Symmetry in non-linear stochastic dynamical systems and its applications

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，对称性是指使随机系统保持不变的单参数连续变换。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的守恒量；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数可以降阶。作为相关课题，我们对随机非线性系统的典型例子--随机Kaldor经济周期模型进行了数值模拟，并对未定权益的定价问题进行了随机分析。在第一个主题中，数值结果表明，模型中的噪声不仅可以掩盖潜在的动力结构，而且可以揭示隐藏的结构，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，股票价格过程分配的概率度量的一个等价鞅度量，它可以看作是一个一般的统计动力系统，对确定未定权益的价格起着重要的作用。如果市场是不完全的，那么有许多等价的鞅度量。在此基础上，采用相对熵最小化原理作为合理鞅度量的判据，将得到的度量称为规范鞅度量(CMM)。研究了CMM的存在性以及CMM与极小鞅测度之间的关系。

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.