基于随机过程的分数阶反常扩散方程及其应用研究

Because of the non-locality and memorability, fractional calculus is suited to describing the anomalous diffusion phenomenon in nature. After building the fractional anomalous diffusion equation from real life, how to solve these equations is a hot problem. In this project, using the properties of fractional calculus and introducing the stochastic representation, we build the relation between fractional diffusion equation and stochastic process from the random walk model. Then, by using stochastic representation, we study the ground water permeation model in porous medium, and simulate the trajectory of the solute in ground water, it is useful for solving the serious environmental problems. At last, we apply fractional calculus to financial market. Considering the abruptness and memory effects of events in market, we get the option pricing formula , which can cover the shortage of classical models and explain the complex phenomena in real market. Above all, the study of this project can not only sovle the anomalous diffusion problems in physics and finance, but also help to promote the development of fractal geometry and fractal dynamical systems.

分数阶微积分具有非局部性及记忆性，从而非常适合用于描述自然界中的反常扩散现象。从现实问题中，抽象出分数阶反常扩散方程后，如何求解这类问题，是当前一个热门问题。本项目结合分数阶微积分算子的性质，从随机游走模型出发，引入随机表示，建立分数阶反常扩散方程与随机过程的联系；利用随机表示，研究多孔介质地下水渗流模型，模拟溶质在地下水中的运动轨迹，有利于解决经济迅猛发展同时引起的严重的环境问题；结合随机表示，将分数阶微积分理论应用到金融市场中，在经典模型基础上考虑了市场的突发性及记忆性，推导期权的定价公式，研究得出的结果不仅可以弥补经典期权定价模型的不足，还可以解释真实市场中出现的复杂经济现象；因此，本项目的研究，不但可以解决物理和金融中的一些反常扩散问题，而且对数学中分形几何和分数维动力学的发展也起到推动作用。

本项目主要是结合分数阶微积分算子的性质，研究了分数阶反常扩散方程解的性质；建立了分数阶微积分与随机过程的联系；并将其应用在金融期权定价问题中，推导出了次扩散下的期权定价公式。基于这些问题，项目组共撰写3篇学术论文，其中2篇已被SCI期刊接收，另外一篇还在审稿阶段。