In this paper, we study two terminal value problems (TVPs) for stochastic bi-parabolic equations perturbed by standard Brownian motion and fractional Brownian motion with Hurst parameter h∈(12,1) separately. For each problem, we provide a representation for the mild solution and find the space where the existence of the solution is guaranteed. Additionally, we show clearly that the solution of each problem is not stable, which leads to the ill-posedness of each problem. Finally, we propose two regularization results for both considered problems by using the filter regularization method.

中文翻译：

---

关于维纳过程和分数布朗运动驱动的双抛物方程的终值问题

在本文中，我们分别研究了由标准布朗运动和分数布朗运动（带有Hurst参数h∈（12,1））扰动的随机双抛物方程的两个终值问题（TVP）。对于每个问题，我们提供温和解决方案的表示形式，并找到可以保证解决方案存在的空间。此外，我们清楚地表明，每个问题的解决方案都是不稳定的，这会导致每个问题的不适定性。最后，我们通过使用过滤器正则化方法针对这两个问题提出了两个正则化结果。