Transition Probability Density Research Articles

Velocity divergence—Generalized adaptive probability density evolution method

AbstractThis study proposes a novel velocity divergence‐generalized adaptive probability density evolution method (VD‐GAPDEM) for calculating the probability density function of the stochastic response process of stochastic structures under stochastic dynamic loads. First, based on the principle of probability conservation, the velocity divergence‐generalized adaptive probability density evolution equation (VD‐GAPDEE) is derived for a stochastic system that can effectively consider the shape and location changes of the joint transitional probability density of representative points (RPs) in the stochastic response process. Second, a novel VD‐GAPDEM is proposed to solve the VD‐GAPDEE directly using the point selection technique based on the generalized F discrepancy and the second‐order Runge–Kutta method with a smoothing kernel method (Runge–Kutta‐SKFAM). Furthermore, the differences and connections between VD‐GAPDEM and the existing probability density evolution method are analyzed. Additionally, the high computational efficiency and accuracy of the proposed VD‐GAPDEM are demonstrated through three typical examples of stochastic response analysis, involving stochastic systems subjected to stochastic dynamic loads.

Deep Learning of Transition Probability Densities for Stochastic Asset Models with Applications in Option Pricing

Transition probability density functions (TPDFs) are fundamental to computational finance, including option pricing and hedging. Advancing recent work in deep learning, we develop novel neural TPDF generators through solving backward Kolmogorov equations in parametric space for cumulative probability functions. The generators are ultra-fast, very accurate and can be trained for any asset model described by stochastic differential equations. These are “single solve,” so they do not require retraining when parameters of the stochastic model are changed (e.g., recalibration of volatility). Once trained, the neural TDPF generators can be transferred to less powerful computers where they can be used for e.g. option pricing at speeds as fast as if the TPDF were known in a closed form. We illustrate the computational efficiency of the proposed neural approximations of TPDFs by inserting them into numerical option pricing methods. We demonstrate a wide range of applications including the Black-Scholes-Merton model, the standard Heston model, the SABR model, and jump-diffusion models. These numerical experiments confirm the ultra-fast speed and high accuracy of the developed neural TPDF generators. This paper was accepted by Kay Giesecke, finance. Funding: H. Su received research funding support from Nottingham Business School at Nottingham Trent University. M. V. Tretyakov was supported by the Engineering and Physical Sciences Research Council [Grant EP/X022617/1]. D. P. Newton received research funding support from the School of Management at Bath University. Supplemental Material: The online appendices and data files are available at https://doi.org/10.1287/mnsc.2022.01448 .

Path integration solutions for stochastic systems with Markovian jumps

A path integration method for solving Markovian jump stochastic dynamical systems is presented. The Markovian jump process and the state vector of the system are combined into a new augmented state vector. The randomness of the Markovian jump is modeled by a stochastic process, which is merged with the stochastic perturbations to form the stochastic source affecting the evolution of the augmented system. Then a probability mapping is constructed, mapping the probability space of the stochastic source to the short-time transition probability density function of the augmented system. By solving this probability mapping, the short-time transition probability density function of the path integration method is obtained, thus a path integration method is customized for Markovian jump stochastic systems. Finally, a hydraulic relief valve system is used as an application example to demonstrate the availability of the path integration algorithm. The results suggest that the pre-compression parameters and coefficient of restitution of the plug can induce stochastic P-bifurcation phenomena. When the above two parameters are small, the system becomes unstable with the parameters taken in this example. Monte Carlo simulations prove that the proposed method effectively captures the transient and stationary responses of Markovian jump systems.

