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

非线性随机动力系统中的守恒量和对称性及其应用

基本信息

批准号：
11640132
负责人：
MISAWA Tetsuya
金额：
$ 2.18万
依托单位：
Nagoya City University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640132/
关键词：
Nonlinear stochastic systems Composition methods Stochastic numerics Geometric Levy processes Mathematical finance Fleming-Biot processes Recurrent Markov chain Bochner-Schwartz' teorem オプション価格付け Flemming-Viot過程 Grusin作用素 Nonlinear s

项目摘要

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

本研究的重点是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以及相关主题，特别是首席研究员三泽深入研究了“组合方法”，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，通过从相关的更简单且精确可积的矢量场算子导出的随机流的组合来近似解。守恒量和对称性的概念有利于在数值上保留随机系统的特殊特征/结构，并且对于解的近似很有用，Misawa 在此基础上进行了多次数值模拟。数学金融中出现的随机系统的拟议方案。作为相关的顶级学科，Misawa 还处理具有噪声效应和时间平滑分析的随机宏观经济模型的随机数值模拟研究者 Miyahara 对不完全市场的期权定价理论进行了研究，假设标的资产的价格过程是几何 Levy 过程，并且期权的价格应该由最小值决定。他将这个定价模型命名为[Geometric Levy Process ＆ MEMM]定价模型，并研究了该模型的属性，研究者 Shimizu 研究了一些与测值扩散过程相关的谱系问题。正循环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。通过这些相关主题，我们发现随机动力学理论和随机数值对于分析更少的随机模型。