BOUMEZOUED Alexandre

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In this paper, we propose a multivariate Hawkes framework for modelling and predicting cyber-attacks frequency. The inference is based on a public dataset containing features of data-breaches targeting the US industry. As a main output of this paper, we demonstrate the ability of Hawkes models to capture self-excitation and interactions of data-breaches depending on their type and targets. In this setting we detail prediction results providing the full joint distribution of future cyber attacks times of occurrence. In addition we show that a non-instantaneous excitation in the multi-variate Hawkes model, which is not the classical framework of the exponential kernel, better fits with our data. In an insurance framework, this study allows to determine quantiles for number of attacks, useful for an internal model, as well as the frequency component for a data breach guarantee.

Motivated by improving mortality tables from human demography databases, we investigate statistical inference of a stochastic age-evolving density of a population alimented by time inhomogeneous mortality and fertility. Asymptotics are taken as the size of the population grows within a limited time horizon: the observation gets closer to the solution of the Von Foerster Mc Kendrick equation, and the difficulty lies in controlling simultaneously the stochastic approximation to the limiting PDE in a suitable sense together with an appropriate parametrisation of the anisotropic solution. In this setting, we prove new concentration inequalities that enable us to implement the Goldenshluger-Lepski algorithm and derive oracle inequalities. We obtain minimax optimality and adaptation over a wide range of anisotropic H\"older smoothness classes.

Following the work of Cairns et al. (2016), we aim at correcting mortality estimates based on fertility data. As already conjectured by Richards (2008), the computation of exposure to risk can suffer from errors for cohorts born in years in which births are fluctuating. In this context, we first point our attention to the Human Mortality Database (HMD), the reference mortality data provider. While comparing period and cohort mortality tables, we highlight the presence of anomalies in period ones in the form of isolated cohort effects. Our investigation of the HMD methodology exhibits a strong assumption of uniform distribution of births that is specific to period tables, therefore likely to be at the core of the asymmetry between both. Based on the idea of Cairns et al. (2016) regarding the construction of kind of "data quality indicator", we make a new and intensive exploitation of the Human Fertility Database (HFD), which is from our point of view a crucial source as it represents the perfect counterpart of the HMD in terms of fertility. This indicator is then used to construct corrected period mortality tables for several countries, which we analyze on both an historical and prospective point of view. Our main conclusions relate to the reduction of volatility of mortality improvement rates, the impact in the use of cohort parameters in stochastic mortality models, as well as a better fit of corrected tables by classical mortality models.

This paper demonstrates the efficiency of using Edgeworth and Gram-Charlier expansions in the calibration of the Libor Market Model with Stochastic Volatility and Displaced Diffusion (DD-SV-LMM). Our approach brings together two research areas. first, the results regarding the SV-LMM since the work of Wu and Zhang (2006), especially on the moment generating function, and second the approximation of density distributions based on Edgeworth or Gram-Charlier expansions. By exploring the analytical tractability of moments up to fourth order, we are able to perform an adjustment of the reference Bachelier model with normal volatilities for skewness and kurtosis, and as a by-product to derive a smile formula relating the volatility to the moneyness with interpretable parameters. As a main conclusion, our numerical results show a 98% reduction in computational time for the DD-SV-LMM calibration process compared to the classical numerical integration method developed by Heston (1993).

We propose a new method to efficiently price swap rates derivatives under the LIBOR Market Model with Stochastic Volatility and Displaced Diffusion (DDSVLMM). This method uses polynomial processes combined with Gram-Charlier expansion techniques. The standard pricing method for this model relies on dynamics freezing to recover an Heston-type model for which analytical formulas are available. This approach is time consuming and efficient approximations based on Gram-Charlier expansions have been recently proposed. In this article, we first discuss the fact that for a class of stochastic volatility model, including the Heston one, the classical sufficient condition ensuring the convergence of the Gram-Charlier series can not be satisfied. Then, we propose an approximating model based on Jacobi process for which we can prove the stability of the Gram-Charlier expansion. For this approximation, we have been able to prove a strong convergence toward the original model. moreover, we give an estimate of the convergence rate. We also prove a new result on the convergence of the Gram-Charlier series when the volatility factor is not bounded from below. We finally illustrate our convergence results with numerical examples.

