Symmetry in non-linear stochastic dynamical systems and its applications

非线性随机动力系统的对称性及其应用

• 批准号: 08640300
• 负责人: MISAWA Tetsuya
• 金额: $ 0.7万
• 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: 1995
• 资助国家: 日本
• 起止时间: 1995 至 1996
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08640300/
• 关键词: non-linear; stochastic systems; symmetry & conserved quantities; similarity method; mathematical finance; martingale measure; relative entropy; wavelet analysis

The present study focuses on the theory of symmetry for stochastic non-linear dynamical systems described by stochastic differential equations. Here symmetry means an one-parameter continuous transformation which leaves the stochastic system invariant. Within the framework, the following results are obtained.1) A method for deriving conserved quantities from symmetry is developed, and thereby the new conserved quantities are obtained for the non-linear stochastic systems.2) The similarity method is formulated to stochastic systems ; that is, if a stochastic dynamical system admits symmetry, it follows that the order of stochastic equations describing the system can be reduced. It is examined that the method is useful to analyze stochastic non-linear systems.As the related topics, numerical simulations of a stochastic Kaldor business cycle model, which is a typical example of stochastic non-linear system, and stochastic analysis of the pricing problem of contingent claims are treated.In the first topic, the numerical results indicate that noise in the model may not only obscure the underlying dynamical structures, but also reveal the hidden structures, for example, the chaotic attractors near a window of chaos or the periodic attractors near a small chaotic parameter region.In the second topic, an equivalent martingale mesure for the probability measure assigned to the price process of stocks, which may be regarded as a genaral stodhastic dynamical system, plays an important role to determine the price of contingent claims. If the market is incomplete, there are many equivalent martingale measures. Hence the minimization principle of relative entropy is adopted for a criterion of reasonable martingale measure ; the obtained measure is called the canonical martingale measure (CMM). The existence of CMM and the relations between CMM and the minimal martingale measure are investgated.

本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对称性理论。这里的对称性意味着一个单参数连续变换，使随机系统不变。在此框架下，得到了如下结果：1）发展了一种由对称性导出守恒量的方法，从而得到了非线性随机系统的新守恒量; 2）将相似方法推广到随机系统，即如果随机动力系统具有对称性，则描述该系统的随机方程的阶可以降低.作为相关的研究课题，本文对随机非线性系统的一个典型例子--随机Kaldor经济周期模型进行了数值模拟，并对未定权益定价问题进行了随机分析，数值结果表明，噪声不仅会掩盖模型的动力学结构，还揭示了隐含的结构，如混沌窗口附近的混沌吸引子或小混沌参数区域附近的周期吸引子.第二部分，股票价格过程的概率测度的等价鞅测度，可以看作是一个一般的随机动力系统，它在未定权益定价中起着重要的作用.如果市场是不完全的，则有许多等价的鞅测度。因此，采用相对熵的最小化原理作为合理鞅测度的一个判据，得到的测度称为典型鞅测度。研究了CMM的存在性以及CMM与极小鞅测度的关系。

项目成果

A. Dohtani: "Chaos, complex transients and noise : illustration with Kaldor model" Chaos Solitons & Fractals. 7・12. 2157-2174 (1996)

A. Dohtani：“混沌、复杂瞬态和噪声：卡尔多模型的插图”混沌孤子和分形 7・12 (1996)。

Y.Miyahara: "Canonical martingale measures and minimal martingale measures of incomplete assets markets" The Australian National Univ.Research Report. vol.007-96. 95-100 (1996)

Y.Miyahara：“不完全资产市场的规范鞅措施和最小鞅措施”澳大利亚国立大学研究报告。

T.Misawa: "Conserved quantities and symmetries in stochastic dynamical systems" Inst.of Statist.Math., Cooperative Research Report (in Japanese). vol.91. 79-88 (1996)

T.Misawa：“随机动力系统中的守恒量和对称性”Inst.of Statist.Math.，合作研究报告（日语）。