  We propose a predictor-corrector adaptive method for the study of hyperbolic
partial differential equations (PDEs) under uncertainty. Constructed around the
framework of stochastic finite volume (SFV) methods, our approach circumvents
sampling schemes or simulation ensembles while also preserving fundamental
properties, in particular hyperbolicity of the resulting systems and
conservation of the discrete solutions. Furthermore, we augment the existing
SFV theory with a priori convergence results for statistical quantities, in
particular push-forward densities, which we demonstrate through numerical
experiments. By linking refinement indicators to regions of the physical and
stochastic spaces, we drive anisotropic refinements of the discretizations,
introducing new degrees of freedom (DoFs) where deemed profitable. To
illustrate our proposed method, we consider a series of numerical examples for
non-linear hyperbolic PDEs based on Burgers' and Euler's equations.
