Stochastic Model Research Articles

Natural gradient hybrid variational inference with application to deep mixed models

Stochastic models with global parameters and latent variables are common, and for which variational inference (VI) is popular. However, existing methods are often either slow or inaccurate in high dimensions. We suggest a fast and accurate VI method for this case that employs a well-defined natural gradient variational optimization that targets the joint posterior of the global parameters and latent variables. It is a hybrid method, where at each step the global parameters are updated using the natural gradient and the latent variables are generated from their conditional posterior. A fast to compute expression for the Tikhonov damped Fisher information matrix is used, along with the re-parameterization trick, to provide a stable natural gradient. We apply the approach to deep mixed models, which are an emerging class of Bayesian neural networks with random output layer coefficients to allow for heterogeneity. A range of simulations show that using the natural gradient is substantially more efficient than using the ordinary gradient, and that the approach is faster and more accurate than two cutting-edge natural gradient VI methods. In a financial application we show that accounting for industry level heterogeneity using the deep mixed model improves the accuracy of asset pricing models. MATLAB code to implement the method and replicate the results can be found at https://github.com/WeibenZhang07/NG-HVI

Optimization based data enrichment using stochastic dynamical system models

We develop a general framework for state estimation in systems modeled with noise-polluted continuous time dynamics and discrete time noisy measurements. Our approach is based on maximum likelihood estimation and employs the calculus of variations to derive optimality conditions for continuous time functions. We make no prior assumptions on the form of the mapping from measurements to state-estimate or on the distributions of the noise terms, making the framework more general than Kalman filtering/smoothing where this mapping is assumed to be linear and the noises Gaussian. The optimal solution that arises is interpreted as a continuous time spline, the structure and temporal dependency of which is determined by the system dynamics and the distributions of the process and measurement noise. Similar to Kalman smoothing, the optimal spline yields increased data accuracy at instants when measurements are taken, in addition to providing continuous time estimates outside the measurement instances. We demonstrate the utility and generality of our approach via illustrative examples that render both linear and nonlinear data filters depending on the particular system. Application of the proposed approach to a Monte Carlo simulation exhibits significant performance improvement in comparison to a common existing method.

A low voltage load balancing distribution method considering street information and V2G technology application

The low-voltage distribution network (LVDN) is the final stage in delivering electric energy from power plants to consumers, and its operational condition greatly impacts many power users. While medium-voltage and high-voltage distribution networks can be managed through intelligent digital systems, load imbalance issues in LVDNs often rely on planners’ experience, leading to significant limitations. With advancements in electric vehicle (EV) charging technology and vehicle-to-grid (V2G) technology, where EVs act as distributed energy storage units, bidirectional energy exchange between vehicles and the grid can now contribute to LVDN operation. This paper proposes a low-voltage load distribution planning method that integrates street information and V2G technology. A two-stage stochastic programming mixed-integer model is developed to tackle load imbalance in LVDNs, with the planning scheme derived from solving this model. A case study is presented to verify the effectiveness of the method, demonstrating that incorporating V2G technology enhances load distribution accuracy and reduces reliance on manual planning, improving network stability and operational efficiency.

ANALYTICALLY PRICING EUROPEAN OPTIONS UNDER A TWO-FACTOR STOCHASTIC INTEREST RATE MODEL WITH A STOCHASTIC LONG-RUN EQUILIBRIUM LEVEL

Abstract We construct a new stochastic interest rate model with two stochastic factors, by introducing a stochastic long-run equilibrium level into the Vasicek interest rate model which follows another Ornstein–Uhlenbeck process. With the interest rate under the Black–Scholes model being assumed to follow the newly proposed model, a closed-form representation of European option prices is successfully presented, when the analytical characteristic function of the underlying log-price under a forward measure is derived. To assess the model performance, a preliminary empirical study is conducted using S&P 500 index and its options, with the Vasicek model and an alternative two-factor Vasicek model taken as benchmarks.

