Sur certains problemes de premier temps de passage motives par des applications financieres
| Autor: | Patie, Pierre |
|---|---|
| Přispěvatelé: | Department of Mathematics [ETH Zurich] (D-MATH), Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich] (ETH Zürich), Universität Zürich, Delbaen Freddy(delbaen@math.ethz.ch) |
| Jazyk: | angličtina |
| Rok vydání: | 2004 |
| Předmět: |
moving boundary
WIENER PROCESSES + BROWNIAN MOTION (PROBABILITY THEORY) DIFFUSION PROCESSES (PROBABILITY THEORY) brownian motion LÉVY PROCESSES (STOCHASTIC PROCESSES) WIENER-PROZESSE + BROWNSCHE BEWEGUNG (WAHRSCHEINLICHKEITSRECHNUNG) 01 natural sciences Bessel processes mouvement brownien 010104 statistics & probability LÉVYPROZESSE (STOCHASTISCHE PROZESSE) options exotiques path-dependent options ornstein-uhlenbeck FOS: Mathematics first passage time 0101 mathematics [MATH]Mathematics [math] DIFFUSIONSPROZESSE (WAHRSCHEINLICHKEITSRECHNUNG) Levy processes markov process processus de Levy 010102 general mathematics premier temps de passage finance mathematique mathematical finance processus de Bessel Mathematics |
| Zdroj: | Mathematics [math]. Universität Zürich, 2004. English |
| Popis: | Alili LarbiNovikov AlexanderSchweizer MartinYor Marc; From both theoretical and applied perspectives, first passagetime problems for random processes are challenging and of greatinterest. In this thesis, our contribution consists on providingexplicit or quasi-explicit solutions for these problems in twodifferent settings.In the first one, we deal with problems related to thedistribution of the first passage time (FPT) of a Brownian motionover a continuous curve. We provide several representations forthe density of the FPT of a fixed level by an Ornstein-Uhlenbeckprocess. This problem is known to be closely connected to the oneof 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 isused to illustrate a relationship which we establish between thelaws of the FPT of a Brownian motion over a twice continuouslydifferentiable curve and the quadratic and linear ones. Finally,we introduce a transformation which maps a continuous functioninto a family of continuous functions and we establish itsanalytical and algebraic properties. We deduce a simple andexplicit relationship between the densities of the FPT over eachelement of this family by a selfsimilar diffusion. In the second setting, we are concerned with the study ofexit problems associated to Generalized Ornstein-Uhlenbeckprocesses. These are constructed from the classicalOrnstein-Uhlenbeck process by simply replacing the drivingBrownian motion by a Lévy process. They are diffusions withpossible jumps. We consider two cases: The spectrally negativecase, that is when the process has only downward jumps and thecase when the Lévy process is a compound Poisson process withexponentially distributed jumps. We derive an expression, in termsof new special functions, for the joint Laplace transform of theFPT of a fixed level and the primitives of theses processes takenat this stopping time. This result allows to compute the Laplacetransform of the price of a European call option on the maximum onthe yield in the generalized Vasicek model. Finally, we study theresolvent density of these processes when the Lévy process is$\alpha$-stable ($1 < \alpha \leq 2$). In particular, weconstruct their $q$-scale function which generalizes theMittag-Leffler function. |
| Databáze: | OpenAIRE |
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