京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第21212号 / 情博第665号 / 新制||情||115(附属図書館) / 京都大学大学院情報学研究科先端数理科学専攻 / (主査)教授 磯 祐介, 教授 木上 淳, 助手 藤原 宏志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Performing detailed simulations of plasma kinetic diffusion is a challenging task and currently requires the largest computational facilities in the world. The reason for this is that, the physics in a confined heated plasma occur on a broad range of temporal and spatial scales. It is therefore of interest to improve the computational algorithms together with the development of more powerful computational resources. Kinetic diffusion processes in plasmas are commonly simulated with the Monte Carlo method, where a discrete set of particles are sampled from a distribution function and advanced in a Lagrangian frame according to a set of stochastic differential equations. The Monte Carlo method introduces computational error in the form of statistical random noise produced by a finite number of particles (or markers) N and the error scales as αN−β where β = 1/2 for the standard Monte Carlo method. This requires a large number of simulated particles in order to obtain a sufficiently low numerical noise level. Therefore it is essential to use techniques that reduce the numerical noise. Such methods are commonly called variance reduction methods. In this thesis, we have developed new variance reduction methods with application to plasma kinetic diffusion. The methods are suitable for simulation of RF-heating and transport, but are not limited to these types of problems. We have derived a novel variance reduction method that minimizes the number of required particles from an optimization model. This implicitly reduces the variance when calculating the expected value of the distribution, since for a fixed error the  optimization model ensures that a minimal number of particles are needed. Techniques that reduce the noise by improving the order of convergence, have also been considered. Two different methods have been tested on a neutral beam injection scenario. The methods are the scrambled Brownian bridge method and a method here called the sorting and mixing method of L´ecot and Khettabi[1999]. Both methods converge faster than the standard Monte Carlo method for modest number of time steps, but fail to converge correctly for large number of time steps, a range required for detailed plasma kinetic simulations. Different techniques are discussed that have the potential of improving the convergence to this range of time steps. / QC 20120314

No description available.

Parameter estimation is a powerful and adaptable framework that addresses the inherent complexities and uncertainties of financial data. We provide an overview of likelihood functions, and likelihood estimations, as well as the essential numerical approximations and techniques. In the financial domain, where unpredictable and non-stationary market dynamics prevail, parameter estimations of relevant SDE models prove highly relevant. We delve into practical applications, showcasing how SDEs can effectively capture the inherent uncertainties and dynamics of financial models related to time evolution of interest rates. We work with the Vašíček model and Cox-Ingersoll-Ross (CIR) model which describes the dynamics of interest rates over time. We incorporate the Maximum likelihood and Quasi-maximum likelihood estimation methods in estimating the parameters of our models.

