On some First Passage Time Problems Motivated by Financial Applications

Abstract

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