Frictional Monotonicity in Optimal Transport

• Frictional Monotonicity Principle is defined as the emergence of piecewise monotonic behavior in optimal transport models under convex friction costs, leading to explicit trade bands and bi-atomic couplings.
• The approach integrates a geometric-duality framework with dynamic programming, balancing frictional costs and continuation values to derive optimal decision rules.
• Parameter sensitivity reveals that increasing friction enlarges the no-trade region while reducing optimal displacement magnitudes, with significant implications for robust pricing and network control.

The Frictional Monotonicity Principle refers to a class of phenomena, mathematical structures, and optimality conditions arising in systems where friction—whether physical, economic, or modeled as a convex cost—imposes monotonic constraints on system evolution or the structure of optimal solutions. In robust pricing, transport theory, and network control, friction alters both the feasible set and the geometry of optimizers, leading to explicit monotonic regions, bi-atomic transport plans, and characterized "trade bands" or "no-action zones." Mathematically, the principle asserts that frictions induce piecewise monotonicity in decision rules, system trajectories, or couplings, with salient transitions governed by explicit conditions that balance frictional costs with dynamic continuation values.

1. Principle Definition and Mathematical Formulation

In the context of martingale optimal transport (MOT) problems under state-dependent frictions (Rai, 9 Oct 2025), the Frictional Monotonicity Principle asserts that optimal couplings maximize penalized expected payoff subject to convex trading cost $f_t^{(a,b)}(x, v)$, where $v = y-x$ is the traded increment. The cost adjustment per time-step takes the form

$$\tilde{c}_t(x, y) = \mathcal{V}_{t+1}(y) - \mathcal{V}_{t+1}(x) - f_t^{(a,b)}(x, y-x),$$

with $\mathcal{V}_{t+1}$ the dynamic continuation value, recursively computed. The optimizer exhibits left-monotonic support, locally disintegrating into a bi-atomic law except within a trade band. Concretely, off the trade band—where trading is optimal—the conditional law puts mass on two points

$T_d^{(t)}(x) \le x \le T_u^{(t)}(x),$

with the endpoints determined by an equal-slope system:

$$\partial_y \mathcal{V}_{t+1}(T_d^{(t)}(x)) - \partial_v f_t^{(a,b)}(x, T_d^{(t)}(x) - x) = \partial_y \mathcal{V}_{t+1}(T_u^{(t)}(x)) - \partial_v f_t^{(a,b)}(x, T_u^{(t)}(x) - x).$$

Within the trade band (the no-transaction region), the dual variable $h_t(x)$ satisfies

$h_t(x) \in \partial_v f_t^{(a,b)}(x, 0),$

so the optimal displacement is zero—identity coupling prevails ($y=x$).

2. Geometric-Duality Framework and Dynamic Programming

A central structure in MOT with frictions is the geometric-duality framework, which integrates convex analysis, optimal transport, and dynamic programming (Rai, 9 Oct 2025). At each time-step, the dual constraint is

$$\varphi(x) + \psi(y) + h(x)(y-x) \leq f_t^{(a,b)}(x, y-x)$$

for all $x, y$, with $h(x)$ a Lagrange multiplier enforcing the martingale property. The dual optimization reads

$$\sum_{t=0}^n \left( \int \varphi_t \, d\mu_t + \int \psi_t \, d\mu_{t+1} \right)$$

subject to the inequality above. Telescoping through time produces the identity

$$\int \mathcal{V}_t(x) d\mu_t(x) = \int \mathcal{V}_{t+1}(y) d\mu_{t+1}(y) + \sup_{\pi_t \in \Pi^m(\mu_t, \mu_{t+1})} \int [\mathcal{V}_{t+1}(y) - \mathcal{V}_{t+1}(x) - f_t^{(a,b)}(x, y-x)] d\pi_t(x, y),$$

allowing local structures (e.g., bi-atomic plans and monotonicity) to propagate into global multi-marginal transport plans.

3. Structure of Optimal Couplings and Trade Bands

The optimizer's support splits into active trading and inaction regimes:

Regime	Law Structure	Endpoint Conditions
Off-band (active set)	Bi-atomic: mass split among $T_d^{(t)}(x), T_u^{(t)}(x)$	Equal-slope (see above), mass-balance
Trade band (no trade)	Identity: mass stays at $x$	$h_t(x) \in \partial_v f_t^{(a,b)}(x, 0)$

For friction costs of linear-quadratic form $f_t(v) = \alpha |v| + \beta v^2$ with $\beta > 0$, the subdifferential at $v=0$ is $[-\alpha, \alpha]$, and the trade band is

$B_t = \{ x : |h_t(x)| \leq \alpha \}.$

Off-band, the optimal law at $x$ takes weights

$$\theta_t(x) = \frac{x - T_d^{(t)}(x)}{T_u^{(t)}(x) - T_d^{(t)}(x)},$$

determined by the martingale property.

4. Parametric and Comparative Statics

The extent and properties of the trade band, as well as the magnitude of optimal displacements, are governed by liquidity parameters in the friction cost:

• Increasing the proportional cost $\alpha$ enlarges $B_t$, so small price movements are less frequently traded.
• Increasing the quadratic penalty $\beta$ reduces the magnitude of optimal trades in the active region.

This suggests liquidity management and execution efficiency can be studied via parametric sensitivity, with direct implications for market-making and inventory control.

5. Robust Applications and Stability

Frictional monotonicity results unify model-independent robust pricing and superhedging for path-dependent derivatives such as lookback, barrier, and Asian options (Rai, 9 Oct 2025). The dual-geometry framework enables explicit characterization of price bounds and optimal hedging strategies:

• In path-dependent payout settings, the trade band determines when trading is suboptimal, thus only significant price excursions trigger rebalancing.
• In the vanishing-friction limit $(\alpha, \beta \to 0)$, the optimal transport recovers the frictionless left-curtain coupling, a canonical structure in frictionless MOT.

Optimal couplings and endpoints are shown to be stable under small perturbations of the underlying marginals and friction parameters, as established using Prokhorov's theorem and Helly's selection principle.

6. Connections to Other Frictional Monotonicity Phenomena

The Frictional Monotonicity Principle realized in robust pricing and martingale transport problems echoes related monotonicity structures in dynamic networks (Zlotnik et al., 2015) and systems exhibiting frictional-jamming behavior (Nath et al., 2018). In dynamic flow networks subject to dissipative friction, increased injections lead monotonically to increased nodal densities, formalized via Metzler-matrix conditions and monotonic optimal control reformulations. In rheology, friction enforces monotonic, non-adjustable transitions in flow regimes, with abrupt changes in flow and alignment suppressed. A plausible implication is that, across domains where friction acts as a convex penalization or dynamical dissipation, monotonicity of state evolution or optimal rules emerges as a generic feature, simplifying robust control and pricing analysis under uncertainty.

7. Summary

The Frictional Monotonicity Principle provides a unified theoretical and computational paradigm for incorporating trading frictions—modeled by convex costs—into martingale optimal transport and robust pricing frameworks. Optimal transport plans exhibit mixed bi-atomic and identity regimes, characterized via equal-slope systems and explicit trade bands defined in terms of the subgradient of frictional cost. Strong duality and stability results guarantee robustness of optimal couplings, and parametric comparative statics reveal tangible effects of liquidity parameters. The principle enables efficient, model-independent computation of price bounds for exotic derivatives, tying together geometric, dynamic, and stability aspects in multi-marginal optimal transport problems under frictions (Rai, 9 Oct 2025).