Frictionless Left-Curtain Coupling

• The frictionless left-curtain coupling is a unique martingale transport plan defined by a sharp local shadow property, ensuring left-monotonicity and minimality.
• It leverages potential theory and quantile parametrization to construct bi-atomic couplings that are stable, robust, and optimal for model-independent pricing and hedging.
• Widely applied in finance and physics, this coupling serves as a benchmark for analyzing martingale optimal transport problems in the frictionless limit.

The frictionless left-curtain coupling is a distinguished martingale transport between probability measures in convex order. It is defined by a sharp local shadow property and serves as a canonical optimizer in martingale optimal transport (MOT) problems without transaction cost or liquidity friction. Its structure encapsulates minimality, left-monotonicity, stability, and precise dual characterizations, and it is instrumental in robust pricing, hedging, and the analysis of martingale inequalities. In finance, physics, and probability, the frictionless left-curtain coupling also provides fundamental insights into the geometry of transport under the martingale constraint and establishes a baseline for the analysis of MOT problems with frictions.

1. Definition, Characterization, and Construction

Given probability measures $\mu$ and $\nu$ on $\mathbb{R}$ in convex order, i.e., $\mu \leq_{cx} \nu$, the frictionless left-curtain coupling is the unique martingale transport plan $\pi_{\text{lc}} \in \Pi_M(\mu, \nu)$ with a shadow property: for every $x \in \mathbb{R}$, the marginal restriction $\pi_{\text{lc}}|_{(-\infty, x] \times \mathbb{R}}$ has first marginal $\mu|_{(-\infty, x]}$ and second marginal $S^{\nu}(\mu|_{(-\infty, x]})$, where $S^{\nu}(\mu')$ denotes the shadow of $\mu'$ in $\nu$ (Juillet, 2014, Hobson et al., 2021, Hobson et al., 2018).

The shadow $S^{\nu}(\mu')$ is the minimal measure (in convex order) among measures with mass and mean matching $\mu'$, satisfying $\mu' \leq_{cx} S^{\nu}(\mu') \leq \nu$. The construction leverages potential theory:

$P_{\eta}(k) := \int (k - x)_+ \, \eta(dx)$

for a finite measure $\eta$, and the shadow is given by

$$P_{S^{\nu}(\mu)}(k) = P_\nu(k) - (P_\nu(k) - P_\mu(k))^c$$

where $(\cdot)^c$ denotes the convex hull (Hobson et al., 2021, Bayraktar et al., 2021).

A key representation is via quantile parametrization. For left-continuous quantile function $G$ of $\mu$, one constructs measurable lower and upper functions $R, S : (0, 1) \to \mathbb{R}$ with $R(u) \leq G(u) \leq S(u)$ forming the coupling support. The conditional law at $x=G(u)$ is bi-atomic, i.e., $Y(u,v) = R(u)$ with probability $\alpha(u)$ and $Y(u,v) = S(u)$ otherwise, where $\mathbb{E}[Y|U = u] = G(u)$ (Hobson et al., 2018, Hobson et al., 2021). This yields a lifted, global construction, robust to atoms in $\mu$.

2. Structural Properties: Monotonicity, Minimality, and Extremality

The frictionless left-curtain coupling has several defining structural properties:

• Left-monotonicity: For a Borel set $\Gamma$ of full $\pi$-measure, if $(x, y^-), (x, y^+), (x', y') \in \Gamma$ with $x < x'$, then $y'$ cannot lie strictly between $y^-$ and $y^+$. No vertical mixing occurs for increasing $x$ (Juillet, 2014).
• Extremality and Minimality: Among all martingale couplings, $\pi_{\text{lc}}$ is an extreme (non-decomposable) point and minimizes a class of costs, e.g., $c(x, y) = [1 + \tanh(-x)] \sqrt{1 + y^2}$. It is the unique optimizer for such costs in $\Pi_M(\mu, \nu)$ (Juillet, 2014, Hobson et al., 2021).
• Binomial Structure: For atom-free $\mu$, the coupling disintegrates $$\pi_{\text{lc}}(dx, dy) = \mu(dx) [\theta(x) \delta_{T_u(x)}(dy) + (1-\theta(x)) \delta_{T_d(x)}(dy)]$$ with $T_d(x) \leq x \leq T_u(x)$ and $\theta(x) = \frac{x-T_d(x)}{T_u(x)-T_d(x)}$. For atomic $\mu$, the lifted construction with $(R,S)$ retains this bi-atomic form (Hobson et al., 2018, Hobson et al., 2021).

