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On inverse initial value problems for the stochastic strongly damped wave equation

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.