Applications of Levy Processes to Mathematical Finance

本文研究了不完全资产市场下的期权定价问题，这是数学金融领域的重要问题之一。我们的目标是构建[几何Levy过程& MEMM]定价模型，其中采用几何Levy过程作为基础资产价格过程，并采用MEMM（=最小熵鞅测度）作为鞅测度。研究了该模型构建的基本理论以及该模型在期权定价中的应用。首先建立了几何列维过程的MEMM存在性定理，然后研究了MEMM的性质以及[几何列维过程& MEMM]定价模型的性质。特别对MEMM与Esscher鞅测度之间的关系进行了比较研究。我们研究了该模型的应用方法，例如Levy过程的估计方法。我们还研究了模型的校准问题。

We have studied the option pricing problems in the incomplete asset market, which is one of the important problems in the field of mathematical finance. Our goal is the construction of the [Geometric Levy process & MEMM] pricing model, in which the geometric Levy processes are adopted as the underlying asset price processes and the MEMM (=minimal entropy martingale measure) is adopted as the martingale measure. And we have investigated the fundamental theories for the construction of this model and the applications of this mode to the option pricing.We first established the existence theorem of MEMM for the geometric Levy processes, and we nest investigated the properties of MEMM and the properties of the [Geometric Levy process & MEMM] pricing model. Especially we have studied the relations between the MEMM and the Esscher martingale measure comparing each other. We investigated the methods for the application of this model, for example the method for the estimation of Levy processes. We also investigated the calibration problems of our model.