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<p>This dissertation focuses on the simulation efficiency of the Markov process for two scenarios: Stochastic differential equations(SDEs) and simulated weather data. </p> <p><br></p> <p>For SDEs, we propose a novel Gibbs sampling algorithm that allows sampling from a particular class of SDEs without any discretization error and shows the proposed algorithm improves the sampling efficiency by orders of magnitude against the existing popular algorithms.  </p> <p><br></p> <p>In the weather data simulation study, we investigate how representative the simulated data are for three popular stochastic weather generators. Our results suggest the need for more than a single realization when generating weather data to obtain suitable representations of climate. </p>

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23682号 / 理博第4772号 / 新制||理||1684(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 矢野 孝次, 教授 泉 正己, 教授 日野 正訓 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Jump-diffusion models can under certain assumptions be expressed as partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a nonlocal integral like for the here considered models of Merton and Kou. We transform the PIDE to eliminate the convection term, discretize it implicitly using finite differences and the second order backward difference formula (BDF2) on a uniform grid. The arising dense linear system is solved by an iterative method, either a splitting technique or a circulant preconditioned conjugate gradient method. Exploiting the Fast Fourier Transform (FFT) yields the solution in only $O(n\log n)$ operations and just some vectors need to be stored. Second order accuracy is obtained on the whole computational domain for Merton's model whereas for Kou's model first order is obtained on the whole computational domain and second order locally around the strike price. The solution for the PIDE with convection term can oscillate in a neighborhood of the strike price depending on the choice of parameters, whereas the solution obtained from the transformed problem is stabilized. / Master of Science

<p><em>Image-to-image regression is an important computer vision research topic. Previous</em></p> <p><em>research works have been concentrating on task-dependent end-to-end regression models. In</em></p> <p><em>this dissertation, we focus on a generative regression framework based on score matching.</em></p> <p><em>Such generative models are called score-based generative models, which learn the data score</em></p> <p><em>functions by gradually adding noise to data using a diffusion process. Images can be generated</em></p> <p><em>with learned score functions through a time-reversal sampling process.</em></p> <p><em>First, we propose a conditional score matching regression framework which targets the</em></p> <p><em>conditional score functions in regression problems. The framework can perform diverse inferences</em></p> <p><em>about conditional distribution by generating samples. We demonstrate its advantages</em></p> <p><em>with various image-to-image regression applications.</em></p> <p><em>Second, we propose a score-based regression model that applies the diffusion process to</em></p> <p><em>both input and response images simultaneously. The proposed method, called synchronized</em></p> <p><em>diffusion, can help stabilize model parameter learning and increase model robustness. In</em></p> <p><em>addition, we develop an effective prediction algorithm based on the Expectation-Maximization</em></p> <p><em>(EM) algorithm which can improve accuracy and computation speed. We illustrate the efficacy</em></p> <p><em>of our proposed approach on high-resolution image datasets.</em></p> <p><em>The last part of the dissertation focuses on analyzing the score-based generative modeling</em></p> <p><em>framework. We conduct a theoretical analysis of the variance exploding behavior observed in</em></p> <p><em>training score-based generative models with denoising score matching objective functions. We</em></p> <p><em>explain the large variance problem from a nonparametric estimation perspective. Furthermore,</em></p> <p><em>we propose a solution to the general score function estimation problem based on Simulation-</em></p> <p><em>Extrapolation (SIMEX), which was originally developed in the measurement error model</em></p> <p><em>literature. We validate our theoretical findings and the effectiveness of the proposed solution</em></p> <p><em>on both synthesized and real datasets.</em></p>

