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E-grāmata: Ruin Probabilities: Smoothness, Bounds, Supermartingale Approach

(Head, Department of Probability, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko Kyiv National University, Kiev, Ukraine), (Taras Shevchenko National University of Kyiv, Ukraine)
  • Formāts: EPUB+DRM
  • Izdošanas datums: 08-Nov-2016
  • Izdevniecība: ISTE Press Ltd - Elsevier Inc
  • Valoda: eng
  • ISBN-13: 9780081020982
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Ruin Probabilities: Smoothness, Bounds, Supermartingale ApproachPalielināt
  • Formāts: EPUB+DRM
  • Izdošanas datums: 08-Nov-2016
  • Izdevniecība: ISTE Press Ltd - Elsevier Inc
  • Valoda: eng
  • ISBN-13: 9780081020982
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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.
  • Provides new original results
  • Detailed investigation of the continuity and differentiability of the infinite-horizon and finite-horizon survival probabilities, as well as possible applications of these results
  • An excellent supplement to current textbooks and monographs in risk theory
  • Contains a comprehensive list of useful references

Papildus informācija

Presents different continuous-time risk models for this branch of insurance mathematics
Preface xi
Part 1 Smoothness of the Survival Probabilities with Applications
1(162)
Chapter 1 Classical Results on the Ruin Probabilities
3(24)
1.1 Classical risk model
3(19)
1.1.1 Description of the model
3(2)
1.1.2 Net profit condition and the behavior of the infinite-horizon ruin probability
5(1)
1.1.3 Integro-differential equations for the survival probabilities
6(2)
1.1.4 Integral equation for the infinite-horizon ruin probability
8(2)
1.1.5 Laplace transform of the infinite-horizon survival probability
10(1)
1.1.6 Analytic expressions for the infinite-horizon survival probability
11(2)
1.1.7 Cramer-Lundberg approximation for the infinite-horizon ruin probability
13(3)
1.1.8 Lundberg inequality for the infinite-horizon ruin probability
16(2)
1.1.9 Bibliographical notes
18(4)
1.2 Risk model with stochastic premiums
22(5)
1.2.1 Description of the model
22(1)
1.2.2 Basic results
23(2)
1.2.3 Bibliographical notes
25(2)
Chapter 2 Classical Risk Model with Investments in a Risk-Free Asset
27(22)
2.1 Description of the model
27(1)
2.2 Continuity and differentiability of the infinite-horizon survival probability
28(4)
2.3 Continuity of the finite-horizon survival probability and existence of its partial derivatives
32(15)
2.3.1 Main theorem
32(8)
2.3.2 Examples
40(7)
2.4 Bibliographical notes
47(2)
Chapter 3 Risk Model with Stochastic Premiums and Investments in a Risk-Free Asset
49(16)
3.1 Description of the model
49(1)
3.2 Continuity and differentiability of the infinite-horizon survival probability
50(9)
3.2.1 Main results
50(6)
3.2.2 Zero interest rate
56(3)
3.3 Continuity of the finite-horizon survival probability and existence of its partial derivatives
59(6)
Chapter 4 Classical Risk Model with a Franchise and a Liability Limit
65(40)
4.1 Introduction
65(3)
4.2 Survival probability in the classical risk model with a franchise
68(16)
4.2.1 Analytic expression for the survival probability
68(12)
4.2.2 Case of small and large enough initial surpluses
80(4)
4.3 Survival probability in the classical risk model with a liability limit
84(9)
4.3.1 Analytic expression for the survival probability
85(6)
4.3.2 Case of small enough and large enough initial surpluses
91(2)
4.4 Survival probability in the classical risk model with both a franchise and a liability limit
93(12)
4.4.1 Analytic expression for the survival probability
94(8)
4.4.2 Case of small initial surpluses
102(3)
Chapter 5 Optimal Control by the Franchise and Deductible Amounts in the Classical Risk Model
105(22)
5.1 Introduction
105(1)
5.2 Optimal control by the franchise amount
106(13)
5.2.1 Problem statement
106(2)
5.2.2 Hamilton--Jacobi--Bellman equation
108(2)
5.2.3 Existence theorem
110(3)
5.2.4 Verification theorem
113(3)
5.2.5 Exponentially distributed claim sizes
116(3)
5.3 Optimal control by the deductible amount
119(5)
5.3.1 Problem statement
119(2)
5.3.2 Hamilton--Jacobi--Bellman equation
121(1)
5.3.3 Existence and verification theorems
122(1)
5.3.4 Exponentially distributed claim sizes
123(1)
5.4 Bibliographical notes
124(3)
Chapter 6 Risk Models with Investments in Risk-Free and Risky Assets
127(36)
6.1 Description of the models
127(2)
6.2 Classical risk model with investments in risk-free and risky assets
129(18)
6.2.1 Upper and lower bounds for the infinite-horizon survival probability
129(7)
6.2.2 Continuity and differentiability of the infinite-horizon survival probability
136(4)
6.2.3 Continuity of the finite-horizon survival probability and existence of its partial derivatives
140(7)
6.3 Risk model with stochastic premiums and investments in risk-free and risky assets
147(3)
6.4 Accuracy and reliability of uniform approximations of the survival probabilities by their statistical estimates
150(9)
6.4.1 Finite-horizon survival probability
150(7)
6.4.2 Infinite-horizon survival probability
157(2)
6.5 Bibliographical notes
159(4)
Part 2 Supermartingale Approach to the Estimation of Ruin Probabilities
163(68)
Chapter 7 Risk Model with Variable Premium Intensity and Investments in One Risky Asset
165(22)
7.1 Description of the model
165(2)
7.2 Auxiliary results
167(8)
7.2.1 Sufficient conditions for the solution to explode
167(5)
7.2.2 Finiteness of the first exit time
172(3)
7.3 Existence and uniqueness theorem
175(1)
7.4 Supermartingale property for the exponential process
176(5)
7.5 Upper exponential bound for the ruin probability
181(3)
7.6 Bibliographical notes
184(3)
Chapter 8 Risk Model with Variable Premium Intensity and Investments in One Risky Asset up to the Stopping Time of Investment Activity
187(18)
8.1 Description of the model
187(2)
8.2 Existence and uniqueness theorem
189(1)
8.3 Redefinition of the ruin time
190(2)
8.4 Supermartingale property for the exponential process
192(4)
8.5 Upper exponential bound for the ruin probability
196(3)
8.6 Exponentially distributed claim sizes
199(1)
8.7 Modification of the model
200(5)
Chapter 9 Risk Model with Variable Premium Intensity and Investments in One Risk-Free and a Few Risky Assets
205(26)
9.1 Description of the model
205(4)
9.2 Existence and uniqueness theorem
209(1)
9.3 Supermartingale property for the exponential process
209(5)
9.4 Upper exponential bound for the ruin probability
214(9)
9.5 Case of one risky asset
223(1)
9.6 Examples
224(7)
Appendix 231(8)
Bibliography 239(16)
Abbreviations and Notation 255(4)
Index 259
Yuliya Mishura is Professor and Head of the Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Ukraine. Her research interests include stochastic analysis, theory of stochastic processes, stochastic differential equations, numerical schemes, financial mathematics, risk processes, statistics of stochastic processes, and models with long-range dependence. Olena Ragulina is Junior Researcher at the Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Ukraine Her research interests include actuarial and financial mathematics.
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