We introduce a new stochastic model for the variations of asset prices at the
tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of
assets). The construction is based on marked point processes and relies on
linear self and mutually exciting stochastic intensities as introduced by
Hawkes. We associate a counting process with the positive and negative jumps of
an asset price. By coupling suitably the stochastic intensities of upward and
downward changes of prices for several assets simultaneously, we can reproduce
microstructure noise (i.e.