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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

本论文主要研究由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，三泽，深入研究“合成方法”，以产生数值逼近计划，这种随机非线性动力系统。在所提出的方法中，该解决方案是近似的组成的随机流来自更简单的和完全可积的向量场运营商，这是相关的守恒量和对称性的概念。新的可获得的计划是有利的，以保持特殊的字符/结构的随机系统的数字和有用的近似的解决方案。为了检验这种优越性，Misawa在所提出的方案的基础上进行了几次数值模拟，这些方案适用于数学金融中出现的随机系统。 ...更多信息 ICS，三泽还处理随机宏观经济模型的随机数值模拟与噪声影响和平滑分析的时间序列数据的小波系统。研究者宫原研究不完全市场的期权定价理论。假设标的资产的价格过程为几何Levy过程，期权的价格由最小相对熵原理确定。他把这个定价模型命名为[几何Levy过程& MEMM]定价模型，并研究了这个模型的性质。调查员，清水，工程与测量值扩散过程相关的一些系谱问题，并检查分数矩的第一次返回时间的正经常马尔可夫链。研究者Hashimoto通过FBI变换证明了Grushin算子的Gevrey亚椭圆性，通过这些相关的主题，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