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A classical result of K. L. Chung and W. Feller deals with the partial sums S_k arising in a fair coin-tossing game. If N_n is the number of "positive" terms among S_1, S_2, ..., S_n then the quantity P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P(N_{2n+1} = r), r = 0, 1, 2, ..., 2n+1. We get to this ansatz by adaptating the Feynman–Kac methodology. [ABSTRACT FROM AUTHOR]

Biased and accelerated molecular simulations (BAMS) are widely used tools to observe relevant molecular phenomena occurring on time scales inaccessible to standard molecular dynamics, but evaluation of the physical time scales involved in the processes is not directly possible from them. For this reason, the problem of recovering dynamics from such kinds of simulations is the object of very active research due to the relevant theoretical and practical implications of dynamics on the properties of both natural and synthetic molecular systems. In a recent paper [A. Rapallo et al., J. Comput. Chem. 42, 586–599 (2021)], it has been shown how the coupling of BAMS (which destroys the dynamics but allows to calculate average properties) with Extended Diffusion Theory (EDT) (which requires input appropriate equilibrium averages calculated over the BAMS trajectories) allows to effectively use the Smoluchowski equation to calculate the orientational time correlation function of the head–tail unit vector defined over a peptide in water solution. Orientational relaxation of this vector is the result of the coupling of internal molecular motions with overall molecular rotation, and it was very well described by correlation functions expressed in terms of weighted sums of suitable time-exponentially decaying functions, in agreement with a Brownian diffusive regime. However, situations occur where exponentially decaying functions are no longer appropriate to capture the actual dynamical behavior, which exhibits persistent long time correlations, compatible with the so called subdiffusive regimes. In this paper, a generalization of EDT will be given, exploiting a fractional Smoluchowski equation (FEDT) to capture the non-exponential character observed in the relaxation of intramolecular distances and molecular radius of gyration, whose dynamics depend on internal molecular motions only. The calculation methods, proper to EDT, are adapted to implement the generalization of the theory, and the resulting algorithm confirms FEDT as a tool of practical value in recovering dynamics from BAMS, to be used in general situations, involving both regular and anomalous diffusion regimes. [ABSTRACT FROM AUTHOR]

Starch-based bio-latex has been widely studied for the purpose of partially substituting petroleum-based latex as coating binder due to its environmental-friendly and sustainable natures. To engineer coating structure, it is essential to understand the bio-latex distribution during coating layer drying. In this work, bio-latex concentration at the surface and along the thickness direction of model coatings on a non-absorbent film consolidated under various conditions was detected with ATR-IR, ESCA and Cryo-SEM techniques in order to understand whether bio-latex redistribute during coating drying and how the migration occurs. The results showed that bio-latex particles moved differentially with respect to kaolin clay pigment toward the coating surface and accumulated at surface during the initial liquid drying period, however, they deprived from coating surface in the end and aggregated at the top region of coating layer. A mechanism was proposed to explain this unexpected finding, which was attributed to the sphere-like morphology with nano or submicron size and highly hydrophilic, water-swollen natures of starch-based bio-latex binder. The small-sized bio-latex particles migrated to the surface due to their higher Brownian mobility relative to pigment initially, and in the later drying stage would move back into the coating pores with water because of its water affinity as the air-water interfaces receded from the surface into the capillaries. [ABSTRACT FROM AUTHOR]

In this paper, we focus on filtering and parameter estimation in stochastic volatility models when observations arise from high-frequency data. We are particularly interested in rough volatility models where spot volatility is driven by fractional Brownian motion with Hurst index $ H < \frac {1}{2} $ H<12. Since volatility is not directly observable, we rely on particle filtering techniques for statistical inference regarding the current level of volatility and the parameters governing its dynamics. In order to obtain numerically efficient and recursive algorithms, we use the fact that a fractional Brownian motion can be represented through a superposition of Markovian semimartingales (Ornstein-Uhlenbeck processes). We analyze the performance of our approach on simulated data and we compare it to similar studies in the literature. The paper concludes with an empirical case study, where we apply our methodology to high-frequency data of a liquid stock. [ABSTRACT FROM AUTHOR]

The transport of molecules and particles across adjacent flow streams is a key process in several operations implemented in microfluidic paper-based analytical devices (μPADs). Here, the transverse dispersion of analytes was quantitatively evaluated by theory and experiments. Different tests were carried out to independently measure the coefficients of both Brownian diffusion and mechanical dispersion under capillary-driven flow. The dispersion width was found to be independent of fluid velocity and analyte properties, and fully determined by the dispersivity coefficient, which is a characteristic of the paper microstructure. This information introduces a change of paradigm for the design of mixers, diluters, and concentration gradient generators on μPADs; therefore, efforts were made to rationalize these operations on paper. The research reveals that mixers and concentration gradient generators can be much more efficient than their counterparts made on conventional microchannels; in contrast, separators such as the H-filter need to be appropriately engineered on paper, because the working principle can be hindered by mechanical dispersion. The knowledge gained throughout this work would contribute to the design of μPADs with a new level of precision and control over the formation of localized concentration profiles. [ABSTRACT FROM AUTHOR]

