  We consider a 2-dimensional marked Hawkes process with increasing baseline
intensity in order to model prices on electricity intraday markets. This model
allows to represent different empirical facts such as increasing market
activity, random jump sizes but above all microstructure noise through the
signature plot. This last feature is of particular importance for practitioners
and has not yet been modeled on those particular markets. We provide analytic
formulas for first and second moments and for the signature plot, extending the
classic results of Bacry et al. (2013) in the context of Hawkes processes with
random jump sizes and time dependent baseline intensity. The tractable model we
propose is estimated on German data and seems to fit the data well. We also
provide a result about the convergence of the price process to a Brownian
motion with increasing volatility at macroscopic scales, highlighting the
Samuelson effect.