Since the conjecture of Richards (2008), the work by Cairns et al. (2016) and subsequent developments by Boumezoued (2016), Boumezoued et al.

In this paper, we propose a multivariate Hawkes framework for modelling and predicting cyber-attacks frequency. The inference is based on a public dataset containing features of data-breaches targeting the US industry. As a main output of this paper, we demonstrate the ability of Hawkes models to capture self-excitation and interactions of data-breaches depending on their type and targets. In this setting we detail prediction results providing the full joint distribution of future cyber attacks times of occurrence. In addition we show that a non-instantaneous excitation in the multi-variate Hawkes model, which is not the classical framework of the exponential kernel, better fits with our data. In an insurance framework, this study allows to determine quantiles for number of attacks, useful for an internal model, as well as the frequency component for a data breach guarantee.

We propose a new inference strategy for general population mortality tables based on annual population and death estimates, completed by monthly birth counts. We rely on a deterministic population dynamics model and establish formulas that links the death rates to be estimated with the observables at hand. The inference algorithm takes the form of a recursive and implicit scheme for computing death rate estimates. This paper demonstrates both theoretically and numerically the efficiency of using additional monthly birth counts for appropriately computing annual mortality tables. As a main result, the improved mortality estimators show better features, including the fact that previous anomalies in the form of isolated cohort effects disappear, which confirms from a mathematical perspective the previous contributions by Richards (2008), Cairns et al. (2016) and Boumezoued (2016).

We propose to take advantage of the common knowledge of the characteristic function of the swap rate process as modelled in the LIBOR Market Model with Stochastic Volatility and Displaced Diffusion (DDSVLMM) to derive analytical expressions of the gradient of swaptions prices with respect to the model parameters. We use this result to derive an efficient calibration method for the DDSVLMM using gradient-based optimization algorithms. Our study relies on and extends the work by (Cui et al., 2017) that developed the analytical gradient for fast calibration of the Heston model, based on an alternative formulation of the Heston moment generating function proposed by (del Baño et al., 2010). Our main conclusion is that the analytical gradient-based calibration is highly competitive for the DDSVLMM, as it significantly limits the number of steps in the optimization algorithm while improving its accuracy. The efficiency of this novel approach is compared to classical standard optimization procedures.

In this paper, we discuss the impact of some mortality data anomalies on an internal model capturing longevity risk in the Solvency 2 framework. In particular, we are concerned with abnormal cohort effects such as those for generations 1919 and 1920, for which the period tables provided by the Human Mortality Database show particularly low and high mortality rates respectively. To provide corrected tables for the three countries of interest here (France, Italy and West Germany), we use the approach developed by Boumezoued (2016) for countries for which the method applies (France and Italy), and provide an extension of the method for West Germany as monthly fertility histories are not sufficient to cover the generations of interest. These mortality tables are crucial inputs to stochastic mortality models forecasting future scenarios, from which the extreme 0,5% longevity improvement can be extracted, allowing for the calculation of the Solvency Capital Requirement (SCR). More precisely, to assess the impact of such anomalies in the Solvency II framework, we use a simplified internal model based on three usual stochastic models to project mortality rates in the future combined with a closure table methodology for older ages. Correcting this bias obviously improves the data quality of the mortality inputs, which is of paramount importance today, and slightly decreases the capital requirement. Overall, the longevity risk assessment remains stable, as well as the selection of the stochastic mortality model. As a collateral gain of this data quality improvement, the more regular estimated parameters allow for new insights and a refined assessment regarding longevity risk.

In the summer of 2013, the term "Big Data" made its appearance and aroused strong interest among companies. This thesis studies the contribution of these methods to actuarial sciences. It addresses both theoretical and practical issues on high-potential topics such as textit{Optical Character Recognition} (OCR), text analysis, data anonymization or model interpretability. Starting with the application of machine learning methods in the calculation of economic capital, we then try to better illustrate the frontality that can exist between machine learning and statistics. Putting forward some advantages and different techniques, we then study the application of deep neural networks in the optical analysis of documents and text, once extracted. The use of complex methods and the implementation of the General Data Protection Regulation (GDPR) in 2018 led us to study the potential impacts on pricing models. By applying anonymization methods on pure premium calculation models in non-life insurance, we explored different generalization approaches based on unsupervised learning. Finally, as the regulation also imposes criteria in terms of model explanation, we conclude with a general study of the methods that allow today to better understand complex methods such as neural networks.

