Stochastic Differential Equation Model Research Articles

The Block-Correlated Pseudo Marginal Sampler for State Space Models

Pseudo Marginal Metropolis-Hastings (PMMH) is a general approach to Bayesian inference when the likelihood is intractable, but can be estimated unbiasedly. Our article develops an efficient PMMH method that scales up better to higher dimensional state vectors than previous approaches. The improvement is achieved by the following innovations. First, a novel block version of PMMH that works with multiple particle filters is proposed. Second, the trimmed mean of the unbiased likelihood estimates of the multiple particle filters is used. Third, the article develops an efficient auxiliary disturbance particle filter, which is necessary when the bootstrap disturbance filter is inefficient, but the state transition density cannot be expressed in closed form. Fourth, a novel sorting algorithm, which is as effective as previous approaches but significantly faster than them, is developed to preserve the correlation between the logs of the likelihood estimates at the current and proposed parameter values. The performance of the sampler is investigated empirically by applying it to nonlinear Dynamic Stochastic General Equilibrium models with relatively high state dimensions and with intractable state transition densities and to multivariate GARCH diffusion-driven volatility in the mean models. Although we only apply the method to state space models, the approach will be useful in a wide range of applications such as large panel data models and stochastic differential equation models with mixed effects.

A Novel Approach for Longitudinal Modeling of Aging Health and Predicting Mortality Rates

Aging is a complex stochastic process that affects healthy functioning through various pathways. In contrast to the more commonly used cross-sectional methods, our research focuses on longitudinal modeling of aging, a less explored but crucial area. We have developed a Stochastic Differential Equation (SDE) model, at the forefront of aging research, designed to accurately forecast the health trajectories and survival rates of individuals. This model adeptly delineates the connections between different health indicators and provides clear, interpretable results. Our approach utilizes the SDE framework to encapsulate the inherent uncertainty in the aging process. Moreover, it incorporates a Recurrent Neural Network (RNN) to integrate past health data into future health projections. We plan to train and test our model using a comprehensive dataset tailored for aging studies. This model is not only computationally cost-effective but also highly relevant in assessing health risks in older populations, particularly for those at high risk. It can serve as an essential tool in anticipating and preparing for challenges like infectious disease outbreaks. Overall, our research aims to improve health equity and global health security significantly, offering substantial benefits to public health and deepening our understanding of the aging process.

Generalised shot-noise representations of stochastic systems driven by non-Gaussian Lévy processes

Abstract We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump Lévy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find application across the physical and biological sciences, mathematics, finance, and engineering. Motivated by important relevant problems in statistical inference, we derive new, generalised shot-noise simulation methods whenever a normal variance-mean (NVM) mixture representation exists for the driving Lévy process, including the generalised hyperbolic, normal-gamma, and normal tempered stable cases. Simple, explicit conditions are identified for the convergence of the residual of a truncated shot-noise representation to a Brownian motion in the case of the pure Lévy process, and to a Brownian-driven SDE in the case of the Lévy-driven SDE. These results provide Gaussian approximations to the small jumps of the process under the NVM representation. The resulting representations are of particular importance in state inference and parameter estimation for Lévy-driven SDE models, since the resulting conditionally Gaussian structures can be readily incorporated into latent variable inference methods such as Markov chain Monte Carlo, expectation-maximisation, and sequential Monte Carlo.

Bifurcation analysis of a delayed stochastic HBV epidemic model: Cell-to-cell transmission

In the present study, we develop a model for hepatitis B virus (HBV) that integrates cell-to-cell transmission of the virus, CTL immunological response, delay effects, and the effect of fluctuating environmental noise. The inheritance randomness of the system shifting the system from a deterministic to another system consists of stochastic differential equations (SDEs). After model formulation, the dynamical aspects of the model are investigated by using the key threshold quantity R0. If the threshold value R0 is assumed less than unity, then it is proven that the solution trajectories are fluctuating around the disease-free state (DFS) of the associated deterministic model. The fluctuation of the trajectories around the DFS explains the phenomenon of extinction of the stochastic models. It is also proved that the trajectories of the SDE model oscillate around the endemic state of the underlying ODE system when R0s>1 explains the sure persistence theory of the proposed model. One notable finding of the present manuscript is the positive association between the fluctuating amplitude and the intensities of the white noises. This understanding can provide useful control tools for managing the worst dynamic(s) of the infection. The exact value of the parameter for time delay determines the stability of the model under consideration. By analyzing the time-delayed parameter, it was identified that the model is stable by keeping the value of the parameter below the crucial value that causes the Hopf bifurcation. This conclusion is supported by the numerical examples having epidemiological significance. Furthermore, the findings suggest that stochastic environmental disturbance can impact the spread of infectious illnesses. Notably, the high levels of noise can help to slow down the spread of epidemics in the population.

