Constrained Density Estimation via Optimal Transport

TL;DR

This paper develops a constrained density-estimation framework based on optimal transport, where the target density $\,\mu$ is chosen to minimize the Wasserstein distance to a prior $\rho$ while satisfying expectation constraints $\int f_k(y)\,d\mu=\bar{f}_k$. It provides analytical insights for simple constraint classes (indicator and RELU) showing how OT shifts mass horizontally, in contrast to KL-based methods that tilt density vertically, and introduces smoothing inequality constraints to mitigate artifacts. A practical, sample-based algorithm with mollification and annealing is presented, along with kernel-based density estimates and adaptive learning-rate schemes, enabling application to empirical data. The framework is demonstrated in finance, recovering risk-neutral pricing measures and pricing exotic options with favorable accuracy relative to KL calibration, underscoring the method's potential for robust inference under uncertainty. These contributions broaden the applicability of OT to density estimation under structured expectation constraints and offer a tractable path for using constrained OT in real-world pricing and inference tasks.

Abstract

A novel framework for density estimation under expectation constraints is proposed. The framework minimizes the Wasserstein distance between the estimated density and a prior, subject to the constraints that the expected value of a set of functions adopts or exceeds given values. The framework is generalized to include regularization inequalities to mitigate the artifacts in the target measure. An annealing-like algorithm is developed to address non-smooth constraints, with its effectiveness demonstrated through both synthetic and proof-of-concept real world examples in finance.

Figures (5)

• Figure 1: Left: the prior, the surrogate distribution, the target distribution restored from KL divergence and the exact target distribution restored from Wasserstein distance without smoothing; Right: the sample-based prior and the target distributions restored from Wasserstein distance with smoothing inequalities. The prior is $\text{Lognormal}(1,1)$. The surrogate measure is $\text{Lognormal}(2,2)$. Single RELU constraint \ref{['func:relu']} is adopted with $\omega=7.3891$ and $\bar{f}_k$ computed via \ref{['eq:guassian_constraint_result']}.
• Figure 2: Left: Target distributions resulting from minimizing KL divergence as well as Wasserstein distance. The prior is $\text{Lognormal}(1,1)$. The surrogate measure is $\text{Lognormal}(2,2)$. We select $K=3$ with $\bar{f}_k$$(k=1,2,\dots, K)$ computed in \ref{['eq:guassian_constraint_result']}. Right: the finite-sampled counterpart with inequality constraints. We choose $N=2000$ samples from the prior distribution and compute the target samples.
• Figure 3: The surface plots and contour plots of the prior, the exact solution and the estimated target measures using the proposed framework without and with regularization. All measures are constructed using samples via kernel density estimation techniques. The exact solution and estimated measure without regularization both contain a delta measure supported on the circle $\{(z_1,z_2)|z_1^2+z^2_2=R^2\}$, denoted as $\alpha(\theta)\delta(R,\theta)$ and are zero outside the circle.
• Figure 4: The surface plots and contour plots of the prior, the exact solution and the estimated target measures using the proposed framework without and with regularization. All measures are constructed using samples via kernel density estimation techniques. The exact solution and estimated measure without regularization both contain a delta measure supported along the line$\{(z_1,z_2): z_1=R\}$ and are zero in $\{(z_1,z_2): z_1<R\}$ with $R=-0.5$.
• Figure 5: Left: the estimated pricing measure of the asset by the proposed framework and the surrogate measure. Right: the estimated histogram of the sampled target measure of the asset by the regularized framework, the surrogate measure and the The prior measure. The prior is $p_\rho$ and $\bar{f}_k$ are computed based on $p^{K}_\mu$ and threshold prices $\omega_1,\dots\omega_K$ with $K=3$.

Theorems & Definitions (17)

• Definition 1: Constrained optimal transport problem, Monge's formulation
• Definition 2: Transportation cost, Kantorovich's formulation
• Definition 3: Constrained optimal transport, Kantorovich's formulation
• Lemma 1: The optimal map is monotone
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Claim 1