Symmetry in non-linear stochastic dynamical systems and its applications

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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.