Evolutionary Probabilistic Analysis of Non-Smoothly Damped Stochastic Oscillator Under Modulated Random and Harmonic Excitations

This paper concerns the evolutionary transitional probability density functions (PDFs) and the mean upcrossing rates (MCRs) of responses of the non-smoothly damped stochastic oscillator under both modulated random and harmonic excitations. The damping expressed by a non-smooth function of velocity is considered in the stochastic dynamical systems. A procedure is presented by making the time-related weighted residue of the formulated Fokker–Planck–Kolmogorov (FPK) equation vanish in the weak sense. Thus, a group of stochastic ordinary differential equations regarding the undetermined parameters in the assumed solution can be formulated and solved numerically. Three cases are studied under different conditions. The achieved evolutionary transitional PDF and MCR solutions attest to the outstanding effectiveness of the presented procedure even in the tail ends of the PDF solutions. Moreover, the achieved solutions of the stochastic dynamical systems are found to have significant non-Gaussian behaviors due to the interaction of the random and harmonic excitations and the system nonlinearity. Notably, the presented procedure greatly improves the running efficiency when compared with Monte Carlo simulation.

A theoretical model for a vane with stochastic rotation

In this paper we propose the mathematical model for a vane with stochastic rotation. By backward stochastic differential equation and the Feynman–Kac formula, we obtain the representation of solution to the proposed convection–diffusion equation with symmetric initial condition through the transition probability density of stochastic differential equation. Moreover the explicit probability density function of solution with t=1 and x=0 is obtained, which shows that the distribution of k<0 is bimodal distribution with double maximum. By comparing to the experiment, it is found that the angular velocity with stochastic bidirectional is the binormal distribution with k<0, thus our model can describe the rotation of a vane with stochastic rotation very well.

Robust Kalman filters under epistemic uncertainty for non‐Gaussian systems with multiplicative noise

AbstractThis article proposes two robust Kalman filters to solve the issue of inaccurate modeling in multiplicative noise systems due to epistemic limitations. First, we construct all conceivable state/measurement transition probability densities as an ambiguity set. This ambiguity set chooses the Wasserstein distance or the moment‐based metric as the distance metric. Besides, this set is an inequality set with a chosen tolerance, which can be seen as a non‐negative radius ball. Then, by combining the robust solution of the least favorable model in that ball with the alternating direction method of multipliers or an efficient direct solution method, we propose two robust Kalman filters based on the minimum mean square error criterion. A classical example is provided to verify the effectiveness of the proposed robust filters in comparison to existing state‐of‐the‐art filters.

An extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems

An extrapolation approach within the Wiener path integral (WPI) technique is developed for determining the stochastic response of diverse nonlinear dynamical systems. Specifically, the WPI technique treats the system response joint transition probability density function (PDF) as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector. Further, the functional integral is evaluated, ordinarily, by considering the contribution only of the most probable path. This corresponds to an extremum of the functional integrand, and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP). This BVP corresponds to a specific grid point of the response PDF domain. Remarkably, the BVPs corresponding to two neighboring grid points not only share the same equations, but also the boundary conditions differ only slightly. This unique aspect of the technique is exploited herein. Specifically, it is shown that solution of a BVP and determination of the response PDF value at a specific grid point can be used for extrapolating and estimating efficiently the PDF values at neighboring points without the need for considering additional BVPs. Notably, the herein developed approach enhances significantly the computational efficiency of the WPI technique without, practically, affecting the associated degree of accuracy. Two numerical examples relating to a Duffing nonlinear oscillator subjected to combined stochastic and deterministic periodic loading, and to an oscillator with asymmetric nonlinearities and fractional derivative elements are considered to demonstrate the reliability of the extrapolation approach. Juxtapositions with pertinent Monte Carlo simulation data are included as well.

A neural-driven stochastic degradation model for state-of-health estimation of lithium-ion battery

Due to usage for a long time, lithium-ion battery degradation issues may endanger security operations for service equipment. As batteries usually operate under uncertain usage profiles and experience random aging processes, accurately estimating the battery state of health (SOH) can be challenging. Most existing SOH estimation methods are either empirical knowledge-based with complicated parameters or serve as a black box neural network-based without stochastic nature. To address this limitation, a neural-driven stochastic degradation model for battery health estimation is proposed to ensure the security and reliability of the operation. Firstly, the parameters of a stochastic process are estimated by martingale methodology to describe battery degradation characteristics. Then the optimized sequence-to-sequence (Seq2Seq) neural network is utilized to develop a mapping model in order to update drift coefficient online. Concurrently, the martingale estimating function is used to obtain the diffusion coefficient function. Finally, the transition probability density function of SOH estimation is derived. Comparative experiments are carried out on different battery datasets to show the effectiveness of the proposed model against traditional Wiener process-based models and existing methods in SOH estimation. The model proposed provides an effective solution for battery SOH estimation of energy storage equipment.

