Decomposition formulas of general exponential operators in a Banach algebra and in a Lie algebra are presented that yield a basis of Monte Carlo simulation of quantum systems. They are applied to study the relaxation and fluctuation from the initially unstable point and to confirm algebraically the scaling theory of transient phenomena. A global approximation method of transient phenomena is also formulated on the basis of decomposition formulas. It is applied to the laser model as a simple example.