Non-life insurance pricing depends on different costs including claim and business acquisition costs, management expenses and other parameters such as margin for fluctuations in claims experience, expected profits etc. Nevertheless, in a competitive insurance market environment, company's premium should respond to changes in the level of premiums being offered by competitors. In this thesis, two major issues are being investigated. Primarily, it is explored how a company's optimal strategy can be determined in a competitive market and secondly a connection between this strategy and market's competition is established. More specifically, two functional equations for the volume of business are proposed. In the first place, the volume of business function is related to the past year's experience, the average premium of the market, the company's premium and a stochastic disturbance. Thus, an optimal premium strategy which maximizes the total expected linear discounted utility of company's wealth over a finite time horizon is defined analytically and endogenously. In the second place, the volume of business function is enriched with company's reputation, for the first time according to the author's knowledge. Moreover, the premium elasticity and reputation elasticity of the volume of business are taking into consideration. Thus, an optimal premium strategy which maximizes the total expected linear discounted utility of company's wealth over a finite time horizon is calculated and for some special cases analytical solutions are presented. Furthermore, an upper bound or a minimum premium excess strategy is found for a company with positive reputation and positive premium elasticity of the volume of business. Thirdly, the calculation of a fair premium in a competitive market is discussed. A nonlinear premium-reserve (P-R) model is presented and the premium is derived by minimizing a quadratic performance criterion concerns the present value of the reserve. The reserve is a stochastic equation, which includes an additive random nonlinear function of the state, premium and not necessarily Gaussian noise which is independently distributed in time, provided only that the mean value and the covariance of the random function is zero and a quadratic function of the state, premium and other parameters, respectively. In this quadratic representation of the covariance function, new parameters are implemented and enriched further the previous linear models, such as the income insurance elasticity of demand, the number of insured and the inflation in addition to the company's reputation. Interestingly, for the very first time, the derived optimal premium in a competitive market environment is also depended on the company's reserve among the other parameters. In each chapter numerical applications show the applicability of the proposed models and their results are further explained and analyzed. Finally, suggestions for further research and summary of the conclusions complete the thesis.
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