This paper analyses cause-of-death mortality changes and its impacts on the whole population evolution. The study combines cause-of-death analysis and population dynamics techniques. Our aim is to measure the impact of cause-of-death reduction on the whole population age structure, and more specifically on the depdendency ratio which is a crucial quantity for pay-as-you-go pension systems. Whereas previous studies on causes of death focused on mortality indicators such as survival curves or life expectancy, our approach provides additional information by including birth patterns. As an important conclusion, our numerical results based on French data show that populations with identical life expectancies can present important differences in their age pyramid resulting from different cause-specific mortality reductions. Sensitivities to fertility level and population flows are also given.

In this paper, we discuss the impact of some mortality data anomalies on an internal model capturing longevity risk in the Solvency 2 framework. In particular, we are concerned with abnormal cohort effects such as those for generations 1919 and 1920, for which the period tables provided by the Human Mortality Database show particularly low and high mortality rates respectively. To provide corrected tables for the three countries of interest here (France, Italy and West Germany), we use the approach developed by Boumezoued (2016) for countries for which the method applies (France and Italy), and provide an extension of the method for West Germany as monthly fertility histories are not sufficient to cover the generations of interest. These mortality tables are crucial inputs to stochastic mortality models forecasting future scenarios, from which the extreme 0,5% longevity improvement can be extracted, allowing for the calculation of the Solvency Capital Requirement (SCR). More precisely, to assess the impact of such anomalies in the Solvency II framework, we use a simplified internal model based on three usual stochastic models to project mortality rates in the future combined with a closure table methodology for older ages. Correcting this bias obviously improves the data quality of the mortality inputs, which is of paramount importance today, and slightly decreases the capital requirement. Overall, the longevity risk assessment remains stable, as well as the selection of the stochastic mortality model. As a collateral gain of this data quality improvement, the more regular estimated parameters allow for new insights and a refined assessment regarding longevity risk.

We propose a new inference strategy for general population mortality tables based on annual population and death estimates, completed by monthly birth counts. We rely on a deterministic population dynamics model and establish formulas that links the death rates to be estimated with the observables at hand. The inference algorithm takes the form of a recursive and implicit scheme for computing death rate estimates. This paper demonstrates both theoretically and numerically the efficiency of using additional monthly birth counts for appropriately computing annual mortality tables. As a main result, the improved mortality estimators show better features, including the fact that previous anomalies in the form of isolated cohort effects disappear, which confirms from a mathematical perspective the previous contributions by Richards (2008), Cairns et al. (2016) and Boumezoued (2016).

In this paper, our aim is to measure mortality rates which are specific to individual observable factors when these can change during life. The study is based on longitudinal data recording marital status and socio-professional features at census times, therefore the observation scheme is interval-censored since individual characteristics are only observed at isolated dates and transition times remain unknown. To this aim, we develop a parametric maximum likelihood estimation procedure for multi-state models that takes into account both interval-censoring and reversible transitions. This method, inspired by recent advances in the statistical literature, allows us to capture characteristic-specific mortality rates, in particular to recover the mortality compensation law at high ages, but also to capture the age pattern of characteristics changes. The dynamics of several population compositions is addressed, and allows us to give explanations on the pattern of aggregate mortality, as well as on the impact on typical life insurance products. Particular attention is devoted to characteristics changes and parameter uncertainty that are both crucial to take into account.

This paper surveys the stochastic modelling of individual claims occurrence and development for reserving purposes in non-life (general) insurance. The paper revisits the continuous time stochastic modelling framework of Norberg (1993) and Hesselager (1994), and provides a consistent presentation of the modelling, inference, and forecasting (with simulation and closed-forms) of individual claims histories as well as aggregate quantities as the overall reserve for both RBNS and IBNR claims. Numerical illustrations are given based on real portfolio datasets, as well as comparisons with classical triangle-based methods.