Cette thèse est consacrée à l'étude des applications de la théorie des erreurs par formes de Dirichlet. Notre travail se divise en trois parties. La première analyse les modèles gouvernés par une équation différentielle stochastique. Après un court chapitre technique, un modèle innovant pour les carnets d’ordres est proposé. Nous considérons que le spread bid-ask n'est pas un défaut, mais plutôt une propriété intrinsèque du marché. L'incertitude est portée par le mouvement Brownien qui conduit l'actif. Nous montrons que l'évolution des spread peut être évaluée grâce à des formules fermées et nous étudions l'impact de l'incertitude du sous-jacent sur les produits dérivés. En suite, nous introduisons le modèle PBS pour le pricing des options européennes. L'idée novatrice est de distinguer la volatilité du marché par rapport au paramètre utilisé par les traders pour se couvrir. Nous assumons la première constante, alors que le deuxième devient une estimation subjective et erronée de la première. Nous prouvons que ce modèle prévoit un spread bid-ask et un smile de volatilité. Les propriétés plus intéressantes de ce modèle sont l’existence de formules fermés pour le pricing, l'impact de la dérive du sous-jacent et une efficace stratégie de calibration. La seconde partie s'intéresse aux modèles décrit par une équation aux dérivées partielles. Les cas linéaire et non-linéaire sont analysés séparément. Dans le premier nous montrons des relations intéressantes entre la théorie des erreurs et celui des ondelettes. Dans le cas non-linéaire nous étudions la sensibilité des solutions à l’aide de la théorie des erreurs. Sauf dans le cas d’une solution exacte, il y a deux approches possibles : on peut d’abord discrétiser l’EDP et étudier la sensibilité du problème discrétisé, soit démontrer que les sensibilités théoriques vérifient des EDP. Les deux cas sont étudiés, et nous prouvons que les sharp et le biais sont solutions d’EDP linéaires dépendantes de la solution de l’EDP originaire et nous proposons des algorithmes pour évaluer numériquement les sensibilités. Enfin, la troisième partie est dédiée aux équations stochastiques aux dérivées partielles. Notre analyse se divise en deux chapitres. D’abord nous étudions la transmission de l’incertitude, présente dans la condition initiale, à la solution de l’EDPS. En suite, nous analysons l'impact d'une perturbation dans les termes fonctionnelles de l’EDPS et dans le coefficient de la fonction de Green associée. Dans le deux cas, nous prouvons que le sharp et le biais sont solutions de deux EDPS linéaires dépendantes de la solution de l’EDPS originaire / This thesis is devoted to the study of the applications of the error theory using Dirichlet forms. Our work is split into three parts. The first one deals with the models described by stochastic differential equations. After a short technical chapter, an innovative model for order books is proposed. We assume that the bid-ask spread is not an imperfection, but an intrinsic property of exchange markets instead. The uncertainty is carried by the Brownian motion guiding the asset. We find that spread evolutions can be evaluated using closed formulae and we estimate the impact of the underlying uncertainty on the related contingent claims. Afterwards, we deal with the PBS model, a new model to price European options. The seminal idea is to distinguish the market volatility with respect to the parameter used by traders for hedging. We assume the former constant, while the latter volatility being an erroneous subjective estimation of the former. We prove that this model anticipates a bid-ask spread and a smiled implied volatility curve. Major properties of this model are the existence of closed formulae for prices, the impact of the underlying drift and an efficient calibration strategy. The second part deals with the models described by partial differential equations. Linear and non-linear PDEs are examined separately. In the first case, we show some interesting relations between the error and wavelets theories. When non-linear PDEs are concerned, we study the sensitivity of the solution using error theory. Except when exact solution exists, two possible approaches are detailed: first, we analyze the sensitivity obtained by taking “derivatives” of the discrete governing equations. Then, we study the PDEs solved by the sensitivity of the theoretical solutions. In both cases, we show that sharp and bias solve linear PDE depending on the solution of the former PDE itself and we suggest algorithms to evaluate numerically the sensitivities. Finally, the third part is devoted to stochastic partial differential equations. Our analysis is split into two chapters. First, we study the transmission of an uncertainty, present on starting conditions, on the solution of SPDE. Then, we analyze the impact of a perturbation of the functional terms of SPDE and the coefficient of the related Green function. In both cases, we show that the sharp and bias verify linear SPDE depending on the solution of the former SPDE itself / Questa tesi é dedicata allo studio delle applicazioni della teoria degli errori tramite forme di Dirichlet, il nostro lavoro si divide in tre parti. Nella prima vengono studiati i modelli descritti da un’equazione differenziale stocastica: dopo un breve capitolo con risultati tecnici viene descritto un modello innovativo per i libri d’ordini. La presenza dei differenziali denarolettera viene considerata non come un’imperfezione, bensi una proprietà intrinseca dei mercati. L’incertezza viene descritta come un rumore sul moto Browniano sottostante all’azione; dimostriamo che l’evoluzione di questi differenziali puó essere valutata attraverso formule chiuse e stimiamo l’impatto dell’incertezza del sottostante sui prodotti derivati. In seguito proponiamo un nuovo modello, chiamato PBS, per il prezzaggio delle opzioni di tipo europeo: l’idea innovativa consiste nel distinguere la volatilità di mercato dal parametro usato dai trader per la copertura. Noi supponiamo la prima constante, mentre il secondo diventa una stima soggettiva ed erronea della prima. Dimostriamo che questo modello prevede dei differenziali lettera-denaro e uno smile di volatilità implicita. Le maggiori proprietà di questo modello sono l’esistenza di formule chiuse per il princing, l’impatto del drift del sottostante e un’efficace strategia per la calibrazione. La seconda parte è dedicata allo studio dei modelli descritti da delle equazioni alle derivate perziali. I casi lineare e non-lineare sono trattati separatamente. Nel primo caso mostriamo interessanti relazioni tra la teoria degli errori e quella delle wavelets. Nel caso delle EDP non-lineari studiamo la sensibilità della soluzione usando la teoria degli errori. Due possibili approcci esistono, salvo quando la soluzione è esplicita. Possiamo prima discretizzare il problema e studiare la sensibilità delle equazioni discretizzate, oppure possiamo dimostrare che le sensibilità teoriche verificano, a loro volta, delle EDP dipendenti dalla soluzione della EDP iniziale. Entrambi gli approcci sono descritti e vengono proposti degli algoritmi per valutare le sensibilità numericamente. Infine, la terza parte è dedicata ai modelli descritti da un’equazione stocastica alle derivate parziali. La nostra analisi é divisa in due capitoli. Nel primo viene studiato l’impatto di un’incertezza, presente nella condizione iniziale, sulla soluzione dell’EDPS, nella seconda si analizzano gli impatti di una perturbazione dei termini funzionali dell’EDPS del coefficiente della funzione di Green associata. In entrambi i casi dimostriamo che lo sharp e la discrepanza sono soluzioni di due EDPS lineari dipendenti dalla soluzione dell’EDPS iniziale

