Finance Mathematics
Seismic Risk Modeling: How Do Stochastic Mathematics Improve the Assessment of Extreme Events?
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By Wiam Belkouche, PhD in Discrete Mathematics from Hassan II University and Scientific Writer at F.initiatives
Recent earthquakes have revealed a significant gap between expected losses and actual observed damages. In 2023, the Kahramanmaraş earthquake generated $5.8 billion in insurance claims, while the 2024 Noto Peninsula earthquake led to ¥74.4 billion paid out within just a few months. These events highlight a rapid accumulation of damage in a context of sustained seismic activity.
These findings show that the final level of losses strongly depends on post-seismic dynamics, particularly aftershocks and the evolving condition of already weakened structures. Yet, these dimensions are still insufficiently captured in traditional models. In this context, spatio-temporal stochastic mathematics emerge as a key lever to improve extreme event assessment and seismic risk modeling.
From Theory to Reality: Why Do Traditional Seismic Models Underestimate Losses?
Traditional models largely rely on Poisson processes, which assume independent events occurring uniformly over time. While simple and interpretable, this approach does not reflect the reality of seismic sequences.
Catalog analyses reveal the presence of residual dependencies, indicating temporal and spatial clustering (Luen & Stark, 2012). Earthquakes follow complex dynamics, characterized by aftershocks, memory effects, and regime changes.
Recent studies confirm that these models underestimate risk in active regions, as they fail to capture event clustering and non-stationarity (Iervolino & Giorgio, 2022).
This limitation is particularly critical in post-seismic phases, where secondary activity alters the distribution of damage over time and space. Ignoring these mechanisms leads to a systematic underestimation of insured losses, even for well-documented portfolios (Iacoletti et al., 2024).
The Contribution of Stochastic Mathematics to Vulnerability Analysis
Contemporary approaches rely on stochastic models capable of capturing the conditional dynamics of seismic sequences and their spatio-temporal complexity.
Self-exciting point processes (such as ETAS models) allow for modeling interactions between events and capturing triggering mechanisms. Bayesian approaches, particularly through methods like INLA, provide robust probabilistic estimates and improved uncertainty quantification (Naylor et al., 2023).
In parallel, Gaussian Cox processes and adaptive smoothing methods better represent spatial heterogeneity and activity gradients (Helmstetter & Werner, 2014; Shirota & Banerjee, 2019).
Non-stationarity is another central challenge. Non-stationary Gaussian fields enable modeling evolving dependencies across time and space, although at the cost of increased computational complexity (Risser & Turek, 2020).
Finally, data quality remains a key issue. Catalog incompleteness—especially in post-seismic phases—biases parameter estimation and aftershock forecasting (Lippiello et al., 2019). Recent model extensions explicitly account for this incompleteness and improve estimation robustness (Kamranzad et al., 2025).
Toward European Risk Management: The Importance of Catalogs and Prevention
Model reliability depends directly on the quality of available data. Seismic catalogs remain incomplete, particularly in the immediate aftermath of an event, where signal overlap and instrumental limitations lead to under-detection.
This incompleteness has direct consequences:
- biased frequency estimation
- misrepresentation of extreme events
- increased uncertainty in models
For insurance stakeholders, these limitations translate into major operational challenges: underpricing of risk, inadequate capital allocation, and increased exposure to default risk.
Conversely, improving catalogs and models enables:
- more accurate loss forecasting
- better capital management
- a more robust structuring of insurable risk
These issues are also relevant in Europe, where seismic risk, although moderate, requires advanced modeling approaches for prevention and territorial planning (Beauval et al., 2020).
FAQ: Understanding Seismic Risk Modeling and Prediction
How can we know if an earthquake will occur?
It is currently impossible to predict an earthquake precisely. Models can only estimate occurrence probabilities based on historical data and statistical approaches.
What are the warning signs of an earthquake?
Certain phenomena such as microseismic activity or ground deformation can be observed, but they do not allow for reliable short-term prediction.
Why is the Poisson model considered insufficient?
Because it assumes independent events, whereas earthquakes exhibit complex temporal and spatial dependencies, including memory and clustering effects.
What is the impact of incomplete seismic catalogs?
It introduces biases in parameter estimation and reduces model reliability, particularly for aftershock forecasting.
How can a natural hazard be transformed into an insurable risk?
By quantifying both the probability and economic impact of events, enabling modeling, risk pooling, and pricing.
Is France concerned by these modeling approaches?
Yes, certain regions present real seismic risk and require advanced modeling tools for prevention and insurance purposes.
Bibliography
[Beauval, C., Bard, P., & Danciu, L. (2020). The influence of source‑ and ground‑motion model choices on probabilistic seismic hazard levels at 6 sites in France. ](https://doi.org/10.1007/s10518-020-00879-z )[](https://doi.org/10.1177/87552930241262044 )[](https://doi.org/10.48048/tis.2025.10064 )
[Helmstetter, A., & Werner, M. J. (2014). Adaptive smoothing of seismicity in time, space, and magnitude. ](https://doi.org/10.1785/0120130105 )
[Lacoletti, S., Cremen, G., & Galasso, C. (2022). Validation of the ETAS models for simulation‑based seismic hazard assessments.]( https://doi.org/10.1785/0220210134 )
[Lervolino, I., & Giorgio, M. (2022). Comment on “How Well Does Poissonian PSHA Approximate ETAS‑based Hazard?”]( https://doi.org/10.1785/0120210299)
Insurance Business Asia. (2024). Insurance claims for 2024 Noto Peninsula earthquake hit ¥74.4 billion by March. Insurance claims for 2024 Noto Peninsula earthquake hit ¥74.4 billion by March | Insurance Business
[Kamranzad, F., Naylor, M., Lindgren, F., Bayliss, K., & Main, I. (2025). Enhancing the ETAS model: incorporating rate‑dependent incompleteness.… ](https://doi.org/10.1093/gji/ggaf156 )[](https://doi.org/10.1186/s40645-025-00694-7 )
Lippiello, E., Cirillo, A., Godano, C., Papadimitriou, E., & Karakostas, V. (2019). Post‑Seismic Catalog Incompleteness and Aftershock Forecasting. [](https://doi.org/10.1093/gji/ggz419 )
Luen, B., & Stark, P. B. (2012). Poisson tests of declustered catalogues.
[Naylor, M., Serafini, F., Lindgren, F., & Main, I. (2023). Bayesian modeling of the temporal evolution of seismicity using ETAS.inlabru. ](https://doi.org/10.3389/fams.2023.1126759 )
[Risser, M. D., & Turek, D. (2020). Bayesian inference for high‑dimensional non‑stationary Gaussian processes. ](https://doi.org/10.1080/00949655.2020.1792472 )
Shirota, S., & Banerjee, S. (2019). Scalable inference for space‑time Gaussian Cox processes. [](https://doi.org/10.1785/0120230074)