Stochastic Analysis of Overlapping Generations Models Under Incomplete Markets
We provide a stochastic analysis of an overlapping generations model under incomplete markets. By casting individual optimization with idiosyncratic income risk into a forward backward stochastic differential equation (FBSDE) system, we (i) establish existence and uniqueness of the dynamic general equilibrium interest rate and (ii) derive analytical and semi explicit formulas for both the equilibrium interest rate path and the natural borrowing limit; defined as the discounted expected shortfall of future income. Our FBSDE based approach yields tractable policy functions and equilibrium mappings without relying on high dimensional PDE methods, offering clear insights into how income dynamics and demographic structure drive interest rate fluctuations and credit constraints.