  Stochastic processes time-changed by an inverse subordinator have been
suggested as a way to model the price of assets in illiquid markets, where the
jumps of the subordinator correspond to periods of time where one is unable to
sell an asset. We develop an excursion theory for time-changed reflected
Brownian motion and use this to express the price of certain European options
with Parisian barrier condition in terms of solutions of a time-fractional PDE.
We provide a general description of the occupation measures of time-changed
processes and use this to prove a Ray-Knight theorem for the occupation measure
of a time-changed Brownian motion with negative drift. We also show that the
duration of the excursions on finite time intervals obey a Poisson-Dirichlet
distribution when a reflected Brownian motion is time-changed by an inverse
stable subordinator.
