Minimizing the payments and borrower risk in a mortgage

We consider a mathematical model of borrower behavior in an annuity mortgage. The price of the apartment, amount of the borrower's own capital, the probability distribution of his income at each time moment, and the interest rate are known. The problem is to find the loan size and schedule of payments that minimize both the total payment and the borrower's risk. We pay the most attention to the problem of optimal reserves of capital. We study a variation of this problem where it is allowed to open an additional credit line.