Learning Lévy density via adaptive RKHS regression with bi-level optimization
We propose a nonparametric method to learn the Lévy density from data consisting of the process’s probability densities. We recast the problem as identifying the kernel of a nonlocal integral operator from discrete or noisy data, which leads to an ill-posed inverse problem. To regularize it, we construct an adaptive reproducing kernel Hilbert space (RKHS) whose kernel is built directly from the data. Under source and spectral decay conditions, we show that the reconstruction error decays with the mesh size at a near-optimal rate. Importantly, we develop a generalized singular value decomposition-based bilevel optimization algorithm to select the regularization parameter, resulting in efficient and robust computation of the regularized estimator. Numerical experiments for several Lévy densities, drift fields and data types (PDE-based densities and sample ensemble-based kernel density estimation reconstructions) demonstrate that our bilevel RKHS method provides a more stable and competitive alternative to classical L-curve and generalized cross-validation strategies and that the adaptive RKHS norm is more accurate and robust than Lρ2- and ℓ2-norms for regularization.