Cylindrical Semi-martingale Optimal Transport and Contiguity: Applications to Large Financial Markets

The motivation for moving beyond classical Optimal Transport (OT)

The theory of Optimal Transport (OT) lies at the crossroads of mathematics, statistics, finance, and machine learning. In the classical setting, OT seeks an optimal coupling between two probability measures on a finite-dimensional space. However, taking as motivation the theory of (LFM) Large Financial Markets^[2], where one encounters infinitely many assets, the relevant objects are infinite-dimensional stochastic processes. Since probability measures on infinite-dimensional spaces are typically only cylindrical, defined through their finite-dimensional marginals rather than on the full space, classical OT must be replaced by its cylindrical, semi-martingale counterpart. The following sections develop this framework and its interplay with the probabilistic notion of contiguity, central in the theory of LFM.

From Finite Dimensions to Cylindrical Semi-martingales

• Classical OT problem: Given two probability measures \(\mu, \nu\) on \(\mathbb{R}^d\), find \[ \inf_{\pi \in \Pi(\mu, \nu)} \int c(x, y)\, \pi(dx, dy) \] where \(c\) is a cost and \(\Pi(\mu, \nu)\) the set of couplings with marginals \(\mu, \nu\).
• Cylindrical Semimartingales:
  Cylindrical semimartingales are infinite-dimensional stochastic objects that are not realized as genuine random variables with values in a Banach or Hilbert space, but are defined entirely through their finite-dimensional projections^[3]. For every finite set of directions, the projected process is a standard real-valued semimartingale, and all such projections are required to be mutually consistent. This approach, crucial in modern stochastic analysis and especially in stochastic partial differential equations (SPDEs), allows stochastic calculus, including Itô integration, to be carried out in locally convex spaces and other infinite-dimensional settings where pathwise processes cannot be defined.
  Financial Perspective:
  In finance, when modeling large or "infinite" markets, such as the theoretical evolution of prices \(\{S^i_t\}_{i \in I}\) for a potentially infinite index set \(I\), the cylindrical semimartingale framework is essential. Each finite subset of assets yields a vector-valued semimartingale, and it is the consistent family of all such projections that defines the market evolution. This enables rigorous modeling, integration, and risk analysis even when the collection of assets is too large for classical Hilbert-space-valued semimartingales to apply.

Optimal Transport of Cylindrical Semi-martingale Laws

Consider cylindrical laws \(\mathbb{P} = (\mathbb{P}^n)_{n \geq 1}\) and \(\mathbb{Q} = (\mathbb{Q}^n)_{n \geq 1}\), where each \(\mathbb{P}^n\), \(\mathbb{Q}^n\) is a measure on \(\mathbb{D}([0,T]; \mathbb{R}^n)\) (vector-valued Skorokhod space)^[4]. The cylindrical OT problem seeks an optimal coupling at every level \(n\) that is consistent as \(n \to \infty\). For cost functionals \(C_n\):

\[ \inf_{\pi^n \in \Pi(\mathbb{P}^n, \mathbb{Q}^n)} \int C_n(\omega, \eta) \;\pi^n(d\omega, d\eta) \] subject to martingale/semi-martingale constraints on projections.

• Costs \(C_n\) may be e.g. sums of pathwise quadratic variation, or supremum norms, depending on applications.
• Martingale constraints: As in martingale OT^[1], one enforces that the canonical processes are (local) martingales under the coupling for each finite block.
• Challenge: To ensure a meaningful infinite-dimensional solution, we need tightness and precise consistency across \(n\). This is where contiguity becomes central.

Contiguity of Probability Measures

Given sequences of probability measures \((\mathbb{P}_n)\), \((\mathbb{Q}_n)\) on a common measurable space, \[ \mathbb{Q}_n \triangleleft \mathbb{P}_n\quad \Longleftrightarrow\quad \forall A_n: \ \mathbb{P}_n(A_n) \to 0 \implies \mathbb{Q}_n(A_n) \to 0. \] That is, any sequence of events negligible under \(\mathbb{P}_n\) is also negligible under \(\mathbb{Q}_n\). This "vanishing together" ensures no sequence of events has probability "leaking" from 1 to 0^[5].

In large financial markets (cf. Kabanov & Kramkov^[2]), contiguity replaces absolute continuity as we move from static to growing, infinite systems. It ensures stability of risk-neutral pricing, the absence of asymptotic arbitrage, and robust duality for hedging/pricing limits.

• Financial Significance: Contiguity is used to rule out "free lunch with vanishing risk," even as the market dimension explodes.
• Technical Role: If families of measures and their couplings are contiguous, limiting results for pricing, superhedging, or risk management persist in the infinite market.

Applications in Large Financial Markets

1. Model-Independent Pricing & Hedging:
   Superhedging prices may be characterized as the values of martingale OT problems indexed by \(n\), with existence and duality persisting when contiguity holds: \[ \mathrm{price}_n = \sup_{\mathbb{Q}_n \triangleleft \mathbb{P}_n, \ \mathbb{Q}_n \text{ martingale}} \mathbb{E}_{\mathbb{Q}_n}[H_n] \] and as \(n \to \infty\), price stability follows from the consistency/contiguity apparatus.
2. No-Free-Lunch Equivalents:
   The Fundamental Theorem of Asset Pricing for large markets is equivalent to the existence of a contiguous family of equivalent martingale measures (\(\mathbb{Q}_n \triangleleft \mathbb{P}_n\)), which excludes asymptotic arbitrage.
3. Risk Management:
   In large/infinite universes (indices, climate, operational risk, etc), we encounter portfolios on spaces where only finite margins are observable. Robust risk bounds rely on cylindrical formulations and the preservation of tail properties via contiguity.

OVanishing Risk and Cylindrical Semimartingale Optimal Transport

Conclusion

In summary: Cylindrical semi-martingale optimal transport and contiguity are fundamental for modern mathematical finance, economics, and probability. They provide the toolbox needed for robust pricing and hedging in infinite-dimensional models, managing risk in sprawling markets, and ensuring that the mathematical structure persists as complexity grows. Their interplay signals new frontiers for theory and practice alike.