一类随机微分方程的随机源反演方法和性质

The Random Source Inverse Method and Properties for a Class of Stochastic Differential Equations

该文考虑了一类由分式Brown运动驱动的随机微分方程的随机源反演方法及其性质,其中分式Brown运动对应的Hurst参数H∈(0,1).该问题可由很多随机模型转化而得,是一种比较广泛的随机问题.对于正问题,通过常数变易法得到方程的温和解,根据温和解的统计性质讨论其适定性.对于反问题,根据终止时刻的随机数据的统计量反演随机源项的部分统计量,证明了反演的唯一性,并讨论了当a(x)在不同范围时反问题的稳定性情况.

The random source inverse method and properties for a class of stochastic differential equations driven by the fractional Brownian motion with Hurst index H∈(0,1).This problem can be obtained from the transform of many stochastic models and is a widely followed problem.For the direct problem,the mild solution to the equation was obtained by means of constant variation,and according to the statistical properties of the mild solution,the well-posedness of the direct problem was discussed.For the inverse problem,some statistics of the random source term were determined from the random data at the final moment,to prove the uniqueness of the inverse problem,and the stability of the inverse problem with a(x) in different ranges was discussed.