top grey wolves to improve convergence. • Lévy flight is integrated into ordinary grey wolves to improve population diversity. • The performance of feature selection is enhanced by more than 5%. • The performance of Improved algorithm is demonstrated by 29 benchmark functions. Feature selection can effectively define the feature subset, remove redundant, irrelevant, and noisy features. In order to adapt the feature selection problem, this paper adopts role-oriented modeling paradigm to make up for the defects of binary grey wolf optimizer such as premature, declining diversity, and insufficient convergence, then comes up with novel V-shaped linear transfer functions. Ultimately, a role-oriented binary grey wolf optimizer is proposed. The top grey wolves combine the foraging-following to enhance convergence, while the ordinary grey wolves integrate Lévy flight to avoid premature and improve diversity. The outstanding ability of role-oriented binary grey wolf optimizer in balance of exploitation and exploration is verified by benchmarks and the effectiveness in solving feature selection is demonstrated through feature selection tasks. The results of benchmark and feature selection task have comprehensively demonstrated the competitiveness of role-oriented modeling paradigm in improving metaheuristic algorithms. [ABSTRACT FROM AUTHOR]

The paper examines the valuation and hedging of life insurance obligations in the presence of mortality risk using the local risk-minimizing hedging approach. Roughly speaking, it is assumed that the lifetime of policyholders in an insurance portfolio is modeled by a point process whose stochastic intensity is controlled by a diffusion process. The stock price process is assumed to be a regime-switching Lévy process with non-zero regime-switching drift, where the parameters are assumed to depend on the economic states. Using the Föllmer–Schweizer decomposition, the main valuation and hedging results for a conditional payment process are determined. Some specific situations have been considered in which the local risk-minimizing strategies for a stream of liability payments or a unit-linked contract are presented. [ABSTRACT FROM AUTHOR]

With the continuous development of rail transportation worldwide, the safety and economy of metro trains have become essential standards. The door system is one of the most frequently used components in the process of train operation, so the fault rate is much higher than other systems. In view of the above problems, a preventive maintenance (PM) model for critical components of metro train door system is established based on reliability in this paper. Firstly, the Pareto chart analysis method and failure mode effect and criticality analysis method are adopted to determine that the lower retaining pin assembly and balanced pressing wheel are the main fault components in the door system. The parameter values of the three-parameter Weibull model of the door components are accurately obtained through the improved dung beetle optimizer algorithm, which provides an essential theoretical basis for optimizing preventive maintenance decisions. Secondly, a PM model of metro door components is established, and Lévy flight is added to PSO to enhance the optimization performance of the algorithm and solve the preventive maintenance model. Finally, the model is verified by sorting out the historical fault maintenance data of the metro door system in a city. The experimental results show that the PM strategy has economic, safety and task performance advantages compared to traditional maintenance methods. [ABSTRACT FROM AUTHOR]

This paper presents a hybrid algorithm based on the slime mould algorithm (SMA) and the mixed dandelion optimizer. The hybrid algorithm improves the convergence speed and prevents the algorithm from falling into the local optimal. (1) The Bernoulli chaotic mapping is added in the initialization phase to enrich the population diversity. (2) The Brownian motion and Lévy flight strategy are added to further enhance the global search ability and local exploitation performance of the slime mould. (3) The specular reflection learning is added in the late iteration to improve the population search ability and avoid falling into local optimality. The experimental results show that the convergence speed and precision of the improved algorithm are improved in the standard test functions. At last, this paper optimizes the parameters of the Extreme Learning Machine (ELM) model with the improved method and applies it to the power load forecasting problem. The effectiveness of the improved method in solving practical engineering problems is further verified. [ABSTRACT FROM AUTHOR]

