Optimal Investments For Insurers With Different Risk Processes In An Incomplete Market

The study of optimal investment for an insurer plays an important role inactuarial science. It combines finance with research on insurance and is highlyvaluable for academic studies and applications. Recently, there has been muchattention to the problem of optimal investment for an insurer. Since somemodels in most papers are not realistic, this paper will be devoted to makingthe problem and model considered more practical. This thesis studies theoptimal investment problems for insurers with di?erent risk processes in anincomplete market. Extensions of the existing works are provided from twoaspects of risk model and financial market structure, respectively.On one hand, optimal investment problems for insurers with di?erentrisk processes are discussed in Chapter 3, Chapter 4 and Chapter 5. Theinsurers can invest in multiple risky assets whose price processes are geometricBrownian motions and a risk-free asset in an incomplete market. In Chapter 3,the Cram′er-Lundberg model is adopted for the risk reserve. After the marketis completed, closed-form solutions to the problems of expected quadratic andexponential utility maximization are obtained via the martingale approach.Finally, computational results are presented for given raw market date. InChapter 4, the risk process is a periodic risk process perturbed by a standardBrownian motion and the correlations between the prices of risky assets andthe risk process are considered. The insurer aims to maximize the adjustmentcoe?cient of the risk model. Exponential bounds for ruin probability andoptimal strategy are obtained explicitly by using martingale approach. Thee?ects of market incompleteness and other parameters on the optimal strategyare discussed and a numerical example is also given. In addition, owing to theinvestment on risky assets and a risk-free asset, we don't need the assumptionson the safety loading. Moreover, two di?erent methods for market completionare applied in Section 4.1-4.4 and Section 4.5, respectively, and the two optimaliii strategies are equivalent. In Chapter 5, the risk model is a modified jump-di?usion risk process and the claim process is supposed to be an increasingpure jump process, which is more general than compound Poisson process.The objective is to maximize the expected exponential utility of the insurer'sreserve at a future time and explicit optimal strategy is obtained. Due to thespecific nature of exponential utility function, the approach in this section ismore concise than martingale approach.On the other hand, Chapter 6 studies the portfolio selection problem foran investor who seeks to maximize the expected utility of terminal wealth ina defined contribution pension plan. The price dynamics of risky assets aredescribed by the constant elasticity of variance (CEV) model, which is anextension of geometric Brownian motion. However, the current researches ofoptimization problem under the CEV model concern only one risky asset anda risk-free asset. Thus, to make the optimization problem even more realis-tic, Chapter 6 deals with the investment problem with a risk-free asset andmultiple risky assets under the CEV model in an incomplete market. Butthe introduction of multiple risky assets does give rise to di?culties and ex-plicit solutions are obtained only for special cases. In Section 6.1 and Section6.2, by applying the approach of Hamilton-Jacobi-Bellman (HJB) equation,we derive the explicit solutions for the exponential and power utility func-tions under special conditions, respectively. In Section 6.3, under assumptionson the correlations between di?erent risky assets, explicit optimal strategiesfor the exponential and power utility functions are derived for general casevia the Legendre transform and variable change technique. Furthermore, theproperties of the optimal strategies are analyzed and the e?ects of elasticitycoe?cient on the optimal strategies are discussed. It is shown that for a port-folio selection problem concerning risky assets with the CEV price processes,the n dimensional case is quite di?erent from one dimensional case.