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.