Formula omitted] control of singularly perturbed systems using deficient hidden semi-Markov model

This paper deals with the H∞ control of a class of stochastic multi-timescale systems, called Markov jump singularly perturbed systems. The hidden semi-Markov model is introduced to handle the situation when system modes are unavailable in semi-Markov systems. Such a model is assumed deficient, that is, it lacks knowledge about the emission probability, transition probability, and probability density function of the sojourn time. It is a more general case compared with works conducted with perfect transition information. Depending on whether a fast or slow sampling rate is used, the resulting discrete-time singularly perturbed system is modeled differently, for both of which the controller design is conducted. Furthermore, criteria expressed in terms of linear matrix inequalities (LMIs) are developed that guarantee the δ-error mean-square stability. An approach to estimate the upper bound on δ-error with incomplete information is provided, meanwhile, the relationship between system performance and the upper of singular perturbation parameter is also presented. Finally, two simulation examples using real-world systems are provided to corroborate the validity as well as the practical merits of the results.

On the existence, uniqueness and calculation of Wiener path integral most probable paths for determining the stochastic response of nonlinear dynamical systems

Various techniques are developed for addressing the existence, uniqueness and numerical calculation of Wiener path integral (WPI) most probable path solutions. Specifically, the WPI technique for determining the stochastic response of diverse nonlinear dynamical systems treats the system response joint transition probability density function as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector. This functional integral is evaluated, ordinarily, by resorting to an approximate approach that considers the contribution only of the most probable path. The most probable path corresponds to an extremum of the functional integrand and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP).In this paper, first, it is shown that for the commonly considered case of the system nonlinearity being of polynomial form, there exist globally optimal solutions corresponding to the most probable path BVP. Further, relying on algebraic geometry concepts and tools, a condition is derived for determining if the BVP for the most probable path exhibits a unique solution over a specific region. Furthermore, a novel solution approach is developed for the BVP by relying on Sylvester’s dialytic method of elimination. Notably, the method reduces the complexity of the BVP system of coupled multivariate polynomial equations by eliminating one or more variables. Various numerical examples pertaining to diverse nonlinear oscillators are included for demonstrating the capabilities of the developed techniques.

Statistical inference for discretely sampled stochastic functional differential equations with small noise

Estimating parameters of drift and diffusion coefficients for multidimensional stochastic delay equations with small noise are considered. The delay structure is written as an integral form with respect to a delay measure. Our contrast function is based on a local-Gauss approximation to the transition probability density of the process. We show consistency and asymptotic normality of the minimum-contrast estimator when a small dispersion coefficient ε→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0$$\\end{document} and sample size n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document} simultaneously.

Bilateral estimates of some pseudo-derivatives of the transition probability density of an isotropic $\alpha$-stable stochastic process

In the paper, the transition probability density of an isotropic $\alpha$-stable stochastic process in a finite dimensional Euclidean space is considered. The results of applying pseudo-differential operators with respect spatial variables to this function are estimated from the both side: above and below. Operators in the consideration are defined by the symbols $|\lambda|^\varkappa$ and $\lambda|\lambda|^{\varkappa-1}$, where $\varkappa$ is some constant. The first operator with negative sign is fractional Laplacian and the second one multiplied by imaginary unit is fractional gradient.