This paper demonstrates the efficiency of using Edgeworth and Gram-Charlier expansions in the calibration of the Libor Market Model with Stochastic Volatility and Displaced Diffusion (DD-SV-LMM). Our approach brings together two research areas. first, the results regarding the SV-LMM since the work of Wu and Zhang (2006), especially on the moment generating function, and second the approximation of density distributions based on Edgeworth or Gram-Charlier expansions. By exploring the analytical tractability of moments up to fourth order, we are able to perform an adjustment of the reference Bachelier model with normal volatilities for skewness and kurtosis, and as a by-product to derive a smile formula relating the volatility to the moneyness with interpretable parameters. As a main conclusion, our numerical results show a 98% reduction in computational time for the DD-SV-LMM calibration process compared to the classical numerical integration method developed by Heston (1993).

Following the work of Cairns et al. (2016), we aim at correcting mortality estimates based on fertility data. As already conjectured by Richards (2008), the computation of exposure to risk can suffer from errors for cohorts born in years in which births are fluctuating. In this context, we first point our attention to the Human Mortality Database (HMD), the reference mortality data provider. While comparing period and cohort mortality tables, we highlight the presence of anomalies in period ones in the form of isolated cohort effects. Our investigation of the HMD methodology exhibits a strong assumption of uniform distribution of births that is specific to period tables, therefore likely to be at the core of the asymmetry between both. Based on the idea of Cairns et al. (2016) regarding the construction of kind of "data quality indicator", we make a new and intensive exploitation of the Human Fertility Database (HFD), which is from our point of view a crucial source as it represents the perfect counterpart of the HMD in terms of fertility. This indicator is then used to construct corrected period mortality tables for several countries, which we analyze on both an historical and prospective point of view. Our main conclusions relate to the reduction of volatility of mortality improvement rates, the impact in the use of cohort parameters in stochastic mortality models, as well as a better fit of corrected tables by classical mortality models.

This thesis focuses on population dynamics models and their applications, on one hand to demography and actuarial science, and on the other hand to Hawkes processes. This work explores through several viewpoints how population structures evolve over time, both in terms of ages and characteristics. In five chapters, we develop a common philosophy which studies the population at the scale of the individual in order to better understand the behavior of aggregate quantities. The first chapter introduces the motivations and details the main contributions in French. In Chapter 2, based on Bensusan et al. (2010–2015), we survey the modeling of characteristic and age-structured populations and their dynamics, as well as several examples motivated by demographic issues. We detail the mathematical construction of such population processes, as well as their link with well known deterministic equations in demography. We illustrate the simulation algorithm on an example of cohort effect, and we also discuss the role of the random environment. The two following chapters emphasize on the importance of the age pyramid. Chapter 3 uses a particular form of the general model introduced in Chapter 2 in order to study Hawkes processes with general immigrants. In this theoretical part based on Boumezoued (2015b) we use the concept of age pyramid to derive new distribution properties for a class of fertility functions which generalize the popular exponential case. Chapter 4 is based on Arnold et al. (2015) and analyses the impact of cause-of- death mortality changes on the population age pyramid, and in particular on the dependency ratio which is crucial to measure population ageing. By including birth patterns, this numerical work based on WHO data gives additional insights compared to the existing literature on causes of death focusing only on mortality indicators. The last two chapters focus on population heterogeneity. The aim of Chapter 5, based on Boumezoued et al. (2015), is to measure mortality heterogeneity on French longitudinal data called Échantillon Démographique Permanent. In this work, inspired by recent advances in the statistical literature, we develop a parametric maximum likelihood method for multi-state models which takes into account both interval censoring and reversible transitions. Finally, Chapter 6, based on Boumezoued (2015a), considers the general model introduced in Chapter 2 in which individuals can give birth, change their characteristics and die. The contribution of this theoretical work is the analysis of the population behavior when individual characteristics change very often. We establish a large population limit theorem for the age pyramid process, whose dynamics is described at the limit by birth and death rates which are averaged over the stable population composition.

