AI 中文总结
本文为带跳半鞅BSDE值过程构造可测聚合器，适用于非支配或不确定控制问题，并建立时间一致非线性条件期望系统、路径正则化及动态规划原理，统一处理扩散、纯跳和离散时间过程。
AI 中文摘要
我们为随机控制问题相关的值过程构造了一个可测聚合器，其中优化准则由带跳的半鞅倒向随机微分方程（BSDEs）的解给出。该结果可应用于半鞅特征三元组在可能非支配情形下受控，或优化中存在关于特征不确定性的控制问题。该构造还在Skorokhod空间上提供了一个时间一致的完全非线性条件期望系统。我们还构造了值函数的适当路径正则化，并证明了相应的动态规划原理。我们所寻求的一般性允许在统一框架下处理受控扩散、纯跳过程和离散时间过程。
英文摘要
We construct a measurable aggregator for the value processes associated with stochastic control problems, where the criterion to optimise is given by solutions to semi-martingale backward stochastic differential equations (BSDEs) with jumps. The results can be applied to control problems where the triplet of semi-martingale characteristics is controlled in a possibly non-dominated case or where uncertainty about the characteristics is present in the optimisation. The construction also provides a time-consistent system of fully nonlinear conditional expectations on the Skorokhod space. We also construct an appropriate path-regularisation of the value function and prove a corresponding dynamic programming principle. The generality we seek allows for the treatment of controlled diffusions, pure-jump processes, and discrete-time processes in a unified setting.
Comments71 pages. This manuscript is Part I of a two-part split of arXiv:2507.01767v1; the existing record will become Part II
