MSPD stress testing demo

Scenario-based stress testing of a cyber MSPD

We consider a cyber portfolio driven by vulnerability disclosures and loss events. Stress testing consists in injecting an adverse scenario (e.g., a cluster of critical vulnerabilities and large claims) and quantifying how key risk quantities such as expected surplus and its variability shift under this scenario.
This widget illustrates the pathwise effect of such a scenario on a simplified two-dimensional MSPD (Multivariate Self-exciting Process with Dependencies) for vulnerabilities (V) and claims (C).

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) ,

In the toy model below, marks represent vulnerability criticality or claim severity and follow a discrete distribution P(M=1)=0.6, P(M=2)=0.3, P(M=3)=0.1; they enter kernels multiplicatively via w(m)=m. The widget focuses on sample paths of intensities; the accompanying research derives closed-form expressions for the corresponding stressed expectations and variances of actuarial quantities (e.g. surplus).
For further details on MSPDS Click Here and to learn more about vulnearabilities in cyber risk Click Here.

MSPD Scenario stress testing Cyber risk Surplus analytics

1. Build a scenario on [0, T]

First, specify a deterministic scenario on the time window [0, T] by clicking in the events panel (bottom part of the plot). Clicking on the Claims band adds a claim event; clicking on the Vulnerabilities band adds a vulnerability event. Each scenario event is treated as a marked jump with medium severity M = 2. Endogenous MSPD events use random marks sampled from P(M=1)=0.6, P(M=2)=0.3, P(M=3)=0.1, with w(m)=m.

Claims (C)
Vulnerabilities (V)
Scenario events
0 scenario events

2. Generate baseline vs stressed paths

Kernel parameters are fixed for this demo: μ_C = 0.10, μ_V = 0.20, α_cc = 0.7, α_vv = 0.5, α_vc = 0.9, β_cc = 1.6, β_vv = 1.3, β_vc = 1.9. Using a common random seed and mark sequence for endogenous events, we simulate in real time a baseline MSPD path (without the scenario) and a stressed path (with the deterministic scenario events added on top).

No simulation yet – define a scenario and click “Simulate”.

The top panel shows λC(t) and λV(t): the baseline MSPD (without scenario) appears as thin dashed lines, while the stressed path (with the scenario) is displayed as solid lines. The bottom panel shows events: endogenous MSPD events (small dots) and scenario events (larger hollow markers). At the theoretical level, such scenarios induce explicit shifts in the distribution of functionals like surplus or portfolio loss.