3. Stability, Robustness, and Frictionless Limit

The mapping $(\mu, \nu) \mapsto \pi_{\text{lc}}$ is Lipschitz continuous (in the suitable semimetric $Z$ for couplings), with the shadow projection satisfying:

$$W(S^{\nu}(\mu), S^{\nu'}(\mu')) \leq W(\mu, \mu') + 2 W(\nu, \nu')$$

where $W$ is the 1-Wasserstein distance (Juillet, 2014). Consequently, the construction is robust to perturbations in the marginals and, by extension, to the stability of robust pricing bounds obtained via this coupling.

In frictional martingale optimal transport (MOT), introduction of convex cost penalizations (e.g., linear–quadratic frictions) produces optimal couplings with a trade band (no-transaction region) and off-band bi-atomic transitions. As friction parameters vanish, the band collapses and the optimal coupling converges in law and $L^1$ to the frictionless left-curtain coupling (Rai, 9 Oct 2025).

4. Duality, Monotonicity, and Geometric Interpretation

A geometric–duality framework underlies the frictionless and frictional MOT problem. In the frictionless setting, the dual variables $(\phi, \psi, h)$ satisfy

$\phi(x) + \psi(y) + h(x)(y-x) \leq 0,$

enforcing the martingale constraint. This dual program attains its maximum, and optimality conditions yield equal-slope relations at the endpoints $T_d(x)$, $T_u(x)$. The left-curtain structure is enforced by a monotonicity principle: for any $x < x'$, $y^- < y^+$,

$$\Delta^\square c((x,x');(y^-,y^+)) := c(x, y^-) + c(x', y^+) - c(x, y^+) - c(x', y^-) > 0$$

(this is the martingale analogue of the Spence–Mirrlees condition) (Rai, 9 Oct 2025, Hobson et al., 2021). In the frictional case, the principle persists for the friction-adjusted costs, and as friction vanishes, the classical left-curtain monotonicity is recovered.

5. Extensions: Atoms, Supermartingales, and Markovian Compositions

The framework extends to measures with atoms by lifting to the unit interval and employing pairwise measurable lower–upper functions $(R, S)$, guaranteeing uniqueness, regularity, and retention of the bi-atomic structure even for atomic initial data (Hobson et al., 2018, Hobson et al., 2021).

The increasing supermartingale coupling generalizes the frictionless left-curtain coupling: for pairs $p \leq_{cd} v$ (convex–decreasing order), the coupling coincides with the left-curtain coupling on one side of a unique regime-switching point $u^*$ and with an antitone (reverse Hoeffding–Fréchet) coupling on the other, preserving "frictionless" gluing without additional cost. The shadow measure concept and potential-theoretic formula $$P_{S^\nu(\mu)}(k) = P_\nu(k) - (P_\nu(k) - P_\mu(k))^c$$ remain central (Bayraktar et al., 2021).

Markov composition of curtain couplings constructs Markovian martingale processes with prescribed convex-order-increasing marginal distributions ("peacocks"). Under reasonable regularity, the Markov composition yields a unique, Markovian limit process; non-uniqueness is possible in degenerate settings (Juillet, 2014).

6. Practical Applications in Finance and Physics

In robust finance, the frictionless left-curtain coupling is foundational for model-independent pricing and superhedging, yielding sharp upper or lower bounds for exotic options such as American, lookback, and barrier options. The coupling's structure produces optimal semi-static hedging strategies and can accommodate atomic marginals without introducing replication friction (Hobson et al., 2018, Rai, 9 Oct 2025).

In the context of frictionless cooling in quantum physics, an analogous "frictionless left-curtain coupling" interpretation emerges in the geometric analysis of Gaussian state evolution: optimal control along one dimension induces partial (but not total) control in orthogonal directions, reflecting the one-sided optimality and limitation seen in left-curtain couplings (Choi et al., 2013).

7. Comparative Statics, Limit Theorems, and Implications

For state-dependent trading frictions, comparative statics reveal that increasing proportional cost parameters enlarges the no-transaction region (trade band), while higher transient impact shrinks off-band jumps. Stability results show small variations in marginals or friction parameters only cause small changes in the optimal coupling, confirming robustness of the frictionless left-curtain regime as the frictionless limit (Rai, 9 Oct 2025).

The potential-theoretic, shadow-based construction, left-monotonicity, stability, and duality properties position the frictionless left-curtain coupling as both a theoretical ideal and a benchmark for analysis, computation, and robust hedging in martingale optimal transport and beyond.