1.1 The exposure law and its risk coordinates

Definition 1 Random functional exposure law

Asset \(i\) carries a random loading \(B_i\sim P_i\in \mathcal P_2(\mathcal H)\) in a real Hilbert space \(\mathcal H\), and the common factor process has covariance operator \(\Gamma \). The risk coordinates and factor-weighted second moments are

\[ Z_i=\Gamma ^{1/2}B_i,\qquad v_i=\mathbb E\lVert Z_i\rVert ^2 . \]

For a coupling \(\pi \in \Pi (P_i,P_j)\) the systematic covariance is \(\kappa ^\pi _{ij}=\mathbb E_\pi \langle Z_i,Z_j\rangle \), and the factor-weighted quadratic transport cost is

\[ W_{2,\Gamma }^2(P_i,P_j) =\inf _{\pi \in \Pi (P_i,P_j)}\mathbb E_\pi \lVert Z_i-Z_j\rVert ^2 . \]

Every object above is now formalized at this level, not only interpreted: risk coordinates as a pushforward under \(\Gamma ^{1/2}\), the second moment and systematic covariance as Bochner integrals, and the transport cost as the infimum defining \(W_2\) on \(\mathcal P_2(\mathcal H)\). The finite instance of ??def:finite-exposure-cloud is the estimator of this object rather than a separate model, and the two agree exactly on finitely supported laws.

One caveat survives and is unchanged: existence of an optimal plan on \(\mathcal P_2(\mathcal H)\) is not proved. It is supplied as a certificate wherever an optimum is needed, so every attainment statement below reads “given such a plan”, never “one exists”.

Definition 2 Finite exposure cloud and coupling

A finite exposure cloud is a finite support together with nonnegative masses summing to one, and a coupling of two clouds is a nonnegative mass matrix with the two clouds as marginals. Over a coupling this fixes the systematic covariance \(\kappa ^\pi _{ij}\), the transport cost \(C(\pi )\), the second moments \(v_i\), and the barycentre, with no measure-theoretic optimal-transport input. A finite optimal-cost certificate is a separate datum: a coupling together with the assertion that its cost is least in the coupling set.

Lemma 3 Polarization

✓

For any \(x,y\) in a real inner-product space,

\[ \langle x,y\rangle =\tfrac 12\left(\lVert x\rVert ^2+\lVert y\rVert ^2-\lVert x-y\rVert ^2\right). \]

Proof ▼

Expand the squared norm of the difference.

Lemma 4 Transport polarization for a coupling

For every coupling \(\pi \) of two finite exposure clouds,

\[ \kappa ^\pi _{ij}=\tfrac 12\left(v_i+v_j-C(\pi )\right). \]

The identity is exact and holds coupling by coupling; no optimality is involved.

Proof ▼

Apply ??lem:polarization inside the coupling expectation and use that the marginals reproduce \(v_i\) and \(v_j\).