Stochastic differential equations (SDEs) are very useful in modeling many problems in biology, economic data, turbulence, and medicine. Fractional Brownian motion (fBm) and variable‐order fractional Brownian motion (vofBm) are suitable alternatives to standard Brownian motion (sBm) for describing and modeling many phenomena, since the increments of these processes are dependent of the past and for H>12$$ \mathcal{H}>\frac{1}{2} $$ these increments have the property of long‐term dependence. Classical mathematical techniques such as Ito's calculus do not work for stochastic computations on fBm and vofBm due to they are not semi‐Martingale for H(ξ)≠12$$ \mathcal{H}\left(\xi \right)\ne \frac{1}{2} $$. Therefore, solving these equations is much more difficult than solving SDEs with sBm. On the other hand, these equations do not have an analytical solution, so we have to use numerical methods to find their solution. In this paper, a computational approach based on hybrid of block‐pulse and parabolic functions (HBPFs) has been introduced for simulating vofBm and solving a modern class of SDEs. The mechanism of this approach is based on stochastic and fractional integration operational matrices, which transform the intended problem to a nonlinear system of algebraic equations. Thus, the complexity of solving the mentioned problem is reduced significantly. Also, convergence analysis of the expressed method has been theoretically examined. Finally, the accuracy and efficiency of the proposed algorithm have been experimentally investigated through some test problems and comparison of obtained results with results of previous papers. High accurate numerical results are obtained by using a small number of basic functions. Therefore, this method deals with smaller matrices and vectors, which is one of the most important advantage of our suggested method. Also, presenting an applicable procedure to construct vofBm is another innovation of this work. To gain this aim, at first, discretized sBm is generated via fundamental features of this process, and afterward, block‐pulse functions (BPFs) and HBPFs are utilized for simulating discretized vofBm. Finally, spline interpolation method has been employed to provide a continuous path of vofBm. [ABSTRACT FROM AUTHOR]

In traditional finance, option prices are typically calculated using crisp sets of variables. However, as reported in the literature novel, these parameters possess a degree of fuzziness or uncertainty. This allows participants to estimate option prices based on their risk preferences and beliefs, considering a range of possible values for the parameters. This paper presents a comprehensive review of existing work on fuzzy fractional Brownian motion and proposes an extension in the context of financial option pricing. In this paper, we define a unified framework combining fractional Brownian motion with fuzzy processes, creating a joint product measure space that captures both randomness and fuzziness. The approach allows for the consideration of individual risk preferences and beliefs about parameter uncertainties. By extending Merton's jump-diffusion model to include fuzzy fractional Brownian motion, this paper addresses the modelling needs of hybrid systems with uncertain variables. The proposed model, which includes fuzzy Poisson processes and fuzzy volatility, demonstrates advantageous properties such as long-range dependence and self-similarity, providing a robust tool for modelling financial markets. By incorporating fuzzy numbers and the belief degree, this approach provides a more flexible framework for practitioners to make their investment decisions. [ABSTRACT FROM AUTHOR]

This paper is concerned with quantum stochastic differential equations driven by the fermion field in noncommutative space Lp(풞) for 2≤p<∞$$ 2\le p<\infty $$. First, we investigate the existence and uniqueness of Lp$$ {L}&#x0005E;p $$‐solutions of quantum stochastic differential equations in an infinite time horizon by using the noncommutative Burkholder–Gundy inequality given by Pisier and Xu and the noncommutative generalized Minkowski inequality. Then, we investigate the stability, self‐adjointness, and Markov properties of Lp$$ {L}&#x0005E;p $$‐solutions and analyze the error of numerical schemes of quantum stochastic differential equations. The results of this paper can be utilized for investigating the optimal control of quantum stochastic systems with infinite dimensions. [ABSTRACT FROM AUTHOR]

We consider the optimal risk transfer from an insurance company to a reinsurer. The problem formulation considered in this paper is closely connected to the optimal portfolio problem in finance, with some crucial distinctions. In particular, the insurance company's surplus is here (as is routinely the case) approximated by a Brownian motion, as opposed to the geometric Brownian motion used to model assets in finance. Furthermore, risk exposure is dialled 'down' via reinsurance, rather than 'up' via risky investments. This leads to interesting qualitative differences in the optimal designs. In this paper, using the martingale method, we derive the optimal design as a function of proportional, non-cheap reinsurance design that maximises the quadratic utility of the terminal value of the insurance surplus. We also consider several realistic constraints on the terminal value: a strict lower boundary, the probability (Value at Risk) constraint, and the expected shortfall (conditional Value at Risk) constraints under the $ \mathbb {P} $ P and $ \mathbb {Q} $ Q measures, respectively. In all cases, the optimal reinsurance designs boil down to a combination of proportional protection and option-like protection (stop-loss) of the residual proportion with various deductibles. Proportions and deductibles are set such that the initial capital is fully allocated. Comparison of the optimal designs with the optimal portfolios in finance is particularly interesting. Results are illustrated. [ABSTRACT FROM AUTHOR]

This paper reports the flow features and distributions of concentration and temperature of a micropolar type nanofluid (water-based) past a stretchable and shrinkable wedge, influenced by variable magnetic force, nonlinear sort thermal radiation and chemical reaction. Along with the consideration of multiple convection, the model of Buongiorno is stated. The Brownian motion and thermophoresis have been kept in the analysis. Suitable similarity alteration is approached to renovate the foremost equations to dimensionless ordinary differential equations (ODEs). Associated conditions became nondimensional forms according to this conversion. Then the numerical solutions of the reduced governing equations with boundary conditions are obtained by adopting the RK-4 technique with shooting criteria. The language MAPLE 17 assisted in developing this solution. Significant upshots of prime parameters on the fluid transmission, mass and heat transport properties are represented with suitable tables and graphs. In tabular form, we have reckoned the physical quantities of heat, mass transfer rates and drag friction coefficients to fulfill the engineering interest. This study acquaints that the material parameter negatively influenced nanofluid's angular velocity. The fluid's temperature improves with thermal and mass Biot numbers, but this response goes opposite for the parameter of wedge angle. Chemical reaction and wedge angle parameters amplify mass transport. This study can be beneficial in the blowing of chilled air by AC panels, the abstraction of crude oils, the nuclear power hub, the working of warships, making flaps on the wings of aeroplanes for advanced lift, submarines, the extraction of polymers and several other sectors in advanced science and industrial developments. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence and uniqueness of fully coupled forward-backward stochastic differential equations (FBSDEs in short) driven by anomalous sub-diffusions B L t under suitable monotonicity conditions on the coefficients. Here B is a Brownian motion on R and L t : = inf ⁡ { r > 0 : S r > t } , t ≥ 0 , is the inverse of a subordinator S with drift κ > 0 that is independent of B. Various a priori estimates on the solutions of the FBSDEs are also presented. [ABSTRACT FROM AUTHOR]