In this thesis, modelling with L&eacute / vy processes is considered in three parts. In the first part, the general geometric L&eacute / vy market model is examined in detail. As such markets are generally incomplete, it is shown that the market can be completed by enlarging with a series of new artificial assets called &ldquo / power-jump assets&rdquo / based on the power-jump processes of the underlying L&eacute / vy process. The second part of the thesis presents two different methods for pricing European options: the martingale pricing approach and the Fourier-based characteristic formula method which is performed via fast Fourier transform (FFT). Performance comparison of the pricing methods led to the fact that the fast Fourier transform produces very small pricing errors so the results of both methods are nearly identical. Throughout the pricing section jump sizes are assumed to have a particular distribution. The third part contributes to the empirical applications of L&eacute / vy processes. In this part, the stochastic volatility extension of the jump diffusion model is considered and calibration on Standard&amp / Poors (S&amp / P) 500 options data is executed for the jump-diffusion model, stochastic volatility jump-diffusion model of Bates and the Black-Scholes model. The model parameters are estimated by using an optimization algorithm. Next, the effect of additional stochastic volatility extension on explaining the implied volatility smile phenomenon is investigated and it is found that both jumps and stochastic volatility are required. Moreover, the data fitting performances of three models are compared and it is shown that stochastic volatility jump-diffusion model gives relatively better results.

We study systems of n dimensional diffusions whose drift and dispersion coefficients depend only on the relative ranking of the processes. We consider the question of how long it takes for a particle to go from one rank to another. It is argued that as n gets large, the distribution of particles satisfies a Porous Medium Equation. Using this, we derive a deterministic limit for the system of particles. This limit allows for direct calculation of the properties of the rank traversal time. The results are extended to the case of asymmetrically colliding particles. These models are of interest in the study of financial markets and economic inequality. In particular, we derive limits for the performance of some Functionally Generated Portfolios originating from Stochastic Portfolio Theory.

General diffusion processes (GDP), or Ito's processes, are potential candidates for the modeling of asset prices, interest rates and other financial quantities to cope with empirical evidence. This thesis considers the applications of general diffusions in finance and potential extensions. In particular, we focus on financial problems involving (optimal) stopping times. A typical example is the valuation of American options. We investigate the use of Laplace-Carson transform (LCT) in valuing American options, and discuss its strengthen and weaknesses. Homotopy analysis from topology is then introduced to derive closed-form American option pricing formulas under GDP. Another example is taken from optimal dividend policies with bankruptcy procedures, which is closely related to excursion time and occupation time of a general diffusion. With the aid of Fourier transform, we further extend the analysis to the case of multi-dimensional GDP by considering the currency option pricing with mean reversion and multi-scale stochastic volatility. / Zhao, Jing. / Adviser: Hoi-Ying Wong. / Source: Dissertation Abstracts International, Volume: 72-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 97-105). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

In this paper, we consider the capture dynamics of a particle undergoing a random walk above a sheet of absorbing traps. In particular, we seek to characterize the distribution in time from when the particle is released to when it is absorbed. This problem is motivated by the study of lymphocytes in the human blood stream; for a particle near the surface of a lymphocyte, how long will it take for the particle to be captured? We model this problem as a diffusive process with a mixture of reflecting and absorbing boundary conditions. The model is analyzed from two approaches. The first is a numerical simulation using a Kinetic Monte Carlo (KMC) method that exploits exact solutions to accelerate a particle-based simulation of the capture time. A notable advantage of KMC is that run time is independent of how far from the traps one begins. We compare our results to the second approach, which is asymptotic approximations of the FPT distribution for particles that start far from the traps. Our goal is to validate the efficacy of homogenizing the surface boundary conditions, replacing the reflecting (Neumann) and absorbing (Dirichlet) boundary conditions with a mixed (Robin) boundary condition.

A Constant Proportion Debt Obligation is a credit derivative which has been introduced to generate a surplus return over a riskless market return. The surplus payments should be obtained by synthetically investing in a risky asset (such as a credit index) and using a linear leverage strategy which is capped for bounding the risk. In this thesis, we investigate two approaches for investigation of constant proportion debt obligations. First, we search for an optimal leverage strategy which minimises the mean-square distance between the final payment and the final wealth of constant proportion debt obligation by the use of optimal control methods. We show that the optimal leverage function for constant proportion debt obligations in a mean-square sense coincides with the one used in practice for geometric type diffusion processes. However, the optimal strategy will lead to a shortfall for some cases. The second approach of this thesis is to develop a pricing formula for constant proportion debt obligations. To do so, we consider both the early defaults and the default on the final payoff features of constant proportion debt obligations. We observe that a constant proportion debt obligation can be modelled as a barrier option with rebate. In this respect, given the knowledge on barrier options, the pricing equation is derived for a particular leverage strategy.