arXiv subjects

Patrick J. Laub

Publications and source records attributed to Patrick J. Laub.

15 recordsLinked to original sources

arXiv2026

Arbitrage-free catastrophe reinsurance valuation for compound dynamic contagion claims

In this paper, we consider catastrophe stop-loss reinsurance valuation for a reinsurance company with dynamic contagion claims. To deal with conventional and emerging catastrophic events, we propose the use of a compound dynamic contagion process for the catastrophic component of the liability. Under the premise that there is an absence of arbitrage opportunity in the market, we obtain arbitrage-free premiums for these contracts. To this end, the Esscher transform is adopted to specify an equivalent martingale probability measure. We show that reinsurers have various ways of levying the security loading on the net premiums to quantify the catastrophic liability in light of the growing challenges posed by emerging risks arising from climate change, cyberattacks, and pandemics. We numerically compare arbitrage-free catastrophe stop-loss reinsurance premiums via the Monte Carlo simulation method. We also compare them with those from generalised compound Hawkes/compound Cox cases. Sensitivity analyses are performed by changing the retention level, the Esscher parameters and the intensity parameters.

q-fin.RM↗

arXiv2025

An Interpretable Deep Learning Model for General Insurance Pricing

This paper introduces the Actuarial Neural Additive Model, an inherently interpretable deep learning model for general insurance pricing that offers fully transparent and interpretable results while retaining the strong predictive power of neural networks. This model assigns a dedicated neural network (or subnetwork) to each individual covariate and pairwise interaction term to independently learn its impact on the modeled output while implementing various architectural constraints to allow for essential interpretability (e.g. sparsity) and practical requirements (e.g. smoothness, monotonicity) in insurance applications. The development of our model is grounded in a solid foundation, where we establish a concrete definition of interpretability within the insurance context, complemented by a rigorous mathematical framework. Comparisons in terms of prediction accuracy are made with traditional actuarial and state-of-the-art machine learning methods using both synthetic and real insurance datasets. The results show that the proposed model outperforms other methods in most cases while offering complete transparency in its internal logic, underscoring the strong interpretability and predictive capability.

cs.LG↗

arXiv2024

Distributional Refinement Network: Distributional Forecasting via Deep Learning

A key task in actuarial modelling involves modelling the distributional properties of losses. Classic (distributional) regression approaches like Generalized Linear Models (GLMs; Nelder and Wedderburn, 1972) are commonly used, but challenges remain in developing models that can (i) allow covariates to flexibly impact different aspects of the conditional distribution, (ii) integrate developments in machine learning and AI to maximise the predictive power while considering (i), and, (iii) maintain a level of interpretability in the model to enhance trust in the model and its outputs, which is often compromised in efforts pursuing (i) and (ii). We tackle this problem by proposing a Distributional Refinement Network (DRN), which combines an inherently interpretable baseline model (such as GLMs) with a flexible neural network-a modified Deep Distribution Regression (DDR; Li et al., 2019) method. Inspired by the Combined Actuarial Neural Network (CANN; Schelldorfer and W{\''u}thrich, 2019), our approach flexibly refines the entire baseline distribution. As a result, the DRN captures varying effects of features across all quantiles, improving predictive performance while maintaining adequate interpretability. Using both synthetic and real-world data, we demonstrate the DRN's superior distributional forecasting capacity. The DRN has the potential to be a powerful distributional regression model in actuarial science and beyond.

stat.ML↗

arXiv2024

Hawkes Models And Their Applications

The Hawkes process is a model for counting the number of arrivals to a system which exhibits the self-exciting property - that one arrival creates a heightened chance of further arrivals in the near future. The model, and its generalizations, have been applied in a plethora of disparate domains, though two particularly developed applications are in seismology and in finance. As the original model is elegantly simple, generalizations have been proposed which: track marks for each arrival, are multivariate, have a spatial component, are driven by renewal processes, treat time as discrete, and so on. This paper creates a cohesive review of the traditional Hawkes model and the modern generalizations, providing details on their construction, simulation algorithms, and giving key references to the appropriate literature for a detailed treatment.

stat.ME↗

arXiv2021

Exact simulation of extrinsic stress-release processes

We present a new and straightforward algorithm that simulates exact sample paths for a generalized stress-release process. The computation of the exact law of the joint interarrival times is detailed and used to derive this algorithm. Furthermore, the martingale generator of the process is derived and induces theoretical moments which generalize some results of Borovkov & Vere-Jones (2000) and are used to demonstrate the validity of our simulation algorithm.

stat.CO↗

arXiv2021

Approximate Bayesian Computations to fit and compare insurance loss models

Approximate Bayesian Computation (ABC) is a statistical learning technique to calibrate and select models by comparing observed data to simulated data. This technique bypasses the use of the likelihood and requires only the ability to generate synthetic data from the models of interest. We apply ABC to fit and compare insurance loss models using aggregated data. A state-of-the-art ABC implementation in Python is proposed. It uses sequential Monte Carlo to sample from the posterior distribution and the Wasserstein distance to compare the observed and synthetic data.

stat.CO↗

arXiv2020

Quickest detection in practice in presence of seasonality: An illustration with call center data

In this chapter, we explain how quickest detection algorithms can be useful for risk management in presence of seasonality. We investigate the problem of detecting fast enough cases when a call center will need extra staff in a near future with a high probability. We illustrate our findings on real data provided by a French insurer. We also discuss the relevance of the CUSUM algorithm and of some machine-learning type competitor for this applied problem.