In this paper we study existence and uniqueness of solution stochastic differential equations involving fractional integrals driven by Riemann-Liouville multifractional Brownian motion and a standard Brownian. Then, we obtain approximate numerical solution of our problem and colon cancer chemotherapy effect model are presented to confirm our results. We show that considering time dependent Hurst parameters play an important role to get more realistic results. [ABSTRACT FROM AUTHOR]

This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of Feynman's path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional Hilbert space of quantum states. This reformulation will undoubtedly have a crucial role in formulating quantum gravity within a mathematically well-defined framework. In fact, this study is fundamentally different from any previous research on the relationship between Feynman's path-integral and the Wiener stochastic process. In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions. Thus, an analytic norm motivated by the role of the fractal functions in the Wilsonian renormalization approach is worked out to properly characterize Brownian motion in the Hilbert space of quantum states on a compact flat manifold. This norm, the so-called fractal norm, pushes the rougher functions (physically the quantum states with higher energies) to the farther points of the Hilbert space until the fractal functions as the roughest ones are moved to infinity. Implementing the Wiener stochastic process with the fractal norm, results in a modified form of the Wiener measure called the Wiener fractal measure, which is a well-defined measure for Feynman's path-integral formulation of quantum fields. Wiener fractal measure has a complicated formula of non-local terms but produces the Klein–Gordon action at the first order of approximation. Using complex integrals to compensate for the removal of non-local terms appearing in higher orders of approximation, the Wiener fractal measure turns into a complex measure and generates Feynman's path-integral formulation of scalar quantum fields. This brings us to the main objective of this study. Finally, some various significant aspects of quantum field theory (such as renormalizability, RG flow, Wick rotation, regularization, etc.) are revisited by means of the analytical aspects of the Wiener fractal measure. [ABSTRACT FROM AUTHOR]

This paper investigates the regularity properties of quantum master equations with unbounded coefficients within the space of square-integrable complex-valued Bernoulli functionals. Initially, we demonstrate the existence of a unique regular solution to the linear stochastic Schrödinger equation driven by cylindrical Brownian motions. Subsequently, we establish the existence of regular solutions for the autonomous linear quantum master equation and provide a probabilistic representation of this solution in terms of the stochastic Schrödinger equation. Finally, by taking the expectation of some observables with respect to the solution of the quantum master equation, we derive a system of differential equations that describe the evolution of the mean values of certain quantum observables. [ABSTRACT FROM AUTHOR]

This paper presents a numerical solution to the problem of time-dependent blood flow via a w-shaped stenotic conduit, driven by pulsatile pressure gradient. The problem is formulated in cylindrical coordinates by employing the theoretical model of tangent hyperbolic fluid. The electro-osmotic effects are also taken into consideration. To simplify the non-dimensional governing equations of the flow problem, a mild stenosis assumption is utilized and the impact of the blood vessel wall is mitigated by employing a radial coordinate transformation. An explicit finite difference method is used to solve the resulting nonlinear system of differential equations, considering the auxiliary conditions specified at the boundary of the blood channel. After obtaining the numerical solution to the problem, an examination is carried out for various flow variables, such as axial velocity, temperature field, mass concentration, skin friction, Nusselt number, and Sherwood number. These results are presented graphically, and a concise explanation is provided using physical facts. An increase in flow rate and blood velocity leads to a rise in response, while an increase in stenosis height, Weissenberg number, and power-law exponent leads to a reverse response. Furthermore, the temperature field is significantly affected by the Brinkman number and the Prandtl number. [ABSTRACT FROM AUTHOR]

This paper examines the activation energy influence in third-grade nanoparticle flow model (TG-NPFM), which is nonlinear mixed convective flow over a spinning disk under the influence of heat sink/source as well as viscous dissipation by utilizing Bayesian Regulation Method with backpropagated Artificial Neural Networks (BRM-BPANN). Nonlinear thermal radiation is also involved in the considered flow dynamics to obtain the approximated numerical solutions. The nonlinear PDEs of TG-NPFM are then transformed into nonlinear ODEs by implementing the corresponding transformation. We solved these ODEs by Optimal Homotopy Analysis Method (OHAM) to explain the dataset used as a reference for BRM-BPANN for different scenarios of TG-NPFM. This reference dataset is then exported to MATLAB to compute the results. The outcomes of TG-NPFM are figured by adopting the procedures of testing, validation and training. Moreover, approximated solution is compared with standard solution and the efficacy examination of TG-NPFM is authenticated by the studies of MSE, error histogram and regression plots. These soft computation frameworks provide incentive to use an efficient and dependable alternative paradigm built on soft computing environments to solve problems by doing a descriptive analysis to mitigate the impacts of different physical features. It is a new implementation of intelligent computational system of artificial intelligence introduced by incorporating the solver BRM-BPANN for interpreting the TG-NPFM. The absolute error values lie between 10 − 8 to 10 − 5 , 10 − 1 0 to 10 − 5 , 10 − 8 to 10 − 4 , 10 − 9 to 10 − 3 , 10 − 9 to 10 − 5 , 10 − 9 to 10 − 3 , 10 − 9 to 10 − 3 and 10 − 9 to 10 − 3 , which show the reliability and accuracy of the technique. The convergence and precision of the algorithm can easily be seen through the results of performance, training state and fitness plot, along with the regression value of R = 1. [ABSTRACT FROM AUTHOR]

