Special Issue "Application of Stochastic Processes in Insurance"

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A special issue of Risks (ISSN 2227-9091).

Deadline for manuscript submissions: closed (1 November 2013)

Special Issue Editors

Guest Editor
Dr. Pierre Patie (Website)

Operations Research and Information Engineering, Cornell University, 220 Rhodes Hall, Ithaca, NY 14853, USA
Interests: insurance mathematics; ruin theory; path dependent options; point processes
Guest Editor
Dr. Angelos Dassios (Website)

Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK
Fax: +44 207 9557 416
Interests: insurance mathematics; ruin theory; path dependent options; point processes
Guest Editor
Prof. Dr. Erhan Bayraktar (Website)

Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043, USA
Interests: mathematical finance; applied probability; stochastic analysis; stochastic control; optimal stopping

Special Issue Information

Dear Colleagues,

Stochastic methods have been intensively used in insurance for a very long time, making the application of stochastic processes in this domain a well-established field both for asset and liability modeling. For example, several kinds of stochastic processes generalizing the classical Cramér-Lundberg model have been successful in the modeling of both the timing and the size of losses. These include dynamics incorporating returns on investments, reinsurance, dividends and taxes as well as stochastic dependence structure between claims amounts and/or arrival times. On the other hand, the management of both pension funds and sophisticated savings products has recently required the use of stochastic processes for modeling several type of risks such as longevity, mortality and policyholders behaviors risks. This volume aim to highlight these diverse applications of stochastic processes in insurance.

Prof. Dr. Pierre Patie
Dr. Angelos Dassios
Prof. Dr. Erhan Bayraktar
Guest Editors

Submission

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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Published Papers (8 papers)

