Stochastic particle models: mean reversion and burgers dynamics. An application to commodity markets

The aim of this study is to propose a stochastic model for commodity markets linked with the Burgers equation from fluid dynamics. We construct a stochastic particles method for commodity markets, in which particles represent market participants. A discontinuity in the model is included through an interacting kernel equal to the Heaviside function and its link with the Burgers equation is given. The Burgers equation and the connection of this model with stochastic differential equations are also studied. Further, based on the law of large numbers, we prove the convergence, for large N, of a system of stochastic differential equations describing the evolution of the prices of N traders to a deterministic partial differential equation of Burgers type. Numerical experiments highlight the success of the new proposal in modeling some commodity markets, and this is confirmed by the ability of the model to reproduce price spikes when their effects occur in a sufficiently long period of time.

Bayesian analysis of SEIR epidemic models

This thesis concerns the analysis of epidemic models. We adopt the Bayesian paradigm and develop suitable Markov Chain Monte Carlo (MCMC) algorithms. This is done by considering an Ebola outbreak in the Democratic Republic of Congo, former Zaïre, 1995 as a case of SEIR epidemic models. We model the Ebola epidemic deterministically using ODEs and stochastically through SDEs to take into account a possible bias in each compartment. Since the model has unknown parameters, we use different methods to estimate them such as least squares, maximum likelihood and MCMC. The motivation behind choosing MCMC over other existing methods in this thesis is that it has the ability to tackle complicated nonlinear problems with large number of parameters. First, in a deterministic Ebola model, we compute the likelihood function by sum of square of residuals method and estimate parameters using the LSQ and MCMC methods. We sample parameters and then use them to calculate the basic reproduction number and to study the disease-free equilibrium. From the sampled chain from the posterior, we test the convergence diagnostic and confirm the viability of the model. The results show that the Ebola model fits the observed onset data with high precision, and all the unknown model parameters are well identified. Second, we convert the ODE model into a SDE Ebola model. We compute the likelihood function using extended Kalman filter (EKF) and estimate parameters again. The motivation of using the SDE formulation here is to consider the impact of modelling errors. Moreover, the EKF approach allows us to formulate a filtered likelihood for the parameters of such a stochastic model. We use the MCMC procedure to attain the posterior distributions of the parameters of the SDE Ebola model drift and diffusion parts. In this thesis, we analyse two cases: (1) the model error covariance matrix of the dynamic noise is close to zero , i.e. only small stochasticity added into the model. The results are then similar to the ones got from deterministic Ebola model, even if methods of computing the likelihood function are different (2) the model error covariance matrix is different from zero, i.e. a considerable stochasticity is introduced into the Ebola model. This accounts for the situation where we would know that the model is not exact. As a results, we obtain parameter posteriors with larger variances. Consequently, the model predictions then show larger uncertainties, in accordance with the assumption of an incomplete model.

New directions in Stochastic Multicriteria Acceptability Analysis

Decisions taken in modern organizations are often multi-dimensional, involving multiple decision makers and several criteria measured on different scales. Multiple Criteria Decision Making (MCDM) methods are designed to analyze and to give recommendations in this kind of situations. Among the numerous MCDM methods, two large families of methods are the multi-attribute utility theory based methods and the outranking methods. Traditionally both method families require exact values for technical parameters and criteria measurements, as well as for preferences expressed as weights. Often it is hard, if not impossible, to obtain exact values. Stochastic Multicriteria Acceptability Analysis (SMAA) is a family of methods designed to help in this type of situations where exact values are not available. Different variants of SMAA allow handling all types of MCDM problems. They support defining the model through uncertain, imprecise, or completely missing values. The methods are based on simulation that is applied to obtain descriptive indices characterizing the problem. In this thesis we present new advances in the SMAA methodology. We present and analyze algorithms for the SMAA-2 method and its extension to handle ordinal preferences. We then present an application of SMAA-2 to an area where MCDM models have not been applied before: planning elevator groups for high-rise buildings. Following this, we introduce two new methods to the family: SMAA-TRI that extends ELECTRE TRI for sorting problems with uncertain parameter values, and SMAA-III that extends ELECTRE III in a similar way. An efficient software implementing these two methods has been developed in conjunction with this work, and is briefly presented in this thesis. The thesis is closed with a comprehensive survey of SMAA methodology including a definition of a unified framework.

