Wiener Process Research Articles

ANALYSIS OF OPTIMAL PORTFOLIO ON FINITE AND SMALL-TIME HORIZONS FOR A STOCHASTIC VOLATILITY MODEL WITH MULTIPLE CORRELATED ASSETS

In this paper, we consider the portfolio optimization problem in a financial market where the underlying stochastic volatility model is driven by [Formula: see text]-dimensional Brownian motions. At first, we derive a Hamilton–Jacobi–Bellman equation including the correlations among the standard Brownian motions. We use an approximation method for the optimization of portfolios. With such approximation, the value function is analyzed using the first-order terms of expansion of the utility function in the powers of time to the horizon. The error of this approximation is controlled using the second-order terms of expansion of the utility function. It is also shown that the one-dimensional version of this analysis corresponds to a known result in the literature. We also generate a close-to-optimal portfolio near the time to horizon using the first-order approximation of the utility function. It is shown that the error is controlled by the square of the time to the horizon. Finally, we provide an approximation scheme to the value function for all times and generate a close-to-optimal portfolio.

New a Priori estimate for stochastic 2D Navier–Stokes equations with applications to invariant measure

AbstractThe paper deals with the stochastic two-dimensional Navier–Stokes equations for homogeneous and incompressible fluids, set in a bounded domain with Dirichlet boundary conditions. We consider additive noise in the form $$G\,\textrm{d}W$$ G d W , where W is a cylindrical Wiener process and G a bounded linear operator with range dense in the domain of $$A^\gamma $$ A γ , A being the Stokes operator. While it is known that existence of invariant measure holds for $$\gamma >1/4$$ γ > 1 / 4 , previous results show its uniqueness only for $$\gamma > 3/8$$ γ > 3 / 8 . We fill this gap and prove uniqueness and strong mixing property in the range $$\gamma \in (1/4, 3/8]$$ γ ∈ ( 1 / 4 , 3 / 8 ] by adapting the so-called Sobolevskiĭ-Kato-Fujita approach to the stochastic N–S equations. This method provides new a priori estimates, which entail both better regularity in space for the solution and strong Feller and irreducibility properties for the associated Markov semigroup.

Power Brownian motion: an Ornstein-Uhlenbeck lookout

Abstract The well-known Ornstein-Uhlenbeck process (OUP) is the central go-to Gaussian model for statistical-equilibrium processes. The recently-introduced power Brownian motion (PBM) is a Gaussian model for diffusive motions, regular and anomalous alike. Using the Lamperti transform, this paper establishes PBM as the ‘diffusion counterpart’ of the OUP. Namely, the paper shows that PBM is for diffusive motions what the OUP is for statistical-equilibrium processes. The intimate parallels between the OUP and PBM are explored and illuminated via four main perspectives. 1) Statistical characterizations. 2) Kernel-integration with respect to Gaussian white noise. 3) Spatio-temporal scaling of the Wiener process. 4) Langevin stochastic dynamics driven by Gaussian white noise. To date, the prominent Gaussian models for anomalous diffusion are fractional Brownian motion (FBM), and scaled Brownian Motion (SBM). Due to its intimate OUP parallels, due to the ‘anomalous features’ it displays, due to the fact that it encompasses SBM, and following a detailed comparison to FBM: this paper argues the case for PBM as a central go-to Gaussian model for regular and anomalous diffusion.

