一类偏积分-微分方程中的反问题

The Inverse Problem in a Partial Integro-Differential Equation

对一类偏积分-微分方程中参数校准的反问题进行研究.在弱解的框架下,原问题可转化为含具体正则化项的最优化问题.文中证明了该最优化问题的解的存在性和稳定性,并考察了最优解存在的一阶必要条件.另外,证明了当正则化参数足够大时,该最优化问题关于参数a的凸性性质.基于偏积分-微分方程反问题的研究对于金融市场中的模型校准问题具有重要的意义.

The paper is concerned with the inverse problem of determining the parameters in a partial integro differential equation. By establishing the problem in the sense of weak solution, the inverse problem is converted into a regularized optimization problem with concrete regularized term. The author derives the existence, stability and the first order optimality condition for the optimization problem. Furthermore, the optimization problem is proved to be convex on function a if the regularization parameter is appropriately chosen. The research has important application to model calibration problem in the financial market.