连续介质物理与力学和金融工程中的若干非线性扩散方程问题

项目名称： 连续介质物理与力学和金融工程中的若干非线性扩散方程问题

项目编号： No.11471175

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 卢国富

作者单位： 莆田学院

项目金额： 78万元

中文摘要： 本项目拟研究：主部为Newton渗流的对流扩散方程在退化和快扩散情形下初始迹问题，解的大时间渐近性和收敛率估计，点源奇性解的存在性和小时间渐近性，解的动力学行为和复杂渐近性，外解的渐近性和解的爆破与熄灭；主部为非Newton渗流的对流扩散方程在退化情形若干相关问题；主部为对数扩散的对流扩散方程的多解性，解的熄灭，质量强制和具测度初值解的正则性和奇性扩展；金融数学中具交易费多资产投资组合问题、双指数跳扩散模型下具有流动性风险公司债券的定价和具随机波动率大头寸股票最优清算策略等三类奇异随机控制HJB方程解的适定性和正则性，最佳实施边界的性质，定价的理论分析和数值模拟. 对金融市场风险管理提供科学依据.

中文关键词： 对流扩散方程；快扩散和对数扩散；自由边界;；渐近性;；收敛率；多解性;；爆破和熄灭；HJB方程;；最优停时

英文摘要： The researches focus on the following topics: Firstly whether porous medium and fast diffusion equations with convection possess the properties of initial trace and Harnack's inequality, large time asymptotic behavior, source-type solution and its short time asymptotic behavior, asymptotic complexity and rate of convergence, asymptotic behavior of exterior solution, blow-up and extinction, known as the equations without convection. And some of similar problems would be discussed for the p-Laplace equations with convection；Secondly whether the logarithmic diffusion equations with convection have the properties of multiple solutions, regularity of solution with a measure as initial data, mass constraint, extinction and blow-down, known as standard logarithmic diffusion equations; Finally the three kinds of HJB equations of singularly stochastic control arising from the portfolio optimization under transaction, a jump-diffusion model of bond pricing and optimal decision for selling an illiquid stock are concerned with. The properties of well-posed, regularity of solution and optimal exercise boundary for these problems are studied. Based on the analysis of mathematical theory and the numerical simulation, the strategy of financial risk management is given.

英文关键词： Convection-Diffusion Equations;Fast Diffusion; Logarithmic Diffusion;Free Boundary; Asymptotic Behavior; Rate of Convergence;Multiple Solution; Blow-up; Extinction;HJB Equations; Optimal Stopping Time