This paper presents an investigation of magnetohydrodynamics (MHD) Casson nanofluid flow along a stretchable surface through a permeable medium. The modeling of the physical phenomena is considered with impact of thermal radiation, heat generation, slip conditions and suction. Transformations of the governing set of mathematical equations for the physical model are carried out into nonlinear ordinary differential equations (ODEs) with appropriate similarity variables. The nonlinear ODE solutions are carried out using the optimal homotopy analysis technique (OHAM), and the findings are presented for determining the influences of the emerging important parameters. The results indicate that velocity field increases in respect of porosity parameter, Casson fluid parameter and magnetic parameter while it declines for enhancing velocity slip and suction parameters. The temperature profile shows rising behavior for heat source, Prandtl number, thermophoresis, radiation and Brownian motion parameters while it declines for enhancing thermal slip parameter. Moreover, the concentration profile enhances for rise in Brownian motion parameter while it reduces for Schmidt number and nanoparticle parameter. We also showed the accuracy of the present results by indicating that skin friction values for varied magnetic parameters agree with earlier findings in the literature. [ABSTRACT FROM AUTHOR]

This paper investigates algorithms for identifying the optimal policy for pricing American Options. The American Option pricing is reformulated as a Sequential Decision-Making problem with two binary actions (Exercise or Continue), transforming it into an optimal stopping time problem. Both the least square Monte Carlo simulation method (LSM) and Reinforcement Learning (RL)-based methods were utilized to find the optimal policy and, hence, the fair value of the American Put Option. Both Classical Geometric Brownian Motion (GBM) and calibrated Stochastic Volatility models served as the underlying uncertain assets. The novelty of this work lies in two aspects: (1) Applying LSM- and RL-based methods to determine option prices, with a specific focus on analyzing the dynamics of "Decisions" made by each method and comparing final decisions chosen by the LSM and RL methods. (2) Assess how the RL method updates "Decisions" at each batch, revealing the evolution of the decisions during the learning process to achieve optimal policy. [ABSTRACT FROM AUTHOR]

The osprey optimization algorithm (OOA) is a metaheuristic algorithm with a simple framework, which is inspired by the hunting process of ospreys. To enhance its searching capabilities and overcome the drawbacks of susceptibility to local optima and slow convergence speed, this paper proposes a modified osprey optimization algorithm (MOOA) by integrating multiple advanced strategies, including a Lévy flight strategy, a Brownian motion strategy and an RFDB selection method. The Lévy flight strategy and Brownian motion strategy are used to enhance the algorithm's exploration ability. The RFDB selection method is conducive to search for the global optimal solution, which is a symmetrical strategy. Two sets of benchmark functions from CEC2017 and CEC2022 are employed to evaluate the optimization performance of the proposed method. By comparing with eight other optimization algorithms, the experimental results show that the MOOA has significant improvements in solution accuracy, stability, and convergence speed. Moreover, the efficacy of the MOOA in tackling real-world optimization problems is demonstrated using five engineering optimization design problems. Therefore, the MOOA has the potential to solve real-world complex optimization problems more effectively. [ABSTRACT FROM AUTHOR]

Aiming at the problem of high failure rate and inconvenient maintenance of wind turbine gearboxes, this paper establishes a stochastic differential equation model that can be used to fit the change of gearbox oil temperature and adopts an iterative computational method and Markov-based modified optimization to fit the prediction sequence in order to realize the accurate prediction of gearbox oil temperature. The model divides the oil temperature change of the gearbox into two parts, internal aging and external random perturbation, adopts the approximation theorem to establish the internal aging model, and uses Brownian motion to simulate the external random perturbation. The model parameters were calculated by the Newton–Raphson iterative method based on the gearbox oil temperature monitoring data. Iterative calculations and Markov-based corrections were performed on the model prediction data. The gearbox oil temperature variations were simulated in MATLAB, and the fitting and testing errors were calculated before and after the iterations. By comparing the fitting and testing errors with the ordinary differential equations and the stochastic differential equations before iteration, the iterated model can better reflect the gear oil temperature trend and predict the oil temperature at a specific time. The accuracy of the iterated model in terms of fitting and prediction is important for the development of preventive maintenance. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic maximum principle for a partially observed optimal control problem of forward-backward stochastic differential equations (FBSDEs for short) of McKean–Vlasov type driven by both a family of Teugels martingales and an independent Brownian motion. The coefficients of the system and the cost functional depending on the state of the solution process as well as its probability law and the control variable. We establish partially observed necessary conditions of optimality for this system under assumption that the control domain is supposed to be convex. Our main result is based on Girsavov's theorem and the derivatives with respect to probability law. As an application of the general theory, a partially observed linear-quadratic control problem of McKean–Vlasov type is studied in terms of stochastic filtering. [ABSTRACT FROM AUTHOR]