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Research

Open AccessArticle Neumann Series on the Recursive Moments of Copula-Dependent Aggregate Discounted Claims
Risks 2014, 2(2), 195-210; doi:10.3390/risks2020195
Received: 1 November 2013 / Revised: 10 April 2014 / Accepted: 16 May 2014 / Published: 27 May 2014
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Abstract
We study the recursive moments of aggregate discounted claims, where the dependence between the inter-claim time and the subsequent claim size is considered. Using the general expression for the m-th order moment proposed by Léveillé and Garrido (Scand. Actuar. J. 2001, [...] Read more.
We study the recursive moments of aggregate discounted claims, where the dependence between the inter-claim time and the subsequent claim size is considered. Using the general expression for the m-th order moment proposed by Léveillé and Garrido (Scand. Actuar. J. 2001, 2, 98–110), which takes the form of the Volterra integral equation (VIE), we used the method of successive approximation to derive the Neumann series of the recursive moments. We then compute the first two moments of aggregate discounted claims, i.e., its mean and variance, based on the Neumann series expression, where the dependence structure is captured by a Farlie–Gumbel–Morgenstern (FGM) copula, a Gaussian copula and a Gumbel copula with exponential marginal distributions. Insurance premium calculations with their figures are also illustrated. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
Open AccessArticle Modeling and Performance of Bonus-Malus Systems: Stationarity versus Age-Correction
Risks 2014, 2(1), 49-73; doi:10.3390/risks2010049
Received: 30 September 2013 / Revised: 17 February 2014 / Accepted: 18 February 2014 / Published: 11 March 2014
Cited by 1 | PDF Full-text (460 KB) | HTML Full-text | XML Full-text
Abstract
In a bonus-malus system in car insurance, the bonus class of a customer is updated from one year to the next as a function of the current class and the number of claims in the year (assumed Poisson). Thus the sequence of [...] Read more.
In a bonus-malus system in car insurance, the bonus class of a customer is updated from one year to the next as a function of the current class and the number of claims in the year (assumed Poisson). Thus the sequence of classes of a customer in consecutive years forms a Markov chain, and most of the literature measures performance of the system in terms of the stationary characteristics of this Markov chain. However, the rate of convergence to stationarity may be slow in comparison to the typical sojourn time of a customer in the portfolio. We suggest an age-correction to the stationary distribution and present an extensive numerical study of its effects. An important feature of the modeling is a Bayesian view, where the Poisson rate according to which claims are generated for a customer is the outcome of a random variable specific to the customer. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
Open AccessArticle Catastrophe Insurance Modeled by Shot-Noise Processes
Risks 2014, 2(1), 3-24; doi:10.3390/risks2010003
Received: 6 November 2013 / Revised: 28 January 2014 / Accepted: 29 January 2014 / Published: 21 February 2014
Cited by 2 | PDF Full-text (343 KB) | HTML Full-text | XML Full-text
Abstract
Shot-noise processes generalize compound Poisson processes in the following way: a jump (the shot) is followed by a decline (noise). This constitutes a useful model for insurance claims in many circumstances; claims due to natural disasters or self-exciting processes exhibit similar features. [...] Read more.
Shot-noise processes generalize compound Poisson processes in the following way: a jump (the shot) is followed by a decline (noise). This constitutes a useful model for insurance claims in many circumstances; claims due to natural disasters or self-exciting processes exhibit similar features. We give a general account of shot-noise processes with time-inhomogeneous drivers inspired by recent results in credit risk. Moreover, we derive a number of useful results for modeling and pricing with shot-noise processes. Besides this, we obtain some highly tractable examples and constitute a useful modeling tool for dynamic claims processes. The results can in particular be used for pricing Catastrophe Bonds (CAT bonds), a traded risk-linked security. Additionally, current results regarding the estimation of shot-noise processes are reviewed. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
Open AccessArticle Ruin Time and Severity for a Lévy Subordinator Claim Process: A Simple Approach
Risks 2013, 1(3), 192-212; doi:10.3390/risks1030192
Received: 6 November 2013 / Revised: 2 December 2013 / Accepted: 5 December 2013 / Published: 13 December 2013
Cited by 1 | PDF Full-text (340 KB) | HTML Full-text | XML Full-text
Abstract
This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a [...] Read more.
This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a subordinator, a simple and direct method for determining the finite time (and ultimate) ruin probabilities, the distribution of the ruin severity, the reserves prior to ruin, and the Laplace transform of the ruin time. Interestingly, the usual net profit condition will be essentially relaxed. Most results generalize those known for the compound Poisson claim process. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
Open AccessArticle Impact of Climate Change on Heat Wave Risk
Risks 2013, 1(3), 176-191; doi:10.3390/risks1030176
Received: 1 November 2013 / Revised: 3 December 2013 / Accepted: 5 December 2013 / Published: 12 December 2013
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Abstract
We study a new risk measure inspired from risk theory with a heat wave risk analysis motivation. We show that this risk measure and its sensitivities can be computed in practice for relevant temperature stochastic processes. This is in particular useful for [...] Read more.
We study a new risk measure inspired from risk theory with a heat wave risk analysis motivation. We show that this risk measure and its sensitivities can be computed in practice for relevant temperature stochastic processes. This is in particular useful for measuring the potential impact of climate change on heat wave risk. Numerical illustrations are given. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
Open AccessArticle A Risk Model with an Observer in a Markov Environment
Risks 2013, 1(3), 148-161; doi:10.3390/risks1030148
Received: 11 October 2013 / Revised: 5 November 2013 / Accepted: 5 November 2013 / Published: 11 November 2013
Cited by 6 | PDF Full-text (311 KB) | HTML Full-text | XML Full-text
Abstract
We consider a spectrally-negative Markov additive process as a model of a risk process in a random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process as negative, where the [...] Read more.
We consider a spectrally-negative Markov additive process as a model of a risk process in a random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process as negative, where the observation rate may depend on the state of the environment. Using an approximation argument and spectral theory, we establish an explicit formula for the resulting survival probabilities in this general setting. We also discuss an efficient evaluation of the involved quantities and provide a numerical illustration. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
Open AccessArticle Optimal Deterministic Investment Strategies for Insurers
Risks 2013, 1(3), 101-118; doi:10.3390/risks1030101
Received: 30 September 2013 / Revised: 28 October 2013 / Accepted: 2 November 2013 / Published: 7 November 2013
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Abstract
We consider an insurance company whose risk reserve is given by a Brownian motion with drift and which is able to invest the money into a Black–Scholes financial market. As optimization criteria, we treat mean-variance problems, problems with other risk measures, exponential [...] Read more.
We consider an insurance company whose risk reserve is given by a Brownian motion with drift and which is able to invest the money into a Black–Scholes financial market. As optimization criteria, we treat mean-variance problems, problems with other risk measures, exponential utility and the probability of ruin. Following recent research, we assume that investment strategies have to be deterministic. This leads to deterministic control problems, which are quite easy to solve. Moreover, it turns out that there are some interesting links between the optimal investment strategies of these problems. Finally, we also show that this approach works in the Lévy process framework. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)
Open AccessArticle Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates
Risks 2013, 1(3), 81-100; doi:10.3390/risks1030081
Received: 8 August 2013 / Revised: 15 October 2013 / Accepted: 17 October 2013 / Published: 25 October 2013
Cited by 1 | PDF Full-text (430 KB) | HTML Full-text | XML Full-text
Abstract
In the actuarial literature, it has become common practice to model future capital returns and mortality rates stochastically in order to capture market risk and forecasting risk. Although interest rates often should and mortality rates always have to be non-negative, many authors [...] Read more.
In the actuarial literature, it has become common practice to model future capital returns and mortality rates stochastically in order to capture market risk and forecasting risk. Although interest rates often should and mortality rates always have to be non-negative, many authors use stochastic diffusion models with an affine drift term and additive noise. As a result, the diffusion process is Gaussian and, thus, analytically tractable, but negative values occur with positive probability. The argument is that the class of Gaussian diffusions would be a good approximation of the real future development. We challenge that reasoning and study the asymptotics of diffusion processes with affine drift and a general noise term with corresponding diffusion processes with an affine drift term and an affine noise term or additive noise. Our study helps to quantify the error that is made by approximating diffusive interest and mortality rate models with Gaussian diffusions and affine diffusions. In particular, we discuss forward interest and forward mortality rates and the error that approximations cause on the valuation of life insurance claims. Full article
(This article belongs to the Special Issue Application of Stochastic Processes in Insurance)

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