Non-negative least-squares variance and covariance component estimation using the positive-valued function for errors-in-variables models

Abstract Although (co)variance component estimation has been widely applied in the errors-in-variables (EIV) model, the occurrence of negative variance components is still a major issue in the estimated variance components. This problem may be due to the following unfavorable factors: 1) unreasonable selection of initial variance values; 2) low redundancy in the EIV functional model; 3) improper design in the EIV stochastic model, and 4) other data quality problems. Many attempts have been made to prevent the appearance of negative variance components. In this contribution, a novel and efficient non-negative least-squares variance component estimation using the PVF (PVF-NLS-VCE) is introduced, which can simultaneously estimate the unknown (co)variance components in the EIV stochastic model and the parameters in the EIV functional model. Its principle is to implicitly impose a non-negative constraint by replacing the variance component with the positive-valued function (PVF) whose range is the set of positive real numbers. Two numerical examples using real and simulated data are presented. The numerical results of linear regression are identical to those obtained based on least-squares variance component estimation (LS-VCE) with positive initial values of variance components. The numerical results of two-dimensional affine transformation are shown to prevent negative variance components and precede those obtained by LS-VCE with a negative initial value of variance component. Both numerical examples verify the effectiveness of the PVF-NLS-VCE method whether the initial values of variance components are positive or negative. The proposed PVF-NLS-VCE is a simple, convenient and flexible method to achieve the non-negative estimates of variance components, which can reduce sensitivity to initial value selection and automatically guarantee a non-negative definite covariance matrix.

Quantifying future carbon emissions uncertainties under stochastic modeling and Monte Carlo simulation: Insights for environmental policy consideration for the Belt and Road Initiative Region

This study critically examines future carbon (CO2) emissions in the Belt & Road Initiative (BRI) region, considering factors such as energy consumption, economic growth, population growth, and population density. The objective of this study is to identify critical areas of higher emissions, which require policy intervention capable of strengthening sustainability in the BRI compact. A combined approach of stochastic modeling and Monte Carlo simulations was employed, utilizing panel data from 45 countries in the BRI region from 1990 to 2021. Results confirm that emissions are higher in all scenarios in direct proportion to electric power consumption, population growth, and Gross Domestic Product (GDP) growth. In scenarios with high emissions, a continuous and significant upward trend in CO2 emissions was observe. The medium emissions scenario exhibited a more moderated rise in emissions, suggesting a balance between economic development and environmental considerations. Critical areas for future environmental policy-making resides in electric power consumption, population growth, and GDP growth. The study strongly recommends for a shift from the current focus on road and railway infrastructure to renewable energy infrastructure, green innovations and efficient technology transfer to member countries. Without this, the BRI region is likely to face increased emissions, posing significant challenges to future sustainable development and global environmental sustainability.

Nonstationary A/B Tests: Optimal Variance Reduction, Bias Correction, and Valid Inference

We develop an analytical framework to appropriately model and adequately analyze A/B tests in presence of nonparametric nonstationarities in the targeted business metrics. A/B tests, also known as online randomized controlled experiments, have been used at scale by data-driven enterprises to guide decisions and test innovative ideas to improve core business metrics. Meanwhile, nonstationarities, such as the time-of-day effect and the day-of-week effect, can often arise nonparametrically in key business metrics involving purchases, revenue, conversions, customer experiences, and so on. First, we develop a generic nonparametric stochastic model to capture nonstationarities in A/B test experiments, where each sample represents a visit or action associated with a time label. We build a practically relevant limiting regime to facilitate analyzing large-sample estimator performances under nonparametric nonstationarities. Second, we show that ignoring or inadequately addressing nonstationarities can cause standard A/B test estimators to have suboptimal variance and nonvanishing bias, therefore leading to loss of statistical efficiency and accuracy. We provide a new estimator that views time as a continuous strata and performs poststratification with a data-dependent number of stratification levels. Without making parametric assumptions, we prove a central limit theorem for the proposed estimator and show that the estimator attains the best achievable asymptotic variance and is asymptotically unbiased. Third, we propose a time-grouped randomization that is designed to balance treatment and control assignments at granular time scales. We show that when the time-grouped randomization is integrated to standard experimental designs to generate experiment data, simple A/B test estimators can achieve asymptotically optimal variance. A brief account of numerical experiments are conducted to illustrate the analysis. This paper was accepted by Baris Ata, stochastic models and simulation. Supplemental Material: The online appendices and data files are available at https://doi.org/10.1287/mnsc.2022.01205 .

