Brémaud, P., & Massoulié, L. (1996). Stability of nonlinear Hawkes processes. The Annals of Probability 24(3), 1563–1588. DOI 10.1214/aop/1065725193
Key takeaways
Read in full (NOTE-728).
- Linearity and excitation are not what make a Hawkes process settle. With rate φ(∫h(t−s)N(ds)), φ α-Lipschitz and h of either sign, α∫|h| < 1 gives a unique stationary law with finite mean intensity, and convergence to it from any past whose influence fades (Theorem 1). For a network the condition is a spectral radius below one of the matrix α_i∫|h_ji| (Theorem 7). In the linear excitatory case these reduce to the branching ratio below one of Hawkes and Oakes (LIT-1028).
- A bounded rate needs no size condition. Bounded Lipschitz φ gives the same with kernels of any size (Theorems 2 and 8); bounded rate and bounded memory give convergence from any past, exponentially fast (Theorems 3 and 5).
- Inhibition is allowed but costs. The conditions use |h|, so inhibitory weight adds to the bound. Mixed networks with monotone, bounded rates and sign-structured cross-connections have a stationary law, with uniqueness open (Theorem 6, Remark 9).
- Sufficient, fixed-kernel, stable side only. No necessary condition, nothing past the threshold, no result on cascade sizes, and kernels do not change with time or state.
- The proofs avoid the branching (cluster) representation, which the authors say is not available in this generality, and use Poisson embedding, a Picard-style iteration and coupling.
Why this is here
The exchange’s model has signed, state-dependent kernels, which A6 and A7 granted do not inherit the linear model’s stability guarantees. This is the standard place where that analysis starts. Method only. The reading (THEORY-215) found three things the filing did not say.
- It covers signed kernels, not state-dependent ones. The rate is a nonlinear function of a fixed linear filter of the past. A6’s influence canvas, which changes the kernels, is outside every theorem.
- Its spectral radius is not a branching ratio except in the linear excitatory case. For QUESTION-091 that matters: in the nonlinear signed case the quantity bounds, and does not measure, how far cascades go; with saturating rates there is no criticality to be near.
- For CLAIM-163 a Hawkes network has a coupling, the kernels, and a control parameter built from them. It has no order parameter in the claim’s sense, and this paper proves nothing about transitions. It meets part of the demand for cascades of events and none of it for valuations, so QUESTION-091’s summary, which calls the branching ratio “the coupling and order parameter”, says more than this reading supports.