DETERMINISTIC AND STOCHASTIC MODEL FOR THE TRANSMISSION OF LASSA FEVER

Many models on the transmission dynamics of Lasser fever were based on purely deterministic approach. This approach does not put into cognizance randomness which is inherent in disease transmission resulting from differences in immunity levels, contact patterns, hygienic practices and mutation rates among so many other possibilities. In this work, we attempt to demonstrate the impact of uncertainties in the mode of transmission of Lassa fever by subjecting the dynamics to some white noise modeled by the Brownian motion as a Wiener process. An existing deterministic model involving the Susceptible, Exposed, Infected and Recovered (SEIR) individuals was transformed into a stochastic differential equation model by applying the procedure proposed by Allen et al (2008). The resulting system of Stochastic Differential Equations (SDE) was solved numerically using the Milstein scheme for SDEs. An algorithm for the method was developed and implemented in Python programming language. Numerical simulations of the model was done using four sets of parameters; , representing the natural birth rate, the natural death rate , the recovering rate from infected to recovered, transmission rate from exposed to infected ,transmission rate from susceptible to exposed are carried out to investigate the transmission dynamic of Lassa fever. The results of the simulations indicate that randomness does affect transmission of Lassa fever. We therefore recommend that factors such as social behavior, hygienic practices, contact patters, mutation rate should be considered while formulating mathematics models of disease transmission.

Condition-based opportunistic maintenance strategy for multi-component wind turbines by using stochastic differential equations

The components of wind turbines are complex in structure and the working environment is harsh, which makes wind turbines face problems such as high failure rates and high maintenance costs. In this paper, the stochastic differential equation model has been established for the harsh operating environment of wind turbines, and used Brownian motion to simulate random disturbances; aiming at the problem of high failure rate of wind turbines, based on Weibull distribution, a new model has been established by combining operating time and equipment state to calculate the failure rate; in the analysis of monitoring data, the Higher-Order Moment method and Bayesian method were used to solve the parameters. The opportunity maintenance threshold curve and preventive maintenance threshold curve were obtained by analyzing Time-Based Maintenance and Condition-Based Maintenance. Therefore, the Condition-Based Opportunistic Maintenance strategy was obtained. The effectiveness of the proposed method was finally verified by arithmetic examples.

Optimal Incentives in a Limit Order Book: A SPDE Control Approach

With the fragmentation of electronic markets, exchanges are now competing in order to attract trading activity on their platform. Consequently, they developed several regulatory tools to control liquidity provision/consumption on their liquidity pool. In this paper, we study the problem of an exchange using incentives in order to increase market liquidity. We model the limit order book as the solution of a stochastic partial differential equation (SPDE) as in [R. Cont and M. S. Müller, 2021, A stochastic partial differential equation model for limit order book dynamics, SIAM Journal on Financial Mathematics 12(2), 744–787]. The incentives proposed to the market participants are functions of the time and the distance of their limit order to the mid-price. We formulate the control problem of the exchange who wishes to modify the shape of the order book by increasing the volume at specific limits. Due to the particular nature of the SPDE control problem, we are able to characterize the solution with a classic Feynman–Kac representation theorem. Moreover, when studying the asymptotic behavior of the solution, a specific penalty function enables the exchange to obtain closed-form incentives at each limit of the order book. We study numerically the form of the incentives and their impact on the shape of the order book, and analyze the sensitivity of the incentives to the market parameters.