Long-time behavior and quasi-density function for a stochastic epidemic model with relapse and reinfection

A stochastic susceptible–infected–recovered–infected (SIRI) epidemic model with relapse and reinfection is established in this paper. First, we prove that the solution to the epidemic model is unique and globally positive. Next, we determine some sufficient conditions for the extinction of the disease when [Formula: see text] and for the persistence in mean in the case of [Formula: see text]. Furthermore, we prove the existence of at least one ergodic stationary distribution of the stochastic model if [Formula: see text]. Additionally, by solving the corresponding three-dimensional Fokker–Planck equation, it is theoretically shown that the epidemic model has a log-normal probability density function when [Formula: see text], then we obtain the exact expression of density function of the stationary distribution. Finally, we give some numerical simulations to support our theoretical results.

Estimation of AMOC transition probabilities using a machine learning based rare-event algorithm

Abstract The Atlantic Meridional Overturning Circulation (AMOC) is an important component of the global climate, known to be a tipping element, as it could collapse under global warming. The main objective of this study is to compute the probability that the AMOC collapses within a specified time window, using a rare-event algorithm called Trajectory-Adaptive Multilevel Splitting (TAMS). However, the efficiency and accuracy of TAMS depend on the choice of the score function. Although the definition of the optimal score function, called “committor function” is known, it is impossible in general to compute it a priori. Here, we combine TAMS with a Next-Generation Reservoir Computing technique that estimates the committor function from the data generated by the rare-event algorithm. We test this technique in a stochastic box model of the AMOC for which two types of transition exist, the so-called F(ast)-transitions and S(low)-transitions. Results for the F-transtions compare favorably with those in the literature where a physically-informed score function was used. We show that coupling a rare-event algorithm with machine learning allows for a correct estimation of transition probabilities, transition times, and even transition paths for a wide range of model parameters. We then extend these results to the more difficult problem of S-transitions in the same model. In both cases of F-transitions and S-transitions, we also show how the Next-Generation Reservoir Computing technique can be interpreted to retrieve an analytical estimate of the committor function.

On the spurious effect of intergroup friendship on outgroup attitudes in schools: The role of social influence and the positive impact of exposure to outgroup peers.

Promoting intergroup friendships in schools is regarded as a powerful strategy for improving outgroup attitudes. However, stochastic actor-oriented modelling (SAOM) studies have often revealed no association between intergroup friendship and outgroup attitudes. I investigated whether SAOM studies reported no effect of intergroup friendship on outgroup attitudes, what was responsible for this surprising finding and whether exposure to outgroup peers was positively related to outgroup attitudes. The meta-analysis of SAOM studies confirms no association between intergroup friendship and outgroup attitudes. Examining longitudinal social network data of 2700 German students, I found that a positive link between intergroup friendship and outgroup attitudes appeared only when I did not control for social influence. This indicates that intergroup friendship did not automatically improve outgroup attitudes. Instead, the development of outgroup attitudes among students depended on social influence and, thus, the quality of outgroup attitudes among their outgroup friends. Exposure to outgroup peers was, however, positively associated with outgroup attitudes. These findings reframe intergroup contact theory by suggesting that intergroup friendship is not essential for improving outgroup attitudes. Rather, social influence and exposure to outgroup peers could be key factors shaping outgroup attitudes.

Optimizing Dry Port Locations: A Comparative Analysis of Deterministic, Stochastic, and Robust Models

The increasing international maritime transportation of freight has caused a huge operation in important ports, which are often established near cities and therefore do not have enough room for expanding the operations. This causes long delays and poor productivity in ports. The construction of dry ports has been proposed as a suitable solution to mitigate the problems. A dry port is an inland intermodal terminal directly connected to a seaport by road or rail, where importers and exporters can clear their cargo through customs and handle other shipping formalities. Selecting the right place for the establishment of dry ports has a significant impact on the time and costs of transportation in the network, as well as environmental consequences. In this research, to determine the optimal location of dry ports among the potential locations, three mathematical models as deterministic, stochastic, and robust with three objectives of minimizing the maximum transportation time in the entire network, environmental pollutants, and transportation costs are presented, which simultaneously specify the required number of dry ports in the network and the optimal place to build them. The presented models have been implemented for data related to 30 million bills of lading in a 10-year period in Iran with four major ports and 400 cities. Both deterministic and non-deterministic approaches reveal a preference not to build a dry port in the first optimal solution. However, in the following desirable answers, different answers have been suggested. According to the sensitivity analysis performed by making changes in the fuel price situation or the existing rail network, the model introduces several potential points for the construction of dry ports. The results of this model show that, with the construction of dry ports, despite the increase in overall costs of the network because of the construction of dry ports, operation, and storage in them, the amount of greenhouse gas emissions will decrease significantly.

