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Abstract
This work is concerned with the study of matrix–analytic methods with novel applications to the area of risk theory.
First, we review some topics of Applied Probability such as phase–type distributions, matrix–exponential distributions, Markovian arrival processes, Rational arrival processes, ﬂuid ﬂow processes and risk models. With these tools in hand, we propose a method to approximate the probability of ruin of any Cramér–Lundberg process using the theory of phase–type distributions, providing an error bound for such an approximation. With the goal of studying risk models with dependencies, we construct a class of bivariate distributions with given phase–type–distributed marginals and given Pearson’s correlation coeﬃcient, which are later used to deﬁne diﬀerent kinds of dependent Sparre–Andersen processes. Later on, we give an explicit formula for the probability of Parisian and cumulative Parisian ruin for a class of risk processes which are based on the theory of ﬂuid ﬂow processes. Next, we study some excursion properties of spectrally negative Lévy processes whenever they are inspected at an independent matrix–exponential time. Finally, inspired by the generalisation of the Markovian arrival process to the Rational arrival process, we construct a novel generalisation of the ﬂuid ﬂow process without Brownian components and study its ﬁrst passage probabilities.
First, we review some topics of Applied Probability such as phase–type distributions, matrix–exponential distributions, Markovian arrival processes, Rational arrival processes, ﬂuid ﬂow processes and risk models. With these tools in hand, we propose a method to approximate the probability of ruin of any Cramér–Lundberg process using the theory of phase–type distributions, providing an error bound for such an approximation. With the goal of studying risk models with dependencies, we construct a class of bivariate distributions with given phase–type–distributed marginals and given Pearson’s correlation coeﬃcient, which are later used to deﬁne diﬀerent kinds of dependent Sparre–Andersen processes. Later on, we give an explicit formula for the probability of Parisian and cumulative Parisian ruin for a class of risk processes which are based on the theory of ﬂuid ﬂow processes. Next, we study some excursion properties of spectrally negative Lévy processes whenever they are inspected at an independent matrix–exponential time. Finally, inspired by the generalisation of the Markovian arrival process to the Rational arrival process, we construct a novel generalisation of the ﬂuid ﬂow process without Brownian components and study its ﬁrst passage probabilities.
Original language  English 

Publisher  DTU Compute 

Number of pages  215 
Publication status  Published  2019 
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Projects
 1 Finished

Matrixexponential methods in finance, risk and queuing theory
Peralta Gutierrez, O., Nielsen, B. F., Bladt, M., Thygesen, U. H., Breuer, L. & Hobolth, A.
01/10/2015 → 13/03/2019
Project: PhD