1st Edition

# Level-Crossing Problems and Inverse Gaussian Distributions Closed-Form Results and Approximations

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**Level-Crossing Problems and Inverse Gaussian Distributions: Closed-Form Results and Approximations*** *focusses on the inverse Gaussian approximation for the distribution of the first level-crossing time in a shifted compound renewal process framework. This approximation, whose name was coined by the author, is a successful competitor of the normal (or Cramér's), diffusion, and Teugels’ approximations, being a breakthrough in its conditions and accuracy.

Since such approximations underlie numerous applications in risk theory, queueing theory, reliability theory, and mathematical theory of dams and inventories, this book is of interest not only to professional mathematicians, but also to physicists, engineers, and economists.

People from industry with a theoretical background in level-crossing problems, e.g., from the insurance industry, can also benefit from reading this book.

Features:

- Primarily aimed at researchers and postgraduates, but may be of interest to some professionals working in related fields, such as the insurance industry
- Suitable for advanced courses in Applied Probability and, as a supplementary reading, for basic courses in Applied Probability

**1. Introduction: Level-Crossing Problem and Related Fields.** 1.1. Sums of independent Random Variables, Gaussian and Inverse Gaussian Distributions. 1.2. Random Walks and Renewal Processes. 1.3. Level-Crossing by a Compound Renewal Process. 1.4. Closed-form Results and Limit Theorems in Level-Crossing. 1.5. Message, Agenda, and Target Audience. **2. Inverse Gaussian and Generalized Inverse Gaussian Distributions. **2.1. Inverse Gaussian Distribution. 2.2. Generalized Inverse Gaussian Distribution. **3. Integral Expressions.** 3.1. Elementary Integral Expressions. 3.2. Core Integral Expression. 3.3. Composite Integral Expressions. **4. Distribution of Compound Renewal Process at a Fixed Time Point. 4.1.** Compound Renewal Process in Continuous Time. 4.2. Closed-form Results. 4.3. Aspects of Renewal Theory. 4.4. Origin of The Method based on Limit Theorems for Sums. 4.5. Approximation for the Mean. 4.6. Approximation for the Distribution. 4.7. Extensions from Renewal to more General Models. **5. Closed-form Results for the Distribution of First Level-Crossing Time.** 5.1. Representations of the Distribution of First Level-Crossing Time. 5.2. Closed-form Results in Exponential Case. 5.3. Closed-form Expression for Conditional Probability. 5.4. Type II Formula and Random Walk with Random Displacements. 5.5. Closed-form Results, When *T *Is Non-Exponentially Distributed. 5.6. A Result, When *Y *Is Mixed Exponential.** 6. The Inverse Gaussian Approximation.** 6.1. Agenda for This Chapter. 6.2. Statement of Main Results. 6.3. Shorthand Notation, Structure Lemmas, Identities Specific to a choice of Arguments, and Centering At *Z *= 0. 6.4. Expressions of The First Kind. 6.5. Expressions of The Second Kind. 6.6. Expressions of the third kind 6.7. Proof of Theorem 6.1. 6.8. Numerical Illustrations. 6.9. Conclusions. **7. Refinement of the Inverse Gaussian Approximation.** 7.1 Asymptotic Expansions: Rigorous Vs. Heuristic. 7.2. Expansion for the Distribution of the First Level-Crossing Time. 7.3. Proof of Theorem 7.1. 7.4. Numerical Illustrations. **8. Derivatives of The First Level-Crossing Time Distribution.** 8.1. The Problem and Its Rationale. 8.2. Approximations for Derivatives. 8.3. Fundamental Identities for Derivatives with Respect to *C *and *U. *8.4. Proof of Theorem 8.1. 8.5. Proof of Theorem 8.2. **9. A Breakthrough in the Level-Crossing Problem.** 9.1. Neyman's Cycles in Level-Crossing Problem. 9.2. Normal Approximation Versus Inverse Gaussian Approximation. 9.3. Diffusion Approximation Versus inverse Gaussian Approximation: Is there A Mix-Up in This Collation? 9.4. Teugels' Approximation Versus Inverse Gaussian Approximation. 9.5. Conclusions. **Appendices. Index.**

### Biography

**Vsevolod K. Malinovskii** graduated from the Moscow State University, earned his Ph.D. in Mathematics from the Steklov Mathematical Institute in 1983, and his D.Sc. in Mathematics from the Central conomics and Mathematics Institute (CEMI) of the Russian Academy of Science in 2000. He joined Probability Theory's Department of Steklov Mathematical Institute in 1982 and worked there until 2006. Since 2009, he has been a Chief research fellow at the CEMI.

He was Visiting Professor at the University of Copenhagen in 1993 and in 1998, and at the University of Montreal in 2001. He has authored *Insurance Planning Models: Price Competition and Regulation of Financial Stability* and *Risk Measures and Insurance Solvency Benchmarks: Fixed-Probability Levels in Renewal Risk Models*.

Professor Malinovskii's main research interests are in Applied Probability and in Mathematical Statistics.