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Stochastic numerical schemes to stochastic dynamical systems with conserved quantities

具有守恒量的随机动力系统的随机数值格式

基本信息

批准号：
09640285
负责人：
MISAWA Tetsuya
金额：
$ 0.45万
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：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

This study is dealt with the stochastic difference schemes and related topics for stochastic dynamical systems governed by stochastic differential equations which have conserved quantities. The head investigator, T.Misawa, focuses on an one-dimensional stochastic Hamilton dynamical system in which the energy function (Hamiltonian) becomes a conserved quantity. For the system, he proposes two energy conservative stochastic difference schemes which leave Hamiltonians numerically invariant. and proves that the local error orders of accuracy of numerical solutions derived from the stochastic schemes are 2 and 4 respectively. Moreover, as a fundamental study for the thema, Nisawa also investigates on a method for deriving conserved quantities from symmetry in stochastic systems, the similarity method (a method of the reduction of the order of stochastic equations), and the relations between conserved quantities and symmetry in stochastic Hamilton systems.As the related topics, Misawa treats the numerical simulations of a stochastic business cycle model in an open economy, a wavelet interpolation method with simulated annealing in time series analysis. A.Shimizu works with a genealogical problem of a stepping stone Fleming-Viot process and examines the fractional moments of the first returning time of positively recurrent Markov.chains. Y.Miyahara is concerned with stochastic analysisof the pricing problem of contingent claims ; he investigates on minimal entropy martingale measures of jump type price processes in incomplete assets markets. Y.Hashimoto deals with in the relation between positive-definite generalized functions and the heat kernel. Through the studies, we find out that stochastic numerical method is useful for the analysis of the several stochastic problems in such topics.

本文研究由具有守恒量的随机微分方程所控制的随机动力系统的随机差分格式及其相关问题。首席研究员T.Misawa专注于一维随机汉密尔顿动力系统，其中能量函数（汉密尔顿）成为守恒量。对于该系统，他提出了两个能量守恒的随机差分格式，使哈密顿量数值不变。并证明了由随机格式得到的数值解的局部精度误差阶分别为2和4。此外，作为该课题的基础研究，Nisawa还研究了从随机系统的对称性导出守恒量的方法--相似性方法（随机方程降阶的一种方法），以及随机汉密尔顿系统中守恒量与对称性之间的关系。作为相关课题，Misawa对开放经济中的随机商业周期模型进行了数值模拟，时间序列分析中的一种模拟退火小波插值方法。A.清水工程的一个垫脚石弗莱明-维奥特过程的系谱问题，并检查分数矩的第一次返回时间的积极经常性的马尔可夫。Miyahara主要研究未定权益定价问题的随机分析，他研究了不完全资产市场中跳跃型价格过程的最小熵鞅测度。Y.Hashimoto研究了正定广义函数与热核的关系。通过研究，我们发现随机数值方法对于这类问题中的几个随机问题的分析是有用的。