中文摘要：

研究了一个重构与空间变量x有关的波动率系数的反问题．通过线性化的方法，将原问题转化为一个热传导方程，在最优控制框架下讨论了控制泛函极小元的存在性及其满足的必要条件，最后证明了最优解的唯一性及稳定性．

英文摘要：

An inverse problem of reconstructing volatility coefficient which depends on the space variable x is investigated. After linearization around constant volatility, the problem is translated into a heat function. Based on the optimal control framework, the existence of the minimizer of the cost functional is established , and the necessary condition which must be satisfied by the minimiz- er is deduced. At last, the uniqueness and stability of the solution are also proved.