A volume‐averaging and stochastic turbulence modeling framework for homogeneous roughness‐induced drag in turbulent flows

AbstractThe goal of this work is the formulation of a model for the parameterization of homogeneous roughness‐induced drag in turbulent flow simulations. We characterize rough surfaces using their surface‐area moments of roughness‐peaks. Additionally, we consider a probabilistic self‐similar power law distribution for the roughness heights, in such a way that an assumed set of discrete roughness elements can behave as a probabilistic fractal set. Upon these assumptions, the surface characterization is complete and yields wall‐normal profiles of porosity, average roughness element diameter, and equivalent pore diameter of the roughness affected porous flow. The profiles are supplied to a (pore‐based) Reynolds number and porosity‐based drag parameterization for staggered cylinder arrays available in the literature, as well as a one‐dimensional map‐based turbulence model, the one‐dimensional turbulence (ODT) model. The purpose of the latter is a way to provide the missing effects of the equivalent dispersion tensor in the porous flow close to the rough surface, as well as that of the nonlinear turbulent transport away from the wall. Simulations are then carried out in order to evaluate the effects of two different rough surfaces on a turbulent channel flow, comparing results with existing direct numerical simulation (DNS) data. Overall, reasonable agreement is obtained, which suggests that, by using the proposed model, it is possible to model the effects of rough surfaces on wall‐bounded turbulent flows, without the explicit need of the full topological representation of the surface on the numerical domain. To that extent, one clear flow control application for the suggested approach would be that of drag‐based surface optimization.

Requirements for the Development and Operation of a Freeze-Up Ice-Jam Flood Forecasting System

This article provides a comprehensive overview of ice-jam flood forecasting methodologies applicable to rivers during freezing. It emphasizes the importance of understanding river ice processes and fluvial geomorphology for developing a freeze-up ice-jam flood forecasting system. The article showcases a stochastic modelling approach, which involves simulating a deterministic river ice model multiple times with varying parameters and boundary conditions. This approach has been applied to the Exploits River at Badger in Newfoundland, Canada, a river that has experienced several freeze-up ice-jam floods. The forecasting involves two approaches: predicting the extent of the ice cover during river freezing and using an ensemble method to determine backwater flood level elevations. Other examples of current ice-jam flood forecasting systems for the Kokemäenjoki River (Pori, Finland), Saint John River (Edmundston, NB, Canada), and Churchill River (Mud Lake, NL, Canada) that are operational are also presented. The text provides a detailed explanation of the processes involved in river freeze-up and ice-jam formation, as well as the methodologies used for freeze-up ice-jam flood forecasting. Ice-jam flood forecasting systems used for freeze-up were compared to those employed for spring breakup. Spring breakup and freeze-up ice-jam flood forecasting systems differ in their driving factors and methodologies. Spring breakup, driven by snowmelt runoff, typically relies on deterministic and probabilistic approaches to predict peak flows. Freeze-up, driven by cold temperatures, focuses on the complex interactions between atmospheric conditions, river flow, and ice dynamics. Both systems require air temperature forecasts, but snowpack data are more crucial for spring breakup forecasting. To account for uncertainty, both approaches may employ ensemble forecasting techniques, generating multiple forecasts using slightly different initial conditions or model parameters. The objective of this review is to provide an overview of the current state-of-the-art in ice-jam flood forecasting systems and to identify gaps and areas for improvement in existing ice-jam flood forecasting approaches, with a focus on enhancing their accuracy, reliability, and decision-making potential. In conclusion, an effective freeze-up ice-jam flood forecasting system requires real-time data collection and analysis, historical data analysis, ice jam modeling, user interface design, alert systems, and integration with other relevant systems. This combination allows operators to better understand ice jam behavior and make informed decisions about potential risks or mitigation measures to protect people and property along rivers. The key findings of this review are as follows: (i) Ice-jam flood forecasting systems are often based on simple, empirical models that rely heavily on historical data and limited real-time monitoring information. (ii) There is a need for more sophisticated modeling techniques that can better capture the complex interactions between ice cover, water levels, and channel geometry. (iii) Combining data from multiple sources such as satellite imagery, ground-based sensors, numerical models, and machine learning algorithms can significantly improve the accuracy and reliability of ice-jam flood forecasts. (iv) Effective decision-support tools are crucial for integrating ice-jam flood forecasts into emergency response and mitigation strategies.