This thesis deals with the modeling of population dynamics and its applications to demography and actuarial science on the one hand, and to the study of Hawkes processes on the other hand. This thesis proposes to explore through different points of view how the structure of a population is deformed, both in terms of age distribution and in terms of its composition in terms of characteristics. Through five chapters, we present the same philosophy which, in order to understand how aggregate quantities evolve, proposes to study the dynamics of the population at a finer scale, that of the individual. After a first introductory chapter in French, detailing the motivations and the main contributions, we first propose in Chapter 2 the description of the general framework of the random dynamic modeling of populations structured in characteristics and ages, based on Bensusan et al. (2010-2015), as well as several examples motivated by demographic and actuarial applications. We detail the mathematical construction of such processes as well as the link to classical deterministic equations in demography. We also discuss the impact of heterogeneity on the example of a cohort effect, as well as the role of the random environment. The next two chapters highlight the importance of the age pyramid. The general population model from Chapter 2 is declined in Chapter 3 to study Hawkes processes with general immigrants, for which we exploit the concept of the age pyramid. In this theoretical study, based on Boumezoued (2015b), we establish new results on their distribution for a class of functions that generalize the exponential case studied so far. In Chapter 4, following Arnold et al. (2015), we analyze the impact of changes in cause-of-death mortality on the dynamics of the population pyramid, and in particular on the dependency ratio which is a crucial indicator of population aging. By including the set of births in the dynamics, this simulation work, based on WHO data, complements the existing literature on causes of death which traditionally focuses on mortality indicators. The last two chapters focus on population heterogeneity. Chapter 5, based on Boumezoued et al. (2015), proposes to measure mortality heterogeneity in INSEE Permanent Demographic Sample data. As part of this contribution of adapting statistical methods and its implementation on real data, we propose a parametric maximum likelihood estimation method for multi-state models that takes into account both interval censoring, characteristic of longitudinal data from the census, and also the return in intermediate states. Finally, Chapter 6, taken from Boumezoued (2015a), repeats the general model from Chapter 2 in which individuals can give birth, change characteristics, and die. The contribution of this theoretical part is to study the behavior of the population when individual characteristics change frequently. We establish a large population limit theorem for the population pyramid process, whose behavior is then described by birth and death rates aggregated over the stable structure in terms of characteristics.

This thesis deals with the modeling of population dynamics and its application to demography and actuarial science on the one hand, and to the study of Hawkes processes on the other hand. This thesis proposes to explore through different points of view how the structure of a population is deformed, both in terms of age distribution and in terms of its composition in terms of characteristics. Through five chapters, we present the same philosophy which, in order to understand how aggregate quantities evolve, proposes to study the dynamics of the population at a finer scale, that of the individual. After a first introductory chapter in French, detailing the motivations and the main contributions, we first propose in Chapter 2 the description of the general framework of random dynamic modeling of populations structured in characteristics and ages, based on Bensusan et al. (2010-2015), as well as several examples motivated by demographic and actuarial applications. We detail the mathematical construction of such processes as well as the link to classical deterministic equations in demography. We also discuss the impact of heterogeneity on the example of a cohort effect, as well as the role of the random environment. The next two chapters highlight the importance of the age pyramid. The general population model from Chapter 2 is declined in Chapter 3 to study Hawkes processes with general immigrants, for which we exploit the concept of the age pyramid. In this theoretical study, based on Boumezoued (2015b), we establish new results on their distribution for a class of functions that generalize the exponential case studied so far. In Chapter 4, following Arnold et al. (2015), we analyze the impact of changes in cause-of-death mortality on the dynamics of the population pyramid, and in particular on the dependency ratio which is a crucial indicator of population aging. By including the set of births in the dynamics, this simulation work, based on WHO data, complements the existing literature on causes of death which traditionally focuses on mortality indicators. The last two chapters focus on the heterogeneity of populations. Chapter 5, based on Boumezoued et al. (2015), proposes to measure mortality heterogeneity in INSEE's Permanent Demographic Sample data. As part of this contribution of adapting statistical methods and its implementation on real data, we propose a parametric maximum likelihood estimation method for multi-state models that takes into account both interval censoring, characteristic of longitudinal data from the census, and also the return in intermediate states. Finally, Chapter 6, taken from Boumezoued (2015a), repeats the general model from Chapter 2 in which individuals can give birth, change characteristics, and die. The contribution of this theoretical part is to study the behavior of the population when individual characteristics change frequently. We establish a large population limit theorem for the population pyramid process, whose behavior is then described by birth and death rates aggregated over the stable structure in terms of characteristics.

