Conserved quantities and symmetries in non-linear stochastic dynamical systems and its applications

本文主要研究用随机微分方程描述的随机非线性动力系统的守恒量和对称性理论及其相关问题。特别是，首席研究员Misawa深入研究了“组成方法”，以便为这种随机非线性动力系统产生数值近似方案。在所提出的方法中，由与守恒量和对称性有关的更简单的精确可积向量场算子导出的随机流组成近似解。新的可得格式有利于在数值上保持随机系统的特殊性质/结构，并有助于解的近似。为了检验其优越性，Misawa在提出的方案的基础上对数学金融中出现的随机系统进行了几个数值模拟。此外，Misawa还研究了具有噪声效应的随机宏观经济模型的随机数值模拟和小波系统对时间序列数据的平滑分析。研究者Miyahara研究了不完全市场下的期权定价理论。假设标的资产的价格过程为几何列维过程，期权的价格由最小相对熵原理决定。他将这种定价模型命名为[几何征费过程& MEMM]定价模型，并研究了该模型的性质。研究者Shimizu研究了一些与测度值扩散过程相关的系谱问题，并研究了正递归马尔可夫链的第一次返回时间的分数矩。研究者Hashimoto通过FBI变换证明了Grushin算子的Gevrey亚椭圆性。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对于几种随机模型的分析是有用的。少

The present study focuses on a theory of conserved quantities and symmetries for stochastic non-linear dynamical systems, which are described by stochastic differential equations, and the related topics. Particularly, the head investigator, Misawa, deeply investigates "composition methods" in order to produce numerical approximation schemes for such stochastic non-linear dynamical systems. In the proposed methods, the solution is approximated by composition of the stochastic flows derived from simpler and exactly integrable vector field operators which are related to the concepts of conserved quantities and symmetries. The new obtainable schemes are advantageous to preserve the special character/structure of the stochastic systems numerically and are useful for approximations of the solutions. To examine the superiority, Misawa carries out several numerical simulations on the basis of the proposed schemes for stochastic systems which arise in the mathematical finance.As the related top … More ics, Misawa also treats the stochastic numerical simulations of stochastic macroeconomic models with noise effects and smoothing analysis of time Series data by wavelet systems. The investigator, Miyahara, studies on the option pricing theory of incomplete markets. The price processes of the underlying assets are assumed to be geometric Levy processes, and the price of options are supposed to be determined as by the minimal relative entropy principle. He has named this pricing model the [Geometric Levy Process ＆ MEMM] Pricing Model, and investigated the properties of this model. The investigator, Shimizu, works with some genealogical problems related to measure-valued diffusion processes and examines the fractional moments of the first returning time of positively recurrent Markov chains. The investigator, Hashimoto, shows Gevrey hypoellipticity for Grushin Operators by FBI transformation.Through these related topics, we find out that stochastic dynamical theory and stochastic numerics are useful for the analysis of the several stochastic models. Less