Extremal Behaviour of Diiusion Models in Nance

We investigate the extremal behaviour of a diiusion (X t) given by the SDE dX t = (X t)dt + (X t)dW t ; t > 0 ; X 0 = x ; where W is standard Brownian motion, is the drift term and is the diiusion coeecient. Under some appropriate conditions on (X t) we prove that the point process of "{upcrossings converges in distribution to a homogeneous Poisson process. As examples we study the ex-tremal behaviour of term structure models or asset price processes such as the Vasicek model, the Cox-Ingersoll-Ross model and the generalised hyperbolic diiusion. We also show how to construct a diiusion with predetermined stationary density which captures any extremal behaviour. As an example we introduce a new model, the generalised inverse Gaussian diffusion .