The paper studies optimal strategies for a borrower who needs to repay his debt, in an infinite time horizon. An instantaneous bankruptcy risk is present, which increases with the size of the debt. This induces a pool of risk-neutral lenders to charge a higher interest rate, to compensate for the possible loss of part of their investment. Solutions are interpreted as Stackelberg equilibria, where the borrower announces his repayment strategy u(t) at all future times, and lenders adjust the interest rate accordingly. This yields a highly non-standard problem of optimal control, where the instantaneous dynamics depend on the entire future evolution of the system. Our analysis shows the existence of optimal open-loop controls, deriving necessary conditions for optimality and characterizing possible asymptotic limits as t +∞.
All Science Journal Classification (ASJC) codes
- Applied Mathematics