In this paper, we focus on fractional stochastic differential equations (FSDEs) with a stochastic forcing term, i.e., to FSDE, we add a stochastic forcing term. Using the implicit scheme of Euler's approximation, the conditions for the existence and uniqueness of the solution of FSDEs with a stochastic forcing term are established. Such equations can be applied to considering FSDEs with a permeable wall. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a stochastic semilinear differential equations driven by both Brownian motion and fractional Brownian motion. Firstly, we establish an inequality for the distance between finite‐dimensional distributions of a random process at two different moments. Then, using the properties of stochastic integrals, fixed point theorems, and based on this inequality, we establish the existence and uniqueness of Besicovich almost automorphic solutions in finite‐dimensional distributions for this type of semilinear equation. Finally, we provide an example to demonstrate the effectiveness of our results. Our results are new to stochastic differential equations driven by Brownian motion or stochastic differential equations driven by fractional Brownian motion. [ABSTRACT FROM AUTHOR]

This paper investigates the stagnation point flow of a second-grade nanomaterial, considering nanofluid effects via thermophoresis and Brownian motion. The study addresses convective heat and mass transfer conditions within the flow induced by a stretching cylinder. The transformed systems are solved using the optimal homotopy solutions (OHAM) method, revealing the impact of various parameters on the quantities of interest. The results indicate that an increase in curvature, viscoelasticity, velocity ratio and injection parameters leads to an enhancement in velocity, while the opposite trend is observed due to suction. The viscoelasticity and curvature parameters also increase skin friction. Additionally, thermophoresis enhances the temperature and concentration fields, while Brownian motion reduces mass diffusion. [ABSTRACT FROM AUTHOR]

This paper studies the optimal asset allocation problem of a defined contribution (DC) pension plan with a stochastic salary and value under a constraint within a stochastic volatility model. It is assumed that the financial market contains a risk-free asset and a risky asset whose price process satisfies the Stein–Stein stochastic volatility model. To comply with regulatory standards and offer a risk management tool, we integrate the dynamic versions of Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), and worst-case CVaR (wcCVaR) constraints into the DC pension fund management model. The salary is assumed to be stochastic and characterized by geometric Brownian motion. In the dynamic setting, a CVaR/wcCVaR constraint is equivalent to a VaR constraint under a higher confidence level. By using the Lagrange multiplier method and the dynamic programming method to maximize the constant absolute risk aversion (CARA) utility of terminal wealth, we obtain closed-form expressions of optimal investment strategies with and without a VaR constraint. Several numerical examples are provided to illustrate the impact of a dynamic VaR/CVaR/wcCVaR constraint and other parameters on the optimal strategy. [ABSTRACT FROM AUTHOR]

Hybrid nanofluids are a type of nanofluid that is created by combining two different types of nanoparticles with a traditional fluid. These nanofluids have unique physicochemical properties that make them more effective at transferring heat than traditional nanofluids. This research paper focuses on predicting thermal and energy transport in non-Newtonian biomagnetic hybrid nanofluids that contain gold and silver nanoparticles, using Gaussian process regression (GPR). The study uses blood as the traditional fluid and incorporates the effects of thermal radiation, thermophoresis, Brownian motion and activation energy into the model equation. The governing nonlinear partial differential equations are simplified to a set of ordinary differential equations using similarity replacements. The shooting method, along with the Runge–Kutta-Fehlberg fourth–fifth-order scheme, is used to solve the transformed equations using MATLAB. The results of the study are presented through figures and tables, which include the coefficient of skin friction, Nusselt number, Sherwood number and motile microbe's flux, illustrated with surface plots. The GPR model is developed using four basic function kernels (squared exponential, exponential, rational quadratic and matern32 functions) and evaluated using statistical indicators such as RMSE, MSE, MAE and R. The predicted results and simulated numerical values are in good agreement with the coefficient of determination (R2) of 0.999999 for all parameters. The study also finds that GPR models with exponential kernel functions outperform other kernel functions in both the Oldroyd-B and Casson hybrid nanofluid data sets. However, the findings indicate that nanofluids and hybrid nanofluids have superior thermal qualities and stability, making them promising candidates for various thermal applications including solar thermal systems, automotive cooling systems, heat sinks, engineering, medical areas and thermal energy storage. [ABSTRACT FROM AUTHOR]

This paper investigates the optimal management of an aggregated defined benefit pension plan in a stochastic environment. The interest rate follows the Ornstein-Uhlenbeck model, the benefits follow the geometric Brownian motion while the contribution rate is determined by the spread method of fund amortization. The pension manager invests in the financial market with three assets: cash, a zero-coupon bond and a stock. Regardless of the initial status of the plan, we suppose that the pension fund may become underfunded or overfunded in the planning horizon. The optimization goal of the manager is to maximize the expected utility in the overfunded region minus the weighted solvency risk in the underfunded region. By introducing an auxiliary process and related equivalent optimization problems and using the martingale method, the optimal wealth process, optimal portfolio and efficient frontier are obtained under four cases (high tolerance towards solvency risk, low tolerance towards solvency risk, a specific lower bound, and high lower bound). Moreover, we also obtain the probabilities that the optimal terminal wealth falls in the overfunded and underfunded regions. At last, we present numerical analyzes to illustrate the manager's economic behaviors. [ABSTRACT FROM AUTHOR]

In this paper, we study optimal stochastic control problems for stochastic systems driven by non-Markov subdiffusion BLt, which have mixed features of deterministic and stochastic controls. Here Bt is the standard Brownian motion on R, and Lt := inf { r > 0 : Sr > t}, t ≥ 0, is the inverse of a subordinator St with drift κ > 0 that is independent of Bt. We obtain stochastic maximum principles (SMPs) for these systems using both convex and spiking variational methods, depending on whether or not the domain is convex. To derive SMPs, we first establish a martingale representation theorem for subdiffusions BLt , and then use it to derive the existence and uniqueness result for the solutions of backward stochastic differential equations (BSDEs) driven by subdiffusions, which may be of independent interest. We also derive sufficient SMPs. Application to a linear quadratic system is given to illustrate the main results of this paper. [ABSTRACT FROM AUTHOR]

