主题词:Random Evolutions,Semi-Markov Chains,Discrete-Time Semi-Markov Chains,Discrete-Time Random Evolution,Limit Theorems for Random Evolutions,Applications in Epidemiology and Finance,Random Evolutions on Banach Spaces,Asymptotic Functional Theory,Average and Diffusion Approximation,Dynamical Systems,Additive Functionals,Merton Problem,Reduced Random Media,Controlled Processes,Optimal Stopping,Markov Renewal Theorem