Using a Lie symmetry group generator and a generalized form of Manale's formula for solving second order ordinary di erential equations, we determine new symmetries for the one and two dimensional heat equations, leading to new solutions. As an application, we test a formula resulting from this approach on thin plate heat conduction. / Applied Mathematics / M.Sc. (Applied Mathematics)

人口統計學家在傳統上習慣將人口的種種變化視為時間的函數，皆試圖以決定型（deterministic）的函數來刻劃，例如：1825年Gompertz提出的死力法則、1838年Verhulst以羅吉斯函數描述人口成長。近年則傾向於逐項（item-by-item）分析各種可能因素，例如：1992年Lee-Carter提出的死亡率模型、目前英國實務上使用的Renshaw與Haberman(2003)提出改善Lee-Carter模型的Reduction Factor模型、加入世代（Cohort）因素的Age-Period-Cohort模型等。但台灣地區近年來生育率與死亡率皆不斷下降，且有隨著時間而變化加劇的傾向，使得以往使用的模型不易捕捉變化。 本文以另一個角度思考生育與死亡變化，將台灣人口視為一隨時間變化的動態系統，使用微分方程來刻劃，找出此動態系統的背後所隱含的規則。人口動態系統的變化，主要來源是出生、死亡與遷移，在建模的過程中，我們先各別針對其中一項，在其他條件不變的情況下，以常微分方程建模，之後再同時考慮各項變動，以偏微分方程建模，找出台灣人口變化的模型。在本文中，我們先介紹使用微分方程模型分別配適與估計出生與死亡。 由台灣地區人口統計資料顯示，不論總生育率或各年齡組的死亡率都有逐漸下降的趨勢，但是每年之間的震盪很大，因此我們提出「二次逼近法」，從出生或死亡對時間的變化率與曲度來估計生育率與死亡率，對於此種震盪幅度較大的資料，可以得到頗精確的估計。唯在連續幾年資料呈現近似線性上升或下降處，非線性的模型容易出現較大的估計誤差，針對此問題我們也提出一些可能的修正方法，以降低整體的模型誤差率。 / Conventionally the change of population is considered as a function of time and described by using deterministic functions. The well-known examples are Gompertz law of mortality (1825) and Verhulst’s logistic growth model (1838). Recently demographers favor stochastic models when analyzing factors in an item-by-item fashion. Since 1992, Lee-Carter model is a most commonly used stochastic model in demographic studies. But empirical studies indicate that the rapid declines in both fertility and mortality rates are against the assumptions of Lee-Carter model. In this study we treat Taiwan population as a dynamic system which changes over time and characterize it by differential equations. Since the changes are from birth, death and migration, we first separately build models using ordinary differential equations. Afterwards the model of Taiwan population can be built by using partial differential equations considering the three main factors simultaneously. Total fertility and age-specific mortality rates in Taiwan decline over time but with shakes between years. Consequently we propose‘parabola approximation method’and apply it to velocity and acceleration of birth or death to solve the differential equations of Taiwan fertility and mortality. Empirical study shows the method allows us to get accurate estimates of mortality and fertility when the data change a lot in a short period of time. But we found the model may over-fit the data at some time point where the function does not seem to be very continuous.

