Hawkes Process
A Hawkes process is a self-exciting point process: the conditional intensity of events at time t depends on the times of past events, λ(t) = μ + Σ φ(t − tᵢ) over events tᵢ < t, with a decaying excitation kernel φ. The arrival of one event raises the probability of further events for a short window, so events cluster in time; a multivariate Hawkes process adds cross-excitation (and empirically inhibition) between event types. It appears in this vault because limit-order-book events — orders, cancellations, trades — empirically cluster and excite one another, so Hawkes processes model real Limit Order Book flow far better than the memoryless Poisson arrivals assumed by Avellaneda-Stoikov 2008.
Crucially, a Hawkes-driven order book is non-Markovian: because intensity is a functional of the entire past event stream, the decision-relevant future cannot be recovered from a finite memoryless state. This is the precise mechanism by which the Markov property — the load-bearing assumption of every Markov chain, hidden Markov model and Markov Decision Process Trading Model — is empirically violated at the microstructure level. Lalor Swishchuk 2025 drive their market-making environment with semi-Markov and Hawkes jump-diffusion dynamics for exactly this reason.
Connections
- Limit Order Book — contradicts (clustered arrivals make the LOB non-Markovian), source: https://www.maths.ox.ac.uk/system/files/attachments/Hawkes%20Process-Driven%20Models%20for%20Limit%20Order%20Book%20Dynamics_0.pdf
- Avellaneda-Stoikov 2008 — contradicts (self-exciting arrivals vs assumed Poisson arrivals), source: https://people.orie.cornell.edu/sfs33/LimitOrderBook.pdf
- Lalor Swishchuk 2025 — part-of (drives the simulated price process), source: https://arxiv.org/html/2410.14504v2
- Markov Decision Process Trading Model — contradicts (history dependence breaks the Markov property), source: https://arxiv.org/html/2410.14504v2