Abstract: This paper studies the random response, stability, correlation functions, and spectral density functions of structures with single degree of freedom subjected to horizontal-vertical random earthquake excitations. According to the relationships between stochastic differential equations and moment differential equations, the moment differential equations of structural response are established. Then, using the Hurwitz’s random stability criterion, this paper obtains the exact formula of stability for the first and second-order moments of structural response. Furtherly, using the complex mode theory, the exact transient and steady-state solutions of the second-order moments are obtained. Lastly, this paper derives the exact solutions to the auto-correlation functions, cross-correlation functions, auto-spectral density functions, cross-correlation functions of structural displacement and velocity responses according to the non-anticipating function characteristics of the solutions of Itô stochastic differential equations. An example is given to analyze the influence of various parameters on structural response, stability, correlation functions and spectral density functions.