### Abstract

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 |
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Publisher | DTU Compute |
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Number of pages | 215 |

Publication status | Published - 2019 |

### Cite this

*Advances of matrix–analytic methods in risk modelling*. DTU Compute.

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*Advances of matrix–analytic methods in risk modelling*. DTU Compute.

**Advances of matrix–analytic methods in risk modelling.** / Peralta Gutierrez, Oscar.

Research output: Book/Report › Ph.D. thesis

TY - BOOK

T1 - Advances of matrix–analytic methods in risk modelling

AU - Peralta Gutierrez, Oscar

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

M3 - Ph.D. thesis

BT - Advances of matrix–analytic methods in risk modelling

PB - DTU Compute

ER -