A Hull and White formula for a general stochastic volatility jump-diffusion model with applications to the study of the short-time behavior of the implied volatility

In this paper, generalizing results in Alòs, León and Vives (2007b), we see that the dependence of jumps in the volatility under a jump-diffusion stochastic volatility model, has no effect on the short-time behaviour of the at-the-money implied volatility skew, although the corresponding Hull and White formula depends on the jumps. Towards this end, we use Malliavin calculus techniques for Lévy processes based on Løkka (2004), Petrou (2006), and Solé, Utzet and Vives (2007).

Variance reduction methods for simulation of densities on Wiener space

We develop a general error analysis framework for the Monte Carlo simulationof densities for functionals in Wiener space. We also study variancereduction methods with the help of Malliavin derivatives. For this, wegive some general heuristic principles which are applied to diffusionprocesses. A comparison with kernel density estimates is made.

On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility

In this paper we use Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be neither a difussion, nor a Markov process as the examples in section 7 show. This expression depends on the derivative of the volatility in the sense of Malliavin calculus.

On the problematic of reserving and stochastic loss models

Abstract Traditionally, the common reserving methods used by the non-life actuaries are based on the assumption that future claims are going to behave in the same way as they did in the past. There are two main sources of variability in the processus of development of the claims: the variability of the speed with which the claims are settled and the variability between the severity of the claims from different accident years. High changes in these processes will generate distortions in the estimation of the claims reserves. The main objective of this thesis is to provide an indicator which firstly identifies and quantifies these two influences and secondly to determine which model is adequate for a specific situation. Two stochastic models were analysed and the predictive distributions of the future claims were obtained. The main advantage of the stochastic models is that they provide measures of variability of the reserves estimates. The first model (PDM) combines one conjugate family Dirichlet - Multinomial with the Poisson distribution. The second model (NBDM) improves the first one by combining two conjugate families Poisson -Gamma (for distribution of the ultimate amounts) and Dirichlet Multinomial (for distribution of the incremental claims payments). It was found that the second model allows to find the speed variability in the reporting process and development of the claims severity as function of two above mentioned distributions' parameters. These are the shape parameter of the Gamma distribution and the Dirichlet parameter. Depending on the relation between them we can decide on the adequacy of the claims reserve estimation method. The parameters have been estimated by the Methods of Moments and Maximum Likelihood. The results were tested using chosen simulation data and then using real data originating from the three lines of business: Property/Casualty, General Liability, and Accident Insurance. These data include different developments and specificities. The outcome of the thesis shows that when the Dirichlet parameter is greater than the shape parameter of the Gamma, resulting in a model with positive correlation between the past and future claims payments, suggests the Chain-Ladder method as appropriate for the claims reserve estimation. In terms of claims reserves, if the cumulated payments are high the positive correlation will imply high expectations for the future payments resulting in high claims reserves estimates. The negative correlation appears when the Dirichlet parameter is lower than the shape parameter of the Gamma, meaning low expected future payments for the same high observed cumulated payments. This corresponds to the situation when claims are reported rapidly and fewer claims remain expected subsequently. The extreme case appears in the situation when all claims are reported at the same time leading to expectations for the future payments of zero or equal to the aggregated amount of the ultimate paid claims. For this latter case, the Chain-Ladder is not recommended.

Data-driven clustering : new methods and applications

Abstract : This work is concerned with the development and application of novel unsupervised learning methods, having in mind two target applications: the analysis of forensic case data and the classification of remote sensing images. First, a method based on a symbolic optimization of the inter-sample distance measure is proposed to improve the flexibility of spectral clustering algorithms, and applied to the problem of forensic case data. This distance is optimized using a loss function related to the preservation of neighborhood structure between the input space and the space of principal components, and solutions are found using genetic programming. Results are compared to a variety of state-of--the-art clustering algorithms. Subsequently, a new large-scale clustering method based on a joint optimization of feature extraction and classification is proposed and applied to various databases, including two hyperspectral remote sensing images. The algorithm makes uses of a functional model (e.g., a neural network) for clustering which is trained by stochastic gradient descent. Results indicate that such a technique can easily scale to huge databases, can avoid the so-called out-of-sample problem, and can compete with or even outperform existing clustering algorithms on both artificial data and real remote sensing images. This is verified on small databases as well as very large problems. Résumé : Ce travail de recherche porte sur le développement et l'application de méthodes d'apprentissage dites non supervisées. Les applications visées par ces méthodes sont l'analyse de données forensiques et la classification d'images hyperspectrales en télédétection. Dans un premier temps, une méthodologie de classification non supervisée fondée sur l'optimisation symbolique d'une mesure de distance inter-échantillons est proposée. Cette mesure est obtenue en optimisant une fonction de coût reliée à la préservation de la structure de voisinage d'un point entre l'espace des variables initiales et l'espace des composantes principales. Cette méthode est appliquée à l'analyse de données forensiques et comparée à un éventail de méthodes déjà existantes. En second lieu, une méthode fondée sur une optimisation conjointe des tâches de sélection de variables et de classification est implémentée dans un réseau de neurones et appliquée à diverses bases de données, dont deux images hyperspectrales. Le réseau de neurones est entraîné à l'aide d'un algorithme de gradient stochastique, ce qui rend cette technique applicable à des images de très haute résolution. Les résultats de l'application de cette dernière montrent que l'utilisation d'une telle technique permet de classifier de très grandes bases de données sans difficulté et donne des résultats avantageusement comparables aux méthodes existantes.

