## Escape time distribution for stochastic flows

##### Resumen

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.