Wiener Process Research Articles

New control variates for pricing basket options

Abstract Accepted by: Giorgio Consigli Accurate pricing of basket options, which are financial derivatives on multiple underlying assets, is a challenging and practically important task for financial institutions. We propose several new control variates for accurate, fast and efficient pricing of basket options. The first approach to deriving new control variates is the use of Hermite polynomial approximation of appropriate function of the underlying asset prices, which leads to a Black–Scholes-like analytic solution. This approach is new in the option pricing context and opens up new possibilities in derivative pricing. Further control variates are analytically derived using Jensen’s inequality in one case, and distributional properties of multivariate Wiener processes in other cases. All the newly proposed control variates are shown to lead to excellent variance reduction in numerical experiments based on realistic data. The proposed methods are novel, computationally simple and have a strong potential to replace more conventional methods, such as the geometric lower bound in simulation-based pricing of basket options and similar products used in financial risk management.

On synchronization of random nonlinear complex networks

Traditionally, stochastic disturbances arising in complex networks are often assumed to be drawn from a Wiener process, potentially limiting their applicability in real engineering scenarios. To address this limitation, we incorporate randomness to quantify the stochastic disturbances within a group of participating individuals, thereby establishing random nonlinear complex networks in a directed interacting setting. Subsequently, we demonstrate that the maximal existence interval of the unique solution to the underlying systems is determined by the properties of the associated noise and the specified Lipschitz constant. Building on this, we further show that, by making use of supermartingale and Lyapunov-based techniques, the almost sure synchronization condition of the investigated random complex system is determined by the communication topology, weight gain, and the number of participating agents. Additionally, we discuss synchronization problems within strongly connected and undirected graphs. Finally, we validate the proposed method using Chen systems.

Fractional Wiener chaos: Part 1

AbstractIn this paper, we introduce a fractional analogue of the Wiener polynomial chaos expansion. It is important to highlight that the fractional order relates to the order of chaos decomposition elements, and not to the process itself, which remains the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as $${\mathcal {H}}_\alpha (x,y)$$ H α ( x , y ) , referred to herein as a power-normalised parabolic cylinder function. Through careful analysis of several fundamental deterministic and stochastic properties, we affirm that this function essentially serves as a fractional extension of the Hermite polynomial. In particular, the power-normalised parabolic cylinder function with the Wiener process and time as its arguments, $${\mathcal {H}}_\alpha (W_t,t)$$ H α ( W t , t ) , demonstrates martingale properties and can be interpreted as a fractional Itô integral with 1 as the integrand, thereby drawing parallels with its non-fractional counterpart. To build a fractional analogue of polynomial Wiener chaos on the real line, we introduce a new function, which we call the extended Hermite function, by smoothly joining two power-normalized parabolic cylinder functions at zero. We form an orthogonal set of extended Hermite functions as a one-parameter family and use tensor products of the extended Hermite functions as building blocks in the fractional Wiener chaos expansion, in the same way that tensor products of Hermite polynomials are used as building blocks in the Wiener chaos polynomial expansion.

Mittag–Leffler Fractional Stochastic Integrals and Processes with Applications

We study Mittag–Leffler (ML) fractional integrals involved in the solution processes of a system of coupled fractional stochastic differential equations. We introduce the ML fractional stochastic process as a ML fractional stochastic integral with respect to a standard Brownian motion. We provide some representation formulas of solution processes in terms of Mittag–Leffler fractional integrals and processes. Computable expressions of the mean functions and of the covariances of such processes are specifically given. The application in neuronal modeling is provided, and all involved functions and processes are specifically determined. Numerical evaluations are carried out and some results are shown and discussed.

Borel control and efficient numerical techniques to solve the Allen–Cahn equation governed by temporal multiplicative noise

In this manuscript, we investigate a stochastic version of the Allen–Cahn equation, which is a nonlinear partial differential equation from mathematical physics. The stochastic model considers temporal multiplicative noise in the form of the derivative of the standard Wiener process. The existence and the uniqueness of solutions for this stochastic model is established rigorously using the theory of distributions. As a corollary from these analytical results, some a priori optimal estimates for the solutions of this model are constructed. In this work, we develop reliable numerical schemes which possess similar features as those of the solutions for the analytical model. Moreover, we establish mathematically the von Neumann stability and the consistency for the schemes proposed in this paper. Both the analytical and numerical results derived in this work are computationally verified through some simulations.

