We consider two final value problem of a stochastic strongly damped wave equation driven by white noise (Section 3) and fractional noise (Section 4). We show that a stochastic integral in the solution is not stable and the problem is not well posed. To regularize the problem in two cases of noise, we apply the Fourier truncated method to control the frequency in such a way that it depends on the noisy level, which is the upper bound of the errors appearing in the input data. Furthermore, the convergence rate of the regularized solution is investigated.

中文翻译：

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随机强阻尼波动方程的逆初值问题

我们考虑由白噪声（第 3 节）和分数噪声（第 4 节）驱动的随机强阻尼波动方程的两个终值问题。我们表明，解中的随机积分是不稳定的，并且问题不是很好提出的。为了在两种噪声情况下规范问题，我们应用傅里叶截断方法来控制频率，使其取决于噪声水平，这是输入数据中出现的错误的上限。此外，研究了正则化解的收敛速度。