On decomposition of the last passage time of diffusions

Abstract

For a regular transient diffusion, we derive the decomposition formula of the Laplace transform of the last passage time to a certain state $\alpha$ explicitly in a simple form in terms of the Green functions, which also leads to the Green function’s decomposition formula. This is accomplished by transforming the original diffusion into two diffusions using the occupation time of the area above and below $\alpha$. We demonstrate applications of the decomposition formulas to various diffusions including a Brownian motion with two-valued drift and present a financial example of the leverage effect caused by the stock price with switching volatility.