A study on discrete Ponzi scheme model through Sturm-Liouville theory (Q2113749)

Summary: In this paper, we introduce a second order self-adjoint difference equation which describes the dynamics of Ponzi schemes: a type of investment fraud that promises more than it can deliver. We use the Sturm-Liouville theory to study the discrete equation with boundary conditions. The model is based on a promised, unrealistic interest rate \(r_p\), a realised nominal interest rate \(r_n\), a growth rate of the deposits \(r_i\), and a withdrawal rate \(r_w\). Giving some restrictions on the rates \(r_p\), \(r_i\), and \(r_w\), we prove some theorems to when the fund will collapse or be solvent. Two examples are given to illustrate the applicability of the main results.