The book is an original monograph on risk theory, which is traditionally considered a branch of insurance mathematics. It deals with different continuous-time risk models and mainly provides results obtained by the authors recently.

The book covers several aspects of risk theory. It provides a detailed investigation of the continuity and differentiability of the infinite-horizon and finite-horizon survival probabilities for different risk models. It gives possible applications of the results concerning the smoothness of the survival probabilities; in particular, models with franchise, deductible and liability limit are considered. The book introduces the supermartingale approach, which generalizes the martingale approach introduced by Gerber, to get upper exponential bounds for the ruin probabilities in models with investments.

The book is an excellent supplement to existing textbooks and monographs in this field. It will be useful for researchers in probability theory, actuarial sciences, and financial mathematics, as well as graduate and postgraduate students. Some its aspects will also definitely be interesting to practitioners.

Part 1. Smoothness of the Survival Probabilities with Applications

1. Classical Results on the Ruin Probabilities.

2. Classical Risk Model with Investments in a Risk-Free Asset.

3. Risk Model with Stochastic Premiums and Investments in a Risk-Free Asset.

4. Classical Risk Model with a Franchise and a Liability Limit.

5. Optimal Control by the Franchise and Deductible Amounts in the Classical Risk Model.

6. Risk Models with Investments in Risk-Free and Risky Assets.

Part 2. Supermartingale Approach to the Estimation of Ruin Probabilities.

7. Risk Model with Variable Premium Intensity and Investments in One Risky Asset.

8. Risk Model with Variable Premium Intensity and Investments in One Risky Asset up to the Stopping Time of Investment Activity.

9. Risk Model with Variable Premium Intensity and Investments in One Risk-Free and a Few Risky Assets.

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.