This paper proposes binary versions of artificial rabbits optimization (ARO) for feature selection (FS) with medical diagnosis data. ARO is a recent swarm-based optimization algorithm that mimics rabbits' natural survival tactics and eating habits. It was modeled in an optimization context to tackle optimization problems of continuous search spaces. In this paper, ARO is improved to deal with the binary domain of FS. The improvements include three additions: First, different alternatives of transfer functions were used to convert ARO from continuous to binary; second, the global-best concept was added to the binary ARO to improve the exploitation capability of the proposed algorithm; and finally, Lévy flight and opposition-based learning strategies were injected into the proposed algorithm to enhance its diversity and thus improve the balance between global exploration and local exploitation during all stages of the search process. Six binary variants of ARO were designed across an extensive set of experiments to study the impact of using the proposed amendments on the performance of the proposed ARO algorithm. These variants are: binary ARO with S-shaped transfer function (BAROS), binary ARO with V-shaped transfer function (BAROV), BAROS with the global-best concept (BGAROS), BGAROV with the global-best concept (BGAROV), BGAROS with Lévy flight and opposition-based learning strategies (BGAROSLO), and BGAROV with Lévy flight and opposition-based learning strategies (BGAROVLO). The proposed binary ARO versions were evaluated using 23 medical FS datasets. In addition, the proposed algorithm was applied to detect coronavirus disease using a real COVID-19 dataset. Five performance measures were used: classification accuracy, sensitivity, specificity, fitness value, and the number of selected features. In a nutshell, the proposed binary ARO versions were able to achieve success rates for these performance metrics as follows: 66.7%, 50%, 33.3%, 66.7%, and 83.3%, respectively. In conclusion, the success of the proposed ARO versions was realized due to the suitable design of the parameters of the proposed ARO version, such as transfer functions, global-best concept, Lévy flight, and opposition-based learning strategies. A comprehensive comparative evaluation was studied against ten well-established methods using the same datasets with a high preference for the proposed ARO versions, especially BGAROSLO which can achieve the best accuracy for the majority of the FS datasets. This is proven using Friedman's statistical test ad-hocked by Holm's test. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of the mean changes in time series with heavy-tailed AR(p) noise. Firstly, it proposes a modified ratio-type test statistic, and the results show that under the null hypothesis of no mean change, the asymptotic distribution of the modified statistic is a functional of Lévy processes and the consistency under the alternative hypothesis is obtained. However, a heavy-tailed index exists in the asymptotic distribution and is difficult to estimate. This paper uses bootstrap sampling, jackknife sampling, and subsampling to approximate the distribution under the null hypothesis, and obtain more accurate critical values and empirical power. In addition, some results from a small simulation study and a practical example give an idea of the finite sample behavior of the proposed statistic. [ABSTRACT FROM AUTHOR]

It is recently shown that a split-step quantum walk possesses a chiral symmetry, and that a certain well-defined index can be naturally assigned to it. The index is a well-defined Fredholm index if and only if the associated unitary time-evolution operator has spectral gaps at both + 1 and − 1. In this paper, we extend the existing index formula for the Fredholm case to encompass the non-Fredholm case (i.e. gapless case). We make use of a natural extension of the Fredholm index to the non-Fredholm case, known as the Witten index. The aim of this paper is to fully classify the Witten index of the split-step quantum walk by employing the spectral shift function for a rank one perturbation of a fourth-order difference operator. It is also shown in this paper that the Witten index can take half-integer values in the non-Fredholm case. [ABSTRACT FROM AUTHOR]

In this paper, we study de Finetti's optimal dividend problem with capital injection under the assumption that the dividend strategies are absolutely continuous. In many previous studies, the process before being controlled was assumed to be a spectrally one-sided Lévy process, however in this paper we use a Lévy process that may have both positive and negative jumps. In the main theorem, we show that a refraction–reflection strategy is an optimal strategy. We also mention the existence and uniqueness of solutions of the stochastic differential equations that define refracted Lévy processes. [ABSTRACT FROM AUTHOR]

