On some First Passage Time Problems Motivated by Financial Applications
Sur certains problemes de premier temps de passage motives par des applications financieres
Résumé
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.
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.
Domaines
Mathématiques [math]
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