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Nonrandom overshoots of Lévy processes
, 2013
"... Abstract. The class of Lévy processes for which overshoots are almost surely constant quantities is precisely characterized. 1. ..."
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Abstract. The class of Lévy processes for which overshoots are almost surely constant quantities is precisely characterized. 1.
MARKOV CHAIN APPROXIMATIONS TO SCALE FUNCTIONS OF LÉVY PROCESSES
"... Abstract. We introduce a general algorithm for the computation of the scale functions of a spectrally negative Lévy process X, based on a natural weak approximation of X via upwards skipfree continuoustime Markov chains with stationary independent increments. The algorithm consists of evaluating ..."
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Abstract. We introduce a general algorithm for the computation of the scale functions of a spectrally negative Lévy process X, based on a natural weak approximation of X via upwards skipfree continuoustime Markov chains with stationary independent increments. The algorithm consists of evaluating a finite linear recursion with, what are nonnegative, coefficients given explicitly in terms of the Lévy triplet of X. Thus it is easy to implement and numerically stable. Our main result establishes sharp rates of convergence of this algorithm providing an explicit link between the semimartingale characteristics of X and its scale functions, not unlike the onedimensional Itô diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE. 1.
A NOTE ON THE TIMES OF FIRST PASSAGE FOR ‘NEARLY RIGHTCONTINUOUS ’ RANDOM WALKS
"... Abstract. A natural extension of a rightcontinuous integervalued random walk is one which can jump to the right by one or two units. First passage times above a given fixed level then admit a tractable Laplace transform (probability generating function). Explicit expressions for the probabilities ..."
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Abstract. A natural extension of a rightcontinuous integervalued random walk is one which can jump to the right by one or two units. First passage times above a given fixed level then admit a tractable Laplace transform (probability generating function). Explicit expressions for the probabilities that the respective overshoots are either 0 or 1, according as the random walk crosses a given level for the first time either continuously or not, also obtain. An interesting nonobvious observation, which follows from the analysis, is that any such (nondegenerate) random walk will, eventually in n ∈ N ∪ {0}, always be more likely to pass over the level n for the first time with overshoot zero, rather than one. Some applications are considered. 1.