Comparison of Deterministic, Stochastic, and Energy-Data-Driven Occupancy Models for Building Stock Energy Simulation

Accurate modelling of occupancy patterns is critical for reliable estimation of building stock energy demand, which is a key input for the design of district energy systems. Aiming to investigate the suitability of different occupancy-modelling approaches for the design of district energy systems, the present study examines a set of standard-based schedules (from the UK National Calculation Methodology), a widely used stochastic occupancy model, and a novel energy-data-driven occupancy model. To this end, a dynamic energy model of a higher education office building developed within a stock model of London’s Bloomsbury district serves as a testbed to implement the occupancy models, explore their implications for the estimation of annual and peak heating and cooling demand, and extrapolate the findings to the computationally demanding building stock stimulations. Furthermore, the simulations were conducted in two years before and after the COVID-19 pandemic to examine the implications of hybrid working patterns after the pandemic. From the results, the energy-data-driven model demonstrated superior performance in annual heating demand estimations, with errors of ±2.5% compared to 14% and 7% for the standard-based and stochastic models. For peak heating demand, the models performed rather similarly, with the data-driven model showing 28% error compared to 29.5% for both the standard-based and stochastic models in 2019. In cooling demand estimations, the data-driven model yielded noticeably higher annual cooling demand and lower peak cooling demand estimations as compared with the standard-based and stochastic occupancy models. Given the adopted building-modelling approach, these findings can be extended to district-level investigations and inform the decision on the choice of occupancy models for building stock energy simulation.

Possibility of the optimum monitoring and evaluation (M&E) production frontier for risk-informed health governance in disaster-prone districts of West Bengal, India

An efficient M&E system in public healthcare is crucial for achieving universal health coverage in low- and middle-income countries, especially when the need for service remains unmet due to the exposure of the population to disaster risks and uncertainties. Current research has conducted exploratory and predictive analyses to estimate the determinants of sustainable M&E solutions for ensuring uninterrupted access during and after disasters. The aim was to estimate the efficiency of reaching a higher M&E production frontier via the Cobb‒Douglas model and stochastic frontier model as the basic theoretical and empirical frameworks. The research followed a deductive approach and used a stratified purposive sampling method to collect data from different layers of health and disaster governance in a flood-prone rural setting in the Malda, South 24 Parganas and Purulia districts in West Bengal, India. The present mixed-method study revealed multiple challenges in healthcare seeking during disasters and how a well-structured M&E system can increase system readiness to combat these challenges. The stochastic frontier model estimated the highest M&E frontier producing the most attainable M&E effectiveness through horizontal convergence between departments, enhanced coordination, the availability of frontline health workers at health centers, the adoption of learned innovation and the outsourcing of the evaluation component to external evaluators to improve M&E process quality. Although the study has several limitations, it shows the potential to increase technical and allocative efficiency through building skills in innovative techniques and applying them in process implementation. In the future, research on strategy improvement followed by real-world evidence-based policy advocacy is needed to increase the impact of M&E on access to healthcare services.

A Study of the Main Mathematical Models Used in Mobility, Storage, Pickup and Delivery in Urban Logistics: A Systematic Review

This systematic review investigates the main mathematical models applied in urban logistics, focusing on routing, location and transshipment problems. The study addresses the growing demand for efficient and sustainable logistics solutions driven by population growth and the expansion of e-commerce. A thorough analysis of 57 scientific articles was carried out, covering deterministic and stochastic methodologies, as well as heuristic and exact solutions. This review revealed that heuristic methods are predominant due to their computational efficiency. Combining exact methods with heuristics has proven effective for complex logistics scenarios, increasing accuracy and efficiency. Synchronization and intermediate stops have also emerged as critical factors in optimizing logistics operations. This review highlights the diversity of methodologies and the need for sustainable and efficient models. The integration of stochastic simulations remains limited, representing a research gap where stochastic models have been shown to provide more robust solutions in addressing the uncertainties inherent in logistics operations. This integration can increase the robustness and applicability of logistics solutions in urban environments. By highlighting the strengths and limitations of current approaches, it paves the way for future research to develop more robust and adaptable solutions to urban logistics challenges, emphasizing interdisciplinary collaboration and the use of real-world data.

