Hazard-selfsimilarity of diffusions’ first passage times

Abstract

A recent study introduced a novel approach to the exploration of diffusions’ first-passage times (FPTs): selfsimilarity. Specifically, consider a general diffusion process that runs over the non-negative half-line; initiating the diffusion at fixed positive levels, further consider the diffusion’s FPTs to the origin. Selfsimilarity means that the FPTs are spanned by an intrinsic scaling of their initial levels. The recent study addressed two types of selfsimilarity: stochastic, scaling the FPTs in ‘real space’; and Laplace, scaling the FPTs in ‘Laplace space’. The Laplace selfsimilarity manifests an underlying sum-like structure. Shifting from the sum-like structure to a max-like structure—a-la the shift from the Central Limit Theorem to Extreme Value Theory—this study addresses a third type of selfsimilarity: hazard, scaling the FPTs in ‘hazard space’. A comprehensive analysis of hazard-selfsimilarity is established here, including: the universal distribution of the FPTs; the dramatically different statistical behaviors that the universal distribution exhibits, and the statistical phase transition between the different behaviors; the characterization of the generative diffusion dynamics, and their universal Langevin representation; and the universal Poissonian statistics that emerge when the initial levels are scattered according to the statistical steady-state of the generative diffusion dynamics. The analysis unveils the following linkages: of the universal distribution to the Gumbel, Gompertz, and Frechet laws; of the universal Langevin representation to diffusion in quadratic and logarithmic potentials; and of the universal Poissonian statistics to non-normalizable densities, to the maxima of the exponential law, and to the harmonic Poisson process.