This paper deals with a class of backward doubly stochastic differential equations driven by fractional Brownian motion with Hurst parameter H greater than 1 2 . We essentially establish the existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and stochastic integral-Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of stability of linear time-varying stochastic perturbed systems. We present sufficient conditions ensuring the global practical uniform exponential stability of a different class of stochastic perturbed systems based on generalized integral inequalities of Gronwall type. Finally, we provide numerical examples to prove the usefulness of the main results of this paper. [ABSTRACT FROM AUTHOR]

This paper deals with a class of backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1/2) with time-delayed generators. In this type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time-delay function, for instance, of the moving average type. We establish an existence and uniqueness result of solutions for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. The stochastic integral used throughout the paper is the divergence operator-type integral. [ABSTRACT FROM AUTHOR]

This paper studies the mixed convective flow of a magnetohydrodynamic micropolar fluid over an extending sheet. The first-order velocity slip condition is taken to observe the slip flow of the fluid. The applications of solar radiation toward the micropolar fluid flow are analyzed in this paper. Furthermore, the Brownian motion, thermophoresis and Joule heating impacts are also studied. Also, the Cattaneo-Christov heat flux model, chemical reaction and activation energy are observed. The leading PDEs have been transformed to ODEs and then solved with the help of homotopy analysis technique. The impacts of different physical parameters have been evaluated theoretically. The outcomes exhibited that the material factors have augmented the microrotation and velocity profiles. Moreover, the velocity slip parameter has a reverse relation with velocity and microrotation profiles, while there is a direct relation of a velocity slip with the energy curve. The velocity profile has increased with higher thermal and mass Grashof numbers. With increasing Brownian motion parameter, the thermal profile is amplified while the concentration profile is declined. On the other hand, the thermal and mass profiles have been boosted with greater thermophoresis parameter. The velocity profile has decreased with higher magnetic parameter, whereas the temperature profile has augmented with higher magnetic parameter. The couple stress and skin friction have been augmented with material parameter, whereas the skin friction has been reduced with thermal and mass Grashof numbers. [ABSTRACT FROM AUTHOR]

The primary aim of this paper is to focus on the stability analysis of an advanced neural stochastic functional differential equation with finite delay driven by a fractional Brownian motion in a Hilbert space. We examine the existence and uniqueness of mild solution of d x a (s) + g (s , x a (s - ω (s))) = I x a (s) + f (s , x a (s - ϱ (s))) d s + ς (s) d ϖ H (s) , 0 ≤ s ≤ T , x a (s) = ζ (s) , - ρ ≤ s ≤ 0. The main goal of this paper is to investigate the Ulam–Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler–Maruyama numerical method through two examples. [ABSTRACT FROM AUTHOR]

The transport phenomena of Casson nanofluid flow between two parallel disks subject to convective boundary conditions are analyzed in this paper. The mathematical model incorporates the impact of thermophoresis and Brownian motion since the Buongiorno's nanoliquid model is adopted to characterize the nanoliquid's transport features. The appropriate similarity transformations are applied to obtain the resulting nondimensional ordinary differential equations from the basic governing equations. The resulting ordinary differential equations and the associated boundary conditions are solved analytically by adopting the homotopy perturbation technique. Further, a statistical experiment is conducted to identify notable flow parameters which cause significant impact on the heat transfer rate. The characteristics of critical pertinent parameters on the flow field are graphically manifested. It is worth noting that the Casson nanofluid velocity escalates by augmenting the magnetic field parameter in the case of injection near the disks. Nanoparticle concentration is considerably diminished with an increment in thermophoresis parameter. In the cases of equal and unequal Biot numbers, the heat transfer rate is promoted with higher values of the Brownian motion parameter. Among the Casson fluid parameter, squeezing parameter and magnetic field parameter, the heat transfer rate discloses the highest positive sensitivity with the lowest value of the Casson fluid parameter. [ABSTRACT FROM AUTHOR]

In this paper, the self-propelled movement on gyrotactic swimming microorganisms into this generalized slip flow by nanoliquid over the stretching cylinder with strong magnetic field is discussed. Constant wall temperature was pretended as well as the Nield conditions of boundary. The intuitive non-Newtonian particulate suspension was included into applying Casson fluid by the base liquid and nanoparticles. This formation on the bio-mathematical model gives the boundary value problem by the nonlinear partial differential equations. Primly, modeled numerical system was converted to nondimensional against this help on acceptable scaling variables and the bvp4c technique was used to acquire the mathematical outcomes on the governing system. This graphical description by significant parameters and their physical performance was widely studied. The Prandtl number has the maximum contribution (112.595%) along the selected physical parameters, whereas the Brownian motion has the least (0.00165%) heat transfer rate. Anyhow, Casson fluid was established for much helpful suspension of this method on fabrication and coatings, etc. Therefore, this magnetic field performs like the resistive force of that fluid motion, and higher energy was enlarged into the structure exhibiting strong thermal radiation. [ABSTRACT FROM AUTHOR]

Non-Gaussian Harmonizable Fractional Stable Motion (HFSM) is a natural and important extension of the well-known Fractional Brownian Motion to the framework of heavy-tailed stable distributions. It was introduced several decades ago; however its properties are far from being completely understood. In our present paper we determine the optimal power of the logarithmic factor in a uniform modulus of continuity for HFSM, which solves an old open problem. The keystone of our strategy consists in Abel transforms of the LePage series expansions of the random coefficients of the wavelets series representation of HFSM. Our methodology can be extended to more general harmonizable stable processes and fields. [ABSTRACT FROM AUTHOR]

