Frictional martingale optimal transport and robust hedging

We study the martingale optimal transport problem with state-dependent trading frictions and develop a geometric and duality framework extending from the one time-step to the multi-marginal setting. Building on the left-monotone structure of frictionless MOT (Beiglböck and Juillet, Ann. Probab., 2016; Henry-Labordère and Touzi, Finance Stoch., 2016; Beiglböck et al., Ann. Probab., 2017), we introduce a convex frictional cost combining proportional bid-ask spreads and quadratic liquidity impacts. The framework extends the martingale Spence-Mirrlees condition to nonlinear frictions and establishes a frictional monotonicity principle. At each time step, the joint distribution between consecutive asset prices exhibits a bi-atomic, monotone geometry: conditional on the current price, the next price lies on one of two monotone branches representing upward and downward rebalancing. A no-transaction region, or trade band, arises where maintaining the position is optimal, while outside the band, transitions follow two monotone graphs whose endpoints satisfy an equal-slope condition balancing continuation value and marginal trading cost. The framework extends dynamically via a recursive identity, ensuring stability and convergence to the frictionless left-curtain limit, and applies to model-independent pricing and robust hedging of path-dependent derivatives.