Flow Direction Algorithm (FDA) has better searching performance than some traditional optimization algorithms. To give the basic Flow Direction Algorithm more effective searching ability and avoid multiple local minima under the searching space, and enable it to obtain better search results, an improved FDA based on the Lévy flight strategy and the self-renewable method (LSRFDA) was proposed in this paper. The Lévy flight strategy and the self-renewable approach were added to the basic Flow Direction Algorithm. Random parameters generated by the Lévy flight strategy can increase the algorithm's diversity of feasible solutions in a short calculation time and greatly enhance the operational efficiency of the algorithm. The self-renewable method lets the algorithm quickly obtain a better possible solution and jump to the local solution space. Then, this paper tested different mathematical testing functions, including low-dimensional and high-dimensional functions, and the test results were compared with those of different algorithms. This paper includes iterative figures, box plots, and search paths to show the different performances of the LSRFDA. Finally, this paper calculated different engineering optimization problems. The test results show that the proposed algorithm in this paper has better searching ability and quicker searching speed than the basic Flow Direction Algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we concentrate on the European option pricing problem in incomplete markets by taking Vasicek stochastic interest rate model into account and expanding it in accordance with a Lévy process named Variance Gamma (VG) process. In doing so, first, we prove the existence and uniqueness of the solution to the stochastic differential equation of this model. Then, by the means of the hedging and replicating techniques, pricing formulas for the bond and, in succession, for European options are established. Finally, from a numerical point of view, we will analyze the value of a bond based on our consideration model and also, we will study a European call option bond price. All these results are reported through the Monte-Carlo simulation method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel hybrid heuristic algorithm, particle swarm optimization, and whale optimization algorithm (PSO–WOA), to solve a multi-objective optimization problem relating to point-to-point trajectory planning of space robots. First of all, the kinematics of the space robot is introduced, and the motion of each revolute joint of the manipulator is parameterized by Bézier curve. Then, contradictory objective functions are proposed, and the trajectory planning problem is transformed into a multi-objective optimization problem. The pose of the end-effector at the end of motion is set as the primary objective. The base disturbance, execution time, and manipulability of the end-effector are also taken into account. Furthermore, self-collision avoidance during the motion is also considered. The trajectory planning problem finally comes down to finding an optimal parameter of the Bézier curve for each joint. We propose a novel hybrid PSO–WOA, which is supposed to take advantages of the best of both methods: the exploration feature of the WOA and exploitation feature of the PSO. In order to enhance the performance of the PSO–WOA, the good point set and lévy flight stochastic steps are employed for the initialization and updating process, respectively. The proposed method is applied to generate an optimal trajectory for a redundant free-floating space robot. The simulation results demonstrate the effectiveness of the PSO–WOA. [ABSTRACT FROM AUTHOR]

This paper studies the stochastic pantograph model, which is considered a subcategory of stochastic delay differential equations. A more general jump process, which is called the Lévy process, is added to the model for better performance and modeling situations, having sudden changes and extreme events such as market crashes in finance. By utilizing the truncation technique, we propose the diffused split-step truncated Euler–Maruyama method, which is considered as an explicit scheme, and apply it to the addressed model. By applying the Khasminskii-type condition, the convergence rate of the proposed scheme is attained in L p (p ≥ 2) sense where the non-jump coefficients grow super-linearly while the jump coefficient acts linearly. Also, the rate of convergence of the proposed scheme in L p (0 < p < 2) sense is addressed where all the three coefficients grow beyond linearly. Finally, theoretical findings are manifested via some numerical examples. [ABSTRACT FROM AUTHOR]

This paper deals with the limit distribution for a stochastic differential equation driven by a non-symmetric cylindrical α -stable process. Under suitable conditions, it is proved that the solution of this equation converges weakly to that of a stochastic differential equation driven by a Brownian motion in the Skorohod space as α → 2. Also, the rate of weak convergence, which depends on 2 − α , for the solution towards the solution of the limit equation is obtained. For illustration, the results are applied to a simple one-dimensional stochastic differential equation, which implies the rate of weak convergence is optimal. [ABSTRACT FROM AUTHOR]

In this paper, we study a smooth approximation of an arbitrary càdlàg Lévy process. Such approximation processes are known as integrated fast oscillating Ornstein–Uhlenbeck (OU) processes. We know that approximating processes are continuous, while the limit of processes may be discontinuous, so convergence in uniform topology or Skorokhod J 1 -topology will not hold in general. Therefore, we establish an approximation in Skorokhod M 1 -topology. Note that the convergence is almost surely, which is an extension result of Hintze and Pavlyukevich. [ABSTRACT FROM AUTHOR]