Microscale robotics has emerged as a transformative field, offering unparalleled opportunities for innovation and advancement in various fields. Owing to the distinctive benefits of wireless operation and a heightened level of safety, magnetic actuation has emerged as a widely adopted technique in the field of microrobotics. However, factors such as Brownian motion, fluid dynamic flows, and various nonlinear forces introduce uncertainties in the motion of micro/nanoscale robots, making it challenging to achieve precise navigational control in complex environments. This paper presents an extensive review encompassing the trajectory from theoretical foundations of the generation and modeling of magnetic fields as well as magnetic field-actuation modeling to motion control methods of magnetic microrobots. We introduce traditional control methods and the learning-based control approaches for robotic systems at the micro/nanoscale, and then these methods are compared. Unlike the conventional navigation methods based on precise mathematical models, the learning-based control and navigation approaches can directly learn control signals for the actuation systems from data and without relying on precise models. This endows the micro/nanorobots with high adaptability to dynamic and complex environments whose models are difficult/impossible to obtain. We hope that this review can provide insights and guidance for researchers interested in automated magnetic microrobot control. [ABSTRACT FROM AUTHOR]

AbstractIn this research paper, a mathematical concept known as the ω-periodic process is discussed, and a new type of processes called the doubly measure pseudo -asymptotically ω-periodic in distribution process is set forward. Additionally, the properties of these processes are identified and invested to tackle the solution of a stochastic differential equation guided by Brownian motion. The basic objective of the current work resides in corroborating the existence and uniqueness of the solution which is doubly measure pseudo -asymptotically ω-periodic in distribution. [ABSTRACT FROM AUTHOR]

This paper deals on the valuation of European call option price in a stochastic environment by employing three factors which are the stochastic model of the asset value, the stochastic interest rate and the transaction cost. We specify that our underlying asset and the stochastic interest rate, particularly Hull-White model, follows a fractional Brownian Motion governed by Hurst parameter H. We used the hedging and replicating technique to established the zero-coupon bond on the European option. Finally, we give a closed-form formula of the European call option price. [ABSTRACT FROM AUTHOR]

Diffusive transport in media with discontinuous properties is a challenging problem that arises in many applications. This paper focuses on one dimensional discontinuous media with generalized permeable boundary conditions at the discontinuity interface. It presents novel analytical expressions from the method of images to simulate diffusive processes, such as mass or thermal transport. The analytical expressions are used to formulate a generalization of the existing Skew Brownian Motion, HYMLA, and Uffink's method, here named as GSBM, GHYMLA, and GUM, respectively, to handle generic interface conditions. The algorithms rely upon the random walk method and are tested by simulating transport in a bimaterial and in a multilayered medium with piecewise constant properties. The results indicate that the GUM algorithm provides the best performance in terms of accuracy and computational cost. The methods proposed can be applied for simulation of a wide range of differential problems. [ABSTRACT FROM AUTHOR]

Clifford analysis is a higher dimensional function theory and a refinement of harmonic analysis. We will extend Clifford analysis to the infinite‐dimensional setting. To do that, we have to introduce infinite‐dimensional Clifford algebras and the infinite‐dimensional Dirac operator. The paper's main result is the Wiener chaos decomposition in Clifford analysis. We apply the decomposition to white noise and Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper, the solution and behaviour of local fractional Fokker–Planck equation (LFFPE) is investigated in fractal media. For this purpose, the local fractional homotopy perturbation Elzaki transform method (LFHPETM) is proposed and utilized to explore the solution of LFFPE. The proposed scheme is a merger of well known local fractional homotopy perturbation technique and recently introduced local fractional Elzaki transform (LFET). The convergence and uniqueness analyses for LFHPETM solution of the general partial differential equation is also presented along with the computational procedure of this new hybrid combination. Three examples of LFFPE are illustrated to depict the applicability of the employed method with graphical simulations on Cantor set. The copulation of LFET with local fractional homotopy perturbation method (LFHPM) efficiently provides the faster solution for LFFPE in a fractal domain as compared to the LFHPM. Furthermore, the achieved solutions are also in a good match with existing solutions. The 3D behavior of solutions of LFFPEs are presented for fractal order ln 2 / ln 3 . Figures illustrate the 3D surface graphics of solutions with respect to input variables. [ABSTRACT FROM AUTHOR]

Multifractality is a concept that extends locally the usual ideas of fractality in a system. Nevertheless, the multifractal approaches used lack a multifractal dimension tied to an entropy index like the Shannon index. This paper introduces a generalized Shannon index (GSI) and demonstrates its application in understanding system fluctuations. To this end, traditional multifractality approaches are explained. Then, using the temporal Theil scaling and the diffusive trajectory algorithm, the GSI and its partition function are defined. Next, the multifractal exponent of the GSI is derived from the partition function, establishing a connection between the temporal Theil scaling exponent and the generalized Hurst exponent. Finally, this relationship is verified in a fractional Brownian motion and applied to financial time series. In fact, this leads us to proposing an approximation called local fractional Brownian motion approximation, where multifractal systems are viewed as a local superposition of distinct fractional Brownian motions with varying monofractal exponents. Also, we furnish an algorithm for identifying the optimal q-th moment of the probability distribution associated with an empirical time series to enhance the accuracy of generalized Hurst exponent estimation. [ABSTRACT FROM AUTHOR]

