Limit theorems for Markov-modulated and reflected diffusion processes

In this thesis, asymptotic properties of two variants of one-dimensional diffusion processes, which are Markov-modulated and reflected Ornstein-Uhlenbeck processes, are studied. Besides the random term of the Brownian motion, the Markov-modulated diffusion process evolves in an extra random environment, namely the finite-state Markov chain. The reflected Ornstein-Uhlenbeck process behaves as an Ornstein-Uhlenbeck process which has instantaneous reflection at boundaries. They are widely used in modeling due to above mentioned their distinctive features and great analytical tractability.
We obtain four limit theorems from the perspectives of weak convergence and large deviations. Firstly, we prove weak convergence of a sequence of Markov-modulated diffusion processes with rapid switching to an ordinary diffusion process by verifying its tightness property. Secondly, a sample-path large deviations principle for the coupling of the Markov-modulated diffusion process with small noise and the occupation measure of the rapid switching Markov chain is obtained. The large deviations principles for each individual term are derived by the contraction principle. Those results reveal interesting behavior of Markov-modulated diffusion process when the modulating Markov chain switches fast. Thirdly, transient asymptotics of large deviations type are acquired for reflected and doubly reflected Ornstein-Uhlenbeck processes. Fourthly, we prove central limit theorems and functional central limit theorems for the centered and scaled loss and idle processes of doubly reflected Ornstein-Uhlenbeck processes.