Stochastic Block Smooth Graphon Model

In this article, we propose combining the stochastic blockmodel and the smooth graphon model, two of the most prominent modeling approaches in statistical network analysis. Stochastic blockmodels are generally used for clustering networks into groups of actors with homogeneous connectivity behavior. Smooth graphon models, on the other hand, assume that all nodes can be arranged on a one-dimensional scale such that closeness implies a similar behavior in connectivity. Both frameworks belong to the class of node-specific latent variable models, entailing a natural relationship. While these two modeling concepts have developed independently, this article proposes their generalization toward stochastic block smooth graphon models. This combined approach enables to exploit the advantages of both worlds. Employing concepts of the EM-type algorithm allows to develop an appropriate estimation routine, where MCMC techniques are used to accomplish the E-step. Simulations and real-world applications support the practicability of our novel method and demonstrate its advantages. The article is accompanied by supplementary material covering details about computation and implementation.

Cost effectiveness analysis comparing varying booster intervals of vaccination policies to address COVID-19 situation in Thailand, 2023.

The COVID-19 booster immunization policy is cost-effective, but evidence on additional booster doses and appropriate strategies is scarce. This research compared the cost-effectiveness of annual, twice-a-year, and biennial booster dose policies. We performed stochastic modeling using compartmental susceptible-exposed-infectious-recovered models and a system dynamic model. We evaluated four policy scenarios: (1) hypothetical no-booster immunization policy; (2) twice-a-year vaccination policy; (3) annual vaccination policy; and (4) biennial vaccination policy. In addition, we conducted a one-way sensitivity analysis by adjusting R0 from 1.8 to 3.0 in all scenarios (epidemic stage) and by decreasing the vaccination cost by 50% at the end of the first year to reflect the current policy direction to enhance domestic vaccine production. Compared to non-booster policies, all three booster strategies reduced the number of cases, hospital admissions, and severe infections remarkably. Without a booster, total cases would reach 16,220,615 (95% confidence interval [CI] 6,726,550-29,661,112) by day 1,460, whereas, with a twice-a-year booster, the total cases would reach 597,901 (95% CI 526,230-694,458) in the same period. Even though the no booster scenario exhibited the lowest cost by approximately the first 500 days, by day 1,460 the biennial booster scenario demonstrated the lowest cost at 72.0 billion baht (95% CI 68.6-79.4 billion). The most cost-saving policy was the biennial booster scenario. The annual booster scenario also stood as a cost-effective option for most outcomes. In the epidemic stage and in an assumption where the vaccination costs dropped, all booster policies became more cost-effective or cost-saving compared with the main assumption. This study underscores the significance of the COVID-19 vaccine booster policy. Implementing policies should take into consideration cost-effectiveness, feasibility, and public communication.

Robustness of Hilbert space-valued stochastic volatility models

In this paper, we show that Hilbert space-valued stochastic models are robust with respect to perturbations, due to measurement or approximation errors, in the underlying volatility process. Within the class of stochastic-volatility-modulated Ornstein–Uhlenbeck processes, we quantify the error induced by the volatility in terms of perturbations in the parameters of the volatility process. We moreover study the robustness of the volatility process itself with respect to finite-dimensional approximations of the driving compound Poisson process and semigroup generator, respectively, when considering operator-valued Barndorff-Nielsen and Shephard stochastic volatility models. We also give results on square root approximations. In all cases, we provide explicit bounds for the induced error in terms of the approximation of the underlying parameter. We discuss some applications to robustness of prices of options on forwards and volatility.

Stationary covariance regime for affine stochastic covariance models in Hilbert spaces

This paper introduces stochastic covariance models in Hilbert spaces with stationary affine instantaneous covariance processes. We explore the applications of these models in the context of forward curve dynamics within fixed-income and commodity markets. The affine instantaneous covariance process is defined on positive self-adjoint Hilbert–Schmidt operators, and we prove the existence of a unique limit distribution for subcritical affine processes, provide convergence rates of the transition kernels in the Wasserstein distance of order p∈[1,2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p \\in [1,2]$\\end{document}, and give explicit formulas for the first two moments of the limit distribution. Our results allow us to introduce affine stochastic covariance models in the stationary covariance regime and to investigate the behaviour of the implied forward volatility for large forward dates in commodity forward markets.