Using time and stochastic models for software reliability forecasting and analysis

Tämä työ luo katsauksen ajallisiin ja stokastisiin ohjelmien luotettavuus malleihin sekä tutkii muutamia malleja käytännössä. Työn teoriaosuus sisältää ohjelmien luotettavuuden kuvauksessa ja arvioinnissa käytetyt keskeiset määritelmät ja metriikan sekä varsinaiset mallien kuvaukset. Työssä esitellään kaksi ohjelmien luotettavuusryhmää. Ensimmäinen ryhmä ovat riskiin perustuvat mallit. Toinen ryhmä käsittää virheiden ”kylvöön” ja merkitsevyyteen perustuvat mallit. Työn empiirinen osa sisältää kokeiden kuvaukset ja tulokset. Kokeet suoritettiin käyttämällä kolmea ensimmäiseen ryhmään kuuluvaa mallia: Jelinski-Moranda mallia, ensimmäistä geometrista mallia sekä yksinkertaista eksponenttimallia. Kokeiden tarkoituksena oli tutkia, kuinka syötetyn datan distribuutio vaikuttaa mallien toimivuuteen sekä kuinka herkkiä mallit ovat syötetyn datan määrän muutoksille. Jelinski-Moranda malli osoittautui herkimmäksi distribuutiolle konvergaatio-ongelmien vuoksi, ensimmäinen geometrinen malli herkimmäksi datan määrän muutoksille.

Stochastic variability of output levels in a pulp mill

Quite often, in the construction of a pulp mill involves establishing the size of tanks which will accommodate the material from the various processes in which case estimating the right tank size a priori would be vital. Hence, simulation of the whole production process would be worthwhile. Therefore, there is need to develop mathematical models that would mimic the behavior of the output from the various production units of the pulp mill to work as simulators. Markov chain models, Autoregressive moving average (ARMA) model, Mean reversion models with ensemble interaction together with Markov regime switching models are proposed for that purpose.

Stochastic approximation methods in stochastic optimization

Stochastic approximation methods for stochastic optimization are considered. Reviewed the main methods of stochastic approximation: stochastic quasi-gradient algorithm, Kiefer-Wolfowitz algorithm and adaptive rules for them, simultaneous perturbation stochastic approximation (SPSA) algorithm. Suggested the model and the solution of the retailer's profit optimization problem and considered an application of the SQG-algorithm for the optimization problems with objective functions given in the form of ordinary differential equation.

Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

Performance measurement model for projects from a strategic management perspective

This research studied the project performance measurement from the perspective of strategic management. The objective was to find a generic model for project performance measurement that emphasizes strategy and decision making. Research followed the guidelines of a constructive research methodology. As a result, the study suggests a model that measures projects with multiple meters during and after projects. Measurement after the project is suggested to be linked to the strategic performance measures of a company. The measurement should be conducted with centralized project portfolio management e.g. using the project management office in the organization. Metrics, after the project, measure the project’s actual benefit realization. During the project, the metrics are universal and they measure the accomplished objectives relation to costs, schedule and internal resource usage. Outcomes of these measures should be forecasted by using qualitative or stochastic methods. Solid theoretical background for the model was found from the literature that covers the subjects of performance measurement, projects and uncertainty. The study states that the model can be implemented in companies. This statement is supported by empirical evidence from a single case study. The gathering of empiric evidence about the actual usefulness of the model in companies is left to be done by the evaluative research in the future.