Hawkes processes have become a very common tool in order to describe jump clustering features in financial markets. Another class of processes naturally exhibiting self-exciting properties is the class of Continuous Branching processes with Immigration (CBI in  the following). Their use in describing the dynamics of financial assets is less common than use of Hawkes processes, but it seems to grow, while their application to population dynamics description has a long tradition.

The main purpose of the present paper is to introduce jump clustering features in the forward dynamics (in contrast to spot) of power prices. The typical approach to forward price modelling is based on the stochastic differential equation proposed by Heath, Jarrow and Morton (HJM from now on) for forward rates in the attempt to capture the evolution in time of the whole forward curve.

By adopting both classes, the Hawkes processes with exponential kernel and the CBI, we extend the HJM approach in the two different directions, both including jump clustering features.

The second purpose of the present investigation is to estimate the parameters of the two classes of processes used to model the forward price dynamics.  For Hawkes processes we apply a well-known methodology based on a Maximum Likelihood approach. For CBI we propose a slightly modified method based on point processes simulation. For both estimation procedures we rely on a data set related to the French power market.

The third purpose of this paper is to verify the hypothesis that Hawkes processes with exponential kernel and/or CBI are really suitable for describing forward power dynamics. In order to verify this hypothesis we performed a test of Kolmogorov-Smirnov type. The main conclusion of this last section is that, based on the same data set mentioned before, a Hawkes-type dynamics cannot be rejected, while both dynamics assuming constant intensity of jumps and a CBI-type dynamics must be rejected.