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Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years, this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property, which hold, in principle, for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with omega-state-dependent killing, and certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W and Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein–Uhlenbeck or Feller branching diffusion with phase-type jumps).

Lévy processes --- non-random overshoots --- skip-free random walks --- fluctuation theory --- scale functions --- capital surplus process --- dividend payment --- optimal control --- capital injection constraint --- spectrally negative Lévy processes --- reflected Lévy processes --- first passage --- drawdown process --- spectrally negative process --- dividends --- de Finetti valuation objective --- variational problem --- stochastic control --- optimal dividends --- Parisian ruin --- log-convexity --- barrier strategies --- adjustment coefficient --- logarithmic asymptotics --- quadratic programming problem --- ruin probability --- two-dimensional Brownian motion --- spectrally negative Lévy process --- general tax structure --- first crossing time --- joint Laplace transform --- potential measure --- Laplace transform --- first hitting time --- diffusion-type process --- running maximum and minimum processes --- boundary-value problem --- normal reflection --- Sparre Andersen model --- heavy tails --- completely monotone distributions --- error bounds --- hyperexponential distribution --- reflected Brownian motion --- linear diffusions --- drawdown --- Segerdahl process --- affine coefficients --- spectrally negative Markov process --- hypergeometric functions --- capital injections --- bankruptcy --- reflection and absorption --- Pollaczek–Khinchine formula --- scale function --- Padé approximations --- Laguerre series --- Tricomi–Weeks Laplace inversion

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Benford's law states that the leading digits of many data sets are not uniformly distributed from one through nine, but rather exhibit a profound bias. This bias is evident in everything from electricity bills and street addresses to stock prices, population numbers, mortality rates, and the lengths of rivers. Here, Steven Miller brings together many of the world's leading experts on Benford's law to demonstrate the many useful techniques that arise from the law, show how truly multidisciplinary it is, and encourage collaboration.Beginning with the general theory, the contributors explain the prevalence of the bias, highlighting explanations for when systems should and should not follow Benford's law and how quickly such behavior sets in. They go on to discuss important applications in disciplines ranging from accounting and economics to psychology and the natural sciences. The contributors describe how Benford's law has been successfully used to expose fraud in elections, medical tests, tax filings, and financial reports. Additionally, numerous problems, background materials, and technical details are available online to help instructors create courses around the book.Emphasizing common challenges and techniques across the disciplines, this accessible book shows how Benford's law can serve as a productive meeting ground for researchers and practitioners in diverse fields.

Distribution (Probability theory) --- Probability measures. --- 1938 paper. --- 2009 presidential elections. --- Benford analysis. --- Benford distribution. --- Benford property. --- Benford test. --- Benford's law geometry. --- Benford's law limit. --- Benford's law. --- Benford-good system. --- Eric Poehlman. --- European statistics. --- FSDs. --- Fourier analysis. --- Frank Benford. --- Fundamental Equivalence. --- Greek statistics. --- Iranian election. --- Iranian presidential elections. --- Lvy processes. --- PV effect. --- Partial Volume effect. --- Poisson Summation Formula. --- Poisson Summation. --- Simon Newcomb. --- Standard Condition. --- VAR. --- Value at Risk. --- accounting programs. --- accounting students. --- accounting. --- auditing. --- authentic data sets. --- behavioral approaches. --- bias. --- complaint data sets. --- computer systems. --- cumulative distribution. --- data sets. --- data-adaptive methods. --- decision-making research. --- densities. --- deterministic processes. --- digit bias. --- direct applications. --- econometric regression. --- economics. --- election fraud. --- elections. --- empirical economics research. --- empirical economics. --- error detection. --- explicit error bounds. --- explicit error estimates. --- exponential Lvy processes. --- finance. --- financial reports. --- financial statistics. --- first significant digits. --- first-digit analysis. --- first-digit frequency. --- fixed odds. --- forecasts. --- fraud detection. --- fraud. --- fraudulent data sets. --- functions. --- gambling. --- generic potential applications. --- geometry. --- government deficit. --- information-theoretic methods. --- local boostrap model. --- logarithms. --- lottery. --- macroeconomic data. --- managing risk. --- mathematical theory. --- meaningful numbers. --- medical tests. --- misreporting. --- natural data. --- natural sciences. --- non-uniformity. --- normalized functionals. --- number lottery games. --- numbers games. --- origins. --- parametric distributions. --- probability distributions. --- psychology. --- random processes. --- replication. --- scale invariance. --- scientific data sets. --- scientific misconduct. --- second digits. --- significand. --- significant digits. --- small number. --- social statistics. --- statistical relationship. --- statistics education. --- statistics. --- tampering. --- tax filing. --- tax fraud. --- total variation. --- uniform distribution. --- vote counts. --- voting.