We prove a general existence result in stochastic optimal control in discrete time where controls take values in conditional metric spaces, and depend on the current state and the information of past decisions through the evolution of a recursively defined forward process. The generality of the problem lies beyond the scope of standard techniques in stochastic control theory such as random sets, normal integrands and measurable selection theory. The main novelty is a formalization in conditional metric space and the use of techniques in conditional analysis. We illustrate the existence result by several examples including wealth-dependent utility maximization under risk constraints with bounded and unbounded wealth-dependent control sets, utility maximization with a measurable dimension, and dynamic risk sharing. Finally, we discuss how conditional analysis relates to random set theory.

中文翻译：

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有限离散时间中参数相关的随机最优控制

我们证明了离散时间中随机最优控制的普遍存在结果，其中控制在条件度量空间中取值，并通过递归定义的前向过程的演变依赖于当前状态和过去决策的信息。该问题的普遍性超出了随机控制理论中标准技术的范围，例如随机集、正态被积函数和可测量选择理论。主要的新颖之处在于条件度量空间的形式化和条件分析中技术的使用。我们通过几个例子来说明存在结果，包括在具有有界和无界财富相关控制集的风险约束下的财富相关效用最大化、具有可测量维度的效用最大化和动态风险分担。最后，