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## Smoothness, Bounds, Supermartingale Approach

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Request a sales quote*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.### Yuliya Mishura

### Olena Ragulina

- 1st Edition - October 10, 2016
- Authors: Yuliya Mishura, Olena Ragulina
- Language: English
- Hardback ISBN:9 7 8 - 1 - 7 8 5 4 8 - 2 1 8 - 2
- eBook ISBN:9 7 8 - 0 - 0 8 - 1 0 2 0 9 8 - 2

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 smooth… Read more

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- 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

Researchers in probability theory, actuarial sciences, and financial mathematics, as well as graduate and postgraduate students, and also accessible to practitioners who want to extend their knowledge in insurance mathematics

- Preface
- Part 1: Smoothness of the Survival Probabilities with Applications
- 1: Classical Results on the Ruin Probabilities
- Abstract
- 1.1 Classical risk model
- 1.2 Risk model with stochastic premiums

- 2: Classical Risk Model with Investments in a Risk-Free Asset
- Abstract
- 2.1 Description of the model
- 2.2 Continuity and differentiability of the infinite-horizon survival probability
- 2.3 Continuity of the finite-horizon survival probability and existence of its partial derivatives
- 2.4 Bibliographical notes

- 3: Risk Model with Stochastic Premiums Investments in a Risk-Free Asset
- Abstract
- 3.1 Description of the model
- 3.2 Continuity and differentiability of the infinite-horizon survival probability
- 3.3 Continuity of the finite-horizon survival probability and existence of its partial derivatives

- 4: Classical Risk Model with a Franchise and a Liability Limit
- Abstract
- 4.1 Introduction
- 4.2 Survival probability in the classical risk model with a franchise
- 4.3 Survival probability in the classical risk model with a liability limit
- 4.4 Survival probability in the classical risk model with both a franchise and a liability limit

- 5: Optimal Control by the Franchise and Deductible Amounts in the Classical Risk Model
- Abstract
- 5.1 Introduction
- 5.2 Optimal control by the franchise amount
- 5.3 Optimal control by the deductible amount
- 5.4 Bibliographical notes

- 6: Risk Models with Investments in Risk-Free and Risky Assets
- Abstract
- 6.1 Description of the models
- 6.2 Classical risk model with investments in risk-free and risky assets
- 6.3 Risk model with stochastic premiums and investments in risk-free and risky assets
- 6.4 Accuracy and reliability of uniform approximations of the survival probabilities by their statistical estimates
- 6.5 Bibliographical notes

- 1: Classical Results on the Ruin Probabilities
- Part 2: Supermartingale Approach to the Estimation of Ruin Probabilities
- 7: Risk Model with Variable Premium Intensity and Investments in One Risky Asset
- Abstract
- 7.1 Description of the model
- 7.2 Auxiliary results
- 7.3 Existence and uniqueness theorem
- 7.4 Supermartingale property for the exponential process
- 7.5 Upper exponential bound for the ruin probability
- 7.6 Bibliographical notes

- 8: Risk Model with Variable Premium Intensity and Investments in One Risky Asset up to the Stopping Time of Investment Activity
- Abstract
- 8.1 Description of the model
- 8.2 Existence and uniqueness theorem
- 8.3 Redefinition of the ruin time
- 8.4 Supermartingale property for the exponential process
- 8.5 Upper exponential bound for the ruin probability
- 8.6 Exponentially distributed claim sizes
- 8.7 Modification of the model

- 9: Risk Model with Variable Premium Intensity and Investments in One Risk-Free and a Few Risky Assets
- Abstract
- 9.1 Description of the model
- 9.2 Existence and uniqueness theorem
- 9.3 Supermartingale property for the exponential process
- 9.4 Upper exponential bound for the ruin probability
- 9.5 Case of one risky asset
- 9.6 Examples

- 7: Risk Model with Variable Premium Intensity and Investments in One Risky Asset
- Appendix: Mathematical Background
- A.1 Differentiability of integrals
- A.2 Hoeffding’s inequality
- A.3 Some results on one-dimensional homogeneous stochastic differential equations
- A.4 Itô’s formula for semimartingales

- Bibliography
- Abbreviations and Notation
- Index

- No. of pages: 276
- Language: English
- Edition: 1
- Published: October 10, 2016
- Imprint: ISTE Press - Elsevier
- Hardback ISBN: 9781785482182
- eBook ISBN: 9780081020982

YM

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.

Affiliations and expertise

Head, Department of Probability, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko Kyiv National University, Kiev, UkraineOR

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.

Affiliations and expertise

Taras Shevchenko National University of Kyiv, UkraineRead *Ruin Probabilities* on ScienceDirect