Bayesian semi-parametric estimation of compound inhomogeneous Poisson processes for ultra-high frequency financial transaction data

Marked point processes provide a flexible framework for studying ultra-high frequency financial data that records the time and price for each transaction. This paper estimates compound, inhomogeneous Poisson processes (CIPP) where trades arrive according to an inhomogeneous Poisson process, and returns are drawn from a distribution after arrival. The intensity functions have Gaussian process priors to model time-of-day (diurnal) effects, which have bursts of activity at the open and close and a midday lull, and lagged returns from the previous trade, which are idiosyncratic and depend on the stock. The nonparametric density estimator for returns imposes a shape constraint to obtain a unimodal distribution that is asymmetric and leptokurtic. A mixture model separates trades into time sensitive and time insensitive trades with different intensity functions. The latter group tends to have larger orders and higher returns. The proposed model is compared to B-splines and geometric Brownian motion.