Data and model synergy-driven rolling bearings remaining useful life prediction approach based on deep neural network and Wiener process

Abstract Various Remaining Useful Life (RUL) prediction methods, encompassing model-based, data-driven, and hybrid methods, have been developed and successfully applied to prognostics and health management for diverse rolling bearing. Hybrid methods that integrate the advantages of model-based and data-driven approaches have garnered significant attention. However, the effective integration of the two methods to address the randomness in rolling bearing full lifecycle processes remains a significant challenge. To overcome the challenge, this paper proposes a data and model synergy-driven RUL prediction framework that includes two data and model synergy approaches. Firstly, a convolutional stacked bidirectional long and short-term memory network with temporal attention mechanism is established to construct health index. The RUL prediction is achieved based on HI and polynomial model. Secondly, a three-phase prediction model based on the Wiener process is established by considering the evolution law of different degradation phases. Then, two synergy methods are designed. Strategy 1: HI is adopted as the observation value for online updating of physics degradation model parameters under Bayesian framework, and the RUL prediction results are obtained from the physics degradation model. Strategy 2: The RUL prediction results from the data-driven and physics-based model are weighted linearly combined to improve the overall prediction accuracy. The effectiveness of the proposed model is verified using two bearing full lifecycle datasets. The results indicate that the proposed approach can accommodate both short-term and long-term RUL predictions, outperforming state-of-the-art single models.

Hybrid genetic algorithm with Wiener process for multi-scale colored balanced traveling salesman problem

Colored traveling salesman problem (CTSP) can be applied to Multi-machine Engineering Systems (MES) in industry, colored balanced traveling salesman problem (CBTSP) is a variant of CTSP, which can be used to model the optimization problems with partially overlapped workspace such as the planning optimization (For example, process planning, assembly planning, productions scheduling). The traditional algorithms have been used to solve CBTSP, however, they are limited both in solution quality and solving speed, and the scale of CBTSP is also restricted. Moreover, the traditional algorithms still have the problems such as lacking theoretical support of mathematical physics. In order to improve these, this paper proposes a novel hybrid genetic algorithm (NHGA) based on Wiener process (ITÖ process) and generating neighborhood solution (GNS) to solve multi-scale CBTSP problem. NHGA firstly uses dual-chromosome coding to construct the solutions of CBTSP, then they are updated by the crossover operator, mutation operator and GNS. The crossover length of the crossover operator and the city number of the mutation operator are controlled by activity intensity based on ITÖ process, while the city keeping probability of GNS can be learned or obtained by Wiener process. The experiments show that NHGA can demonstrate an improvement over the state-of-art algorithms for multi-scale CBTSP in term of solution quality.

Characterization of a wiener process taking values in a Hilbert space

This paper characterize wiener process by taking values in a Hilbert space. A standard wiener process is stochastic process indesed by nonnegative real numbers t with the following properties: W0=0 With probability 1, the function t àWt is continuous in t. The process has stationary, independent increments. The increments - Ws has the NORMAL (0,t) distribution

Moderate deviations for a stochastic Schrödinger equation with linear drift

Moderate deviation principle is achieved by the weak convergence approach for a stochastic Schrödinger type equation with linear drift term and noise driven by a [Formula: see text]-Wiener process. The central limit theorem is also shown for the equation to further analyze its asymptotic behavior.

A Method for Predicting the Remaining Service Life of Finite Data Products Based on Gaussian Stochastic Processes

There is a problem of poor predictive performance when predicting the remaining service life of products. Therefore, this paper proposes a finite data product remaining service life prediction method based on Gaussian stochastic processes. Analyze the types of product failure and degradation under limited data, and determine the factors affecting the degradation performance of product life through constant stress accelerated degradation tests, step stress accelerated degradation tests, and sequential stress accelerated aging tests; using the Wiener process to analyze the performance degradation characteristics of products, setting product failure thresholds, defining the remaining life of the product based on the degradation process, and using equal interval partitioning methods to determine the product health index, and clarifying the product failure process under limited data; by calculating the mathematical expected value, the numerical characteristics of the remaining life of the product are defined. Variance and moments are used to determine the predicted second-order central moment and origin moment. Based on the monotonic increasing characteristics of the inverse Gaussian process, the probability distribution of the random variable of product failure is determined after the Gaussian process. Under limited data, the actual working time of the product and the degradation amount at the time scale are clarified, and a preset threshold is set for the degradation amount. Build a residual life prediction model for product degradation and output the prediction results. The experimental results show that this method can accurately predict the remaining service life of the product and has good predictive performance.