stat.AP↗

arXiv2019

Orthogonal polynomial expansions to evaluate stop-loss premiums

A numerical method is proposed to evaluate the survival function of a compound distribution and the stop-loss premiums associated with a non-proportional global reinsurance treaty. The method relies on a representation of the probability density function in terms of Laguerre polynomials and the gamma density. We compare the method against a well established Laplace transform inversion technique at the end of the paper.

math.PR↗

arXiv2018

Monte Carlo Estimation of the Density of the Sum of Dependent Random Variables

We study an unbiased estimator for the density of a sum of random variables that are simulated from a computer model. A numerical study on examples with copula dependence is conducted where the proposed estimator performs favourably in terms of variance compared to other unbiased estimators. We provide applications and extensions to the estimation of marginal densities in Bayesian statistics and to the estimation of the density of sums of random variables under Gaussian copula dependence.

math.ST↗

arXiv2018

Rare tail approximation using asymptotics and $L^1$ polar coordinates

In this work, we propose a class of importance sampling (IS) estimators for estimating the right tail probability of a sum of continuous random variables based on a change of variables to $L^1$ polar coordinates in which the radial and angular components of the IS distribution are considered separately. When the asymptotic behaviour of the sum is known we exploit it for the radial change of measure, and the resulting estimator has the appealing form of the (known) asymptotic multiplied by a random multiplicative correction factor. Given we assume knowledge of the asymptotic behaviour of the sum in this framework, traditional notions of efficiency that appear in the rare-event literature hold little practical meaning here. Instead, we focus on the practical behaviour of the proposed estimator in the pre-asymptotic regime for right tail probabilities between roughly $10^{-3}$ and $10^{-7}$. The proposed estimator and procedure are applicable in both the heavy- and light-tailed settings, as well as for independent and dependent summands. In the case of independent summands, we find that our estimator compares favourably with exponential tilting (iid light-tailed summands) and the Asmussen--Kroese method (independent subexponential summands). However, for dependent subexponential summands using the same simple angular distribution as for the independent case, the performance of our estimator rapidly degenerates with increasing dimension, suggesting an open avenue for further research.

stat.ME↗

arXiv2017

Tail asymptotics of light-tailed Weibull-like sums

We consider sums of $n$ i.i.d. random variables with tails close to $\exp\{-x^β\}$ for some $β>1$. Asymptotics developed by Rootzén (1987) and Balkema, Klüppelberg & Resnick (1993) are discussed from the point of view of tails rather of densities, using a somewhat different angle, and supplemented with bounds, results on a random number $N$ of terms, and simulation algorithms.

math.PR↗

arXiv2016

Efficient simulation for dependent rare events with applications to extremes

We consider the general problem of estimating probabilities which arise as a union of dependent events. We propose a flexible series of estimators for such probabilities, and describe variance reduction schemes applied to the proposed estimators. We derive efficiency results of the estimators in rare-event settings, in particular those associated with extremes. Finally, we examine the performance of our estimators in a numerical example.

math.PR↗

arXiv2016

Orthonormal polynomial expansions and lognormal sum densities

Approximations for an unknown density $g$ in terms of a reference density $f_ν$ and its associated orthonormal polynomials are discussed. The main application is the approximation of the density $f$ of a sum $S$ of lognormals which may have different variances or be dependent. In this setting, $g$ may be $f$ itself or a transformed density, in particular that of $\log S$ or an exponentially tilted density. Choices of reference densities $f_ν$ that are considered include normal, gamma and lognormal densities. For the lognormal case, the orthonormal polynomials are found in closed form and it is shown that they are not dense in $L_2(f_ν)$, a result that is closely related to the lognormal distribution not being determined by its moments and provides a warning to the most obvious choice of taking $f_ν$ as lognormal. Numerical examples are presented and comparisons are made to established approaches such as the Fenton--Wilkinson method and skew-normal approximations. Also extensions to density estimation for statistical data sets and non-Gaussian copulas are outlined.

math.PR↗

arXiv2015

Approximating the Laplace transform of the sum of dependent lognormals

Let $(X_1, \dots, X_n)$ be multivariate normal, with mean vector $\boldsymbolμ$ and covariance matrix $\boldsymbolΣ$, and $S_n=\mathrm{e}^{X_1}+\cdots+\mathrm{e}^{X_n}$. The Laplace transform ${\cal L}(θ)=\mathbb{E}\mathrm{e}^{-θS_n} \propto \int \exp\{-h_θ(\boldsymbol{x})\} \,\mathrm{d} \boldsymbol{x}$ is represented as $\tilde{\cal L}(θ)I(θ)$, where $\tilde{\cal L}(θ)$ is given in closed-form and $I(θ)$ is the error factor ($\approx 1$). We obtain $\tilde{\cal L}(θ)$ by replacing $h_θ(\boldsymbol{x})$ with a second order Taylor expansion around its minimiser $\boldsymbol{x}^*$. An algorithm for calculating the asymptotic expansion of $\boldsymbol{x}^*$ is presented, and it is shown that $I(θ)\to 1$ as $θ\to\infty$. A variety of numerical methods for evaluating $I(θ)$ are discussed, including Monte Carlo with importance sampling and quasi-Monte Carlo. Numerical examples (including Laplace transform inversion for the density of $S_n$) are also given.

math.PR↗

arXiv2015

Hawkes Processes

Hawkes processes are a particularly interesting class of stochastic process that have been applied in diverse areas, from earthquake modelling to financial analysis. They are point processes whose defining characteristic is that they 'self-excite', meaning that each arrival increases the rate of future arrivals for some period of time. Hawkes processes are well established, particularly within the financial literature, yet many of the treatments are inaccessible to one not acquainted with the topic. This survey provides background, introduces the field and historical developments, and touches upon all major aspects of Hawkes processes.

math.PR↗