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| 研究生： |
潘柏樺 Pan, Po-Hua |
|---|---|
| 論文名稱： |
Crámer-Lundberg 風險模型及其擴散近似之最適再保險策略 Optimal Proportional Reinsurance Strategies for Classical Crámer-Lundberg Risk Model and It’s Corresponding Diffusion Approximations |
| 指導教授： |
許順吉
Sheu, Shuenn-Jyi |
| 口試委員: |
許順吉
Sheu, Shuenn-Jyi 姜祖恕 Chiang, Tzuu-Shuh 孫立憲 Sun, Li-Hsien |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2023 |
| 畢業學年度： | 111 |
| 語文別： | 英文 |
| 論文頁數： | 55 |
| 中文關鍵詞： | 部分再保險 、隨機控制 、隨機過程 、哈密頓-雅可比-貝爾曼方程式 |
| 外文關鍵詞： | Proportional reinsurance, Stochastic control, Stochastic process, HJB equation |
| 相關次數： | 點閱：105 下載：21 |
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再保險是保險公司管理風險的有力工具。如果我們將保險公司隨時間變化的淨利潤建模為一個隨機過程，將再保險策略視為一個控制過程，那麼最小化這種隨機過程的破產機率就是一個隨機控制問題。本文旨在尋找最適再保險策略，使破產機率最小化。與許多其他隨機控制問題一樣，我們使用哈密頓-雅可比-貝爾曼方程來求解該問題。
Reinsurance is a powerful tool for insurance company to manage the risk. If we model the net profit of insurance company over time as a stochastic process and view the reinsurance strategy as a control process, then to minimize the ruin probability of such stochastic process is a stochastic control problem. This article aims to find the optimal reinsurance strategy so that the ruin probability to be minimized. As many other stochastic control problem, we use the Hamilton-Jacobi-Bellman (HJB) equation to solve the problem.
Chapter 1 Introduction 1
Chapter 2 The Risk Model and Ruin Probability 3
Section 2.1 The Classical Cramér-Lundberg Model Risk Model 3
Section 2.2 The Ruin Probability 5
Section 2.3 Upper Bound of the Ruin Probability 7
Chapter 3 Reinsurance Strategies 11
Chapter 4 Optimal Proportional Reinsurance Strategy under Classic Cramér-Lundberg Risk Model 14
Section 4.1 HJB Formulation 16
Section 4.2 Solution of the HJB equation 18
Section 4.3 Verification Theorem 24
Chapter 5 Optimal Proportional Reinsurance Strategy under Diffusion Approximation Model 27
Section 5.1 Diffusion Approximation of Cramér-Lundburg Process 28
Section 5.2 Properties of Controlled Ruin Probability under Diffusion Approximation Model 30
Section 5.3 HJB Formulation 35
Section 5.4 Solution of the HJB equation 38
Section 5.5 Verification Theorem 41
Chapter 6 Numerical Results 44
Section 6.1 Path Simulation and the Monte Carlo Method 44
Section 6.2 Estimation of the Ruin Probability in Finite Time Interval 46
Section 6.3 Conclusions 52
Bibliography 53
Appendix A Lévy Process and Strong Markov Property 54
[1] Hanspeter Schmidli (2007): Stochastic Control in Insurance, 2007.
[2] Hanspeter Schmidli (2017): Risk Theory, 2017.
[3] Jan Grandell (1977): A class of approximations of ruin probabilities, Scandinavian Actuarial Journal, 1977:sup1, 37-52.
[4] Donald L. Iglehart (1969): Diffusion Approximations in Collective Risk Theory, Journal of Applied Probability, 1969: 285-289.
[5] Bernt Øksendal (2000): Stochastic Differential Equations, 2000.
[6] Jean-François Le Gall (2016): Brownian motion, Martingales, and Stochastic Calculus, 2016.
[7] Jean Jacod, Philip Protter (2004): Probability Essential, 2004.