No summary available.

This paper focuses on a class of linear Hawkes processes with general immigrants. These are counting processes with shot noise intensity, including self-excited and externally excited patterns. For such processes, we introduce the concept of age pyramid which evolves according to immigration and births. The virtue if this approach that combines an intensity process definition and a branching representation is that the population age pyramid keeps track of all past events. This is used to compute new distribution properties for a class of linear Hawkes processes with general immigrants which generalize the popular exponential fertility function. The pathwise construction of the Hawkes process and its underlying population is also given.

This paper analyses cause-of-death mortality changes and its impacts on the whole population evolution. The study combines cause-of-death analysis and population dynamics techniques. Our aim is to measure the impact of cause-of-death reduction on the whole population age structure, and more specifically on the depdendency ratio which is a crucial quantity for pay-as-you-go pension systems. Whereas previous studies on causes of death focused on mortality indicators such as survival curves or life expectancy, our approach provides additional information by including birth patterns. As an important conclusion, our numerical results based on French data show that populations with identical life expectancies can present important differences in their age pyramid resulting from different cause-specific mortality reductions. Sensitivities to fertility level and population flows are also given.

This paper analyses cause-of-death mortality changes and its impacts on the whole population evolution. The study combines cause-of-death analysis and population dynamics techniques. Our aim is to measure the impact of cause-of-death reduction on the whole population age structure, and more specifically on the depdendency ratio which is a crucial quantity for pay-as-you-go pension systems. Whereas previous studies on causes of death focused on mortality indicators such as survival curves or life expectancy, our approach provides additional information by including birth patterns. As an important conclusion, our numerical results based on French data show that populations with identical life expectancies can present important differences in their age pyramid resulting from different cause-specific mortality reductions. Sensitivities to fertility level and population flows are also given.

In this paper, our aim is to measure mortality rates which are specific to individual observable factors when these can change during life. The study is based on longitudinal data recording marital status and socio-professional features at census times, therefore the observation scheme is interval-censored since individual characteristics are only observed at isolated dates and transition times remain unknown. To this aim, we develop a parametric maximum likelihood estimation procedure for multi-state models that takes into account both interval-censoring and reversible transitions. This method, inspired by recent advances in the statistical literature, allows us to capture characteristic-specific mortality rates, in particular to recover the mortality compensation law at high ages, but also to capture the age pattern of characteristics changes. The dynamics of several population compositions is addressed, and allows us to give explanations on the pattern of aggregate mortality, as well as on the impact on typical life insurance products. Particular attention is devoted to characteristics changes and parameter uncertainty that are both crucial to take into account.

In this paper, we consider the large population limit of an age and characteristic-structured stochastic population model evolving according to individual birth, death and fast characteristics changes during life. Both the large population framework and the fast characteristics changes assumption are motivated by demographic patterns of human populations at the scale of a given country. When rescaling the population process, and under some invariance assumption about the characteristics changes dynamics, the classical determin-istic transport-renewal McKendrick-Von Foerster equation appears, that describes the time evolution of the age pyramid driven by equivalent birth and death rates. The proof follows the work of Méléard and Tran (2012) and Gupta et al. (2014) in which analogous mathematical issues are encountered. We further prove that the sequence of processes taking track of the characteristics distribution is not tight even in the presence of age-independent demographic rates. To illustrate the use of the limiting model, a set of computable invariant distributions is given, as well as numerical implementation of equivalent birth and death rates which mimics real demographic data. These results highlight the fact that characteristics changes frequencies are crucial to understand aggregate demographic rates at the macroscopic scale.