Asymptotic expansions for the first hitting times of Bessel processes

We study a precise asymptotic behavior of the tail probability of the first hitting time of the Bessel process. We deduce the order of the third term and decide the explicit form of its coefficient.

On the Laplace Transforms of the First Hitting Times for Drawdowns and Drawups of Diffusion-Type Processes

We obtain closed-form expressions for the value of the joint Laplace transform of therunning maximum and minimum of a diffusion-type process stopped at the first time at which theassociated drawdown or drawup process hits a constant level before an independent exponentialrandom time. It is assumed that the coefficients of the diffusion-type process are regular functionsof the current values of its running maximum and minimum. The proof is based on the solution tothe equivalent inhomogeneous ordinary differential boundary-value problem and the applicationof the normal-reflection conditions for the value function at the edges of the state space of theresulting three-dimensional Markov process. The result is related to the computation of probabilitycharacteristics of the take-profit and stop-loss values of a market trader during a given time period.