Before revision:  We model these two price processes as `` bouncing geometric Brownian motions ( GBM)'': that is, the price processes evolve according to two independent GBMs between trading times . We show that, under this model,  the inter-trading times follow an inverse Gaussian distribution , and the logarithmic returns between consecutive trading times follow a normal inverse Gaussian distribution.
After revision:  We model these two price processes as `` bouncing geometric Brownian motions ( GBMs)", which are defined as exponentials of two mutually reflected Brownian motions. We then modify these bouncing GBMs to construct a discrete time stochastic process of trading times and trading prices, which is parameterized by a positive parameter \delta. Under this model, the inter-trading times follow an inverse Gaussian distribution , and the logarithmic returns between consecutive trading times follow a normal inverse Gaussian distribution.