This paper provides a real application of a popular swarm-intelligence optimisation method. The aim is to analyse the efficiency of various settings of the marine predator algorithm (MPA). Four crucial numerical parameters of the MPA are statistically analysed to propose the most efficient setting for solving engineering problems. Besides population size, particle velocity parameter P, Lévy flight parameter β , and fish aggregating device (FAD) probabilities are studied. Finally, 193 various settings, including fixed values and dynamic changes of the MPA parameters, are experimentally compared when solving 13 engineering problems. Standard statistical approaches are employed to highlight significant differences in various MPA settings. The setting of two MPA parameters (P, FADs) significantly influences MPA performance. Three newly proposed MPA settings outperform the original variant significantly. The best results provide the MPA variant with the dynamic linear change of P from 0.5 to 0. These parameters influence the velocity of prey and predator individuals in all three stages of the MPA search process. Decreasing the value of P showed that decreasing the velocity of individuals during the search provides good performance. Further, lower efficiency of the MPA with higher FAD values was detected. It means that more occasional use of fish aggregating devices (FADs) can increase the solvability of engineering problems. Regarding population size, lower values ( N = 10 ) provided significantly better results compared with the higher values ( N = 500 ). [ABSTRACT FROM AUTHOR]

To fully harness the burgeoning array of in-orbit satellite resources and augment the efficacy of dynamic surveillance of densely clustered terrestrial targets, this paper delineates the following methodologies. Initially, we leverage the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) clustering algorithm to aggregate the concentrated terrestrial targets, taking into account the field-of-view peculiarities of agile staring satellites. Subsequently, we architect a model for a synergistic multiple angle earth observation satellites (AEOSs) mission planning with the optimization objectives of observational revenue, minimal energy expenditure, and load balancing, factoring in constraints such as target visibility time window, AEOSs maneuverability, and satellite storage. To tackle this predicament, we propose an improved heuristic ant colony optimization (ACO) algorithm, utilizing the task interval, task priority, and the length of time a task can start observation as heuristic information. Furthermore, we incorporate the notion of the max–min ant system to regulate the magnitude of pheromone concentration, and we amalgamate global and local pheromone update strategies to expedite the convergence rate of the algorithm. We also introduce the Lévy flight improved pheromone evaporation coefficient to bolster the algorithm's capacity to evade local optima. Ultimately, through a series of simulation experiments, we substantiate the significant performance improvements achieved by the improved heuristic ant colony algorithm compared to the standard ant colony algorithm. We furnish proof of its efficacy in resolving the planning of multiple AEOS staring observation missions. [ABSTRACT FROM AUTHOR]

Meta-heuristics have been successfully applied to many complex optimization problems. One of the main reasons for its success is its ability to handle non-convex, nonlinear, multimodal, multi-variable, and multi-objective problems with easy implementation. However, the quality of the response of these algorithms to an optimization problem is highly susceptible to the control parameters, and few works aim to tune them or find tools that can improve the algorithms. The literature is rich in proposals for new algorithms, but not for improving existing ones. This paper presents different strategies for tuning and accelerating meta-heuristics using the first hybrid algorithm in the literature. The Lichtenberg algorithm is inspired by lightning and Lichtenberg figures and has been increasingly successfully applied to various optimization problems. However, a study of its best parameters has never been presented until now. After a discussion of the best tuning tools, its tuning parameters are performed using response surface methodology. Then, 14 versions are studied through 10 test functions using chaos theory and Lévy flights scenarios. After 13,500 simulations, the chaotic Lichtenberg algorithm equipped with the piecewise function and tuned parameters proved the best version with only 16% similarity to the original algorithm. Then, it was compared to the genetic algorithm, particle swarm optimization, gray wolf optimizer, salp swarm optimization, whale optimization algorithm, and dragonfly algorithm. The proposed algorithm had both the best average accuracy, lower computational cost, and the smallest standard deviation. [ABSTRACT FROM AUTHOR]

