Demand for catastrophe insurance under the path-dependent effects
Pith reviewed 2026-05-18 22:13 UTC · model grok-4.3
The pith
Path-dependent effects raise demand for catastrophe insurance and reduce underinsurance.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We show that demand for catastrophe insurance rises when path-dependent effects are included via a rough volatility model for the stock market and a Hawkes process with power kernel for catastrophes. Adding auxiliary state variables degenerates the non-Markovian problem to a Markovian one, after which an explicit solution to the path-dependent extended Hamilton-Jacobi-Bellman equation is derived by extending the functional Ito formula for fractional Brownian motion to general path-dependent processes. Numerical results calibrated to earthquake data from Sichuan Province indicate that individuals become more risk-averse under rough volatility but more risk-seeking with catastrophic shocks, so
What carries the argument
The path-dependent extended Hamilton-Jacobi-Bellman equation solved after augmenting the state with auxiliary variables to capture memory from rough volatility and Hawkes-process event clustering.
If this is right
- Optimal stock trading becomes more conservative when volatility is rough.
- Individuals take more investment risk when path-dependent catastrophic shocks are modeled but simultaneously purchase more insurance.
- Explicit closed-form strategies for investment and insurance are available once the problem is Markovianized.
- Ignoring path dependence in pricing or regulation produces measurable underinsurance relative to the calibrated benchmark.
Where Pith is reading between the lines
- Insurers could incorporate observed clustering statistics into premium calculations to better match revealed demand.
- The same state-augmentation technique might apply to other insurance products that exhibit event memory, such as health or flood coverage.
- Regulators in disaster-prone areas could test whether mandating path-dependent models reduces aggregate underinsurance exposure.
Load-bearing premise
The extension of the functional Ito formula from fractional Brownian motion to general path-dependent processes, including the Hawkes process, remains valid and yields explicit solutions for comparing insurance demand with and without path dependence.
What would settle it
Compute optimal catastrophe insurance coverage in the full path-dependent model versus the same model with path dependence removed, using the fitted Sichuan earthquake parameters; the claim fails if the path-dependent version does not produce strictly higher demand.
read the original abstract
This paper investigates optimal investment and insurance strategies under a mean-variance criterion with path-dependent effects. We use a rough volatility model and a Hawkes process with a power kernel to capture the path dependence of the market. By adding auxiliary state variables, we degenerate a non-Markovian problem to a Markovian problem. Next, an explicit solution is derived for a path-dependent extended Hamilton-Jacobi-Bellman (HJB) equation. Then, we derive the explicit solutions of the problem by extending the Functional Ito formula for fractional Brownian motion to the general path-dependent processes, which includes the Hawkes process. In addition, we use earthquake data from Sichuan Province, China, to estimate parameters for the Hawkes process. Our numerical results show that the individual becomes more risk-averse in trading when stock volatility is rough, while more risk-seeking when considering catastrophic shocks. Moreover, an individual's demand for catastrophe insurance increases with path-dependent effects. Our findings indicate that ignoring the path-dependent effect would lead to a significant underinsurance phenomenon and highlight the importance of the path-dependent effect in the catastrophe insurance pricing.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper investigates optimal investment and insurance strategies under a mean-variance criterion incorporating path-dependent effects, modeled via a rough volatility process and a Hawkes process with power kernel for catastrophe risks. Auxiliary state variables are introduced to reduce the non-Markovian problem to Markovian form, after which an explicit solution is claimed for the resulting path-dependent extended HJB equation. The authors extend the Functional Itô formula from fractional Brownian motion to general path-dependent processes (including the Hawkes process) to obtain explicit feedback controls. Parameters of the Hawkes process are estimated from Sichuan earthquake data, and numerical results indicate that rough volatility increases risk aversion in trading while catastrophic shocks increase risk-seeking behavior; crucially, path-dependent effects raise demand for catastrophe insurance, with the claim that ignoring them produces significant underinsurance.
Significance. If the derivations hold, the work would offer explicit solutions for a class of path-dependent stochastic control problems in insurance and finance, together with empirical calibration from real catastrophe data. The quantitative warning about underinsurance when path dependence is neglected could inform pricing and regulatory practice. The combination of rough volatility, Hawkes-driven jumps, and mean-variance optimization with auxiliary variables is technically ambitious and, if rigorously justified, would strengthen the literature on non-Markovian risk management.
major comments (2)
- [Derivation of explicit solutions] Derivation of explicit solutions (around the extension of the Functional Itô formula): The manuscript states that explicit solutions are obtained by extending the Functional Itô formula from fractional Brownian motion to general path-dependent processes that include the Hawkes process with power kernel. No self-contained proof, verification of regularity conditions, or citation to a standard reference is supplied. Because the Hawkes intensity is a non-semimartingale jump process whose quadratic variation and path dependence differ materially from fBM, the applicability of the formula (and therefore the validity of the claimed explicit feedback controls for the extended HJB equation) is not established.
