A Ponzi scheme exposed to volatile markets

The PGBM model for a couple of counteracting, exponentially growing capital flows is presented: the available capital stock $X(t)$ evolves according to a variant of inhomogeneous geometric Brownian motion (GBM) with time-dependent drift, in particular, to the stochastic differential equation $dX(t)=[pX(t)+\rho_1\exp(q_1 t)+\rho_2\exp(q_2 t)]dt+\sigma X(t) dW(t)$, where $W(t)$ is a Wiener process. As a paragon, we study a continuous-time model for a nine-parameter Ponzi scheme with an exponentially growing number of investors. Investors either maintain their investment or withdraw it after some fixed investment span and quit the system. The first two moments of the process and hence a closed-form solution for the mean path are given. The capital stock exhibits a dynamic lognormal probability distribution as long as the system remains solvent. The assumed stochastic performance allows for earlier or later collaps of the investment system as compared to the deterministic analogy ($\sigma = 0$). Allowing also for negative capital values the system's default probability can be calculated at any time by numerically solving the corresponding Kolmogorov forward equation. We use the finite difference method and obtain results in accordance with those of simple Monte-Carlo simulations. Finally, a minor modification of the payout function provides a toy model for a social security system exhibiting critical behaviour. Depending on whether some parameter value violates a weak no-Ponzi game condition or not, the system represents either a non-lasting Ponzi game or a lasting no-Ponzi game in the weak sense.