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On a partial integrodifferential equation of Seal’s type
Institution:1. Institute for Methods and Instrumentation in Synchrotron Radiation Research G-ISRR, Helmholtz-Zentrum für Materialien und Energie GmbH, Albert-Einstein-Strasse 15, 12489 Berlin, Germany;2. Fakultät für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Strasse 24-25, 14476 Potsdam, Germany;1. Department of Surgery, Children’s Mercy Hospital, Kansas City, Missouri;2. Department of Surgery, Lucille-Packard Children’s Hospital Stanford, Palo Alto, California;3. Division of Pediatric Surgery, University of California San Francisco, 513 Parnassus Ave, San Francisco, California 94143;1. Department of Anesthesiology, Chang Gung Memorial Hospital, Linkou, Taipei, Taiwan, ROC;2. Transgenic & Molecular Immunogenetics Laboratory, Chang Gung Memorial Hospital, Linkou, Taipei, Taiwan, ROC;3. Department of Medicine, Chang Gung University, Linkou, Taipei, Taiwan, ROC;4. Department of Medical Research and Development, Chang Gung Memorial Hospital, Linkou, Taipei, Taiwan, ROC;5. Chang Gung University of Science and Technology, Gueishan, Taoyuan, Taiwan, ROC;1. Department of Pediatrics (AK, NRM), Division of Pediatric Hematology-Oncology, King George’s Medical University, Department of Microbiology (JK), Division of Virology, Sanjay Gandhi Post-Graduate Institute of Medical Sciences, Department of Hematopathology (AK), Department of Microbiology (AJ), King George’s Medical University, Lucknow, Uttar Pradesh, India;1. Faculty of Political Science and Economics, Waseda University, Tokyo, 169-8050, Japan;2. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan
Abstract:In this paper we generalize a partial integrodifferential equation satisfied by the finite time ruin probability in the classical Poisson risk model. The generalization also includes the bivariate distribution function of the time of and the deficit at ruin. We solve the partial integrodifferential equation by Laplace transforms with the help of Lagrange’s implicit function theorem. The assumption of mixed Erlang claim sizes is then shown to result in tractable computational formulas for the finite time ruin probability as well as the bivariate distribution function of the time of and the deficit at ruin. A more general partial integrodifferential equation is then briefly considered.
Keywords:Finite time ruin probability  Deficit at ruin  Time of ruin  Lagrange’s implicit function theorem  Lundberg’s fundamental equation  Mixed Erlang distribution
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