Reaction-diffusion for fish populations with realistic mobility

Nonlinear reaction-diffusion equations, with Fisher logistic growth and constant diffusion coefficient, have been used in fisheries research to estimate sustainable harvesting rates and critical domain sizes of no-take areas. However, constant diffusivity in a population density corresponds to standard Brownian motion of individuals, with a normal distribution for displacement over a fixed time interval. For available good data sets on mobile fish populations, the distribution is certainly not normal. The data can be fitted with a long-tailed stable Lévy distribution that results from diffusion by fractional Laplacian. Exact multidimensional solutions are developed here for realistic Fisher–Kolmogorov–Petrovski-Piscounov models with diffusion by fractional Laplacian. These can also account for a delay in the reaction term. It is then shown how to modify critical domain sizes of protected areas with Dirichlet and Robin boundary conditions for populations.

Dynamic Programming-Based Approach to Model Antigen-Driven Immune Repertoire Synthesis

This paper presents a novel approach to modeling the repertoire of the immune system and its adaptation in response to the evolutionary dynamics of pathogens associated with their genetic variability. It is based on application of a dynamic programming-based framework to model the antigen-driven immune repertoire synthesis. The processes of formation of new receptor specificity of lymphocytes (the growth of their affinity during maturation) are described by an ordinary differential equation (ODE) with a piecewise-constant right-hand side. Optimal control synthesis is based on the solution of the Hamilton–Jacobi–Bellman equation implementing the dynamic programming approach for controlling Gaussian random processes generated by a stochastic differential equation (SDE) with the noise in the form of the Wiener process. The proposed description of the clonal repertoire of the immune system allows us to introduce an integral characteristic of the immune repertoire completeness or the integrative fitness of the whole immune system. The quantitative index for characterizing the immune system fitness is analytically derived using the Feynman–Kac–Kolmogorov equation.

Remaining useful life prediction method for jointless track circuits based on multivariate feature fusion and nonlinear Wiener process

Abstract Predicting the Remaining Useful Life (RUL) of track circuits is essential to ensure the safe and reliable operation of high-speed railways. In response to the challenges faced by current machine-learning-based RUL prediction methods, which struggle to represent the uncertainty in the probability distribution of RUL predictions, this paper suggests a hybrid-driven method for estimating remaining life. Firstly, the track circuit Health Index (HI) is constructed by feature dimensionality reduction and fusion of the original multivariate monitoring data through Kernel Principal Component Analysis (KPCA) and Autoencoder (AE); Secondly, the degraded state of the rail circuit is modelled using a nonlinear Wiener degradation model. Finally, the principle of First Hitting Time (FHT) is used to derive the Probability Density Function (PDF) of the anticipated RUL. The efficacy and superiority of the approach presented in this paper are validated by experimental research on the track circuit monitoring dataset. The method enhances forecast accuracy and reduces prediction uncertainty, offering robust technical support for track circuit maintenance decision-making.

Design of extended dissipative-based composite anti-disturbance reliable tracking control for stochastic Takagi–Sugeno fuzzy systems

This study focuses on examining the issues of extended dissipative-based composite anti-disturbance reliable tracking control for stochastic Takagi–Sugeno fuzzy systems afflicted by sensor faults and multiple disturbances. Explicitly, the multiple disturbances include the autonomous Wiener process and non random noises with partly known information that are generated by an exogenous plant. To address the difficulty of estimating the non random disturbance while considering the partially known information, the study incorporates a fuzzy stochastic disturbance observer. Furthermore, to solve the output tracking issue of the system under consideration, a modified repetitive tracking control is implemented. Then, combining the methodologies of disturbance observer-based control technique with modified repetitive control, a composite anti-disturbance reliable tracking control strategy is developed for achieving the intended tracking goal despite the multiple disturbances and sensor faults encountered. Additionally, the proposed approach involves extended dissipativity scheme, which includes H ∞ , passivity, dissipativity, and L 2 − L ∞ performances in a unified framework. In particular, by employing Lyapunov–Krasovskii functional approach, the essential stability criteria for the proposed closed-loop system are articulated in the form of linear matrix inequalities. Additionally, numerical illustrations are presented to exemplify the application and efficiency of the developed control protocol.