The threshold dividend strategy, under which dividends are paid only when the insurer's surplus exceeds a pre-determined threshold, has received considerable attention in risk theory. However, in practice, it seems rather unlikely that an insurer will immediately pull back the dividend payments as soon as its surplus level drops below the dividend threshold. Hence, in this paper, we propose a refracted Lévy risk model with delayed dividend pullbacks triggered by a certain Poissonian observation scheme. Leveraging the extensive literature on fluctuation identities for spectrally negative Lévy processes, we obtain explicit expressions for two-sided exit identities of the proposed insurance risk process. Also, penalties are incorporated into the analysis of dividend payouts as a mechanism to penalize for the volatility of the dividend policy and account for an investor's typical preference for more stable cash flows. An explicit expression for the expected (discounted) dividend payouts net of penalties is derived. The criterion for the optimal threshold level that maximizes the expected dividend payouts is also discussed. Finally, several numerical examples are considered to assess the impact of dividend delays on ruin-related quantities. We numerically show that dividend strategies with more steady dividend payouts can be preferred (over the well-known threshold dividend strategy) when penalty fee become too onerous. [ABSTRACT FROM AUTHOR]

We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution, with the same mass, of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a "projection" of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one. Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally Hölder, has not been considered before in the nonlocal diffusion setting. [ABSTRACT FROM AUTHOR]

In the paper, we prove probabilistic analogs of the universal approximation theorems and link continuous random variables of a certain type and monotonic feedforward artificial neural networks with one-dimensional input, output and one hidden layer. In particular, we show that any continuous infinitely divisible random variable can be successfully approximated with a mix of logistic distributions. Based on the theorems proved in the current paper, we develop a new approach for developing Monte Carlo methods combined with artificial neural networks for pricing options in Lévy models. In contrast to straightforward incorporation of neural networks into Monte Carlo methods, we approximate the cumulative distribution function, but not its inverse. Moreover, we give a clear probabilistic interpretation of the constructed approximator that helps us simulate the Lévy process by using only separate components of our neural network. [ABSTRACT FROM AUTHOR]

In this paper we obtain a Wiener–Hopf type factorization for a real-valued arithmetic Brownian motion with time-dependent drift and volatility. To the best of our knowledge, this paper is the very first step towards realizing the objective of deriving Wiener–Hopf type factorizations for (real-valued) time-inhomogeneous Lévy processes. In order to prove our main theorem, we derive some new results regarding time-inhomogeneous noisy Wiener–Hopf factorization. We demonstrate that in the special case of the arithmetic Brownian motion with constant drift and volatility our main result agrees with classical Wiener–Hopf factorization for this particular time-homogenous Lévy process. [ABSTRACT FROM AUTHOR]

In recent years, due to the growing complexity of real-world problems, researchers have been favoring stochastic search algorithms as their preferred method for problem solving. The slime mould algorithm is a high-performance, stochastic search algorithm inspired by the foraging behavior of slime moulds. However, it faces challenges such as low population diversity, high randomness, and susceptibility to falling into local optima. Therefore, this paper presents an enhanced slime mould algorithm that combines multiple strategies, called the ESMA. The incorporation of selective average position and Lévy flights with jumps in the global exploration phase improves the flexibility of the search approach. A dynamic lens learning approach is employed to adjust the position of the optimal slime mould individual, guiding the entire population to move towards the correct position within the given search space. In the updating method, an improved crisscross strategy is adopted to reorganize the slime mould individuals, which makes the search method of the slime mould population more refined. Finally, the performance of the ESMA is evaluated using 40 well-known benchmark functions, including those from CEC2017 and CEC2013 test suites. It is also recognized by Friedman's test as statistically significant. The analysis of the results on two real-world engineering problems demonstrates that the ESMA presents a substantial advantage in terms of search capability. [ABSTRACT FROM AUTHOR]

In the face of fire in buildings, people need to quickly plan their escape routes. Intelligent optimization algorithms can achieve this goal, including the sparrow search algorithm (SSA). Despite the powerful search ability of the SSA, there are still some areas that need improvements. Aiming at the problem that the sparrow search algorithm reduces population diversity and is easy to fall into local optimum when solving the optimal solution of the objective function, a hybrid improved sparrow search algorithm is proposed. First, logistic-tent mapping is used to initialize the population and enhance diversity in the population. Also, an adaptive period factor is introduced into the producer's update position equation. Then, the Lévy flight is introduced to the position of the participant to improve the optimization ability of the algorithm. Finally, the adaptive disturbance strategy is adopted for excellent individuals to strengthen the ability of the algorithm to jump out of the local optimum in the later stage. In order to prove the improvement of the optimization ability of the improved algorithm, the improved sparrow algorithm is applied to five kinds of maps for evacuation path planning and compared with the simulation results of other intelligent algorithms. The ultimate simulation results show that the optimization algorithm proposed in this paper has better performance in path length, path smoothness, and algorithm convergence, showing better optimization performance. [ABSTRACT FROM AUTHOR]