Разработка математической модели оптимизации процесса обучения курсантов силовых структур в высших учебных заведениях как сложной системы

This article discusses the problem of developing a mathematical model to describe the process of training cadets of law enforcement agencies in higher educational institutions, which are a complex system. Currently, reality places high demands on the qualifications and personal qualities of law enforcement officers, which in turn requires the relevant educational organizations to carry out advanced professional training of specialists. The functioning of higher educational institutions of law enforcement agencies is influenced by objective and subjective factors. To this end, in order to increase the efficiency of managerial decision-making in the process of training cadets of law enforcement agencies in higher educational institutions, it is proposed to use a mathematical model of training and education of a group of cadets, developed on the basis of the method of operations research, specifically the theory of Markov random processes that occur during continuously changing time and use discrete states of the system. The mathematical model of training and education of a group of cadets in its functioning uses streams of events that have the properties of stationarity and ordinariness. The process of transition from one state to another is described in the form of a marked graph of states with a transition probability density. Setting the intensity of the event flows will allow you to find the limiting probabilities and predict the states in which the study group is located. This approach will reduce the level of subjectivity of university staff and will allow structuring and optimizing the process of assessing the level of knowledge of cadets.

A new path integration method for the stochastic system under Poisson white noise excitation based on a probability mapping

In this paper, a new path integration method is proposed for stochastic dynamical systems excited by Poisson white noise. An efficient one-step transition probability density function (TPDF) matrix is constructed based on a decoupling probability mapping. The new method can handle the case of multiple impulses occurring in a one-step transition time interval and considers the randomness of the impulse instant, which compensates for a drawback of previous path integration methods. The probability mapping realizes the decoupling of randomness and the one-step TPDF matrix, which can be extended to general stochastic systems satisfying the Markov property. The stochastic responses of two dynamical systems excited by Poisson white noises for different mean arrival rates are obtained by using the new path integration method, and MC simulations prove that the new method is very effective, and it maintains good accuracy even for large mean arrival rates.

A stochastic log-logistic diffusion process: Statistical computational aspects and application to real data

This article introduces a new stochastic diffusion process based on the theory of diffusion processes whose mean function is proportional to the log-logistic growth curve. The main characteristics of the process are analyzed, including the transition probability density function, the mean functions and in particular, the auto-correlation function between two times of the process. The parameters of the process are estimated by maximum likelihood method using discrete sampling. The simulated annealing algorithm is applied after bounding the parametric space by a strategy procedure to solve the likelihood equations. The behavior of the diffusion process here derived is finally applied to study an example for the growth data of a microorganism culture.

Path integration method based on a decoupling probability mapping for fast solving the stochastic response of dynamical systems

An efficient path integration method is proposed for obtaining the stochastic response of dynamical systems. This method benefits from a new short-time transition probability density function (TPDF) matrix. First, a decoupling probability mapping is revealed, which achieves the decoupling of the short-time TPDF from stochastic processes that the system suffers. Then, the short-time transition paths carrying known probabilities can be determined by solving the probability mapping. Thus, the new TPDF matrix can be obtained simply and efficiently. The transient and steady responses of two 4-dimensional systems for different noise parameters and system parameters are explored by using the efficient path integration method. The results confirm that our method requires a cheap time cost to achieve good results that fit well with the Monte Carlo simulations.

Modeling orientation perception adaptation to altered gravity environments with memory of past sensorimotor states.

Transitioning between gravitational environments results in a central reinterpretation of sensory information, producing an adapted sensorimotor state suitable for motor actions and perceptions in the new environment. Critically, this central adaptation is not instantaneous, and complete adaptation may require weeks of prolonged exposure to novel environments. To mitigate risks associated with the lagging time course of adaptation (e.g., spatial orientation misperceptions, alterations in locomotor and postural control, and motion sickness), it is critical that we better understand sensorimotor states during adaptation. Recently, efforts have emerged to model human perception of orientation and self-motion during sensorimotor adaptation to new gravity stimuli. While these nascent computational frameworks are well suited for modeling exposure to novel gravitational stimuli, they have yet to distinguish how the central nervous system (CNS) reinterprets sensory information from familiar environmental stimuli (i.e., readaptation). Here, we present a theoretical framework and resulting computational model of vestibular adaptation to gravity transitions which captures the role of implicit memory. This advancement enables faster readaptation to familiar gravitational stimuli, which has been observed in repeat flyers, by considering vestibular signals dependent on the new gravity environment, through Bayesian inference. The evolution and weighting of hypotheses considered by the CNS is modeled via a Rao-Blackwellized particle filter algorithm. Sensorimotor adaptation learning is facilitated by retaining a memory of past harmonious states, represented by a conditional state transition probability density function, which allows the model to consider previously experienced gravity levels (while also dynamically learning new states) when formulating new alternative hypotheses of gravity. In order to demonstrate our theoretical framework and motivate future experiments, we perform a variety of simulations. These simulations demonstrate the effectiveness of this model and its potential to advance our understanding of transitory states during which central reinterpretation occurs, ultimately mitigating the risks associated with the lagging time course of adaptation to gravitational environments.