Bearing remaining life prediction method based on ARAD -ELN and multi-stage wiener process

Abstract Stochastic process-based models are extensively utilized in health assessments and Remaining Useful Life (RUL) predictions of bearings. Nevertheless, bearings in actual operation undergo multiple degradation stages, each characterized by its unique trend of degradation. The application of a singular stochastic process for RUL prediction falls short of achieving optimal performance. Consequently, this paper introduces a multi-stage Wiener process-based approach for the prediction of bearings' RUL. Initially, to tackle the challenge of imbalanced sample sizes across different degradation stages of bearings, an ensemble learning-based neural network (ELN) enhanced by ARIMA Residual Anomaly Detection (ARAD) for the identification of bearing degradation stages is proposed. Subsequently, taking into account the temporal, unit-to-unit, and nonlinear variabilities of the degradation process at each stage, a Wiener process-based multi-stage degradation model for bearings is developed. A method for parameter estimation and updating, utilizing Kalman filtering and Maximum Likelihood Estimation (K-M) is introduced. Finally, the proposed model undergoes validation using both simulated data and the XJTU-SY bearing dataset. Experimental results of three RUL predictions show that the proposed method leads the benchmark model with RMSE values of 3.61, 2.92 and 7.24 respectively, affirming that the proposed model can precisely classify equipment degradation stages and predict RUL with high accuracy and stability.

On partially observed jump diffusions III: regularity of the filtering density

AbstractThe filtering equations associated to a partially observed jump diffusion model $$(Z_t)_{t\in [0,T]}=(X_t,Y_t)_{t\in [0,T]}$$ ( Z t ) t ∈ [ 0 , T ] = ( X t , Y t ) t ∈ [ 0 , T ] , driven by Wiener processes and Poisson martingale measures are considered. Building on results from two preceding articles on the filtering equations, the regularity of the conditional density of the signal $$X_t$$ X t , given observations $$(Y_s)_{s\in [0,t]}$$ ( Y s ) s ∈ [ 0 , t ] , is investigated, when the conditional density of $$X_0$$ X 0 given $$Y_0$$ Y 0 exists and belongs to a Sobolev space, and the coefficients satisfy appropriate smoothness and growth conditions.

Reliability Analysis for Degradation-Shock Processes with State-Varying Degradation Patterns Using Approximate Bayesian Computation (ABC) for Parameter Estimation

The degradation of products is an integral part of their life-cycle, often following predictable trajectories. However, sudden, unexpected events, termed ’shocks’, can substantially alter these degradation paths. Shocks can significantly influence the pace of degradation, leading to accelerated system failure. Moreover, they may initiate changes in degradation patterns, transitioning from linear to non-linear or random trajectories. To address this challenge, we present a novel multi-state reliability model for competing failure processes that account for degradation-shock dependencies by considering the state-varying degradation pattern. The degradation process is divided into s-states, with each state treated according to its pattern based on the time-transform Wiener process. The reliability function is derived based on soft failure caused by continuous degradation involving the s-states, the sudden increase in degradation caused by random shocks, and hard failure due to some shock processes. Additionally, we performed a sensitivity analysis to determine which parameters have the most significant impact on product reliability. Due to the complexity of the likelihood function, we adopted the ABC method to estimate the model parameters. A simulation study and a practical application with micro-electro-mechanical systems (MEMS) degradation results are used to demonstrate the efficiency and effectiveness of the proposed approach.

