Duality and DeepMartingale for High-Dimensional Optimal Switching: Computable Upper Bounds and Approximation-Expressivity Guarantees

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 $\epsilon >0$, there exist neural networks of size at most $c{d}^q{\epsilon }^{-r}$ whose induced dual upper bound approximates the true value within $\epsilon$, where $c$, $q$, and $r$ are independent of $d$ and $\epsilon$. 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.