Extremal Behavior of Diffusion Models in Finance

AbstractWe investigate the extremal behavior of a diffusion Xt given by the SDE $$dX_t=\mu\left({X_t }\right)dt+{\sigma }\left({X_t}\right)dW_t ,t > 0,X_0=x$$ , where W is standard Brownian motion, μ is the drift term and σ is the diffusion coefficient. Under some appropriate conditions on Xt we prove that the point process of ε -upcrossings converges in distribution to a homogeneous Poisson process. As examples we study the extremal behavior of term structure models or asset price processes such as the Vasicek model, the Cox–Ingersoll–Ross model and the generalized hyperbolic diffusion. We also show how to construct a diffusion with pre-determined stationary density which captures any extremal behavior. As an example we introduce a new model, the generalized inverse Gaussian diffusion.