[Paper Review] Stability, convergence to equilibrium and simulation of non-linear Hawkes Processes with memory kernels given by the sum of Erlang kernels

Aline Duarte, Eva Löcherbach+1 more|arXiv (Cornell University)|Oct 11, 2016

Point processes and geometric inequalities4 citations

TL;DR

This paper establishes the stability and exponential convergence to equilibrium for non-linear Hawkes processes with memory kernels composed of sums of Erlang kernels by modeling them as Markovian cascades—piecewise deterministic Markov processes (PDMPs)—and proving positive Harris recurrence via integration by parts and invertibility of a Vandermonde matrix. It further introduces a modified thinning algorithm for efficient simulation of such processes.

ABSTRACT

Non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels are considered. It is shown that their stability properties can be studied in terms of an associated class of piecewise deterministic Markov processes, called Markovian cascades of successive memory terms. Explicit conditions implying the positive Harris recurrence of these processes are presented. The proof is based on integration by parts with respect to the jump times. A crucial property is the non-degeneracy of the transition semigroup which is obtained thanks to the invertibility of an associated Vandermonde matrix. For Lipschitz continuous rate functions we also show that these Markovian cascades converge to equilibrium exponentially fast with respect to the Wasserstein distance. Finally, an extension of the classical thinning algorithm is proposed to simulate such Markovian cascades.

Motivation & Objective

• To analyze the longtime behavior and stability of non-linear Hawkes processes with memory kernels given as sums of Erlang kernels.
• To establish conditions under which the associated Markovian cascades are positive Harris recurrent.
• To prove exponential convergence to equilibrium in the Wasserstein distance for Lipschitz rate functions.
• To develop a new simulation algorithm based on thinning for these Markovian cascades.
• To demonstrate that Erlang-kernel-based Hawkes processes can approximate general integrable memory kernels over compact time intervals.

Proposed method

• Model the intensity process of the Hawkes process as a system of piecewise deterministic Markov processes (PDMPs), referred to as Markovian cascades of successive memory terms.
• Represent each memory term using auxiliary processes that evolve deterministically between jumps and are updated at jump times via a flow governed by the Erlang kernel parameters.
• Apply integration by parts with respect to jump times to analyze the transition density and establish non-degeneracy of the semigroup.
• Leverage the invertibility of an associated Vandermonde matrix to ensure the transition density is strictly positive, enabling recurrence analysis.
• Use coupling techniques and the Wasserstein distance to prove exponential convergence to equilibrium under Lipschitz continuity of the rate function.
• Extend the classical thinning algorithm to simulate the Markovian cascade by exploiting the Markovian structure induced by the Erlang kernels.

Experimental results

Research questions

• RQ1 Under what conditions is the Markovian cascade associated with a non-linear Hawkes process with Erlang-sum memory kernels positive Harris recurrent?
• RQ2 How does the structure of the Erlang kernel decomposition enable the reduction of the Hawkes process to a PDMP framework?
• RQ3 What conditions ensure exponential convergence to equilibrium in the Wasserstein distance for such processes?
• RQ4 Can the transition density of the Markovian cascade be shown to be non-degenerate, and how is this established?
• RQ5 How can the classical thinning algorithm be adapted to simulate non-linear Hawkes processes with Erlang-sum memory kernels?

Key findings

• The Markovian cascades associated with non-linear Hawkes processes with Erlang-sum memory kernels are positive Harris recurrent under the standard sub-criticality condition ‖f′‖L∞ < 1.
• Exponential convergence to equilibrium in the Wasserstein distance is established for Lipschitz continuous rate functions, with the convergence rate depending on the parameters of the Erlang kernels.
• The non-degeneracy of the transition semigroup is proven via the invertibility of a Vandermonde matrix constructed from the kernel parameters, ensuring the existence of a strictly positive density.
• The simulation of such processes is made feasible through an extension of the thinning algorithm that exploits the Markovian structure of the cascaded system.
• The class of Erlang-sum memory kernels is dense in L¹(ℝ₊), meaning any integrable memory kernel can be approximated arbitrarily well over compact time intervals.
• The proof of recurrence relies on a recursive induction argument using diffeomorphisms and change of variables, with the key step being the non-vanishing Jacobian determinant of the transformation.