AI 中文总结
该研究针对带离散路径依赖的McKean-Vlasov控制问题，建立了随机最大值原理，推导了最优性条件，在线性二次情形证明了正倒向系统的全局可解性，并将其应用于基于核偏差的时间序列生成。
AI 中文摘要
我们研究一类具有离散路径依赖的McKean-Vlasov控制问题，其系数和代价泛函可依赖于受控状态在固定确定性时刻观测到的有限个过去值及其联合分布。我们建立了受控状态方程的适定性，并通过随机庞特里亚金最大值原理推导了最优性的必要和充分条件；伴随过程由带固定观测时刻跳跃的倒向随机微分方程刻画，每个跳跃对应未来代价关于对应观测状态及其分布的条件敏感性。在线性二次情形下，我们结合延拓论证和平均场Riccati约化证明了所得正倒向系统的全局可解性。最后，我们讨论了其在时间序列生成中的应用，其中终端代价由采样路径的分布与目标路径分布之间基于核的偏差给出。
英文摘要
We study a class of McKean-Vlasov control problems with discrete path dependence. The coefficients and cost functional may depend on finitely many past values of the controlled state, observed at fixed deterministic times, and on their joint law. We establish well-posedness of the controlled state equation and derive necessary and sufficient optimality conditions through a stochastic Pontryagin maximum principle. The adjoint process is characterized by a backward stochastic differential equation with jumps at fixed observation times. Each jump represents the conditional sensitivity of future costs with respect to the corresponding observed state and its distribution. In the linear-quadratic case, we prove global solvability of the resulting forward-backward system by combining a continuation argument with a mean-field Riccati reduction. We finally discuss an application to time-series generation, where the terminal cost is given by a kernel-based discrepancy between the law of the sampled path and a target path distribution.
