ICNPAA 2010: Mathematical Problems in
Engineering, Aerospace and Sciences

This paper considers a financial contract where the borrower makes a continuous payment to the lender at each moment when the contract is in effect. We study the optimal financial decision from the borrower's point of view when he has the choice of early settlement of debt and when alternative maket investment is available. Assume market interest follows the CIR model, an infinite horizon problem is formulated with an unknown optimal early settlement interest rate to be determined together with the contract values at all interest rate levels. The well posedness of the problem is established. The analytical solution to the problem is obtained using variational method. Finally, a finite element scheme is designed for finding numerical solutions.