Stochastic Areas of Diffusions and Applications in Risk Theory
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).