Comparison of semimartingales and Lévy processes with applications to financial mathematics [Elektronische Ressource] / vorgelegt von Jan Bergenthum

Comparison of semimartingalesand Levy processes withapplications to nancialmathematicsDissertationzur Erlangung des Doktorgradesder Fakult at fur Mathematik und Physikder Albert-Ludwigs-UniversitatFreiburg im Breisgauvorgelegt vonJan BergenthumJuli 2005Dekan: Prof. Dr. J. Honerkamp1. Referent: Prof. Dr. L. Rusc hendorf2. Referent: Prof. Dr. J. KallsenDatum der Promotion: 07. Oktober 2005ContentsIntroduction i1 Comparison of semimartingales 11.1 Multivariate comparison results . . . . . . . . . . . . . . . . 41.1.1 Comparison in terms of local characteristics of thestochastic logarithms . . . . . . . . . . . . . . . . . . 41.1.2 Comparison in terms of local characteristics . . . . . 111.2 Propagation of order property . . . . . . . . . . . . . . . . . 161.2.1 Monotone convexity for diusions . . . . . . . . . . . 171.2.2 Monotone convexity for diusions with jumps . . . . 211.2.3 PO(g) for processes with independent increments . . 262 Comparison of Levy processes 292.1 Compound Poisson processes. . . . . . . . . . . . . . . . . . 312.2 Levy processes with in nite Levy measures . . . . . . . . . . 412.3 Extension to PII . . . . . . . . . . . . . . . . . . . . . . . . 543 Applications 593.1 Non-trivial bounds for European option prices . . . . . . . . 603.2 Comparison of martingale measures . . . . . . . . . . . . . . 713.2.1 Jump models . . . . . . . . . . . . . . . . . . . . . . 733.2.2 Stochastic volatility models . . . . . . . . . .