From both theoretical and applied perspectives, first passage<br />time problems for random processes are challenging and of great<br />interest. In this thesis, our contribution consists on providing<br />explicit or quasi-explicit solutions for these problems in two<br />different settings.<br /><br />In the first one, we deal with problems related to the<br />distribution of the first passage time (FPT) of a Brownian motion<br />over a continuous curve. We provide several representations for<br />the density of the FPT of a fixed level by an Ornstein-Uhlenbeck<br />process. This problem is known to be closely connected to the one<br />of the FPT of a Brownian motion over the square root boundary.<br />Then, we compute the joint Laplace transform of the $L^1$ and<br />$L^2$ norms of the $3$-dimensional Bessel bridges. This result is<br />used to illustrate a relationship which we establish between the<br />laws of the FPT of a Brownian motion over a twice continuously<br />differentiable curve and the quadratic and linear ones. Finally,<br />we introduce a transformation which maps a continuous function<br />into a family of continuous functions and we establish its<br />analytical and algebraic properties. We deduce a simple and<br />explicit relationship between the densities of the FPT over each<br />element of this family by a selfsimilar diffusion.<br /><br /> In the second setting, we are concerned with the study of<br />exit problems associated to Generalized Ornstein-Uhlenbeck<br />processes. These are constructed from the classical<br />Ornstein-Uhlenbeck process by simply replacing the driving<br />Brownian motion by a Lévy process. They are diffusions with<br />possible jumps. We consider two cases: The spectrally negative<br />case, that is when the process has only downward jumps and the<br />case when the Lévy process is a compound Poisson process with<br />exponentially distributed jumps. We derive an expression, in terms<br />of new special functions, for the joint Laplace transform of the<br />FPT of a fixed level and the primitives of theses processes taken<br />at this stopping time. This result allows to compute the Laplace<br />transform of the price of a European call option on the maximum on<br />the yield in the generalized Vasicek model. Finally, we study the<br />resolvent density of these processes when the Lévy process is<br />$\alpha$-stable ($1 < \alpha \leq 2$). In particular, we<br />construct their $q$-scale function which generalizes the<br />Mittag-Leffler function.