Multi-Stage Wiener Process Based Remaining Useful Life Prediction for Load Cells

Abstract Nowadays, sensor technology is used in a wide range of fields, and the demand for its reliability continues to increase. Accurate estimation of the remaining useful life of sensors is critical. Since the performance degradation of sensors involves multiple stages, the single-stage Wiener degradation model is difficult to accurately characterize. To resolve this issue, a multi-stage Wiener process degradation model is established. The model utilizes the cumulative sum algorithm to identify change points among stages. Then, the performance of the multi-stage Wiener model for characterizing degradation trends is validated with load cell accelerated aging test data. Furthermore, the results of experiment demonstrate that the proposed method provides a more accurate remaining useful life estimation compared to the single-stage Wiener model, and the relative error is reduced by 18.82%.

Stochastic analysis and soliton solutions of the Chaffee–Infante equation in nonlinear optical media

The Chaffee–Infante (CI) equation is a nonlinear equation that may be used to predict the complex dynamics of soliton propagation in a nonlinear optical medium. In this study, we elucidate soliton solutions of the CI equation by stochastic differential equations (SDEs) with the Wiener process. In excess of the modified auxiliary equation (MAE) method, we obtain new exact soliton solutions. By combining stochastic differential equations with the Wiener process, we explain the stochastic processes directing the magnitude of solitons within the basis of the CI equation, providing dynamic views on their behavior. The acquired solitary waves are depicted via 3D and 2D graphs to show their dynamics with and without Brownian motion.

A novel dynamic predictive maintenance framework for gearboxes utilizing nonlinear Wiener process

Abstract In the context of advancing industrial automation, gearboxes, as pivotal components in power transmission systems, have a direct bearing on the operational efficiency and safety of the entire machinery. This study introduces a novel dynamic predictive maintenance framework for gearboxes using a nonlinear Wiener process. Comprehensive experiments validate the framework, demonstrating significant reductions in maintenance costs and improvements in reliability. First, a full-life degradation experiment was executed on the gearbox, leveraging the root mean square (RMS) value of the vibration signal as an indicator of system degradation. Subsequently, the signals from four vibration sensors were synthesized and normalized through Kernel Principal Component Analysis (KPCA), thereby enabling a more nuanced representation of the gearbox's degradation profile. The degradation trajectory was then modeled using a nonlinear Wiener process framework. The Wiener process's parameters and state variables were iteratively refined utilizing an online filtering algorithm grounded in Bayesian inference. This facilitated the derivation of the probability density function for the remaining useful life (RUL), thereby enabling a robust prediction of the gearbox's RUL. Finally, to minimize maintenance costs per unit of time, an optimization model for dynamic maintenance decision-making was formulated. The optimal maintenance timing was ascertained by solving this model. The empirical findings of this investigation demonstrate the efficacy of the proposed approach in executing dynamic predictive maintenance for gearboxes. This research endeavors to furnish novel theoretical underpinnings and pragmatic directives for the field of predictive maintenance in the context of gearboxes.

Laplacian operator and its square lattice discretization: Green function vs. Lattice Green function for the flat 2-torus and other related 2D manifolds

Abstract The paper investigates the truncation error between the Green function and the lattice Green function (LGF) for the Laplacian operator defined on the 2-torus and its discretization on a regular square lattice. Extensions to the cylinder and the rectangular domain with free (or Neumann) boundary conditions are also proposed. In each of these instances, the Green function and its discrete analog are given in exact analytical closed-form allowing to infer accurate estimates as the lattice spacing tends to zero. As expected, it is shown that the continuum limit of the LGF coincides well with the Green function in every case. In particular, the issue of logarithmic singularity regularization of the Green function by the lattice discretization is addressed through two related application examples regarding the rectangular domain, and devoted to the computation of corner-to-corner resistance of an electrical conducting square and the mean first-passage time between the diagonally opposite vertices of a square for a standard Brownian motion, both derived considering the continuum limit.