A study of stochastic epidemic model driven by liouville fractional brownian motion coupled with seasonal air pollution

Air pollution can cause and provoke respiratory diseases. It is an important topic to the public, particularly in developing countries. Since there are many uncertain factors in the environment, stochastic differential equation model is a powerful tool to study the changes of air pollution and the transmission of infectious diseases. The removal of air pollutants as well as the transmission of diseases can be influenced by random perturbations with memories. In this research, we develop a mathematical model in the form of a system of stochastic differential equations driven by fractional Brownian motion of Liouville-type, coupled with seasonal air pollution, to study the dynamics of infectious respiratory disease spread. As a result , by using stochastic calculus techniques, we derive the equation for the level of air pollution.

Stochastic epidemic model for the dynamics of novel coronavirus transmission

<abstract><p>Stochastic differential equation models are important and provide more valuable outputs to examine the dynamics of SARS-CoV-2 virus transmission than traditional models. SARS-CoV-2 virus transmission is a contagious respiratory disease that produces asymptomatically and symptomatically infected individuals who are susceptible to multiple infections. This work was purposed to introduce an epidemiological model to represent the temporal dynamics of SARS-CoV-2 virus transmission through the use of stochastic differential equations. First, we formulated the model and derived the well-posedness to show that the proposed epidemiological problem is biologically and mathematically feasible. We then calculated the stochastic reproductive parameters for the proposed stochastic epidemiological model and analyzed the model extinction and persistence. Using the stochastic reproductive parameters, we derived the condition for disease extinction and persistence. Applying these conditions, we have performed large-scale numerical simulations to visualize the asymptotic analysis of the model and show the effectiveness of the results derived.</p></abstract>

Multi-channel Optimization Generative Model for Stable Ultra-Sparse-View CT Reconstruction.

Score-based generative model (SGM) has risen to prominence in sparse-view CT reconstruction due to its impressive generation capability. The consistency of data is crucial in guiding the reconstruction process in SGM-based reconstruction methods. However, the existing data consistency policy exhibits certain limitations. Firstly, it employs partial data from the reconstructed image of iteration process for image updates, which leads to secondary artifacts with compromising image quality. Moreover, the updates to the SGM and data consistency are considered as distinct stages, disregarding their interdependent relationship. Additionally, the reference image used to compute gradients in the reconstruction process is derived from intermediate result rather than ground truth. Motivated by the fact that a typical SGM yields distinct outcomes with different random noise inputs, we propose a Multi-channel Optimization Generative Model (MOGM) for stable ultra-sparse-view CT reconstruction by integrating a novel data consistency term into the stochastic differential equation model. Notably, the unique aspect of this data consistency component is its exclusive reliance on original data for effectively confining generation outcomes. Furthermore, we pioneer an inference strategy that traces back from the current iteration result to ground truth, enhancing reconstruction stability through foundational theoretical support. We also establish a multi-channel optimization reconstruction framework, where conventional iterative techniques are employed to seek the reconstruction solution. Quantitative and qualitative assessments on 23 views datasets from numerical simulation, clinical cardiac and sheep's lung underscore the superiority of MOGM over alternative methods. Reconstructing from just 10 and 7 views, our method consistently demonstrates exceptional performance.

Heavy Tail and Long-Range Dependence for Skewed Time Series Prediction Based on a Fractional Weibull Process

In this paper, a fractional Weibull process is utilized in a predictive stochastic differential equation model to allow for skewness and heavy-tailed characteristics. To this aim, a fractional Weibull process with non-Gaussian characteristics and a long memory effect is proposed to drive the predictive stochastic differential equation. The difference iterative forecasting model is proposed as its stochastic difference scheme. The consistency, stability, and convergence of the model are analyzed. In the proposed model, variational mode decomposition is utilized as the data preprocessing approach to separate the stationary and non-stationary components. Actual wind speed data and stock price data are employed in two separate case studies.

