MHP-DDP: Multivariate Hawkes Process based on Dependent Dirichlet Process

Music City Center 

Multivariate Hawkes Processes (MHPs) model complex temporal dynamics among event sequences on multiple dimensions. Typically, strong parametric assumptions are made about the excitation functions of MHP, motivating the need for modeling flexible excitation patterns. Further, different excitation functions across dimensions often have strong similarities. Motivated by reasons above, we propose MHP based on dependent Dirichlet process (MHP-DDP), a hierarchical nonparametric Bayesian modeling approach for MHP. MHP-DDP flexibly estimates the excitation function via a mixture of scaled Beta distributions, and borrows strengths across dimensions by modeling such mixing distribution as a mixture of a shared Dirichlet process (DP) and a group-specific idiosyncratic DP. We develop a Markov chain Monte Carlo (MCMC) algorithm for implementation. We also conduct simulations to compare MHP-DDP to benchmark methods where total or no information is borrowed. We show that MHP- DDP outperforms the benchmark methods in terms of lower estimation error for the MCMC algorithm. Finally, we apply our method to an example in finance, where we study the order flow in the limit order book (LOB) records.