BadiRate: Estimating Family Turnover Rates by Likelihood-Based Methods

Motivation: The comparative analysis of gene gain and loss rates is critical for understanding the role of natural selection and adaptation in shaping gene family sizes. Studying complete genome data from closely related species allows accurate estimation of gene family turnover rates. Current methods and software tools, however, are not well designed for dealing with certain kinds of functional elements, such as microRNAs or transcription factor binding sites. Results: Here, we describe BadiRate, a new software tool to estimate family turnover rates, as well as the number of elements in internal phylogenetic nodes, by likelihood-based methods and parsimony. It implements two stochastic population models, which provide the appropriate statistical framework for testing hypothesis, such as lineage-specific gene family expansions or contractions. We have assessed the accuracy of BadiRate by computer simulations, and have also illustrated its functionality by analyzing a representative empirical dataset.

BadiRate: estimating family turnover rates by likelihood-based methods

Motivation: The comparative analysis of gene gain and loss rates is critical for understanding the role of natural selection and adaptation in shaping gene family sizes. Studying complete genome data from closely related species allows accurate estimation of gene family turnover rates. Current methods and software tools, however, are not well designed for dealing with certain kinds of functional elements, such as microRNAs or transcription factor binding sites. Results: Here, we describe BadiRate, a new software tool to estimate family turnover rates, as well as the number of elements in internal phylogenetic nodes, by likelihood-based methods and parsimony. It implements two stochastic population models, which provide the appropriate statistical framework for testing hypothesis, such as lineage-specific gene family expansions or contractions. We have assessed the accuracy of BadiRate by computer simulations, and have also illustrated its functionality by analyzing a representative empirical dataset.

Statistical methods for river runoff prediction

Methods used to analyze one type of nonstationary stochastic processes?the periodically correlated process?are considered. Two methods of one-step-forward prediction of periodically correlated time series are examined. One-step-forward predictions made in accordance with an autoregression model and a model of an artificial neural network with one latent neuron layer and with an adaptation mechanism of network parameters in a moving time window were compared in terms of efficiency. The comparison showed that, in the case of prediction for one time step for time series of mean monthly water discharge, the simpler autoregression model is more efficient.

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.

Monotonic continuous-time random walks with drift and stochastic reset events

In this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes. We focus our attention on monotonic continuous-time random walks with a constant drift: The process increases between the reset events, either by the effect of the random jumps, or by the action of the deterministic drift. As a result of all these combined factors interesting properties emerge, like the existence (for any drift strength) of a stationary transition probability density function, or the faculty of the model to reproduce power-law-like behavior. General formulas for two extreme statistics, the survival probability, and the mean exit time, are also derived. To corroborate in an independent way the results of the paper, Monte Carlo methods were used. These numerical estimations are in full agreement with the analytical predictions.

Bayesian methods for estimation, optimization and experimental design

Mathematical models often contain parameters that need to be calibrated from measured data. The emergence of efficient Markov Chain Monte Carlo (MCMC) methods has made the Bayesian approach a standard tool in quantifying the uncertainty in the parameters. With MCMC, the parameter estimation problem can be solved in a fully statistical manner, and the whole distribution of the parameters can be explored, instead of obtaining point estimates and using, e.g., Gaussian approximations. In this thesis, MCMC methods are applied to parameter estimation problems in chemical reaction engineering, population ecology, and climate modeling. Motivated by the climate model experiments, the methods are developed further to make them more suitable for problems where the model is computationally intensive. After the parameters are estimated, one can start to use the model for various tasks. Two such tasks are studied in this thesis: optimal design of experiments, where the task is to design the next measurements so that the parameter uncertainty is minimized, and model-based optimization, where a model-based quantity, such as the product yield in a chemical reaction model, is optimized. In this thesis, novel ways to perform these tasks are developed, based on the output of MCMC parameter estimation. A separate topic is dynamical state estimation, where the task is to estimate the dynamically changing model state, instead of static parameters. For example, in numerical weather prediction, an estimate of the state of the atmosphere must constantly be updated based on the recently obtained measurements. In this thesis, a novel hybrid state estimation method is developed, which combines elements from deterministic and random sampling methods.

