Symmetry 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对称性理论。这里对称是指使随机系统保持不变的单参数连续变换。在该框架内，得到了以下结果。1)提出了一种由对称性推导守恒量的方法，从而得到了非线性随机系统的新守恒量。2)将相似方法应用于随机系统；也就是说，如果一个随机动力系统允许对称，那么描述该系统的随机方程的阶数可以降低。验证了该方法对随机非线性系统的分析是有用的。本文对随机非线性系统的典型例子——随机卡尔多经济周期模型进行了数值模拟，并对或有债权定价问题进行了随机分析。在第一个主题中，数值结果表明，模型中的噪声不仅会掩盖潜在的动力学结构，而且会揭示隐藏的结构，例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。在第二个主题中，分配给股票价格过程的概率测度的等效鞅测度可以看作是一个一般的随机动力系统，它在确定或有债权的价格方面起着重要的作用。如果市场是不完全的，则有许多等效的鞅度量。因此，采用相对熵最小化原理作为合理鞅测度的判据；得到的测度称为正则鞅测度（CMM）。研究了CMM的存在性以及CMM与最小鞅测度的关系。

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.