[Paper Review] Precise deviations for Hawkes processes

Fuqing Gao, Lingjiong Zhu|arXiv (Cornell University)|Feb 9, 2017

Point processes and geometric inequalities44 references3 citations

TL;DR

This paper establishes precise deviations for linear Hawkes processes using mod-ϕ convergence theory, deriving higher-order asymptotic expansions beyond the standard large and moderate deviation principles. The key contribution is a refined Laplace approximation for the tail probability P(N_t ≥ xt), incorporating higher cumulants and yielding sharper exponential estimates than prior Donsker-Varadhan type results.

ABSTRACT

Hawkes process is a class of simple point processes with self-exciting and clustering properties. Hawkes process has been widely applied in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we study precise deviations for Hawkes processes for large time asymptotics, that strictly extends and improves the existing results in the literature. Numerical illustrations will also be provided.

Motivation & Objective

• To extend existing large and moderate deviation results for Hawkes processes by deriving precise deviations with higher-order corrections.
• To address the limitation of existing Donsker-Varadhan type rate functions, which only capture leading-order behavior.
• To develop a refined asymptotic approximation for the tail probability P(N_t ≥ xt) using moment generating function analysis.
• To apply mod-ϕ convergence theory to linear Hawkes processes under integrability conditions on the kernel h.
• To provide explicit formulas for higher cumulants and asymptotic expansions of the Laplace transform of the counting process.

Proposed method

• Utilizes mod-ϕ convergence theory to analyze the moment generating function of the Hawkes process N_t.
• Derives a semi-explicit formula for the moment generating function F(t;θ) = E[exp(θN_t)] via integral equations.
• Establishes recursive differential equations for F^{(k)}(t;θ) with respect to θ, enabling higher-order derivative computation.
• Applies change-of-measure techniques to express P(N_t ≥ xt) in terms of a tilted measure involving the Radon-Nikodym derivative.
• Employs the Legendre transform of the cumulant generating function to identify the optimal tilting parameter θ*.
• Derives asymptotic expansions for the tail probability using the saddle-point method and higher cumulants up to order four.

Experimental results

Research questions

• RQ1 How can higher-order asymptotics be derived for the tail probability P(N_t ≥ xt) in linear Hawkes processes?
• RQ2 What is the precise exponential rate of decay for P(N_t ≥ xt) beyond the leading-order large deviation principle?
• RQ3 Can mod-ϕ convergence theory be effectively applied to point processes with stochastic intensity like the Hawkes process?
• RQ4 How do higher cumulants (up to order four) contribute to refining the Laplace approximation of the tail probability?
• RQ5 What is the role of the kernel h in shaping the precise deviation behavior, particularly when ∫th(t)dt < ∞?

Key findings

• The paper derives a precise Laplace approximation for P(N_t ≥ xt) that includes higher-order corrections beyond the standard large deviation rate function.
• The asymptotic expansion of the tail probability involves the fourth cumulant of the counting process, with explicit expressions derived for x^{(3)}(θ*) and x^{(4)}(θ*) in terms of ν, ‖h‖_{L^1}, and x.
• The optimal tilting parameter θ* satisfies θ* = argmax_θ {θx − ν(x(θ) − 1)}, and x(θ*) = x / (ν + ‖h‖_{L^1}x).
• The second-order correction term in the expansion is proportional to x^{(4)}(θ*) / (1 − ‖h‖_{L^1}x(θ*)) with a coefficient involving (γ − 1)^2 and H(t − u).
• The result improves upon the Donsker-Varadhan large deviation principle by capturing the next-order term in the exponential rate of decay.
• Numerical illustrations confirm the accuracy of the precise deviation approximation, especially in the moderate to large deviation regime.