The similarity method in stochastic dynamical systems

Abstract

In the present article, the similarity method is formulated to stochastic dynamical systems described by stochastic differential equations of Stratonovich type. When a stochastic dynamical system admits symmetries, it follows that the order of stochastic equations describing the system can be reduced. Here a symmetry means an one-parameter continuous transformation which leaves the stochastic system invariant. Several examples are given, in which a nonlinear stochastic system, stochastic Hamiltonian dynamical systems related to the harmonic oscillator, and a stochastic version of the neo-classical optimal growth model suggested by Samuelson are contained.