### Book

## АКТУАРНЫЕ РАСЧЕТЫ в 2 ч. Часть1

The textbook is a complete introductory course in actuarial mathematics. The basic concepts of insurance and actuarial calculations, financial mathematics and demographic statistics are revealed. The principles and definitions of insurance, the tasks of actuarial calculations, the structure of the insurance premium and the main approaches to its calculation, methods of forming insurance reserves in insurance other than life insurance, life insurance and pensions are considered. The theoretical material is illustrated with numerous examples and tasks, in addition, an extensive set of tasks for seminars and independent work of students is offered. The textbook contains questions and tasks for self-control on all topics. The appendices include mathematical-statistical and demographic tables that are necessary for performing independent calculations and solving problems.

The paper studies a problem of optimal insurer’s choice of a risk-sharing policy in a dynamic risk model, so-called Cramer-Lundberg process, over infinite time interval. Additional constraints are imposed on residual risks of insureds: on mean value or with probability one. An optimal control problem of minimizing a functional of the form of variation coefficient is solved. We show that: in the first case the optimum is achieved at stop loss insurance policies, in the second case the optimal insurance is a combination of stop loss and deductible policies. It is proved that the obtained results can be easily applied to problems with other optimization criteria: maximization of long-run utility and minimization of probability of a deviation from mean trajectory.

The chapter studies a dynamic risk model defined on infinite time interval, where both insurance and per-claim reinsurance policies are chosen by the insurer in order to minimize a functional of the form of variation coefficient under constraints imposed with probability one on insured's and reinsurer's risks. We show that the optimum is achieved at constant policies, the optimal reinsurance is a partial stop loss reinsurance and the optimal insurance is a combination of stop loss and deductible policies. The results are illustrated by a numerical example involving uniformly distributed claim sizes.

A singular boundary value problem for a second order linear integrodifferential equation with Volterra and nonVolterra integral operators is formulated and analyzed. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer–Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market.

The concept of economic equilibrium under uncertainty is applied to a model of insurance market where, in distinction to the classic Borch's model of a reinsurance market, risk exchanges are allowed between the insurer and each insured only, not among insureds themselves. Conditions characterizing an equilibrium are found. A variant of the conditions, based on the Pareto optimality notion and involving risk aversion functions of the agents, is derived. An existence theorem is proved. Computation of the market premiums and optimal indemnities is illustrated by an example with exponential utility functions.