Identify the meta-distributions governing standardized order arrivals

Determine the meta-distributions Q0+ and Q0− of standardized buy and sell order arrivals, which are assumed to have zero mean, given access to a sequence of independent samples from these distributions. The observed arrival distributions Q±(ε±) are related to the meta-distributions by the transformation ΔN±(ε±) = h±(ε±) Δѱ + f±(ε±), where the shift f±(ε±) and scale h±(ε±) are known continuous functions of the bid/ask spreads ε±.

Background

The paper models buy and sell order arrivals as random variables whose distributions depend on bid and ask spreads. To handle model uncertainty, the authors posit that each spread-dependent distribution Q±(ε±) is obtained by shifting and scaling draws from an underlying meta-distribution Q0± via known functions f± and h±, with Q0± assumed to have zero mean.

While the robust optimization framework accommodates this uncertainty through Wasserstein balls around empirical distributions, the authors explicitly note that the underlying meta-distributions Q0± are not known a priori and only sample data are available. Identifying these meta-distributions is thus an unresolved component underlying the stochastic policy design.

References

We assume that the meta-distributions \mathbb{Q}_0\pm have zero mean. We do not know the distributions \mathbb{Q}_0\pm in advance, but we assume that we have access to a sequence of independent samples of these distributions.

— Wasserstein Robust Market Making via Entropy Regularization (2503.04072 - Fang et al., 6 Mar 2025) in Assumption 1 (Model Uncertainty), Section 2 (Model Setup)