随机指标分枝过程的大偏差及其应用

In recent years, Markov process in random environment is one of the important research topics, especially, branching process in random environment and random walk in random environment have attracted the attentions of many researchers. The problems arised in these fields not only have profound theoretical significance, but also have a broad application background...In this project, we consider the large deviation of the renewal random indexed branching process and its applications in finance and physics. We mainly discuss the following three problems. ...Firstly, we consider the large deviation and the moderate deviation of the renewal random indexed branching process. We use the generating function and the asymptotic properties of the harmonic moments to study the large deviation and the moderate deviation of the ratio and the logarithm of the renewal random indexed branching process...Secondly, we study the large deviation and the moderate deviation for the multitype random indexed branching process. We use the asymptotic theory of the matrix arised from the partial derivative of the generating function to extend the large deviation and the moderate deviation of the single type to the case of multitype. ..Finally, we use the the renewal random indexed branching process to price the options and to study the mean reversion in the stock market, and obtain the fair pricing formula of European call options and the barrier options, the convergent rate with respect to the mean reversion in the stock market is also derived.

近年来，随机环境中的马氏过程是国内外概率论界及相关领域的研究热点之一，而随机环境中的分枝过程和随机游动尤其得到众多概率论学者的关注。这些领域的问题不仅有深刻的理论意义，而且有广泛的应用背景。.本项目拟研究随机指标分枝过程的大偏差理论。主要考虑以下三个问题: .一、更新随机指标分枝过程的大偏差和中偏差问题。利用矩母函数与调和矩的渐近性质研究更新随机指标分枝过程的比率以及对数函数的大偏差和中偏差理论；.二、多物种随机指标分枝过程的大偏差和中偏差问题。利用矩母函数的偏导数产生的随机矩阵的渐近理论将单物种情形的大偏差和中偏差理论推广到多物种情形；.三、利用更新随机指标分枝过程研究期权定价理论和股票市场的均值回归现象，得到欧式看涨期权和障碍期权的公平定价公式以及股票市场中关于均值回归的收敛速度。

随机指标分枝过程的主要用途之一是描述股票价格波动情况。相较于此前的几何布朗运动模型，随机指标分枝过程具有如下两个优点。一、现实中股票价格是离散的，而几何布朗运动是连续取值的，因此分枝过程更为贴切。二、分枝过程模型允许公司破产，而几何布朗运动只取正值。. 本项目研究随机指标分枝过程的长期行为。我们得到的主要结果有以下几个方面。.1．.分枝过程的统计推断主要关心的问题之一就是分枝律均值的估计，常用的统计量为Lotka-Negaev 估计和 Heyde 估计，我们得到了两种估计量的大偏差结果。另外、对于Heyde 估计，我们还得到了它的 Berry-Esseen 型不等式。.2．.分枝过程产生的鞅是研究的重要内容之一。我们得到了鞅收敛的速度，其中包括正态偏差、中偏差、大偏差概率的衰减速度。.3．.我们得到了随机指标分枝过程跳时刻的极限理论，得到了大偏差、中偏差和渐近正态性。