Closed-form likelihood for discretely observed Hawkes processes

Derive an explicit, closed-form expression for the likelihood function L_dis(θ) = Pr_θ(N((t_{i-1}, t_i]) = n_i for i = 1, …, m) of a Hawkes process when only discrete-time observations of the sample path are available at times 0 = t_0 < t_1 < … < t_m, with background intensity ν(t) and excitation kernel g(t; θ_g) specified parametrically.

Background

The paper considers parameter estimation for Hawkes processes when only discrete observations (counts in disjoint time intervals) are available. In this setting, the standard continuous-time likelihood involving exact event times is not applicable, and the authors note that the likelihood for the discretely observed case lacks an explicit formula.

Because of this intractability, the paper develops a sequential Monte Carlo (SMC) method to obtain an unbiased estimator of the likelihood and then uses a pseudo-marginal Metropolis–Hastings scheme for inference. A closed-form expression for the discrete-observation likelihood would obviate the need for such approximations and enable direct maximum likelihood estimation and standard error computation.