马尔可夫过程在Girsanov变换下的性质及其应用

项目名称： 马尔可夫过程在Girsanov变换下的性质及其应用

项目编号： No.11201221

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 宋瑞丽

作者单位： 南京财经大学

项目金额： 22万元

中文摘要： Girsanov变换是马尔可夫过程理论中重要的变换之一，Girsanov定理在金融数学特别是期权定价理论中起着重要的作用，研究马尔可夫过程在Girsanov变换下的性质及其应用具有重要的理论意义和应用价值。虽已有很多学者研究了马尔可夫过程在Girsanov变换下的性质，但大都是关于对称马尔可夫过程的研究，采用狄氏型的研究方法。本项目以一般（非对称）的马尔可夫过程为研究对象，同时假设此马尔可夫过程是一个半鞅，研究此马尔可夫过程在局部绝对连续测度下的转移密度函数的表达形式以及位势分析（包括：Revuz测度，能量泛函，容量等），采用马尔可夫桥和乘泛函的研究方法，然后将这些结果应用于具体的期权定价模型中。本项目旨在揭示一般的马尔可夫过程在一般的Girsanov变换下的性质，特别是位势分析方面。本项目的研究结果将对马尔可夫过程的理论与Girsanov定理在各个领域的应用起到重要的补充作用。

中文关键词： Girsanov定理；转移密度函数；Revuz测度；能量泛函；Levy系

英文摘要： Girsanov transform is one of the important transforms in Markov theory, Girsanov theorems play very important roles in financial mathematics, especially in option pricing theory. That will have great theoretical significance and application value to study the properties of Markov processes under Girsanov transform and their applications. Although many scholars studied the properties of Markov processes under Girsanov transform, most of their works were only about the symmetric Markov processes where Dirichlet forms methods were used. Our work mainly focused on general (non-symmetric) Markov processes and assumed they were semi-martingales. We study the transition density functions and potential analysis (including Revuz measure,enegy functional,capacity,etc.) of the Markov processes under locally absolutely continuity measures. In order to solve this problem, we adopt Markov Bridge and multiplicative functional methods. Then we will apply the results which we obtained to various option pricing models. We aim to reveal the properties of general Markov processes under general Girsanov transform, especially for the potential analysis. The results will play effectively complementary roles in Markov theory and the applications of Girsanov theorem in various fields.

英文关键词： Girsanov theorem；transition density function；Revuz measure；energy functional；Levy system