Warnings about future jumps: properties of the exponential Hawkes model

Having observed a cluster of jumps produced by an exponential Hawkes process, we study and quantify the residual length of the cluster. We then formalize the stochastic increasingness property of the durations between two consecutive jumps, which strengthens their positive correlation. Finally we consider the case where the process is only observed discretely and provide bounds for the probability of observing a given number of consecutive jumps. As an empirical exercise, we apply our results to a record of JPM's asset prices. First, we show that the identified jumps display dependence and clustering behavior. Second, we find that, under the exponential Hawkes model delivering the best QQ-plot, our formulas indicate a very high probability that an observed cluster of more than 1 jump did not exhaust yet.