Εκθετικές ανισότητες για πιθανότητες χρεοκοπίας διαδικασιών κινδύνου με όρο διάχυσης
Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion
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Keywords
Θεωρία κινδύνου ; Markov processes ; Θεωρία πιθανοτήτων ; ΧρεοκοπίαAbstract
The classic model of Risk Theory has been extended by Gerber by adding a
diffusion process to the Poisson compound process. The study of this model was at first
made by Dufresne & Gerber in 1991. It turns out that the probabilities of bankruptcy
satisfy specific renewal equations. The type of convolution and a Lundberg inequality for
probability of bankruptcy in a similar way as in the classic case is also demonstrated.
Then it is studied the construction of a class of diffusion processes followed locally
by a vector field and the expanded generator is calculated for a subset of the generator
definition field. Using this theory, martingales are constructed for a diffusion-based risk
process. This leads us to exponential inequalities for the probability of bankruptcy in
infinity as well as in finite time.