Brownian excursions in mathematical finance

The Brownian excursion is defined as a standard Brownian motion conditioned on starting and ending at zero and staying positive in between. The first part of the thesis deals with functionals of the Brownian excursion, including first hitting time, last passage time, maximum and the time it is achieved. Our original contribution to knowledge is the derivation of the joint probability of the maximum and the time it is achieved. We include a financial application of our probabilistic results on Parisian default risk of zero-coupon bonds. In the second part of the thesis the Parisian, occupation and local time of a drifted Brownian motion is considered, using a two-state semi-Markov process. New versions of Parisian options are introduced based on the probabilistic results and explicit formulae for their prices are presented in form of Laplace transforms. The main focus in the last part of the thesis is on the joint probability of Parisian and hitting time of Brownian motion. The difficulty here lies in distinguishing between different scenarios of the sample path. Results are achieved by the use of infinitesimal generators on perturbed Brownian motion and applied to innovative equity exotics as generalizations of the Barrier and Parisian option with the advantage of being highly adaptable to investors’ beliefs in the market.