对偶性与深度马尔可夫过程在高维最优切换中的应用：可计算的上界和近似-表达性保证
Duality and DeepMartingale for High-Dimensional Optimal Switching: Computable Upper Bounds and Approximation-Expressivity Guarantees
AI总结：
本文研究有限时间最优切换问题，提出基于深度学习的对偶框架，通过马尔可夫惩罚项推导对偶表示，并建立收敛性保证，同时提供神经网络的表达性分析，证明在高维情况下可避免维度灾难。
摘要：
我们研究有限时间最优切换问题，在一般过滤下允许离散干预日期，并在决策日期之间进行连续时间观测，开发基于深度学习的对偶框架并建立可计算的上界。我们首先通过引入一系列马尔可夫惩罚项推导多重切换的对偶表示。最小惩罚由延续值的Doob马尔可夫量刻画，从而得到完全可计算的上界。然后将DeepMartingale从最优停止扩展到最优切换，并在上界损失和L²替代损失下建立收敛性。我们还提供了表达性分析：在所陈述的结构假设下，对于任何目标精度ε>0，存在大小不超过c d^{q} ε^{-r}的神经网络，其诱导的对偶上界可将真实值近似至ε以内，其中c、q和r与d和ε无关。因此，在所陈述的结构假设下，对偶求解器可避免维度灾难。在数值评估中，我们还实现了一种深度策略方法以生成可行的下界和经验上界差距。在布朗和布朗-泊松模型上的数值实验显示，上界与下界差距较小且在高维情况下表现良好。所学的对偶马尔可夫过程还产生了一种实用的delta对冲策略。
英文摘要：
We study finite-horizon optimal switching with discrete intervention dates on a general filtration, allowing continuous-time observations between decision dates, and develop a deep-learning-based dual framework with computable upper bounds. We first derive a dual representation for multiple switching by introducing a family of martingale penalties. The minimal penalty is characterized by the Doob martingales of the continuation values, which yields a fully computable upper bound. We then extend DeepMartingale from optimal stopping to optimal switching and establish convergence under both the upper-bound loss and an $L^2$-surrogate loss. We also provide an expressivity analysis: under the stated structural assumptions, for any target accuracy $\varepsilon>0$, there exist neural networks of size at most $c d^{q}\varepsilon^{-r}$ whose induced dual upper bound approximates the true value within $\varepsilon$, where $c$, $q$, and $r$ are independent of $d$ and $\varepsilon$. Hence, the dual solver avoids the curse of dimensionality under the stated structural assumptions. For numerical assessment, we additionally implement a deep policy-based approach to produce feasible lower bounds and empirical upper--lower gaps. Numerical experiments on Brownian and Brownian--Poisson models demonstrate small upper--lower gaps and favorable performance in high dimensions. The learned dual martingale also yields a practical delta-hedging strategy.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
