Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs

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Lévy Processes and Stochastic Calculus

(2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

Modelling Extremal Events for Insurance and Finance

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

/pdf/quantitative-risk-management-concepts-techniques-and-tools-afazrwn4wm.pdf

Quantitative Risk Management: Concepts, Techniques, and Tools

Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.

Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

Deviance sociology Wikipedia April 22nd, 2019 In sociology deviance describes an action or behavior that violates social norms including a formally enacted rule e g crime as well as informal violations of social norms e g rejecting folkways and mores Although deviance may have a negative connotation the violation of social norms is not always a negative action positive deviation exists in some situations

An introduction to mathematical risk theory

This paper studies the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transforms, which can naturally be interpreted as discounting. Hence the classical risk theory model is generalized by discounting with respect to the time of ruin. We show how to calculate an expected discounted penalty, which is due at ruin and may depend on the deficit at ruin and on the surplus immediately before ruin. The expected discounted penalty, considered as a function of the initial surplus, satisfies a certain renewal equation, which has a probabilistic interpretation. Explicit answers are obtained for zero initial surplus, very large initial surplus, and arbitrary initial surplus if the claim amount distribution is exponential or a mixture of exponentials. We generalize Dickson’s formula, which expresses the joint distribution of the surplus immediately prior to and at ruin in terms of the probability of ult...

On the Time Value of Ruin

The Esscher transform is a time-honored tool in actuarial science. This paper shows that the Esscher transform is also an efficient technique for valuing derivative securities if the logarithms of the prices of the primitive securities are governed by certain stochastic processes with stationary and independent increments. This family of processes includes the Wiener process, the Poisson process, the gamma process, and the inverse Gaussian process. An Esscher transform of such a stock-price process induces an equivalent probability measure on the process. The Esscher parameter or parameter vector is determined so that the discounted price of each primitive security is a martingale under the new probability measure. The price of any derivative security is simply calculated as the expectation, with respect to the equivalent martingale measure, of the discounted payoffs. Straightforward consequences of the method of Esscher transforms include, among others, the celebrated Black-Scholes optionpricing formula, the binomial option-pricing formula, and formulas for pricing options on the maximum and minimum of multiple risky assets. Tables of numerical values for the prices of certain European call options (calculated according to four different models for stock-price movements) are also provided.

https://pages.stern.nyu.edu/~dbackus/Disasters/Gerber_Shiu_94.pdf

Option pricing by Esscher transforms.

1 The Mathematics of Compound Interest.- 2 The Future Lifetime of a Life Aged x.- 3 Life Insurance.- 4 Life Annuities.- 5 Net Premiums.- 6 Net Premium Reserves.- 7 Multiple Decrements.- 8 Multiple Life Insurance.- 9 The Total Claim Amount in a Portfolio.- 10 Expense Loadings.- 11 Estimating Probabilities of Death.- Appendix A. Commutation Functions.- A.1 Introduction.- A.2 The Deterministic Model.- A.3 Life Annuities.- A.4 Life Insurance.- A.5 Net Annual Premiums and Premium Reserves.- Appendix B. Simple Interest.- Appendix C. Exercises.- C.0 Introduction.- C.1 Mathematics of Compound Interest: Exercises.- C.1.1 Theory Exercises.- C.1.2 Spreadsheet Exercises.- C.2.1 Theory Exercises.- C.2.2 Spreadsheet Exercises.- C.3 Life Insurance.- C.3.1 Theory Exercises.- C.3.2 Spreadsheet Exercises.- C.4 Life Annuities.- C.4.1 Theory Exercises.- C.4.2 Spreadsheet Exercises.- C.5 Net Premiums.- C.5.1 Notes.- C.5.2 Theory Exercises.- C.5.3 Spreadsheet Exercises.- C.6 Net Premium Reserves.- C.6.1 Theory Exercises.- C.6.2 Spreadsheet Exercises.- C.7 Multiple Decrements: Exercises.- C.7.1 Theory Exercises.- C.8 Multiple Life Insurance: Exercises.- C.8.1 Theory Exercises.- C.8.2 Spreadsheet Exercises.- C.9 The Total Claim Amount in a Portfolio.- C 9.1 Theory Exercises.- C.10 Expense Loadings.- C.10.1 Theory Exercises.- C.10.2 Spreadsheet Exercises.- C.11 Estimating Probabilities of Death.- C.11.1 Theory Exercises.- Appendix D. Solutions.- D.0 Introduction.- D.1 Mathematics of Compound Interest.- D.1.1 Solutions to Theory Exercises.- D.1.2 Solutions to Spreadsheet Exercises.- D.2.1 Solutions to Theory Exercise.- D.2.2 Solutions to Spreadsheet Exercises.- D.3 Life Insurance.- D.3.1 Solutions to Theory Exercises.- D.3.2 Solution to Spreadsheet Exercises.- D.4 Life Annuities.- D.4.1 Solutions to Theory Exercises.- D.4.2 Solutions to Spreadsheet Exercises.- D.5 Net Premiums: Solutions.- D.5.1 Theory Exercises.- D.5.2 Solutions to Spreadsheet Exercises.- D.6 Net Premium Reserves: Solutions.- D.6.1 Theory Exercises.- D.6.2 Solutions to Spreadsheet Exercises.- D.7 Multiple Decrements: Solutions.- D.7.1 Theory Exercises.- D.8 Multiple Life Insurance: Solutions.- D 8.1 Theory Exercises.- D.8.2 Solutions to Spreadsheet Exercises.- D.9 The Total Claim Amount in a Portfolio.- D.9.1 Theory Exercises.- D.10 Expense Loadings.- D.10.1 Theory Exercises.- D.10.2 Spreadsheet Exercises.- D.11 Estimating Probabilities of Death.- D.11.1 Theory Exercises.- Appendix E. Tables.- E.0 Illustrative Life Tables.- E.1 Commutation Columns.- E.2 Multiple Decrement Tables.- References.

Life Insurance Mathematics

In the absence of dividends, the surplus of a company is modeled by a Wiener process (or Brownian motion) with positive drift. Now dividends are paid according to a barrier strategy: Whenever the (modified) surplus attains the level b, the “overflow” is paid as dividends to shareholders. An explicit expression for the moment-generating function of the time of ruin is given. Let D denote the sum of the discounted dividends until ruin. Explicit expressions for the expectation and the moment-generating function of D are given; furthermore, the limiting distribution of D is determined when the variance parameter of the surplus process tends toward infinity. It is shown that the sum of the (undiscounted) dividends until ruin is a compound geometric random variable with exponentially distributed summands. The optimal level b* is the value of b for which the expectation of D is maximal. It is shown that b* is an increasing function of the variance parameter; as the variance parameter tends toward infini...

Hans U. Gerber

Papers

An introduction to mathematical risk theory

On the Time Value of Ruin

Option pricing by Esscher transforms.

Life Insurance Mathematics

Optimal Dividends: Analysis with Brownian Motion. .