A simplified version of Spitzer’s formula for semicontinuous and almost semicontinuous processes
Author:
D. V. Gusak
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 85 (2012), 61-71
MSC (2000):
Primary 60G50; Secondary 60K10
DOI:
https://doi.org/10.1090/S0094-9000-2013-00874-6
Published electronically:
January 11, 2013
MathSciNet review:
2933703
Full-text PDF Free Access
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Abstract: Let $\{\xi (t), \zeta (0)=0, t\geq 0\}$ be a process with stationary independent increments. We establish simplified versions of relations for spectral functions in Spitzer’s formulas in terms of the exponential function whose index is determined by the corresponding root of Lundberg’s equation for the case where $\xi (t)$ is a semicontinuous or almost semicontinuous process.
References
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- Frank Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc. 82 (1956), 323–339. MR 79851, DOI https://doi.org/10.1090/S0002-9947-1956-0079851-X
- D. V. Gusak, Granichnī zadachī dlya protsesīv z nezalezhnimi prirostami v teorīï riziku, Pratsī Īnstitutu Matematiki Natsīonal′noï Akademīï Nauk Ukraïni. Matematika ta ïï Zastosuvannya [Proceedings of Institute of Mathematics of NAS of Ukraine. Mathematics and its Applications], vol. 65, Natsīonal′na Akademīya Nauk Ukraïni, Īnstitut Matematiki, Kiev, 2007 (Ukrainian, with English and Ukrainian summaries). MR 2382816
- Julian Keilson, The first passage time density for homogeneous skip-free walks on the continuum, Ann. Math. Statist. 34 (1963), 1003–1011. MR 153060, DOI https://doi.org/10.1214/aoms/1177704023
- V. M. Zolotarev, The moment of first passage of a level and the behaviour at infinity of a class of processes with independent increments, Teor. Verojatnost. i Primenen. 9 (1964), 724–733 (Russian, with English summary). MR 0171315
- A. A. Borovkov, On the first passage time for a class of processes with independent increments, Teor. Verojatnost. i Primenen. 10 (1965), 360–364 (Russian, with English summary). MR 0182052
- Søren Asmussen, Ruin probabilities, Advanced Series on Statistical Science & Applied Probability, vol. 2, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1794582
References
- F. A. Spitzer, Principles of Random Walk, D. Van Nostrand, Princeton, 1964. MR 0171290 (30:1521)
- F. A. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc. 82 (1956), 323–329. MR 0079851 (18:156e)
- D. V. Gusak, Boundary Problems for Stochastic Processes with Independent Increments in the Risk Theory, Proc. Inst. Math. Nat. Acad. Sci. Ukraine, vol. 67, Kyiv, 2007 (Ukrainian) MR 2382816 (2009e:60002)
- J. H. B. Keilson, The first passage time density for homogeneous skip-free walks on the continuum, Ann. Math. Statist. 34 (1963), no. 3, 1001–1011. MR 0153060 (27:3029)
- V. M. Zolotarev, The moment of first passage of a level and the behaviour at infinity of a class of processes with independent increments, Teor. Verojatnost. i Primenen. 9 (1964), no. 4, 724–733; English transl. in Theory Probab. Appl. 9 (1964), no. 4, 653–661. MR 0171315 (30:1546)
- A. A. Borovkov, On the first passage time for a class of processes with independent increments, Teor. Verojatnost. i Primenen. 10 (1965), no. 2, 360–364; English transl. in Theory Probab. Appl. 10 (1965), no. 2, 331–334. MR 0182052 (31:6276)
- S. Asmussen, Ruin Probabilities, World Scientific, Singapore, 2000. MR 1794582 (2001m:62119)
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Additional Information
D. V. Gusak
Affiliation:
Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 3, 252601 Kyiv–4, Ukraine
Email:
random@imath.kiev.ua
Keywords:
Semicontinuous and almost semicontinuous processes with independent increments,
Spitzer’s formula,
Lundberg’s equation,
Spitzer’s spectral functions
Received by editor(s):
December 10, 2010
Published electronically:
January 11, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