Stochastic differential equations mixed model for individual growth with the inclusion of genetic characteristics

In early work we have studied a class of stochastic differential equation (SDE) models, for which the Gompertz and the Bertalanffy-Richards stochastic models are particular cases, to describe individual growth in random environments, and applied it to model cattle weight evolution using real data. We have started to work on these type of models considering that the model parameters are fixed, i.e. the same for all the animals. Aiming to incorporate variability among individuals, we consider that the model parameters can be random variables, resulting in SDE mixed models. In additon, here we consider SDE mixed models, allowing the parameters to be random and propose to incorporate each animal's genetic characteristics considering the transformed animal's weight at maturity to be a function of its genetic values. The main objective is to extend the SDE mixed model to the more realistic case where the individual genetic value becomes an important component in the estimated growth curve. For the estimation of the model parameters we have used maximum likelihood estimation theory. Estimates and asymptotic confidence intervals of the parameters are presented. A comparison with SDE non-mixed model and SDE mixed model without the inclusion of genetic characteristics is shown with the conclusion that the incorporation of some genetic characteristics in the model parameters improves the estimation of the animal's growth curve.

Comparative summer thermal performance analysis between open ventilated facade and modular living wall

In recent years, passive solutions for building envelopes have become much more common due to their capacity to decrease the heat flux through the envelope during summer time. Vertical greenery systems (VGS) are emerging as an interesting method of decreasing the thermal demand of cities, and also improving the quality of urban life. Open ventilated facades (OVF) have gained popularity due to their capacity to enhance the thermal resistance of the building envelope. As part of a project carried out in a Paslink cell in Vitoria-Gasteiz, an experimental campaign with full-scale VGS and OVF was carried out during the summer season to assess the thermal performance of a modular living wall (MLW) with respect to an OVF. The objective is to demonstrate that a stochastic differential equations (SDE) model can be used to assess the cooling requirements of an MLW and an OVF. An analysis was carried out to evaluate how different characteristics of the main facade affect performance, such as thermal resistance, solar absorption coefficient and convection coefficient. The results of these experiments show that both MLW (46 %) and OVF (67 %) configurations significantly minimize solar heat loads compared to non-passive bare wall (BW) facades, which are the reference configurations.

A stochastic differential equation model for predator-avoidance fish schooling

This paper presents a mathematical model based on stochastic differential equations (SDEs) to depict the dynamics of a predator–prey system in an aquatic environment characterized by schooling behavior among the prey. The model employs a particle-like approach, incorporating attractive and repulsive forces, akin to phenomena observed in molecular physics, to capture the interactions among the constituent units. Two hunting tactics of the predator, center-attacking and nearest-attacking strategies, are integrated into the model. Numerical simulations of this model unveil four distinct predator-avoidance patterns exhibited by schooling prey: Split and Reunion, Split and Separate into Two Groups, Scattered, and Maintain Formation and Distance. Our results also confirm the effectiveness of large groups of schooling prey in mitigating predation risk, consistent with real-life observations in natural aquatic ecosystems. These findings validate the accuracy and applicability of our model.

Continuous‐time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations

AbstractWe develop a new continuous‐time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an estimate for the gradient of the stationary distribution. The gradient estimate is simultaneously updated using forward propagation of the SDE state derivatives, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of the online forward propagation algorithm for linear SDE models (i.e., the multidimensional Ornstein–Uhlenbeck process) and present its numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations (PDEs), which are then used to analyze the parameter fluctuations in the algorithm. Our algorithm is applicable to a range of mathematical finance applications involving statistical calibration of SDE models and stochastic optimal control for long time horizons where ergodicity of the data and stochastic process is a suitable modeling framework. Numerical examples explore these potential applications, including learning a neural network control for high‐dimensional optimal control of SDEs and training stochastic point process models of limit order book events.

Modelling French and Portuguese Mortality Rates with Stochastic Differential Equation Models: A Comparative Study

