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We focus on planar Random Walks and some related stochastic processes. The discrete models are introduced and some of their core properties examined. We then turn to the question of continuous analogues, starting with the well-known convergence of the Random Walk to Brownian Motion. For the Harmonic Explorer and the Loop Erased Random Walk, we discuss the idea for convergence to SLE(\kappa) and carry out parts of the proof in the former case using a martingale observable to pin down the Loewner driving process.

A stochastic method is developed, implemented and investigated here for solving Laplace, Poisson's, and standard parabolic wave equations. This method is based on the properties of random walk, diffusion process, Ito formula, Dynkin formula and Monte Carlo simulations. The developed method is a local method i:e: it gives the value of the solution directly at an arbitrary point rather than extracting its value from complete field solution and thus is inherently parallel. Field computation by this method is demonstrated for electrostatic and electrodynamic propagation problems by considering simple examples and numerical results are presented to validate this method. Numerical investigations are carried out to understand efficacy and limitations of this method and to provide qualitative understanding of various parameters involved in this method.

This dissertation studies information economics in strategic and decision settings. In Chapter 1, I introduce flexible endogenous monitoring into dynamic moral hazard. A principal can commit to not only an employment plan but also the monitoring technology to incentivize dynamic effort from an agent. Optimal monitoring follows a Poisson process that produces rare informative signals, and the optimal employment plan features decreasing turnover. To incentivize persistent effort, the Poisson monitoring takes the form of "bad news'' that leads to immediate termination. Monitoring is non-stationary: the bad news becomes more precise and less frequent. In Chapter 2, which is joint work with Qingmin Liu, we analyze a model of strategic exploration in which competing players independently explore a set of alternatives. The model features a multiple-player multiple-armed bandit problem and captures a strategic trade-off between preemption---covert exploration of alternatives that the opponent will explore in the future---and prioritization---exploration of the most promising alternatives. Our results explain how the strategic trade-off shapes equilibrium behaviors and outcomes, e.g., in technology races between superpowers and R&D competitions between firms. We show that players compete on the same set of alternatives, leading to duplicated exploration from start to finish, and they explore alternatives that are a priori less promising before more promising ones are exhausted. In Chapter 3, I study how a forward-looking decision maker experiments on unknown alternatives of spatially correlated utilities, modeled by a Brownian motion so that similar alternatives yield similar utilities. For example, a firm experiments on its size that yields unknown, spatially correlated profitability. Experimentation trades off the opportunity cost of exploitation for the indirect inference from the explored alternatives to unknown ones. The optimal strategy is to explore unknown alternatives and then exploit the best known alternative when the explored becomes sufficiently worse than the best. The decision maker explores more quickly as the explored alternative worsens. My model predicts the conditional Gibrat's law and linear relation between firm size and profitability.

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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>

Movement is an essential feature of life. It allows organisms to move towards a more favorable environment and to search for food. There are many biological systems that fall under the category active matter, from molecular motors walking on microtubules inside cells to flocks of birds. What these systems have in common is that each of its constituents converts energy into directed motion, that is, they propel themselves forward. Besides the many biological examples, there is also synthetic active matter, these are self-propelled particles made in a laboratory. These are typically colloidal sized particles that can propel themselves forward by self-phoresis. In this work the focus is on the low Reynolds number regime, meaning that the typical size of the constituents is less than a few micrometers. The models that are used to describe such active matter are can be viewed as nonequilibrium extensions to Brownian motion (the thermal motion of small particles dissolved in a fluid). In many systems the self-propulsion speed (activity) is not homogeneous in space: the particles swim faster in some areas than in others. The main topic of this dissertation is how a single active particle, or a few active particles tied together by a potential, behave in such systems. It is known that a single active particle without any steering mechanism spends most time in the regions where it moves slowly, or in other words, they spend most time in regions where they are less active. However, here it is shown that, even though they spend most time in the less active regions, dynamical properties, such as the probability to move towards the more active regions is higher than moving towards the less active regions. Furthermore, when the active particles are connected to a passive Brownian 'cargo' particle, chained together to form a colloidal sized polymer, or fixed to another active particle, the resulting active dimers or polymers either accumulate in the high activity regions or the low activity regions, depending on the friction of the cargo particle, the number of monomers in the polymer, or the relative orientation of active particles. Lastly, when the activity is both time- and space-dependent, a steady drift of active particles can be induced, without any coupling between the self-propulsion direction and the gradient in the activity. This phenomenon can be used to position the particles depending on their size.:1. Brownian Motion 2. Active Matter 3. Modeling Active Matter 4. Introduction: Inhomogeneous activity 5. Pseudochemotaxis 6. Cargo-Carrying Particles 7. Active Colloidal Molecules 8. Time-Varying Activity Fields Appendix: Hydrodynamics

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第23717号 / 理博第4807号 / 新制||理||1688(附属図書館) / 京都大学大学院理学研究科化学専攻 / (主査)教授 谷村 吉隆, 教授 林 重彦, 教授 渡邊 一也 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Stochastic delay differential equations (SDDEs) are systems of differential equations with a time lag in a noisy or random environment. There are many nonlinear SDDEs where a linear growth condition is not satisfied, for example, the stochastic delay Lotka-Volterra model of food chain discussed by Xuerong Mao and Martina John Rassias in 2005. Much research has been done using discrete delay where the dynamics of a process at time t depend on the state of the process in the past after a single fixed time lag \tau. We are researching processes with continuously distributed delay which depend on weighted averages of past states over the entire time lag interval [t-\tau,t].By using martingale concepts, we prove sufficient conditions for the existence of a unique solution, ultimate boundedness, and non-extinction of one-dimensional nonlinear SDDE with continuously distributed delay. We give generalized Khasminskii-type conditions which along with local Lipschitz conditions are sufficient to guarantee the existence of a unique global solution of certain n-dimensional nonlinear SDDEs with continuously distributed delay. Further, we give conditions under which Euler-Maruyama numerical approximations of such nonlinear SDDEs converge in probability to their exact solutions.We give some examples of one-dimensional and 2-dimensional stochastic differential equations with continuously distributed delay which satisfy the sufficient conditions of our theorems. Moreover, we simulate their solutions and analyze the error of approximation using MATLAB to implement the Euler-Maruyama algorithm.