Effects of fractional derivative and Wiener process on approximate boundary controllability of differential inclusion

One of the core concepts of contemporary control theory is the idea that a dynamical system can be controlled. Several abstract settings have been developed to describe the distributed control systems in a domain in which the control is acted through the boundary. In this manuscript, we investigate the boundary approximate controllability (ABC) of stochastic differential inclusion (SDI) with Hilfer fractional derivative (HFD) and nonlocal condition by implementing the principle of stochastic analysis, the fixed-point theorem, fractional calculus and multi-valued map. Moreover, an example is offered to define the primary results.

A failure-dependence related stochastic crack growth modeling approach of competing cracking mode

Competing crack-induced fatigue fracture has been a typical failure mode especially in the very high cycle fatigue regime. However, the stochastic crack growth modeling related to failure-dependence has not been fully investigated. Here, a failure-dependence related stochastic crack growth modeling approach for competing cracking mode is proposed to address this issue. Firstly, copulas are used to model the dependent competing relationships among multiple fatigue fracture modes. The reliability analysis of multiple fatigue fracture modes is conducted from a copula perspective. Besides, the fatigue crack growth is modeled based on a nonlinear Wiener process, with unit-to-unit variability addressed by introducing random effects. Marginal reliability expressions are derived based on the Wiener process. Furthermore, a two-stage Bayesian inference method based on Hamiltonian Monte Carlo sampling is proposed to estimate model parameters. A Monte Carlo simulation study is conducted to validate the accuracy and robustness of the proposed inference method. Finally, the effectiveness of the proposed approach is proven through a real case study. It turns out that the ignorance of the competing relationships among multiple cracking modes leads to an underestimation of overall reliability. The accuracy of the model can be further improved with random effects considered.

Inference of a Susceptible-Infectious stochastic model.

We considered a time-inhomogeneous diffusion process able to describe the dynamics of infected people in a susceptible-infectious (SI) epidemic model in which the transmission intensity function was time-dependent. Such a model was well suited to describe some classes of micro-parasitic infections in which individuals never acquired lasting immunity and over the course of the epidemic everyone eventually became infected. The stochastic process related to the deterministic model was transformable into a nonhomogeneous Wiener process so the probability distribution could be obtained. Here we focused on the inference for such a process, by providing an estimation procedure for the involved parameters. We pointed out that the time dependence in the infinitesimal moments of the diffusion process made classical inference methods inapplicable. The proposed procedure were based on the generalized method of moments in order to find a suitable estimate for the infinitesimal drift and variance of the transformed process. Several simulation studies are conduced to test the procedure, these include the time homogeneous case, for which a comparison with the results obtained by applying the maximum likelihood estimation was made, and cases in which the intensity function were time dependent with particular attention to periodic cases. Finally, we applied the estimation procedure to a real dataset.

Discussion on the existence and controllability of Hilfer fractional stochastic differential equation with non-dense domain

This article studies the approximate controllability for a class of fractional control system with semigroup operators governed by differential equations with Hilfer fractional derivatives of order α ∈ ( 0 , 1 ) and type β ∈ [ 0 , 1 ] in Hilbert spaces. First, we establish a set of sufficient conditions for the approximate controllability for Hilfer fractional stochastic differential equation with non-dense domain in Hilbert spaces. The existence results are obtained by using the fractional calculus, semi-group theory, Wiener process and fixed point technique to prove our main results. Further, we extend the result to study the approximate controllability concept. An example is provided to illustrate the application of the obtained theory.

Random walk in random permutation set theory.

Random walk is an explainable approach for modeling natural processes at the molecular level. The random permutation set theory (RPST) serves as a framework for uncertainty reasoning, extending the applicability of Dempster-Shafer theory. Recent explorations indicate a promising link between RPST and random walk. In this study, we conduct an analysis and construct a random walk model based on the properties of RPST, with Monte Carlo simulations of such random walk. Our findings reveal that the random walk generated through RPST exhibits characteristics similar to those of a Gaussian random walk and can be transformed into a Wiener process through a specific limiting scaling procedure. This investigation establishes a novel connection between RPST and random walk theory, thereby not only expanding the applicability of RPST but also demonstrating the potential for combining the strengths of both approaches to improve problem-solving abilities.

Mathematical analysis of the Wiener processes with time-delayed feedback

It is known that time delays generally make a system unstable. However, it is numerically observed that the diffusion coefficients of the Wiener processes with time-delayed feedback decrease while increasing the time delay τ. In particular, the decay of the diffusion coefficients with the form (11+τ)2 has been confirmed by numerical simulations [Ando et al., Phys. Rev. E 96, 012148 (2017)]. In this paper, we present two analytical derivations for the relation (11+τ)2 by dynamical system approaches using the Laplace transform and stochastic differential equations.

