A growth-collapse model: Lévy inflow, geometric crashes, and generalized Ornstein-Uhlenbeck dynamics

We introduce and study a stochastic growth-collapse model. The growth process is a steady random inflow with stationary, independent, and non-negative increments. Crashes occur according to an arbitrary renewal process, they are geometric, and their magnitudes are random and are governed by an arbitrary distribution on the unit interval. If the system's pre-crash level is X>0, and the crash magnitude is 0<C<1, then an 'avalanche' of size CX takes place - after which the system collapses to the post-crash level (1-C)X. This results in a stochastic growth → collapse → growth → collapse → ⋯ system evolution, governed by generalized, non-Markovian, Ornstein-Uhlenbeck dynamics. We explore the behavior of this growth-collapse model and compute various system statistics including: mean, variance, auto-correlation, Laplace transform, and - in the case of heavy-tailed inflows - probability tails. Special emphasis is devoted to 'scale-free' systems where the crash magnitudes are governed by a power-law distribution. We show that (i) there is a hidden regular Ornstein-Uhlenbeck structure underlying all scale-free systems (ii) when crash-rates tend to infinity these systems yield regular Ornstein-Uhlenbeck limits, and, (iii) when the inflow is selfsimilar these systems yield Linnik-distributed equilibria.