  In this paper we study the stochastic area swept by a regular
time-homogeneous diffusion till a stopping time. This unifies some recent
literature in this area. Through stochastic time change we establish a link
between the stochastic area and the stopping time of another associated
time-homogeneous diffusion. Then we characterize the Laplace transform of the
stochastic area in terms of the eigenfunctions of the associated diffusion. We
also explicitly obtain the integer moments of the stochastic area in terms of
scale and speed densities of the associated diffusion. Specifically we study in
detail three stopping times: the first passage time to a constant level, the
first drawdown time and the Azema-Yor stopping time. We also study the total
occupation area of the diffusion below a constant level. We show applications
of the results to a new structural model of default (Yildirim 2006), the Omega
risk model of bankruptcy in risk analysis (Gerber, Shiu and Yang 2012), and a
diffusion risk model with surplus-dependent tax (Albrecher and Hipp 2007, Li,
Tang and Zhou 2013).
