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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 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.
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Contributor : Pierre Patie <>
Submitted on : Saturday, April 23, 2005 - 6:52:02 PM
Last modification on : Thursday, October 10, 2019 - 1:14:01 AM
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  • HAL Id : tel-00009074, version 1



Pierre Patie. On some First Passage Time Problems Motivated by Financial Applications. Mathematics [math]. Universität Zürich, 2004. English. ⟨tel-00009074⟩



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