The classical Cramér–Lundberg risk process is commonly used to model the surplus of an insurer; it characterizes the claim arrival process and the claim size random variable Y through a compound Poisson process, along with a constant rate of premium income. When E (Y 2) < ∞ , the process can be approximated by a diffusion process, but that requirement eliminates many heavy-tailed claim models, such as the Pareto (α , θ) with α ≤ 2. In this paper, we generalize the well known diffusion approximation by assuming that Y lies in the domain of attraction of an α-stable random variable, for 0 < α ≤ 2. Then, we construct a sequence of classical Cramér–Lundberg risk processes and show that the sequence converges to an α-stable Lévy motion in the Skorokhod J 1 -topology. We prove this convergence by proving the pointwise convergence of the corresponding Laplace exponents of our processes, which to our knowledge, is a new result. To apply this convergence result, we show the convergence of a sequence of Gerber–Shiu distributions of exponential Parisian ruin, and we show the convergence of a sequence of payoff functions for barrier dividend strategies. Both of these applications provide new proofs of the stated limits. [ABSTRACT FROM AUTHOR]

Feature selection is a knowledge discovery tool to understand the problem by analysing features. In particular, the application of feature selection in data mining can not only improve the quality of extracted patterns and knowledge but also decrease computational costs. Various techniques have been applied to this complex optimization problem, in which metaheuristics have been validated to be superior. This study introduces a new metaheuristic known for having lean and fast programming, inspired by the sunflower's motions for feature selection for the first time. It is equipped with a v‐shaped transfer function and associated with the KNN classifier to become the binary sunflower optimization (BSFO). A total of 12 variants of BSFO are designed based on the chaos theory and Lévy flights, called improved binary sunflower optimization (IBSFO). A discussion between these improvement theories for feature selection has also not been made yet, and it is performed in this paper using 15 benchmark datasets from the UCI repository. The experimental results show that all variants can advance the fitness value of BSFO, and nine of them considerably decrease the computational costs. Furthermore, the chaotic BSFO with the Chebyshev function, taking replacement to normal rand, has the lowest fitness value (−11.37%) and execution time (−9.31%) than the original BSFO. Further, IBSFO is compared with another eight metaheuristics and outperforms these competitors on average fitness value and execution time. Overall, IBSFO proved to find subsets with reduced dimension and high accuracy with meagre computational cost due to its robust explorative and exploitative capacities. [ABSTRACT FROM AUTHOR]

In this paper, a two‐way relay‐aided cognitive satellite terrestrial network (TR‐CSTN) model is proposed, where primary users are located at the edge of the base station. In the TR‐CSTN, one of satellite terminal users (STUs) is selected by the fusion center as the TR to forward information between two edge primary users with power of the TR. Meanwhile, these edge primary users share the licensed frequency band with the selected TR to send information to the satellite. Then, given the limited spectrum utilization and energy efficiency (EE) of the communication system, the cooperative spectrum sensing is employed to realize green communication. Specifically, the fusion center threshold, energy detection threshold, sensing duration and number of STUs are jointly optimized to enhance EE. Furthermore, considering that the node's energy shortage results in a short network lifetime, absolute EE gets improved. In detail, a power allocation scheme named normalized power aided Lévy flight trajectory‐based whale optimization algorithm (NP‐LWOA) is provided, which fulfills effective energy compensation among STUs to prolong the network lifetime notably. Finally, numerical results confirm the theoretical analysis and show the effectiveness of the TR‐CSTN and the NP‐LWOA in efficiently achieving the concept of green communication compared with other methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence/non-existence of ground states for the following type of elliptic equations with mixed local and nonlocal operators and general nonlinearity:  (− ▵) s u − ▵ u + λ u = f (u) , x ∈ R N , which is driven by the superposition of Brownian and Lévy processes. By considering a constrained variational problem, under suitable assumptions on f, we manage to establish a sharp existence of the ground state solutions to the equation considered. These results improve the ones in the existing reference. [ABSTRACT FROM AUTHOR]

