Escape time distribution for stochastic flows

Abstract

The model is based on models developed at the Federal Reserve Board of Governors by Robert Martin, PhD. His models were used to model data arising from subprime mortgages. They are very simple but capture data very well. In this thesis we used his model and derived the partial differential equations describing the time history of the corresponding distributions. In the case of Brownian motion this reduced to just the Fokker-Planck equation and in the case of the jump process we followed the derivation in the notes by Roger Brockett. In doing this, a deep understanding of how to use and manipulate the It\^{o} formula and other aspects of stochastic differential equations is gained.

We assume x, as a weighted variable, to evaluate the borrower's ability to continue making payments, refinance, default or pay off. It is scaled so that 0 represents default and 1 represents paid. For each treatment we assume the approximation difference equation $x_{n+1}=(1+r)x_n-s\epsilon_{n}$ as the model where the parameters $r$ and $s$ are two positive constants to be determined. $r$ stands for the growth rate which is a positive real number in $(0,1)$. The $s\epsilon_{n}$ term as the bad accidents such as divorce, job loss, career moves, etc. which can dramatically affect the ability to pay. After 10,000 treatments, we will find the histograms which are obtained by recording the frequency of those jump time points. We will then analyze and explain our results of simulation based on the histograms of the escape time distributions.