Theses

On some First Passage Time Problems Motivated by Financial Applications

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 Lévy 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 Lévy 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 Lévy process is

$\alpha$-stable ($1 < \alpha \leq 2$). In particular, we

construct their $q$-scale function which generalizes the

Mittag-Leffler function.

Cited literature [119 references]

https://tel.archives-ouvertes.fr/tel-00009074

Contributor : Pierre Patie <>

Submitted on : Saturday, April 23, 2005 - 6:52:02 PM

Last modification on : Thursday, October 10, 2019 - 1:14:01 AM

Long-term archiving on: : Friday, September 14, 2012 - 12:40:19 PM

Contributor : Pierre Patie <>

Submitted on : Saturday, April 23, 2005 - 6:52:02 PM

Last modification on : Thursday, October 10, 2019 - 1:14:01 AM

Long-term archiving on: : Friday, September 14, 2012 - 12:40:19 PM