The $W,Z$ scale functions kit for first passage problems of spectrally negative Levy processes, and applications to control problems

Avram, Florin
Grahovac, Danijel
Vardar Acar, Ceren
In the last years there appeared a great variety of identities for first passage problems of spectrally negative Levy processes, which can all be expressed in terms of two "q-harmonic functions" (or scale functions)WandZ. The reason behind that is that there are two ways of exiting an interval, and thus two fundamental "two-sided exit" problems from an interval (TSE). Since many other problems can be reduced to TSE, researchers developed in the last years a kit of formulas expressed in terms of the "W,Zalphabet". It is important to note - as is currently being shown - that these identities apply equally to other spectrally negative Markov processes, where however theW,Zfunctions are typically much harder to compute. We collect below our favorite recipes from the "W,Zkit", drawing from various applications in mathematical finance, risk, queueing, and inventory/storage theory. A small sample of applications concerning extensions of the classic de Finetti dividend problem is offered. An interesting use of the kit is for recognizing relationships between problems involving behaviors apparently unrelated at first sight (like reflection, absorption, etc.). Another is expressing results in a standardized form, improving thus the possibility to check when a formula is already known.
ESAIM-Probability and Statistics


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Gueven, Bilgehan (Informa UK Limited, 2006-12-01)
We consider a linear regression model with an unbalanced 1-fold nested error structure, where group effect and error are from nonnormal universes. The limiting distribution of the F-statistic in this model is derived, as the sample size is large and group sizes take values from a finite set of distinct integers. The result is used to approximate the F-distribution quantile and to test the significance of the random effect variance component. Results are also applicable to the F-statistic in the one-way rand...
An iterative approximation scheme for repetitive Markov processes
Tüfekçi, Tolga; Güllü, Refik (Cambridge University Press (CUP), 1999-09-01)
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Maximum Drawdown and Drawdown Duration of Spectrally Negative Levy Processes Decomposed at Extremes
Vardar Acar, Ceren; Avram, Florin (Springer Science and Business Media LLC, 2020-06-06)
Path decomposition is performed to characterize the law of the pre-/post-supremum, post-infimum and the intermediate processes of a spectrally negative Levy process taken up to an independent exponential time T. As a result, mainly the distributions of the supremum of the post-infimum process and the maximum drawdown of the pre-/post-supremum, post-infimum processes and the intermediate processes are obtained together with the law of drawdown durations.
Multiple linear regression model with stochastic design variables
İslam, Muhammed Qamarul (Informa UK Limited, 2010-01-01)
In a simple multiple linear regression model, the design variables have traditionally been assumed to be non-stochastic. In numerous real-life situations, however, they are stochastic and non-normal. Estimators of parameters applicable to such situations are developed. It is shown that these estimators are efficient and robust. A real-life example is given.
Minimum variance quadratic unbiased estimation for the variance components in simple linear regression with onefold nested error
Gueven, Ilgehan (Informa UK Limited, 2006-01-01)
The explicit forms of the minimum variance quadratic unbiased estimators (MIVQUEs) of the variance components are given for simple linear regression with onefold nested error. The resulting estimators are more efficient as the ratio of the initial variance components estimates increases and are asymptotically efficient as the ratio tends to infinity.
Citation Formats
F. Avram, D. Grahovac, and C. Vardar Acar, “The $W,Z$ scale functions kit for first passage problems of spectrally negative Levy processes, and applications to control problems,” ESAIM-Probability and Statistics, pp. 1–65, 2019, Accessed: 00, 2020. [Online]. Available: