  We study time reversal, last passage time, and $h$-transform of linear
diffusions. For general diffusions with killing, we obtain the probability
density of the last passage time to an arbitrary level and analyze the
distribution of the time left until killing after the last passage time. With
these tools, we develop a new risk management framework for companies based on
the leverage process (the ratio of a company asset process over its debt) and
its corresponding alarming level. We also suggest how a company can determine
the alarming level for the leverage process by constructing a relevant
optimization problem.

Mathematical Finance