Cyber MSPD demo

MSPD dynamics for cyber vulnerabilities and claims

This interactive module illustrates a simplified two-dimensional MSPD (Multivariate Self-exciting Process with Dependencies) used in cyber-risk modeling. For further details on MSPDS Click Here and to learn more about vulnearabilities in cyber risk Click Here.
We consider two marked components: vulnerabilities (V) and claims (C). Each event carries a mark representing its severity, sampled from a discrete distribution P(M=1)=0.6, P(M=2)=0.3, P(M=3)=0.1. Marks enter multiplicatively through w(m)=m. Vulnerabilities self-excite and also excite claims, while claims self-excite, producing clustered dynamics consistent with event-driven cyber systems.

λC(t) = μC + ∑s<t [ αcc e−βcc(t−s) w(MsC) + αvc e−βvc(t−s) w(MsV) ]
λV(t) = μV + ∑s<t αvv e−βvv(t−s) w(MsV) .
MSPD Marked processes Cyber risk Cross-excitation

Process & kernel parameters

μ_C baseline claims intensity 0.10
μ_V baseline vulnerability intensity 0.20
α_cc claims self-excitation 0.70
α_vv vulnerability self-excitation 0.50
α_vc vuln → claims excitation 0.90
β_cc decay of claims self-kernel 1.20
β_vv decay of vuln self-kernel 1.00
β_vc decay of vuln → claims kernel 1.50

Claims (loss severity mark)
Vulnerabilities (criticality mark)

Simulation

Paused

Each dot is a marked event. Bigger dots correspond to higher criticality (vulnerabilities) or larger losses (claims). Intensities λC and λV are driven by exponential kernels with separate amplitudes and decays for self- and cross-excitation.

Toy stability conditions (for this simple parametrisation) suggest keeping α_cc < β_cc, α_vv < β_vv and α_vc < β_vc, but those aren't the "true" stability conditions associate to this model.