This thesis develops efficient modeling frameworks via a Partial Differential Equation (PDE) approach for multi-factor financial derivatives, with emphasis on three-factor models, and studies highly efficient implementations of the numerical methods on novel high-performance computer architectures, with particular focus on Graphics Processing Units (GPUs) and multi-GPU platforms/clusters of GPUs. Two important classes of multi-factor financial instruments are considered: cross-currency/foreign exchange (FX) interest rate derivatives and multi-asset options. For cross-currency interest rate derivatives, the focus of the thesis is on Power Reverse Dual Currency (PRDC) swaps with three of the most popular exotic features, namely Bermudan cancelability, knockout, and FX Target Redemption. The modeling of PRDC swaps using one-factor Gaussian models for the domestic and foreign interest short rates, and a one-factor skew model for the spot FX rate results in a time-dependent parabolic PDE in three space dimensions. Our proposed PDE pricing framework is based on partitioning the pricing problem into several independent pricing subproblems over each time period of the swap's tenor structure, with possible communication at the end of the time period. Each of these subproblems requires a solution of the model PDE. We then develop a highly efficient GPU-based parallelization of the Alternating Direction Implicit (ADI) timestepping methods for solving the model PDE. To further handle the substantially increased computational requirements due to the exotic features, we extend the pricing procedures to multi-GPU platforms/clusters of GPUs to solve each of these independent subproblems on a separate GPU. Numerical results indicate that the proposed GPU-based parallel numerical methods are highly efficient and provide significant increase in performance over CPU-based methods when pricing PRDC swaps. An analysis of the impact of the FX volatility skew on the price of PRDC swaps is provided. In the second part of the thesis, we develop efficient pricing algorithms for multi-asset options under the Black-Scholes-Merton framework, with strong emphasis on multi-asset American options. Our proposed pricing approach is built upon a combination of (i) a discrete penalty approach for the linear complementarity problem arising due to the free boundary and (ii) a GPU-based parallel ADI Approximate Factorization technique for the solution of the linear algebraic system arising from each penalty iteration. A timestep size selector implemented efficiently on GPUs is used to further increase the efficiency of the methods. We demonstrate the efficiency and accuracy of the proposed GPU-based parallel numerical methods by pricing American options written on three assets.

Stochastic Integrals Driven by Isonormal Gaussian Processes and Applications Master Thesis - Petr Čoupek Abstract In this thesis, we introduce a stochastic integral of deterministic Hilbert space valued functions driven by a Gaussian process of the Volterra form βt = t 0 K(t, s)dWs, where W is a Brownian motion and K is a square integrable kernel. Such processes generalize the fractional Brownian motion BH of Hurst parameter H ∈ (0, 1). Two sets of conditions on the kernel K are introduced, the singular case and the regular case, and, in particular, the regular case is studied. The main result is that the space H of β-integrable functions can be, in the strictly regular case, embedded in L 2 1+2α ([0, T]; V ) which corresponds to the space L 1 H ([0, T]) for the fractional Brownian mo- tion. Further, the cylindrical Gaussian Volterra process is introduced and a stochastic integral of deterministic operator-valued functions, driven by this process, is defined. These results are used in the theory of stochastic differential equations (SDE), in particular, measurability of a mild solution of a given SDE is proven.

This thesis uses a simple deterministic model represented by an ordinary di- fferential equation with two equilibrium points - depending on the initial state the illness either vanishes or persists forever. This model is expanded by adding some diffusion coefficients leading to different stochastic differential equations. They are analyzed to show how the choice of diffusion coefficients changes be- havior of the model in proximity of its equilibria and near the boundary of area with biological meaning. The theoretical results are than illustrated by computer simulations. 1