Convergence of the robust convex expectation to the classical conditional expectation in filtering

Determine the convergence behavior of the robust convex expectation operator ℰ(ϕ(S_t) | 𝒴_t) = ess sup_{(γ, μ_0, Σ_0)} { E^{γ, μ_0, Σ_0}[ϕ(S_t) | 𝒴_t] − (β(γ, μ_0, Σ_0 | 𝒴_t)/k_1)^{k_2} }, where β is a negative log-likelihood-based penalty and γ denotes the controlled model parameters, to the classical conditional expectation E[ϕ(S_t) | 𝒴_t] in the pathwise robust filtering framework. Ascertain the conditions and sense in which such convergence holds.

Background

The paper reformulates robust filtering pathwise by introducing a convex expectation that penalizes misspecified model parameters through a negative log-likelihood term. This yields a deterministic optimization perspective for inference under uncertainty.

While the construction provides robustness, the authors explicitly highlight that the relationship between this convex expectation and the classical conditional expectation is not settled. Clarifying whether, and in what sense, the convex expectation converges to the conditional expectation is important for interpreting robustness and fidelity of the filter.