MCMC Analysis for Optimization of Stochastic Models

In any decision making under uncertainties, the goal is mostly to minimize the expected cost. The minimization of cost under uncertainties is usually done by optimization. For simple models, the optimization can easily be done using deterministic methods.However, many models practically contain some complex and varying parameters that can not easily be taken into account using usual deterministic methods of optimization. Thus, it is very important to look for other methods that can be used to get insight into such models. MCMC method is one of the practical methods that can be used for optimization of stochastic models under uncertainty. This method is based on simulation that provides a general methodology which can be applied in nonlinear and non-Gaussian state models. MCMC method is very important for practical applications because it is a uni ed estimation procedure which simultaneously estimates both parameters and state variables. MCMC computes the distribution of the state variables and parameters of the given data measurements. MCMC method is faster in terms of computing time when compared to other optimization methods. This thesis discusses the use of Markov chain Monte Carlo (MCMC) methods for optimization of Stochastic models under uncertainties .The thesis begins with a short discussion about Bayesian Inference, MCMC and Stochastic optimization methods. Then an example is given of how MCMC can be applied for maximizing production at a minimum cost in a chemical reaction process. It is observed that this method performs better in optimizing the given cost function with a very high certainty.

Computational methods for tactical simulations

Tämä taktiikan tutkimus keskittyy tietokoneavusteisen simuloinnin laskennallisiin menetelmiin, joita voidaan käyttää taktisen tason sotapeleissä. Työn tärkeimmät tuotokset ovat laskennalliset mallit todennäköisyyspohjaisen analyysin mahdollistaviin taktisen tason taistelusimulaattoreihin, joita voidaan käyttää vertailevaan analyysiin joukkue-prikaatitason tarkastelutilanteissa. Laskentamallit keskittyvät vaikuttamiseen. Mallit liittyvät vahingoittavan osuman todennäköisyyteen, jonka perusteella vaikutus joukossa on mallinnettu tilakoneina ja Markovin ketjuina. Edelleen näiden tulokset siirretään tapahtumapuuanalyysiin operaation onnistumisen todennäköisyyden osalta. Pienimmän laskentayksikön mallinnustaso on joukkue- tai ryhmätasolla, jotta laskenta-aika prikaatitason sotapelitarkasteluissa pysyisi riittävän lyhyenä samalla, kun tulokset ovat riittävän tarkkoja suomalaiseen maastoon. Joukkueiden mies- ja asejärjestelmävahvuudet ovat jakaumamuodossa, eivätkä yksittäisiä lukuja. Simuloinnin integroinnissa voidaan käyttää asejärjestelmäkohtaisia predictor corrector –parametreja, mikä mahdollistaa aika-askelta lyhytaikaisempien taistelukentän ilmiöiden mallintamisen. Asemallien pohjana ovat aiemmat tutkimukset ja kenttäkokeet, joista osa kuuluu tähän väitöstutkimukseen. Laskentamallien ohjelmoitavuus ja käytettävyys osana simulointityökalua on osoitettu tekijän johtaman tutkijaryhmän ohjelmoiman ”Sandis”- taistelusimulointiohjelmiston avulla, jota on kehitetty ja käytetty Puolustusvoimien Teknillisessä Tutkimuslaitoksessa. Sandikseen on ohjelmoitu karttakäyttöliittymä ja taistelun kulkua simuloivia laskennallisia malleja. Käyttäjä tai käyttäjäryhmä tekee taktiset päätökset ja syöttää nämä karttakäyttöliittymän avulla simulointiin, jonka tuloksena saadaan kunkin joukkuetason peliyksikön tappioiden jakauma, keskimääräisten tappioiden osalta kunkin asejärjestelmän aiheuttamat tappiot kuhunkin maaliin, ammuskulutus ja radioyhteydet ja niiden tila sekä haavoittuneiden evakuointi-tilanne joukkuetasolta evakuointisairaalaan asti. Tutkimuksen keskeisiä tuloksia (kontribuutio) ovat 1) uusi prikaatitason sotapelitilanteiden laskentamalli, jonka pienin yksikkö on joukkue tai ryhmä; 2) joukon murtumispisteen määritys tappioiden ja haavoittuneiden evakuointiin sitoutuvien taistelijoiden avulla; 3) todennäköisyyspohjaisen riskianalyysin käyttömahdollisuus vertailevassa tutkimuksessa sekä 4) kokeellisesti testatut tulen vaikutusmallit ja 5) toimivat integrointiratkaisut. Työ rajataan maavoimien taistelun joukkuetason todennäköisyysjakaumat luovaan laskentamalliin, kenttälääkinnän malliin ja epäsuoran tulen malliin integrointimenetelmineen sekä niiden antamien tulosten sovellettavuuteen. Ilmasta ja mereltä maahan -asevaikutusta voidaan tarkastella, mutta ei ilma- ja meritaistelua. Menetelmiä soveltavan Sandis -ohjelmiston malleja, käyttötapaa ja ohjelmistotekniikkaa kehitetään edelleen. Merkittäviä jatkotutkimuskohteita mallinnukseen osalta ovat muun muassa kaupunkitaistelu, vaunujen kaksintaistelu ja maaston vaikutus tykistön tuleen sekä materiaalikulutuksen arviointi.

