Almost all home buyers have mortgages and it is quite common to have mortgage refinanced. There are two main reasons that make people decide to refinance the mortgage: (i) need some cash for urgent purposes, and (ii) lower the monthly payment. In this dissertation, we are not going to discuss (i), and we are investigating problems related to (ii). To begin with, let us intuitively make the following observations: If the interest rate remains the same as the current mortgage interest rate, then the monthly payment will automatically lower if you start a new mortgage with the same term, say, 30-year, because the loan amount is lower than the previous one. It is not hard to see that although the monthly payment is lowered, your overall payment is higher since the overall term is longer. From this, we see that rational people will not refinance the mortgage if the interest rate is not lower than the current one. Now, the subtle question is how much lower the interest rate than the current one, people should start to think about refinance. Actually, besides interest rate, one should also take into account the mortgage size and closing cost. Mathematically, this can be formulated as an optimal impulse control problem, with some interesting features that make this problem significantly from the classical problems. Let us now make the above a little more precise. We will formulate an optimal impulse control problem for stochastic differential equations with the running cost and the terminal being changed at the time that an impulse of the control is applied. Because of these, unlike the classical impulse control problems, a control with some zero impulses might be optimal. On the other hand, these features bring some technical difficulties to the problem. Our idea of solving the problem is as follows. First of all, we will prove that the number of impulses must be finite, and optimal impulse control must exist. Second, by using a backward method, we can solve an optimal impulse control problem with given number of impulses. These problems are parameterized by the number of impulses. Finally, we solve the original problem by optimizing the number of the impulses.
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Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Cao, Yuchen, "Optimal Impulse Controls With Changing Running Cost And Applications In Mortgage Refinance" (2022). Electronic Theses and Dissertations, 2020-. 1529.