An interactive exploration of MSPD — a unifying framework for modelling contagion, clustering, and cascading risk in finance and insurance.
Hillairet, Peyrat & Réveillac (2026) · Cherkaoui,Hillairet, Peyrat & Réveillac (2026)
In classical actuarial models, claims arrive independently — each event has no memory of the past. Real risks, especially cyber, don't work this way: one breach triggers others, and the size of an event shapes how likely future events are. MSPDs are built to capture exactly this.
The Poisson random measure N(dk,dt,dθ,dy) is a random cloud of points on a product space. Each point carries four coordinates: a component index k (which dimension of the risk), a time t, a uniform threshold θ, and a mark y (the severity).
The thinning trick is the key mechanism: a point in the cloud becomes a real event only when its threshold satisfies θ ≤ λ(t). Since the intensity λ(t) jumps upward each time an event is accepted, the acceptance probability increases — this is how self-excitation is encoded at the measure level, without needing to keep track of a list of past events explicitly.
The background rate of events in dimension i, in the absence of any past excitation. Can be time-varying to model seasonality or a known trend.
How much a past event of size y in dimension k raises the rate in dimension i, as a function of elapsed time. Off-diagonal entries encode cross-contagion between lines of business.
Determines what the process measures. Set to 1{i=k} to count events, to y for cumulative losses, or to y·e−τt for actualized losses.
The random size attached to each event (claim amount, vulnerability score…). Its distribution enters both the observation and the excitation, creating a frequency–severity dependency absent from standard Hawkes processes.
The decay rate δ in φ governs how long a past event keeps influencing future ones. Small δ = long memory. Power-law alternatives give even heavier tails, relevant for cyber incidents that propagate over months.
When φi,k(t, y) depends non-trivially on y, a large loss doesn't just count as one event — it contributes proportionally more to future excitation. This breaks the classical independence between N and the mark sequence Yi.
Off-diagonal entries φi,k with i ≠ k model spillovers across lines of business or risk types — a ransomware event in one sector triggering claims in another.
The spectral radius of the branching matrix K = ∫φ̄(t)dt must be below 1. If it exceeds 1, each event produces on average more than one offspring and the process explodes — an actuarially non-viable regime.
The matrix encodes self- and cross-excitation. The observation kernel determines what is measured. Marks enter multiplicatively through both.
Available kernels below: Exponential and Erlang-2 . Marks follow distributions.
The explorer and stress-testing tools below let you configure this exact setup — tune μ, α, δ, and the severity distribution to observe how each parameter shapes the trajectory, the intensity path, and the correlation structure.
Configure a -dimensional MSPD step by step. Click on matrix cells to set kernels and parameters. Check "slider" to get a live control.
d dimensionModeClick a cell to configure.
0 = ignore, or .
Top: Intensities. Middle: Events (dot size = mark). Bottom: Processes .
Run simulations using the model above (requires fixed maturity). Compare empirical distributions with quantile markers.
A cyber loss model where vulnerability disclosures (homogeneous Poisson of rate ) excite the claims process through exponential kernels.
with claims and vulnerability scores . Vulnerabilities arrive as a homogeneous Poisson process of intensity .
λ₀ baseline 0.50ρ vuln rate 0.20δ decay 1.7θ_C claim rate 1.0θ_V vuln score rate 0.20T horizon 40Click in the event panel to add events. Set severity and type below.