This thesis deals with models driven by so-called semi-Markov processes, and studies some limit theorems and financial applications in this context. Given a system whose dynamics are governed by various regimes, a semi-Markov process is simply a process that keeps track" of the system regime at each time. It becomes fully Markovian if we add" to it the process keeping track of how long the system has been in its current regime.

Chapter 1 consists of a global introduction to the thesis. We introduce the concepts of semi-Markov and Markov renewal processes, and give a brief overview of each chapter, together with the main results obtained.

Chapter 2 introduces a semi-Markovian model of high frequency price dynamics: as suggested by empirical observations, it extends recent results to arbitrary distributions for limit order book events inter-arrival times, and both the nature of a new limit order book event and its corresponding inter-arrival time depend on the nature of the previous limit order book event.

Chapter 3 establishes strong law of large numbers and central limit theorem results for time-inhomogeneous semi-Markov processes, for which the kernel is time-dependent.

Chapter 4 develops a rigorous treatment of so-called inhomogeneous semi-Markov driven random evolutions, and extends already existing results related to the time-homogeneous case. Random evolutions allow to model a situation in which the dynamics of a system are governed by various regimes, and the system switches from one regime to another at random times. This phenomenon will be modeled by using semi-Markov processes. The notion of ``time-inhomogeneity" appears twice in our framework: random evolutions will be driven by inhomogeneous semi-Markov processes (using results from chapter 3), and constructed with propagators, which are time-inhomogeneous counterparts of semigroups.

Chapter 5 presents a drift-adjusted version of the well-known Heston model - the delayed Heston model - which allows us to improve the implied volatility surface fitting. Pricing and hedging of variance and volatility swaps is also considered.

Finally, chapter 6 concludes the thesis and presents some possible future research directions.