The search for hidden targets is a fundamental problem in many areas of science, engineering, and other fields. Studies of search processes often adopt a probabilistic framework, in which a searcher randomly explores a spatial domain for a randomly located target. There has been significant interest and controversy regarding optimal search strategies, especially for superdiffusive processes. The optimal search strategy is typically defined as the strategy that minimizes the time it takes a given single searcher to find a target, which is called a first hitting time (FHT). However, many systems involve multiple searchers, and the important timescale is the time it takes the fastest searcher to find a target, which is called an extreme FHT. In this paper, we study extreme FHTs for any stochastic process that is a random time change of Brownian motion by a Lévy subordinator. This class of stochastic processes includes superdiffusive Lévy flights in any space dimension, which are processes described by a Fokker–Planck equation with a fractional Laplacian. We find the short-time distribution of a single FHT for any Lévy subordinate Brownian motion and use this to find the full distribution and moments of extreme FHTs as the number of searchers grows. We illustrate these rigorous results in several examples and numerical simulations. [ABSTRACT FROM AUTHOR]

We describe the mathematical modelling of the group behaviour of workers in the labor market. The worker receives the salary and seeks to improve his qualifications in order to receive higher wages. The worker enlarges his qualification by the investments in human capital. At a random moment of time, a vacancy appears that provides a jump in the worker's salary. The mathematical model of the worker's behaviour in the labor market is presented as an optimal control problem on an infinite time horizon. The paper presents the derivation of the Kolmogorov–Fokker–Planck equation for the Lévy process, which describes the behaviour of a large amount of workers within a social layer. The numerical solution of the Kolmogorov–Fokker–Planck equation and the calculation results are presented. [ABSTRACT FROM AUTHOR]

The aim of this paper is threefold. Firstly, we prove the existence and uniqueness of a global strong (in both the probabilistic and the PDE senses) H¹2-valued solution to the 2D stochastic Navier--Stokes equations (SNSEs) driven by a multiplicative Lévy noise under the natural Lipschitz condition on balls and linear growth assumptions on the jump coefficient. Secondly, we prove a Girsanov-type theorem for Poisson random measures and apply this result to a study of the wellposedness of the corresponding stochastic controlled problem for these SNSEs. Thirdly, we apply these results to establish a Freidlin--Wentzell-type large deviation principle for the solutions of these SNSEs by employing the weak convergence method introduced by Budhiraja et al. (2011, 2013). [ABSTRACT FROM AUTHOR]

The wireless sensor network (WSN) is an essential technology of the Internet of Things (IoT) but has the problem of low coverage due to the uneven distribution of sensor nodes. This paper proposes a novel enhanced whale optimization algorithm (WOA), incorporating Lévy flight and a genetic algorithm optimization mechanism (WOA-LFGA). The Lévy flight technique bolsters the global search ability and convergence speed of the WOA, while the genetic optimization mechanism enhances its local search and random search capabilities. WOA-LFGA is tested with 29 mathematical optimization problems and a WSN coverage optimization model. Simulation results demonstrate that the improved algorithm is highly competitive compared with mainstream algorithms. Moreover, the practicality and the effectiveness of the improved algorithm in optimizing wireless sensor network coverage are confirmed. [ABSTRACT FROM AUTHOR]

In this paper, we consider possibly misspecified stochastic differential equation models driven by Lévy processes. Regardless of whether the driving noise is Gaussian or not, Gaussian quasi-likelihood estimator can estimate unknown parameters in the drift and scale coefficients. However, in the misspecified case, the asymptotic distribution of the estimator varies by the correction of the misspecification bias, and consistent estimators for the asymptotic variance proposed in the correctly specified case may lose theoretical validity. As one of its solutions, we propose a bootstrap method for approximating the asymptotic distribution. We show that our bootstrap method theoretically works in both correctly specified case and misspecified case without assuming the precise distribution of the driving noise. [ABSTRACT FROM AUTHOR]

