We study linear backward stochastic partial differential equations (BSPDEs) of parabolic type. We consider a new boundary value problem where a Cauchy condition is replaced by a prescribed average of the solution over time. We establish well-posedness, existence, uniqueness, and regularity, for the solutions of this new problem. In particular, this can be considered as a possibility to recover a solution of a BSPDE in a setting where its values at the terminal time are unknown, and where the average of the solution over time is preselected.

中文翻译：

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在没有适当柯西条件的反向 SPDE 上

我们研究抛物线型的线性向后随机偏微分方程 (BSPDE)。我们考虑一个新的边值问题，其中柯西条件随着时间的推移被解的规定平均值所取代。我们为这个新问题的解决方案建立适定性、存在性、唯一性和规律性。特别是，这可以被认为是在终端时间的值未知的情况下恢复 BSPDE 解的可能性，并且解随时间的平均值是预选的。