<p style='text-indent:20px;'>A variant of the optimal control problem is considered which is nonstandard in that the performance index contains "stochastic" integrals, that is, integrals against very irregular functions. The motivation for considering such performance indices comes from dynamical estimation problems where observed time series need to be "fitted" with trajectories of dynamical models. The observations may be contaminated with white noise, which gives rise to the nonstandard performance indices. Problems of this kind appear in engineering, physics, and the geosciences where this is referred to as data assimilation. The fact that typical models in the geosciences do not satisfy linear growth nor monotonicity conditions represents an additional difficulty. Pathwise existence of minimisers is obtained, along with a maximum principle as well as preliminary results in dynamic programming. The results also extend previous work on the maximum aposteriori estimator of trajectories of diffusion processes.</p>

中文翻译：

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连续时间变分数据同化的存在性和唯一性

<p style='text-indent:20px;'>考虑了最优控制问题的变体，它是非标准的，因为性能指标包含“随机”积分，即针对非常不规则函数的积分。考虑此类性能指标的动机来自动态估计问题，其中观察到的时间序列需要与动态模型的轨迹“拟合”。观察结果可能被白噪声污染，从而产生非标准的性能指标。这类问题出现在工程、物理学和地球科学中，这被称为数据同化。地球科学中的典型模型不满足线性增长或单调性条件这一事实代表了一个额外的困难。获得最小化器的路径存在，连同一个最大值原理以及动态规划的初步结果。该结果还扩展了之前关于扩散过程轨迹的最大后验估计量的工作。</p>