This essay analyses a key motif in geographical scholarship: the most basic form of mobility achieved by an abled-bodied person engaging in acts of walking. By embedding "walking" firmly within a phenomenological tradition, the essay places "being mobile" qua walking within a field of enquiry that conceptualises an embodied form of mobility as both enabling and limiting. Building furthermore from a growing body of literature that has differentiated between "walking" as an active form of engagement and a host of different geographically relevant modes of being, the paper adds a specifically epistemological set of considerations in an attempt critically to contribute to existing literatures and to interrogate the embodied practice of walking. Key in this endeavour is the contribution mobile modes of existence make to the construction of knowledge about the social world. The paper concludes with a prolegomena that recasts walking in the form of a geographically informed pedagogical practice. [ABSTRACT FROM AUTHOR]

The object of the present paper is the study of the joint lifetime of d components subject to a common stressful external environment. Out of the stressing environment, the components are independent and the lifetime of each component is characterized by its failure (hazard) rate function. The impact of the external environment is modelled through an increase in the individual failure rates of the components. The failure rate increments due to the environment increase over time and they are dependent among components. The evolution of the joint failure rate increments is modelled by a non negative multivariate additive process, which include Lévy processes and non-homogeneous compound Poisson processes, hence encompassing several models from the previous literature. A full form expression is provided for the multivariate survival function with respect to the intensity measure of a general additive process, using the construction of an additive process from a Poisson random measure (or Poisson point process). The results are next specialized to Lévy processes and other additive processes (time-scaled Lévy processes, extended Lévy processes and shock models), thus providing simple and easily computable expressions. All results are provided under the assumption that the additive process has bounded variations, but it is possible to relax this assumption by means of approximation procedures, as is shown for the last model of this paper. [ABSTRACT FROM AUTHOR]

This paper provides a novel, unique, and improved optimization algorithm called the modified Orca Predation Algorithm (mOPA). The mOPA is based on the original Orca Predation Algorithm (OPA), which combines two enhancing strategies: Lévy flight and opposition-based learning. The mOPA method is proposed to enhance search efficiency and avoid the limitations of the original OPA. This mOPA method sets up to solve the global optimization issues. Additionally, its effectiveness is compared with various well-known metaheuristic methods, and the CEC'20 test suite challenges are used to illustrate how well the mOPA performs. Case analysis demonstrates that the proposed mOPA method outperforms the benchmark regarding computational speed and yields substantially higher performance than other methods. The mOPA is applied to ensure that all load demand is met with high reliability and the lowest energy cost of an isolated hybrid system. The optimal size of this hybrid system is determined through simulation and analysis in order to service a tiny distant location in Egypt while reducing costs. Photovoltaic panels, biomass gasifier, and fuel cell units compose the majority of this hybrid system's configuration. To confirm the mOPA technique's superiority, its outcomes have been compared with the original OPA and other well-known metaheuristic algorithms. [ABSTRACT FROM AUTHOR]

We consider three independent objects: a two-sided wide-sense stationary random sequence ξ := (... , ξ−1, ξ0, ξ1,...) with zero mean and finite variance, a standard Poisson process Π and a subordinator S, that is a nondecreasing Lévy process. By means of reflection about zero we extend Π and S to the negative semi-axis and define a random time change Π(S(t)), t ∈ ℝ. Then we define a so-called PSI-process ψ(t) := ξΠ(S(t)), t ∈ ℝ, which is wide-sense stationary. Notice that PSI-processes generalize pseudo-Poisson processes. The main aim of the paper is to express spectral properties of the process ψ in terms of spectral characteristics of the sequence ξ and the Lévy measure of the subordinator S. Using complex analytic techniques, we derive a general formula for the spectral measure G of the process ψ. We also determine exact spectral characteristics of ψ for the following examples of ξ: almost periodic sequence; finite-order moving average; finite order autoregression. These results can find their applications in all areas where L2-theory of stationary processes is used. [ABSTRACT FROM AUTHOR]