  The main focus of this paper is to explore how much similarity between two
stochastic differential systems. Motivated by the conjugate theory of
stochastic dynamic systems, we study the relationship between two systems by
finding homeomorphic mappings $K$. Particularly, we use the minimizer $K^*$ to
measure the degree of similarity. Under appropriate assumptions, we give
sufficient and necessary conditions for the existence of the minimizer $K^*$.
The former result can be regarded as a strong law of large numbers, while the
latter is a stochastic maximum principle. Finally, we provide different
examples of stochastic systems and an application to stochastic Hartman Grobman
theorem. Thus, the results illustrate what is the similarity, extending the
conjugacy in stochastic dynamical systems.