The Safety Assessment of Civil Ships against Capsizing Based on the Probability Analysis of Multiple Threshold-crossings of Stochastic Sea Waves

Abstract This paper investigates the impact of threshold-crossing events on ship capsizing through a probabilistic model that predicts random wave heights. Utilizing statistical data from wave height observations, this paper proposes that random waves can be approximated and modeled using the Wiener process, employing autocorrelation function identification and probabilistic statistical verification methods. The threshold-crossing duration of a random wave is just the period of the wave exceeds a given threshold, which can reflect the frequency property of the stochastic sea waves. And the probability distribution of the period can theoretically be determined using the derived probability density function for the time interval between any two adjacent crossings of the Wiener process and a threshold. Based on the above theories, the capsizing probabilities of a civil ship sailing in different wave areas are analyzed under different safety thresholds considering certain ratios of the ship's intrinsic period and wave period. The research results can provide a reference for the anti-overturning design of the ships under the action of random waves.

Degradation Modeling and RUL Prediction of Hot Rolling Work Rolls Based on Improved Wiener Process.

Hot rolling work rolls are essential components in the hot rolling process. However, they are subjected to high temperatures, alternating stress, and wear under prolonged and complex working conditions. Due to these factors, the surface of the work rolls gradually degrades, which significantly impacts the quality of the final product. This paper presents an improved degradation model based on the Wiener process for predicting the remaining useful life (RUL) of hot rolling work rolls, addressing the critical need for accurate and reliable RUL estimation to optimize maintenance strategies and ensure operational efficiency in industrial settings. The proposed model integrates pulsed eddy current testing with VMD-Hilbert feature extraction and incorporates a Gaussian kernel into the standard Wiener process to effectively capture complex degradation paths. A Bayesian framework is employed for parameter estimation, enhancing the model's adaptability in real-time prediction scenarios. The experimental results validate the superiority of the proposed method, demonstrating reductions in RMSE by approximately 85.47% and 41.20% compared to the exponential Wiener process and the RVM model based on a Gaussian kernel, respectively, along with improvements in the coefficient of determination (CD) by 121% and 19.76%. Additionally, the model achieves reductions in MAE by 85.66% and 42.61%, confirming its enhanced predictive accuracy and robustness. Compared to other algorithms from the related literature, the proposed model consistently delivers higher prediction accuracy, with most RUL predictions falling within the 20% confidence interval. These findings highlight the model's potential as a reliable tool for real-time RUL prediction in industrial applications.

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.

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.

A novel performance prediction method for harmonic reducers based on the trend-enhanced Fisher’s discriminant ratio-Wiener process

Abstract Harmonic reducers, as core components of industrial robots, play a critical role in maintaining robot health. Performance degradation assessment (PDA) and remaining useful life (RUL) prediction are essential for ensuring the operational reliability of such robots. To address the multistage characteristics and stochastic uncertainties in the performance degradation of harmonic reducers, a performance prediction method based on Fisher’s discriminant ratio and Wiener process is proposed. Firstly, the health indicator (HI) construction is defined as an optimization problem based on Fisher’s discriminant ratio and trend monotonicity constraints. By leveraging the sum of the multiplication of frequency amplitudes and optimized weights, precise segmentation of the multistage degradation process over the entire lifecycle is achieved. Notably, the optimized weights can automatically identify the resonance frequency bands caused by damage. Subsequently, a nonlinear Wiener process model with a drift coefficient in the form of a power function is established. The probability density function (PDF) expression for RUL is derived based on the concept of first hit time (FHT). Additionally, a logarithmic likelihood function (Log-LF) for the unknown parameters in the degradation model is constructed. The experimental results indicate that the proposed method surpasses the other two Wiener process models in terms of root mean square error (RMSE), mean absolute percentage error (MAPE), and cumulative relative accuracy (CRA). This provides robust support for preventive maintenance decision-making for harmonic reducers.