Lie algebra & group methods to non-linear stochastic dynamical systems and its applications

本研究主要研究随机微分方程数值格式的李代数和群论方法，以及由随机微分方程控制的非线性随机动力系统的相关课题。本研究的主要研究者三泽先生利用基于李代数和群论的“合成方法“，深入研究了随机Hamilton动力系统的“辛积分器”。新格式在数值上有利于保持随机哈密顿系统的辛结构，有利于产生稳定的数值解。为了检验其优越性，Misawa还给出了一个说明性的例子来说明所提出的方案。为了研究随机数值分析的实用性，研究人员在各个研究领域研究了非线性随机动力系统。三泽研究随机宏观经济模型的数值模拟，马尔可夫链算法， ...更多信息 在平滑时间序列数据和金融工程中的随机和统计模型。特别是，他主要研究与电力市场有关的回归模型。Miyahara研究了不完全资产市场中的期权定价问题，这是数学金融领域的重要问题之一。以几何Levy过程为基础资产价格过程，以最小熵鞅测度（MEMM）为基础资产价格过程，构造了[Geometric Levy过程& MEMM]定价模型，并研究了该模型的性质。调查员，清水，工作与以下主题;结合过程与波动的人口规模及其有效的人口规模，泊松随机措施和无限可分的分布在人口遗传学，和汉达模型纳入对称选择。研究者Notohara研究了由称为结构化合并模型的马尔可夫过程描述的地理结构化群体中采样基因的系谱。他研究了地理结构对聚结时间分布的影响，聚结时间是样本最近的共同祖先的时间，以及在采样的DNA序列中检测到的分离位点数量的分布。研究员桥本研究了Grushin算子的非各向同性Gevrey亚椭圆性和非线性椭圆型偏微分方程解的解析性。通过这些相关的课题，我们发现随机动力学理论和随机数值方法对于分析几种随机模型是有用的。少

The present study focuses on Lie algebra & group methods to formulate numerical schemes of stochastic differential equations and the related topics on non-linear stochastic dynamical systems governed by such stochastic equations.The head investigator, Misawa, deeply investigates "symplectic integrators" for stochastic Hamiltonian dynamical systems by using "composition methods"which are formulated on the basis of Lie algebra & group theory. The new schemes are advantageous to preserve symplectic structure of the stochastic Hamiltonian systems numerically and are useful for producing the stable numerical solutions. To examine the superiority, Misawa also gives an illustrative example on the proposed schemes for such stochastic systems.In order to investigate the utility of stochastic numerical analysis, the investigators study on non-linear stochastic dynamical systems in various research fields. Misawa studies numerical simulations of stochastic macroeconomic models, Markov chain algor … More ithm in smoothing time series data and stochastic and statistical models in financial engineering. Particularly, he mainly works with investigating some regression models concerning with electricity markets. The investigator, Miyahara, studies the option pricing problems in the incomplete asset market, which is one of the important problems in the field of mathematical finance. He also constructs the [Geometric Levy process & MEMM] pricing model, in which the geometric Levy processes are adopted as the underlying asset price processes and the MEMM(=minimal entropy martingale measure), and investigates the properties of this model. The investigator, Shimizu, works with the following topics ; coalescent process with fluctuating population size and its effective population size, Poisson random measures and infinitely divisible distributions in population genetics, and the Handa model incorporating symmetric selections. The investigator, Notohara studies the genealogy of sampled genes from geographically structured population which is described by a Markov process called Structured Coalescent Model. He investigates effects of the geographical structure on the distribution of the coalescence time, which is the time to the most recent common ancestor of samples, and the distribution of the number of segregating sites detected in sampled DNA sequences. The investigator, Hashimoto, studies non-isotropic Gevrey hypoellipticity for Grushin operators and the analyticity of the solutions of the non-linear elliptic partial differential equations. Through these related topics, we find out that stochastic dynamical theory and stochastic numerical methods are useful for the analysis of the several stochastic models. Less