Author
Jan Grandell
Other affiliations: Stockholm University
Bio: Jan Grandell is an academic researcher from Royal Institute of Technology. The author has contributed to research in topics: Ruin theory & Cox process. The author has an hindex of 17, co-authored 36 publications receiving 2074 citations. Previous affiliations of Jan Grandell include Stockholm University.
Papers
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Book•
01 Sep 1992
TL;DR: The classical risk model generalizations of the classical renewal models Cox models stationary models appendix - finite time ruin probabilities as mentioned in this paper, which is a generalization of the risk model renewal model.
Abstract: The classical risk model generalizations of the classical risk model renewal models Cox models stationary models appendix - finite time ruin probabilities
837 citations
Book•
01 May 1997
TL;DR: In this paper, the Mixed Poisson Distributions (MPD) is defined as a mixture of Cox Processes, Gauss-Poisson Processes and Mixed Renewal Processes.
Abstract: Preface Introduction The Mixed Poisson Distributions Some Basic Concepts The Mixed Poisson Process Some Related Processes Cox Processes Gauss-Poisson Processes Mixed Renewal Processes Characterization of Mixed Poisson Processes Reliability Properties of Mixed Poisson Processes Characterization within Birth Processes Characterization within Stationary Point Processes Characterization within General Point Processes Compound Mixed Poisson Distributions Compound Distributions Exponential Bounds Asymptotic Behaviour Recursive Evaluation The Risk Business The Claim Process Ruin Probabilities
360 citations
Book•
01 Jan 1976
TL;DR: In this article, the second order properties of stationary doubly stochastic Poisson sequences are estimated. But the second-order properties of random variables are not considered in this paper.
Abstract: Definitions and basic properties.- Some miscellaneous results.- Characterization and convergence of non-atomic random measures.- Limit theorems.- Estimation of random variables.- Linear estimation of random variables in stationary doubly stochastic Poisson sequences.- Estimation of second order properties of stationary doubly stochastic Poisson sequences.
281 citations
TL;DR: In this article, a statistical investigation is presented on the distribution of dry and precipitation periods at a Swedish station, and it is shown that the expected length of time from an arbitrary moment to the onset of the next precipitation period is about 90 and 35 hours for summer and winter conditions respectively.
Abstract: A statistical investigation is presented on the distribution of dry and precipitation periods at a Swedish station. It is shown that the expected length of time from an arbitrary moment to the onset of the next precipitation period is about 90 and 35 hours for summer and winter conditions respectively. Based on the results of these investigations we have derived expressions for the expected length of life of an aerosol particle in the lower layers of the troposphere. Estimates of effective precipitation scavenging coefficients give expected lengths of life of the order 100–300 hours in summer and 35–80 hours in winter. The connection between the distribution of the length of life of particles in the atmosphere and the distribution of particles on the ground away from a source, is discussed and the importance of studying such age distributions is emphasized. DOI: 10.1111/j.2153-3490.1972.tb01571.x
125 citations
TL;DR: In this paper, the authors considered the approximation of a risk reserve process by a Wiener process and showed that weak convergence can be applied to the problem of risk reserve processes in probability measures on metric spaces.
Abstract: We shall in this paper consider approximation of a risk reserve process by a Wiener process. Our main mathematical tool is the theory of weak convergence of probability measures on metric spaces. Today Billingsley (1968) is the standard reference for that theory. We hope that this paper is readable also for those not acquainted with Billingsley's book. To our knowledge the first application of weak convergence in risk theory is due to Iglehart (1969). In chapter 5 of Beekman (1974) some further discussions are found.
91 citations
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TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.
5,689 citations
Book•
01 Jan 1997
TL;DR: In this article, the authors discuss the relationship between Markov Processes and Ergodic properties of Markov processes and their relation with PDEs and potential theory. But their main focus is on the convergence of random processes, measures, and sets.
Abstract: * Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices
4,562 citations
TL;DR: In this article, Modelling Extremal Events for Insurance and Finance is discussed. But the authors focus on the modeling of extreme events for insurance and finance, and do not consider the effects of cyber-attacks.
Abstract: (2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.
2,729 citations
Book•
16 Oct 2005
TL;DR: The most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management can be found in this paper, where the authors describe the latest advances in the field, including market, credit and operational risk modelling.
Abstract: This book provides the most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management. Whether you are a financial risk analyst, actuary, regulator or student of quantitative finance, Quantitative Risk Management gives you the practical tools you need to solve real-world problems. Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book's methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives. Fully revised and expanded to reflect developments in the field since the financial crisis Features shorter chapters to facilitate teaching and learning Provides enhanced coverage of Solvency II and insurance risk management and extended treatment of credit risk, including counterparty credit risk and CDO pricing Includes a new chapter on market risk and new material on risk measures and risk aggregation
2,580 citations
Book•
18 Oct 2012TL;DR: This rigorous introduction to stochastic geometry will enable you to obtain powerful, general estimates and bounds of wireless network performance and make good design choices for future wireless architectures and protocols that efficiently manage interference effects.
Abstract: Covering point process theory, random geometric graphs and coverage processes, this rigorous introduction to stochastic geometry will enable you to obtain powerful, general estimates and bounds of wireless network performance and make good design choices for future wireless architectures and protocols that efficiently manage interference effects. Practical engineering applications are integrated with mathematical theory, with an understanding of probability the only prerequisite. At the same time, stochastic geometry is connected to percolation theory and the theory of random geometric graphs and accompanied by a brief introduction to the R statistical computing language. Combining theory and hands-on analytical techniques with practical examples and exercises, this is a comprehensive guide to the spatial stochastic models essential for modelling and analysis of wireless network performance.
2,327 citations