Abstracts

We study functionals of Brownian excursion, including first hitting time, last passage time, maximum and the time it is achieved. Our method is based on the analogy of Brownian excursion, Brownian motion and Bessel process, which can be related using conditional martingales. We derive the first hitting time of Brownian excursion in a closed form and provide three proofs using elementary arguments from probability theory emphasizing the nature of excursions. Relying on Pitman's Bessel bridge representation we deduce time reversibility and derive the last passage time of Brownian excursion. From the law of hitting time we implicate the law of the maximum and conclude with our main result, studying the joint probability of maximum and time it is achieved. These results are applied to address problems in option pricing. Since Madan, Roynette and Yor discovered that European option prices can be written in terms of last passage times where they allow great flexibility to the local martingale modeling the stock price, they came into focus of financial mathematics. We also discuss the pricing of options depending on the running maximum and the time it is achieved and being triggered when the drawdown of the underlying price exceeds a certain level within a prespecified time period.