关于粘性斜扩散过程及其应用的研究

As a class of diffusion processes which possess special trajectory properties, sticky skew diffusion processes have important applications in many scientific fields. This project is aimed at studying this kind of process from three aspects: the theoretical properties, the financial application and the parameter estimations. Firstly, we consider the properties of the sticky skew diffusion, including the transition density, the probability mass at the sticky point, the first hitting time distribution and the Laplace transform of the symmetric local time at the sticky point. This offers a theoretical foundation for applications. Secondly, many countries nowadays try to stimulate their economies by lowering the interest rate, which leads to the zero interest rate environment. For this, we use the diffusion process with a sticky boundary to characterize the sticky phenomenon at the zero point for the interest rate, and focus on the pricing problem of European interest rate options under such a model. Thirdly, we make use of tools like stochastic analysis and statistics to study the method for estimating the parameters of above processes so as to calibrate the practical models. The results of this project will enrich and develop the theory of sticky skew diffusion processes, and will also be an exploration as well as an innovation on financial models and methods.

粘性斜扩散过程作为一类具有特殊轨道性质的扩散过程，在许多学科领域中有着重要的应用。本项目拟从理论性质、金融应用和参数估计三方面对此类过程展开研究。第一，我们考虑粘性斜扩散过程的转移密度、粘点处概率质量、首达时分布和粘点处对称局部时的Laplace变换等性质，为应用提供理论基础。第二，当前，许多国家通过降低利率来刺激经济发展，这导致了零利率环境的出现。为此，我们用粘性边界扩散过程刻画利率在零点的粘性现象，主要研究该模型下欧式利率期权的定价问题。第三，我们综合运用随机分析和统计工具讨论以上过程的参数估计方法，从而对实际模型进行校准。本项目的结果将进一步丰富和发展粘性斜扩散过程的相关理论，同时也是对金融模型和方法的一次探索和创新。

近几年来，粘性斜扩散过程在理论物理、金融经济等领域的应用逐渐成为研究的热点，而弄清此类过程的概率理论性质是解决应用问题的关键所在。本项目主要研究了粘性斜扩散过程的基本性质，并将结果推广到带跳的情形，同时讨论了与之相关的期权定价问题。具体如下：. 1）考虑了几类典型粘性斜扩散过程的首达时、占位时、转移密度、积分过程的Lapalce变换等理论性质，并得到了一系列显式结果；. 2）将粘性扩散过程的概念推广到带有复合Poisson跳的情形，给出了其基本的轨道性质，并用这类模型刻画金融市场中的价格聚集现象；. 3）将斜扩散过程的概念推广到带有复合Poisson跳的情形，证明了解的存在唯一性，并计算了首出时的Laplace变换；. 4）用粘性扩散过程及门限扩散过程描述资产价格动态，得到了期权定价公式。