- [Numerical results and parameter estimation] Numerical results and parameter estimation (Sichuan data section): The reported increase in catastrophe-insurance demand is obtained after fitting the Hawkes-process parameters to the Sichuan earthquake data and then feeding those fitted values into the optimal-control expressions. Consequently the quantitative finding is conditional on the specific parameter estimates rather than a parameter-free or robustness-checked prediction; this weakens the central claim that path-dependent effects generically produce higher insurance demand.
minor comments (2)
- [Model formulation] Notation for the auxiliary state variables introduced to Markovianize the problem should be defined more explicitly before the extended HJB equation is written, to avoid ambiguity when the Functional Itô formula is applied.
- [Introduction] The abstract and introduction would benefit from a brief statement of the precise regularity conditions assumed on the power kernel of the Hawkes process.
Simulated Author's Rebuttal
We thank the referee for the thorough review and valuable comments on our paper. We address each of the major comments in detail below and outline the revisions we intend to make to strengthen the manuscript.
read point-by-point responses
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Referee: [Derivation of explicit solutions] Derivation of explicit solutions (around the extension of the Functional Itô formula): The manuscript states that explicit solutions are obtained by extending the Functional Itô formula from fractional Brownian motion to general path-dependent processes that include the Hawkes process with power kernel. No self-contained proof, verification of regularity conditions, or citation to a standard reference is supplied. Because the Hawkes intensity is a non-semimartingale jump process whose quadratic variation and path dependence differ materially from fBM, the applicability of the formula (and therefore the validity of the claimed explicit feedback controls for the extended HJB equation) is not established.
Authors: We acknowledge that the current version of the manuscript does not provide a self-contained proof or detailed verification of the regularity conditions for extending the Functional Itô formula to the Hawkes process. In the revised manuscript, we will include an appendix with a rigorous derivation of the extended Functional Itô formula applicable to general path-dependent processes, including those driven by the Hawkes process with a power kernel. We will verify the necessary conditions for the formula to hold, such as appropriate integrability and differentiability requirements on the path-dependent functionals. Additionally, we will cite relevant literature on functional Itô calculus for processes with jumps and path dependence to support the extension. revision: yes
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Referee: [Numerical results and parameter estimation] Numerical results and parameter estimation (Sichuan data section): The reported increase in catastrophe-insurance demand is obtained after fitting the Hawkes-process parameters to the Sichuan earthquake data and then feeding those fitted values into the optimal-control expressions. Consequently the quantitative finding is conditional on the specific parameter estimates rather than a parameter-free or robustness-checked prediction; this weakens the central claim that path-dependent effects generically produce higher insurance demand.
Authors: We agree that the numerical illustrations rely on the specific parameter estimates obtained from the Sichuan earthquake data. The central claim regarding increased demand for catastrophe insurance under path-dependent effects is demonstrated through the explicit solutions and the role of the auxiliary state variables in the model. To address the concern about generality, we will add a robustness analysis in the revised version by varying the key Hawkes process parameters (such as the branching ratio and decay rate) around the estimated values and showing that the qualitative increase in insurance demand persists. This will support that the underinsurance phenomenon when ignoring path dependence is not an artifact of the particular calibration but arises from the path-dependent structure itself. revision: yes
Circularity Check
No significant circularity; derivation is self-contained
full rationale
The paper's core derivation proceeds by introducing auxiliary state variables to convert the non-Markovian control problem into a Markovian one, then solving the resulting path-dependent extended HJB equation via an extension of the Functional Ito formula applied to general path-dependent processes (including the Hawkes process with power kernel). This sequence is presented as a direct mathematical construction within the paper itself, without reducing to a prior fitted quantity or self-citation chain by definition. Parameter estimation from Sichuan earthquake data occurs separately and is used only for subsequent numerical illustrations of insurance demand; the explicit feedback controls and the qualitative claim that path dependence increases demand are not statistically forced by those fitted values. No load-bearing step collapses to an input by construction, and external benchmarks or independent verification of the extension are not required for the circularity assessment.
Axiom & Free-Parameter Ledger
free parameters (1)
- Hawkes process parameters (power kernel) =
fitted to real data
axioms (2)
- domain assumption Adding auxiliary state variables correctly degenerates the non-Markovian problem to a Markovian one.
- domain assumption The rough volatility model and Hawkes process with power kernel adequately capture the relevant path dependence.
discussion (0)
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