Activation energy is the fundamental unit of energy required to initiate chemical reactions. Some instances of initiation energy use include liquid dispersions bite dust fashioning, polymer ejection, food handling, and paper manufacturing. In view of this, the present study investigates the impact of steady, incompressible magnetized Casson–Maxwell non-Newtonian nanofluid between two stationary porous disks. The fluid movement is created by uniform injection in the direction of the axial of stationary disks. It is noticed that the thermal conductivity of the fluid varies with varying temperature. By using Buongiorno nanofluid concept the Cattaneo–Christov thermal expression is implemented. Along with this, activation energy is considered. By introducing the suitable similarity variables, the fluid model is reduced in terms of ordinary differential equations and numerical solutions are generated by using inbuild bvp4c function in MATLAB. The various dimensionless parameters impact on velocity, temperature, and concentration are presented graphically. A numerical table is presented to show the variation of local-Nusselt and -Sherwood numbers on various values of different parameters. The study reveals that improvement in Casson parameter will incline the fluid velocity, but an opposite trend is seen in the case of a magnetic field. A rise in the Brownian motion and thermophoresis parameter will enhance both temperature profiles. Improved values of activation energy will increase the concentration. The rate of heat transfer is observed more in the case of the upper disk than the lower disk. [ABSTRACT FROM AUTHOR]

The novelty of this paper is examining the impacts of Hall currents on Sisko nanofluid slip flow generated by an exponentially stretching sheet in the existence of entropy generation. Entropy generation assessment, Brownian, thermophoresis and Bejan numbers during convection flow are explored in the presence of the Hall effect. The pertinent parameters are Hall parameter m, a magnetic field parameter M, Brownian motion parameter N b , thermophoresis parameter N t , generalized thermal Biot number γ 1 , generalized velocity slip parameter λ and material parameter of Sisko fluid A. The changes in pertinent parameters on principal and secondary velocities, entropy generation and temperature are discussed. The coefficients of local skin friction, Brownian, thermophoresis and Nusselt/Sherwood numbers are expressed in a tabular structure. Two different kinds of power-law nanofluid, Pseudoplastic (n = 0. 7) and dilatant fluid (n = 1. 5) are considered. The current results exposed that growth in magnetic field parameter diminishes the nanofluid axial velocity and boosts the profiles of temperature, secondary nanofluid velocity and nanoparticle volume fraction. The coefficients of local skin friction are enhanced according to a growth in the magnetic field factor. The Pseudoplastic fluid supports the velocity, temperature and nanoparticle volume fraction profiles compared to the dilatant fluid. The local entropy generation number, Bejan number, local skin friction coefficients and local Nusselt/Sherwood numbers are influenced by the pertinent parameters. [ABSTRACT FROM AUTHOR]

The contributions of this paper are twofold: we define and investigate the properties of a short rate model driven by a general Gaussian Volterra process and, after defining precisely a notion of convexity adjustment, derive explicit formulae for it. [ABSTRACT FROM AUTHOR]

Tool wear prediction is of great significance in industrial production. Current tool wear prediction methods mainly rely on the indirect estimation of machine learning, which focuses more on estimating the current tool wear state and lacks effective quantification of random uncertainty factors. To overcome these shortcomings, this paper proposes a novel method for predicting cutting tool wear. In the offline phase, the multiple degradation features were modeled using the Brownian motion stochastic process and a SVR model was trained for mapping the features and the tool wear values. In the online phase, the Bayesian inference was used to update the random parameters of the feature degradation model, and the future trend of the features was estimated using simulation samples. The estimation results were input into the SVR model to achieve in-advance prediction of the cutting tool wear in the form of distribution densities. An experimental tool wear dataset was used to verify the effectiveness of the proposed method. The results demonstrate that the method shows superiority in prediction accuracy and stability. [ABSTRACT FROM AUTHOR]

Controlling infectious diseases has become an increasingly complex issue, and vaccination has become a common preventive measure to reduce infection rates. It has been thought that vaccination protects the population. However, there is no fully effective vaccine. This is based on the fact that it has long been assumed that the immune system produces corresponding antibodies after vaccination, but usually does not achieve the level of complete protection for undergoing environmental fluctuations. In this paper, we investigate a stochastic SIS epidemic model with incomplete inoculation, which is perturbed by the Ornstein–Uhlenbeck process and Brownian motion. We determine the existence of a unique global solution for the stochastic SIS epidemic model and derive control conditions for the extinction. By constructing two suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process, we establish sufficient conditions for the existence of stationary distribution, which means the disease will prevail. Furthermore, we obtain the exact expression of the probability density function near the pseudo-equilibrium point of the stochastic model while addressing the four-dimensional Fokker–Planck equation under the same conditions. Finally, we conduct several numerical simulations to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Energy is vital to international trade, social security, and financial markets. Crude oil, as a non-renewable resource, is affected by complex factors. To better capture this influence, we introduce stochastic differential equations (SDEs) to depict crude oil prices. This paper establishes time-dependent linear and space-dependent nonlinear SDEs respectively. Time-dependent linear SDEs are established by recovering the drift and diffusion terms based on point estimation and sliding window. Space-dependent nonlinear SDEs are established by sparse Bayesian learning. Empirical results show that time-dependent linear SDE can better describe the actual fluctuations of historical crude oil prices, and can predict more accurately compared with constant coefficient linear SDE, namely geometric Brownian motion. The space-dependent nonlinear SDE can achieve the effect of time-dependent linear SDEs on historical data, while the former is more accurate in predicting. In addition, the former gained an in-depth understanding of the intrinsic dynamics. In summary, the models proposed in this study provide powerful tools for better understanding and predicting crude oil prices, which make a positive impact on the stability of the global economy and energy markets. [ABSTRACT FROM AUTHOR]