Study of non-life insurance markets using Nash equilibrium and dependence risk models.

Summary Non-life actuarial science studies the various quantitative aspects of the insurance business. This thesis aims to explain from different perspectives the interactions between the different economic agents, the insured, the insurer and the market, in an insurance market. Chapter 1 highlights the importance of taking into account the market premium in the policyholder's decision to renew or not to renew his insurance contract with his current insurer. The need for a market model is established. Chapter 2 addresses this issue by using non-cooperative game theory to model competition. In the current literature, models of competition are always reduced to a simplistic optimization of premium volume based on a view of one insurer against the market. Starting from a one-period market model, a set of insurers is formulated, where the existence and uniqueness of the Nash equilibrium are verified. The properties of equilibrium premiums are studied to better understand the key factors of a dominant position of one insurer over the others. Then, the integration of the one-period game in a dynamic framework is done by repeating the game over several periods. A Monte-Carlo approach is used to evaluate the probability of an insurer being ruined, remaining leader, or disappearing from the game due to a lack of policyholders in its portfolio. This chapter aims at better understanding the presence of cycles in non-life insurance. Chapter 3 presents in depth the actual Nash equilibrium calculation for n players under constraints, called generalized Nash equilibrium. It provides an overview of optimization methods for solving the n optimization subproblems. This solution is done using a semi-smooth equation based on the Karush-Kuhn-Tucker reformulation of the generalized Nash equilibrium problem. These equations require the use of the generalized Jacobian for the locally Lipschitzian functions involved in the optimization problem. A convergence study and a comparison of the optimization methods are performed. Finally, chapter 4 deals with the calculation of the probability of ruin, another fundamental theme in non-life insurance. In this chapter, a risk model with dependence between the amounts or waiting times of claims is studied. New asymptotic formulas for the probability of ruin in infinite time are obtained in a broad framework of risk models with dependence between claims. In addition, explicit formulas for the probability of ruin in discrete time are obtained. In this discrete model, the dependence structure analysis allows to quantify the maximum deviation on the joint distribution functions of the amounts between the continuous and the discrete version.