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.

Application and development of numerical methods for the modelling of innovative gas cooled fission reactors

Innovative gas cooled reactors, such as the pebble bed reactor (PBR) and the gas cooled fast reactor (GFR) offer higher efficiency and new application areas for nuclear energy. Numerical methods were applied and developed to analyse the specific features of these reactor types with fully three dimensional calculation models. In the first part of this thesis, discrete element method (DEM) was used for a physically realistic modelling of the packing of fuel pebbles in PBR geometries and methods were developed for utilising the DEM results in subsequent reactor physics and thermal-hydraulics calculations. In the second part, the flow and heat transfer for a single gas cooled fuel rod of a GFR were investigated with computational fluid dynamics (CFD) methods. An in-house DEM implementation was validated and used for packing simulations, in which the effect of several parameters on the resulting average packing density was investigated. The restitution coefficient was found out to have the most significant effect. The results can be utilised in further work to obtain a pebble bed with a specific packing density. The packing structures of selected pebble beds were also analysed in detail and local variations in the packing density were observed, which should be taken into account especially in the reactor core thermal-hydraulic analyses. Two open source DEM codes were used to produce stochastic pebble bed configurations to add realism and improve the accuracy of criticality calculations performed with the Monte Carlo reactor physics code Serpent. Russian ASTRA criticality experiments were calculated. Pebble beds corresponding to the experimental specifications within measurement uncertainties were produced in DEM simulations and successfully exported into the subsequent reactor physics analysis. With the developed approach, two typical issues in Monte Carlo reactor physics calculations of pebble bed geometries were avoided. A novel method was developed and implemented as a MATLAB code to calculate porosities in the cells of a CFD calculation mesh constructed over a pebble bed obtained from DEM simulations. The code was further developed to distribute power and temperature data accurately between discrete based reactor physics and continuum based thermal-hydraulics models to enable coupled reactor core calculations. The developed method was also found useful for analysing sphere packings in general. CFD calculations were performed to investigate the pressure losses and heat transfer in three dimensional air cooled smooth and rib roughened rod geometries, housed inside a hexagonal flow channel representing a sub-channel of a single fuel rod of a GFR. The CFD geometry represented the test section of the L-STAR experimental facility at Karlsruhe Institute of Technology and the calculation results were compared to the corresponding experimental results. Knowledge was gained of the adequacy of various turbulence models and of the modelling requirements and issues related to the specific application. The obtained pressure loss results were in a relatively good agreement with the experimental data. Heat transfer in the smooth rod geometry was somewhat under predicted, which can partly be explained by unaccounted heat losses and uncertainties. In the rib roughened geometry heat transfer was severely under predicted by the used realisable k − epsilon turbulence model. An additional calculation with a v2 − f turbulence model showed significant improvement in the heat transfer results, which is most likely due to the better performance of the model in separated flow problems. Further investigations are suggested before using CFD to make conclusions of the heat transfer performance of rib roughened GFR fuel rod geometries. It is suggested that the viewpoints of numerical modelling are included in the planning of experiments to ease the challenging model construction and simulations and to avoid introducing additional sources of uncertainties. To facilitate the use of advanced calculation approaches, multi-physical aspects in experiments should also be considered and documented in a reasonable detail.