Ruin Probabilities: Smoothness, Bounds, Supermartingale Approach deals with continuous-time risk models and covers several aspects of risk theory. The first of them is the smoothness of the survival probabilities. In particular, the book provides a detailed investigation of the continuity and differentiability of the infinite-horizon and finite-horizon survival probabilities for different risk models. Next, it gives some possible applications of the results concerning the smoothness of the survival probabilities. Additionally, the book introduces the supermartingale approach, which generalizes the martingale one introduced by Gerber, to get upper exponential bounds for the infinite-horizon ruin probabilities in some generalizations of the classical risk model with risky investments.
1. 4.1 Introduction - p. 65 2. 4.2 Survival probability in the classical risk model with a franchise - p. 67 3. 4.3 Survival probability in the classical risk model with a liability limit - p. 84 4. 4.4 Survival probability in the classical risk model with both a franchise and a liability limit - p. 93 5. 5: Optimal Control by the Franchise and Deductible Amounts in the Classical Risk Model - p. 105 6. 6: Risk Models with Investments in Risk-Free and Risky Assets - p. 127 7. Abstract - p. 127 8. 6.1 Description of the models - p. 127 9. 6.2 Classical risk model with investments in risk-free and risky assets - p. 128 10. 6.3 Risk model with stochastic premiums and investments in risk-free and risky assets - p. 147 11. 6.4 Accuracy and reliability of uniform approximations of the survival probabilities by their statistical estimates - p. 150 12. 6.5 Bibliographical notes - p. 159 13. Part 2: Supermartingale Approach to the Estimation of Ruin Probabilities - p. 162 14. 7: Risk Model with Variable Premium Intensity and Investments in One Risky Asset - p. 165 15. 8: Risk Model with Variable Premium Intensity and Investments in One Risky Asset up to the Stopping Time of Investment Activity - p. 187 16. 9: Risk Model with Variable Premium Intensity and Investments in One Risk-Free and a Few Risky Assets - p. 205 17. Abstract - p. 205 18. 9.1 Description of the model - p. 205 19. 9.2 Existence and uniqueness theorem - p. 208 20. 9.3 Supermartingale property for the exponential process - p. 209 21. 9.4 Upper exponential bound for the ruin probability - p. 214 22. 9.5 Case of one risky asset - p. 222 23. 9.6 Examples - p. 224 24. Appendix: Mathematical Background - p. 231 25. Bibliography - p. 239 26. Abbreviations and Notation - p. 255 27. Index - p. 259
Ruin Probabilities: Smoothness, Bounds, Supermartingale Approach deals with continuous-time risk models and covers several aspects of risk theory. The first of them is the smoothness of the survival probabilities. In particular, the book provides a detailed investigation of the continuity and differentiability of the infinite-horizon and finite-horizon survival probabilities for different risk models. Next, it gives some possible applications of the results concerning the smoothness of the survival probabilities. Additionally, the book introduces the supermartingale approach, which generalizes the martingale one introduced by Gerber, to get upper exponential bounds for the infinite-horizon ruin probabilities in some generalizations of the classical risk model with risky investments.
1. 4.1 Introduction - p. 65 2. 4.2 Survival probability in the classical risk model with a franchise - p. 67 3. 4.3 Survival probability in the classical risk model with a liability limit - p. 84 4. 4.4 Survival probability in the classical risk model with both a franchise and a liability limit - p. 93 5. 5: Optimal Control by the Franchise and Deductible Amounts in the Classical Risk Model - p. 105 6. 6: Risk Models with Investments in Risk-Free and Risky Assets - p. 127 7. Abstract - p. 127 8. 6.1 Description of the models - p. 127 9. 6.2 Classical risk model with investments in risk-free and risky assets - p. 128 10. 6.3 Risk model with stochastic premiums and investments in risk-free and risky assets - p. 147 11. 6.4 Accuracy and reliability of uniform approximations of the survival probabilities by their statistical estimates - p. 150 12. 6.5 Bibliographical notes - p. 159 13. Part 2: Supermartingale Approach to the Estimation of Ruin Probabilities - p. 162 14. 7: Risk Model with Variable Premium Intensity and Investments in One Risky Asset - p. 165 15. 8: Risk Model with Variable Premium Intensity and Investments in One Risky Asset up to the Stopping Time of Investment Activity - p. 187 16. 9: Risk Model with Variable Premium Intensity and Investments in One Risk-Free and a Few Risky Assets - p. 205 17. Abstract - p. 205 18. 9.1 Description of the model - p. 205 19. 9.2 Existence and uniqueness theorem - p. 208 20. 9.3 Supermartingale property for the exponential process - p. 209 21. 9.4 Upper exponential bound for the ruin probability - p. 214 22. 9.5 Case of one risky asset - p. 222 23. 9.6 Examples - p. 224 24. Appendix: Mathematical Background - p. 231 25. Bibliography - p. 239 26. Abbreviations and Notation - p. 255 27. Index - p. 259
