Discrete Formulation¶

This page records the discrete formulation implemented by the package and explains how the linear constraints correspond to the underlying martingale problem.

Continuous Statement¶

The starting point is a pair of marginals mu1 and mu2 and the class of couplings P such that:

S1 ~ mu1
S2 ~ mu2
E[S2 | S1] = S1

Given a payoff c(S1, S2), the objective is to compute extremal values of E_P[c(S1, S2)] over this admissible class.

Discretization¶

The package uses midpoint-style uniform grids for the built-in uniform examples and arbitrary discrete atoms for the general discrete interface.

Let:

x_1, ..., x_m denote the atoms of the first marginal
y_1, ..., y_n denote the atoms of the second marginal
alpha_i denote the weights of the first marginal
beta_j denote the weights of the second marginal
pi_ij denote the coupling weights

The coupling is represented as a nonnegative matrix Pi = (pi_ij).

Constraint Structure¶

The matrix Pi must satisfy three sets of linear constraints.

First Marginal¶

For each i:

sum_j pi_ij = alpha_i

This enforces the distribution of S1.

Second Marginal¶

For each j:

sum_i pi_ij = beta_j

This enforces the distribution of S2.

Martingale Rows¶

For each i:

sum_j pi_ij y_j = alpha_i x_i

Equivalently,

sum_j pi_ij (y_j - x_i) = 0

This is the discrete form of the martingale condition E[S2 | S1 = x_i] = x_i whenever alpha_i > 0.

Linear Objective¶

For a payoff matrix c_ij = c(x_i, y_j), the exact optimization problem is:

maximize or minimize   sum_{i,j} pi_ij c_ij
subject to             Pi satisfies the marginal and martingale constraints
                       pi_ij >= 0

The package solves this problem with scipy.optimize.linprog using the HiGHS backend.

Why Convex Order Appears¶

A martingale coupling between two one-dimensional laws requires matching means and convex order. In the discrete setting used here, convex order is checked through call-price inequalities on a finite strike grid.

For strikes K, the comparison is:

E[(S2 - K)+] - E[(S1 - K)+] >= 0

up to numerical tolerance, together with equality of means.

This check does not replace the LP solve, but it is a useful feasibility diagnostic before optimization begins.

Exact Versus Regularized Problems¶

The package contains two computational layers.

Exact Layer¶

The exact layer solves the discrete LP directly. This is the benchmark for the discretized problem.

Regularized Layer¶

The regularized layer solves an entropy-penalized approximation indexed by eps.

This layer is useful for:

tracing approximation paths
studying convergence behavior
producing smoother computational experiments

The regularized objective is not identical to the raw payoff expectation. Both are recorded explicitly in the output artifacts.

Interpretation¶

The exact solution should be interpreted as exact for the chosen discrete problem, not as a direct theorem about the underlying continuous problem without additional analysis. The discretization is therefore part of the model used for computation, and not merely an implementation detail.