Ruin Theory

You walk into a casino with a certain amount of surplus money.  Every hour you spend in the casino, there is a certain probability of winning or losing some given amount of money.  A ruin theory question might ask, what is the probability that you go bankrupt in x number of hours.  The bankruptcy state is an absorbing state, so once you enter into that state, the probability of leaving that state is 0.  The solution to these types of problems usually require calculating an exhaustive list of probabilities for ruin by the convolution method.  But first, familiarity with the notation should be developed.

Definitions:

1. $\psi(u)$ — Starting with surplus $u$, this is the probability of ruin as $t$ goes to infinity with $t$ defined on $\mathbb{R}$.
2. $\bar{\psi}(u)$ —  Same as above except $t \in \mathbb{N}_+$, positive integers.
3. $\psi(u,t)$ — Probability of ruin from time 0 to time $t$ with $t \in \mathbb{R}$.
4. $\bar\psi(u,t)$ — Same as above except $t \in \mathbb{N}_+$.
The analogous survival probabilities follow the same conventions except they are denoted by $\phi$.
The following relationships are useful:
1. $\psi(u) \geq \psi(u,t) \geq \bar\psi(u,t)$
2. $\psi(u) \geq \bar\psi(u) \geq \bar\psi(u,t)$
3. $\psi(u) \geq \psi(u+k)$ for $k \geq 0$
4. $\phi(u) \leq \phi(u,t) \leq \bar\phi(u,t)$
5. $\phi(u) \leq \bar\phi(u) \leq \bar\phi(u,t)$
6. $\phi(u) \leq \phi(u+k)$ for $k \geq 0$