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Hawkes Process

Algorithm description

A Hawkes process is a self-exciting point process: past events increase the probability of future events. In its multivariate form, events in one stream can excite other streams, making it well-suited to modelling the mutual dependence between bid and offer order flows in a limit order book.

The conditional intensity (instantaneous event rate) for process \(k\) at time \(t\):

\[\lambda^k(t) = \mu^k + \sum_{j=1}^{K} \sum_{t_i^j < t} \phi^{kj}(t - t_i^j)\]

where: - \(\mu^k > 0\) is the background (baseline) rate for stream \(k\). - \(\phi^{kj}(\cdot) \geq 0\) is the excitation kernel: how a past event in stream \(j\) affects stream \(k\). - The sum runs over all past event times \(t_i^j\) in stream \(j\).

Kernel choices

Kernel Form Notes
Exponential \(\alpha e^{-\beta t}\) Simple; fast decay; closed-form likelihood
Sum of Exponentials \(\sum_m \alpha_m e^{-\beta_m t}\) Multi-scale memory; best in forecasting-high-frequency-ofi
Power law \(\alpha (t + c)^{-p}\) Slower decay; models long memory
Non-parametric Estimated from data Flexible but harder to fit

Stability condition

The process is stationary (doesn't explode) if and only if the spectral radius of the branching matrix \(\|\int_0^\infty \Phi(t)\,dt\|_2 < 1\), where \(\Phi_{kj} = \int_0^\infty \phi^{kj}(t)\,dt\).

Fitting

Maximum likelihood estimation (MLE). The log-likelihood for a realisation on \([0, T]\):

\[\mathcal{L} = \sum_k \left[ -\int_0^T \lambda^k(t)\,dt + \sum_{t_i^k \leq T} \log \lambda^k(t_i^k) \right]\]

Optimised by gradient descent or EM algorithms. For Sum of Exponentials kernels, the integral \(\int \lambda^k(t)\,dt\) has an efficient recursive form.

Computational complexity

  • Fitting: \(O(n^2)\) naively; \(O(n)\) with recursive likelihood for exponential kernels.
  • Simulation: \(O(n \log n)\) via thinning algorithm.

Applications in this wiki

Paper Use
forecasting-high-frequency-ofi Model bid/offer order flow; forecast OFI distribution
order-flow-filtration Measure Hawkes excitation between OBI and return regimes

When to use / when not to use

Use when: - Modelling arrival processes where past events cluster future events (order flow, earthquakes, social media cascades). - You need a causal, interpretable model (not a black box). - Distributional forecasts of arrival counts matter (not just point estimates).

Avoid when: - Events are approximately independent (Poisson suffices). - Very non-stationary environments (e.g., intraday seasonality requires careful handling).

Implementations

  • tick (Python): tick.hawkes
  • PtProcess (R)
  • hawkeslib (Python)
  • Custom MLE in PyTorch/JAX for differentiable fitting.

Connections