In recent times, there has been a notable global phenomenon characterized by a double predicament arising from the concomitant rise in worldwide life expectancy and a significant decrease in birth rates. The emergence of this phenomenon has posed a significant challenge for governments worldwide. It not only poses a threat to the continued viability of state-funded welfare programs, such as social security, but also indicates a potential decline in the future workforce and tax revenue, including contributions to social benefits. Given the anticipated escalation of these issues in the forthcoming decades, it is crucial to comprehensively examine the extension of the human lifespan to evaluate the magnitude of this matter. Recent research has focused on utilizing stochastic differential equations as a helpful means of describing the dynamic nature of mortality rates, in order to tackle this intricate issue. The usage of these models proves to be superior to deterministic ones due to their capacity to incorporate stochastic variations within the environment. This enables individuals to gain a more comprehensive understanding of the inherent uncertainty associated with future forecasts. The most important aims of this study are to fit and compare stochastic differential equation models for mortality (the geometric Brownian motion and the stochastic Gompertz model), conducting separate analyses for each age group and sex, in order to generate forecasts of the central mortality rates in France up until the year 2030. Additionally, this study aims to compare the outcomes obtained from fitting these models to the central mortality rates in Portugal. The results obtained from this work are quite promising since both stochastic differential equation models manage to replicate the decreasing central mortality rate phenomenon and provide plausible forecasts for future time and for both populations. Moreover, we also deduce that the performances of the models differ when analyzing both populations under study due to the significant contrast between the mortality dynamics of the countries under study, a consequence of both external factors (such as the effect of historical events on Portuguese and French mortality) and internal factors (behavioral effect).

Stability and control in a stochastic model of malaria population dynamics

This article proves a stability theorem for the disease-free equilibrium of a stochastic differential equations model of malaria disease dynamics. The theorem is formulated in terms of an invariant which is similar to the basic reproduction number of a related deterministic model. Compared to the deterministic model, stability of the disease-free equilibrium holds more generally for the stochastic model. The optimal control theory is applied to the stochastic model, revealing some important new insights. Theoretical results are illustrated by way of simulations.

Assessment of Stochastic Numerical Schemes for Stochastic Differential Equations with “White Noise” Using Itô’s Integral

Stochastic Differential Equations (SDEs) model physical phenomena dominated by stochastic processes. They represent a method for studying the dynamic evolution of a physical phenomenon, like ordinary or partial differential equations, but with an additional term called “noise” that represents a perturbing factor that cannot be attached to a classical mathematical model. In this paper, we study weak and strong convergence for six numerical schemes applied to a multiplicative noise, an additive, and a system of SDEs. The Efficient Runge–Kutta (ERK) technique, however, comes out as the top performer, displaying the best convergence features in all circumstances, including in the difficult setting of multiplicative noise. This result highlights the importance of researching cutting-edge numerical techniques built especially for stochastic systems and we consider to be of good help to the MATLAB function code for the ERK method.

Strong convergence of an explicit numerical approximation for [formula omitted]-dimensional superlinear SDEs with positive solutions

For a stochastic differential equation (SDE) with a unique positive solution, a rational numerical method is expected to be structure preserving. However, most existing methods are not, as far as we know. Some characteristics of the SDE models including the multi-dimension and super-linearity make it even more challenging. In this work, we fill the gap by proposing an explicit numerical method which is not only structure preserving but also cost effective. The strong convergence framework is set up by moment convergence analysis. We use the Lotka–Volterra system to elaborate our theory, nevertheless, the method works for a wide range of multi-dimensional superlinear SDE models.

Estimation of nonlinear mixed-effects continuous-time models using the continuous-discrete extended Kalman filter.

Many intensive longitudinal measurements are collected at irregularly spaced time intervals, and involve complex, possibly nonlinear and heterogeneous patterns of change. Effective modelling of such change processes requires continuous-time differential equation models that may be nonlinear and include mixed effects in the parameters. One approach of fitting such models is to define random effect variables as additional latent variables in a stochastic differential equation (SDE) model of choice, and use estimation algorithms designed for fitting SDE models, such as the continuous-discrete extended Kalman filter (CDEKF) approach implemented in the dynr R package, to estimate the random effect variables as latent variables. However, this approach's efficacy and identification constraints in handling mixed-effects SDE models have not been investigated. In the current study, we analytically inspect the identification constraints of using the CDEKF approach to fit nonlinear mixed-effects SDE models; extend a published model of emotions to a nonlinear mixed-effects SDE model as an example, and fit it to a set of irregularly spaced ecological momentary assessment data; and evaluate the feasibility of the proposed approach to fit the model through a Monte Carlo simulation study. Results show that the proposed approach produces reasonable parameter and standard error estimates when some identification constraint is met. We address the effects of sample size, process noise variance, and data spacing conditions on estimation results.