Modelling the Dependence between a Wiener Process and Its Running Maxima and Running Minima Processes

Modelling the Dependence between a Wiener Process and Its Running Maxima and Running Minima Processes

Uncovering the stochastic dynamics of solitons of the Chaffee–Infante equation

In this paper, we apply stochastic differential equations with the Wiener process to investigate the soliton solutions of the Chaffee–Infante (CI) equation. The CI equation, a fundamental model in mathematical physics, explains concepts such as wave propagation and diffusion processes. Exact soliton solutions are obtained through the application of the modified extended tanh (MET) method. The obtained wave figures in 3D, 2D, and contour are highly localized and determine an individual frequency shift under the behavior of sharp peak, periodic wave, and singular soliton. The MET method shows to be a valuable analytical tool for obtaining soliton solutions, essential for understanding the dynamics of nonlinear wave phenomena. Numerical simulations enable us to explore soliton solutions in two and three dimensions, shedding light on their properties over time. Our results have wide applications in various domains, including stochastic processes and nonlinear dynamics, impacting advancements in physics, engineering, finance, biology, and beyond.

Remaining useful life prediction based on multi-stage Wiener process and Bayesian information criterion

Equipment will go through multiple degradation stages under complex operating conditions, and the single-stage degradation model cannot accurately describe the degradation process of the equipment at different stages, resulting in inaccurate remaining service life prediction results and reliability analysis. Therefore, this paper establishes a multi-stage Wiener degradation process model that considers measurement errors and includes three different forms of drift functions. First, by calculating the Bayesian information criterion (BIC) values of these three degradation models separately and analysing the variation trends of the BIC values, a method for detecting change-points is proposed to achieve stage division. Next, by comparing the BIC values of the three models, a method for adaptively selecting the optimal model for each stage is proposed. Then, based on the results of stage division and optimal model selection, approximate analytical expressions for the RUL of each stage are derived, and parameter estimation is performed using maximum likelihood estimation (MLE). Finally, the RUL prediction study using the proposed method is carried out through simulation cases and practical cases. The results show that the accuracy of the proposed method is higher than the existing research methods, verifying the effectiveness of the proposed method.

Degradation and life prediction of mechanical equipment based on multivariate stochastic process

IntroductionAccurately predicting the remaining mechanical equipment is of great significance for ensuring the safe operation of the equipment and improving economic efficiency.MethodsTo accurately assess the mechanical equipment degradation, predict its remaining useful life, and ensure efficient, stable, and safe operation, a degradation and life prediction model for mechanical equipment based on multivariate stochastic processes is studied. The study innovatively predicts the remaining life of mechanical equipment using multivariate stochastic processes, and facilitates the correlation analysis between performance indicators based on the characteristics of Copula functions.Results and discussionThe results showed that the Root Mean Squared Error value of the prediction results based on the trivariate Wiener process was 2.58, which decreased by 46.91% and 35.82% compared with the univariate and bivariate Wiener processes, respectively. The prediction value based on the trivariate gamma process was 3.49, which decreased by 44.95% and 40.54% compared with the univariate and bivariate gamma processes, respectively. In conclusion, the degradation and life prediction model with multivariate stochastic processes can effectively assess the mechanical equipment degradation and predict its remaining useful life. This provides an important reference for the maintenance and management of mechanical equipment, improving equipment efficiency and extend its service life.

The impact of standard Wiener process on the optical solutions of the stochastic nonlinear Kodama equation using two different methods

The stochastic nonlinear Kodama (SNLK) equation forced in the Stratonovich sense by multiplicative noise is considered here. New elliptic, hyperbolic, trigonometric, and rational stochastic solutions are acquired using ( G′/ G)-expansion method and mapping method. Because the SNLK equation is extensively used in is extensively used in nonlinear optics, fluid dynamics, and plasma physics, the obtained solutions may be used to study a broad range of relevant physical phenomena. In order to interpret the effects of multiplicative noise, the dynamic performances of the various obtained solutions are displayed utilizing 3D and 2D graphs. We infer that multiplicative noise impacts and stabilizes the solutions of SNLK equation.