Large Deviations And Minimal Entropy Martingale Measures For Several Models In Finance And Insurance

In the first chapter, we elaborate the background of several models in finance and insurance, give out our main results, finally, we list some knowledge we need in this article.In the second chapter, we consider three kinds of risk processes:risk pro-cess with reinsurance, risk process with delayed claims and Cox risk process with Poisson shot noise intensity. We give out the sample path large and moderate deviation principles for these risk processes on finite interval by em-ploying the method of constructing exponential martingales which solve not only the finite dimensional large and moderate deviations but also correspond-ing exponential tightness. Furthermore, for the Cox risk process with Poisson shot noise intensity, we obtain the sample path large and moderate deviation principles on infinite interval, and the large and moderate deviation principles with perturbation on both finite and infinite interval are also achieved. As the applications of sample path large and moderate deviation principles, we get the asymptotic behaviors of ruin probabilities under two different speeds for the risk process with delayed claims and the Cox risk process with Poisson shot noise intensity. We also get the exponential inequalities instead of the Lundberg asymptotic estimations for the ruin probabilities of the three risk processes mentioned above by applying the corresponding exponential mar-tingales associated with the risk processes. Finally, we give out numerical examples for every risk process to support our results.In the third chapter, we consider the problem of the minimal entropy martingale measures for the price process with Poisson shot noise jumps. We give out the minimal entropy martingale measure for the price process and the corresponding expression of relative entropy by the exponential martin-gale method. Finally, we give out a pricing formula of a kind of catastrophe insurance options by constructing an exponential martingale.