Variational adaptive Kalman filter for unknown measurement loss and inaccurate noise statistics

Considering a common situation that the measurements are obtained from independent sensors and the accurate noise statistics are not available, we propose a novel variational adaptive Kalman filter (KF), which can selectively treat measurement loss and adaptively estimate inaccurate state and measurement noise covariance matrices. Firstly, a multiple inverse-Wishart mixture (MIWM) distribution is utilized to modeled state transition probability density function (PDF), which reduces the dependence on the pre-selected nominal state noise covariance matrix (SNCM). Then, a modified measurement model is constructed and a new measurement likelihood PDF is provided, where the measurement losses from different sensors are considered independently and the measurement noise covariance matrix (MNCM) is modeled as an inverse Gamma distribution. Finally, based on the modified state transition and measurement likelihood PDFs, a novel variational adaptive KF is derived by variational Bayesian method, and the feasibility and superiority of the filter are demonstrated by the numerical simulation.

Survival probability determination of nonlinear oscillators subject to combined deterministic periodic and non-stationary stochastic loads

This paper presents a technique based on Wiener path integrals (WPI) for computing the survival probability of nonlinear oscillators under combined nonstationary stochastic and periodic excitation. Specifically, the equation of motion is equivalently decomposed into a correlated combination of a nonlinear stochastic differential equation (SDE) and a nonlinear deterministic differential equation. The nonstationary response probability density function (PDF) of the stochastic component is obtained using the Wiener path integral concept in conjunction with stochastic averaging/linearization. The approach involves utilizing stochastic averaging/linearization techniques to transform the nonlinear SDE into a linear one with varying equivalent stiffness and damping. Afterwards, using the WPI method and the concept of the most probable path, a closed-form approximate analytical expression of the joint transition probability density function (PDF) is obtained for small intervals corresponding to the equivalent linear SDE. Additionally, the survival probability of the original nonlinear oscillator subject to combined excitation with fixed barriers can be approximated equivalently by using the survival probability with time-varying barriers of the equivalent linear system with time-varying stiffness and damping subject only to the nonstationary stochastic excitation. Finally, by using the analytical expression of the transition PDF of the equivalent linear system and a discrete version of the Chapman–Kolmogorov (C–K) equation, it is possible to determine the response and survival probability of the original oscillator under combined excitation step by step. Examples involving a Duffing oscillator and a vibro-impact oscillator are analyzed to showcase the precision and efficacy of the suggested technique in contrast to relevant Monte Carlo simulations.

Inferring Ocean Transport Statistics With Probabilistic Neural Networks

AbstractUsing a probabilistic neural network and Lagrangian observations from the Global Drifter Program, we model the single particle transition probability density function (pdf) of ocean surface drifters. The transition pdf is represented by a Gaussian mixture whose parameters (weights, means, and covariances) are continuous functions of latitude and longitude determined to maximize the likelihood of observed drifter trajectories. This provides a comprehensive description of drifter dynamics allowing for the simulation of drifter trajectories and the estimation of a wealth of dynamical statistics without the need to revisit the raw data. As examples, we compute global estimates of mean displacements over 4 days and lateral diffusivity. We use a probabilistic scoring rule to compare our model to commonly used transition matrix models. Our model outperforms others globally and in three specific regions. A drifter release experiment simulated using our model shows the emergence of concentrated clusters in the subtropical gyres, in agreement with previous studies on the formation of garbage patches. An advantage of the neural network model is that it provides a continuous‐in‐space representation and avoids the need to discretize space, overcoming the challenges of dealing with nonuniform data. Our approach, which embraces data‐driven probabilistic modeling, is applicable to many other problems in fluid dynamics and oceanography.