Markovian risk model
- 주제(키워드) risk model
- 발행기관 고려대학교 대학원
- 지도교수 김바라
- 발행년도 2021
- 학위수여년월 2021. 2
- 학위구분 박사
- 학과 대학원 수학과
- 원문페이지 122 p
- UCI I804:11009-000000235606
- DOI 10.23186/korea.000000235606.11009.0001203
- 본문언어 영어
초록/요약
In this thesis, we investigate risk problems in a Markovian environment. We begin by reviewing relevant literatures for Markov risk models. First, we introduced transform approaches about Markov risk models. Next, we searched for papers about ruin problems in a discrete risk model in a Markovian environment. Finally, we introduced particular ruin problems for general risk models. We continue by considering a transform approach for discounted aggregate claims in a risk model with descendant claims. We consider a risk model with three types of losses: ordinary, leading, and descendant losses. We derive an expression for the Laplace-Stieltjes transform of the distribution of the discounted aggregate claims. By using this expression, we can then obtain the mean and variance of the discounted aggregate claims. For actuarial applications, the VaR and CTE are computed by numerical inversion of the Laplace transforms for the tail probability and the conditional tail expectation of the discounted aggregate claims. The net premium for stop-loss reinsurance contract is also computed. Next, we studied ruin problems in a discrete risk model in a Makorvian environment. We find that the derivations of Yang et al. [91] are erroneous and then correctly analyze the model. Lastly, we consider the Parisian ruin problems in a discrete-time Markov- modulated dual risk model, where the gain process is governed by the underlying Markov process with a finite state space. By using the strong Markov property of the risk process, we derive recursive expressions for the conditional probability generating functions of the classical ruin time and the Parisian ruin time. From this we can obtain the infinite-time ruin probabilities, but it also enables us to compute the finite- time ruin probabilities by numerical inversion. In addition, in the case when the gain amounts have discrete phase-type distributions, we obtain specialized expressions for the probability generating functions of the classical and Parisian ruin times, which can be used to reduce computational works for the numerical computation of the ruin probabilities. Finally, we present numerical examples for the computations of the finite-time and infinite-time ruin probabilities.
more목차
1 Introduction 12
1.1 Background and motivation 12
1.2 Overview of the thesis 21
2 Literature review 25
2.1 Transform approach for Markovian risk model 30
2.2 Discrete risk model in Markovian environment 33
2.3 Particular ruin problems 40
3 Transform approach for discounted aggregate claims in a risk model with descendant claims 44
3.1 Introduction 44
3.2 The model 47
3.2.1 Ordinary losses 47
3.2.2 Leading losses 48
3.2.3 Descendant losses 48
3.3 The discounted value of aggregate losses 51
3.3.1 The LST of Lδ(t) 52
3.3.2 The mean and variance of Lδ(t) 57
3.4 Actuarial applications 52
3.4.1 VaR and CTE risk measures 54
3.4.2 Stop-loss reinsurance premium 56
3.5 Conclusion 57
4 Ruin problems in a discrete risk model in a Markovian environment 71
4.1 Introduction and background 71
4.2 The discrete risk model with Markov environment 72
4.3 Analysis of the risk model 74
5 Parisian ruin in a discrete-time Markov-modulated dual risk model 81
5.1 Introduction 81
5.2 The mode 85
5.3 The transforms of the classical and Parisian ruin times 87
5.3.1 The transform of the classical ruin time 87
5.3.2 The transform of the Parisian ruin time 90
5.4 Markov-modulated model with phase-type distributed gains 95
5.5 Numerical results 102
