We propose a new approach to the application of the Korolyuk potential method for the investigation of limit functionals for processes with independent increments. The formulas for the joint distribution of functionals related to crossing a level by the process are obtained and their asymptotic analysis is performed. The possibility of crossing a level by the process in a continuous way is also investigated.
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References
J. Bertoin, Lévy Processes, Cambridge Univ. Press, Cambridge (1996).
L. Takács, Combinatorial Methods in the Theory of Stochastic Processes, Wiley, New York (1967).
R. Doney, “Hitting probabilities for spectrally positive Levy processes,” J. London Math. Soc., 44, No. 3, 566–576 (1991).
V. S. Korolyuk, “Limit problems for complex Poisson processes,” Teor. Ver. Primen., 19, No. 1, 3–14 (1974).
V. S. Korolyuk, V. N. Suprun, and V. M. Shurenkov, “Potential method in the limit problems for processes with independent increments and jumps of the same sign,” Teor. Ver. Primen., 22, No. 2, 419–425 (1976).
D. V. Gusak and V. S. Korolyuk, “Distribution of functionals of a homogeneous process with independent increments,” Teor. Ver. Mat. Statist., No. 1, 55–73 (1970).
D. V. Gusak, “Factorization method in the limit problems for processes with independent increments,” in: Distribution of Some Functionals for Processes with Independent Increments and Semi-Markov Processes [in Russian], Preprint 85.43, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1985), pp. 12–42.
B. A. Rogozin, “Distribution of some functionals connected with the process with independent increments,” Teor. Ver. Primen., 11, No. 4, 656–670 (1966).
M. S. Bratiichuk, “Resolvent of a stopping process with independent increments,” Ukr. Mat. Zh., 30, No. 1, 96–100 (1978); English translation: Ukr. Math. J., 30, No. 1, 71–74 (1978).
E. B. Dynkin, Markov Processes [in Russian], Fizmatgiz, Moscow (1963).
M. S. Bratiichuk and V. S. Korolyuk, “Resolvent of a homogeneous process with independent increments terminating on the semi-axis,” Teor. Ver. Primen., 30, No. 2, 368–372 (1985).
M. S. Bratiichuk, An Approach to the Investigation of Limit Functionals for the Processes with Independent Increments [in Russian], Preprint 89.38, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1989).
M. S. Bratiichuk, Limit Theorems for the Limit Functionals of the Process with Independent Increments [in Russian], Preprint 89.39, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1989).
N. S. Bratiichuk and D. V. Gusak, Limit Problems for Processes with Independent Increments [in Russian], Naukova Dumka, Kiev (1990).
P. Millar, “Exit properties of stochastic process with stationary independent increments,” Trans. Amer. Math. Soc., 178, 459–479 (1973).
D. V. Husak, Limit Problems for Processes with Independent Increments [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2007).
D. V. Husak, Processes with Independent Increments in the Risk Theory [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2011).
D. V. Gusak, “Intersection of a level by a homogeneous process with independent increments and a nondegenerate Wiener component,” Ukr. Mat. Zh., 32, No. 3, 373–378 (1980); English translation: Ukr. Math. J., 32, No. 3, 247–250 (1980).
W. Feller, An Introduction to Probability Theory and Its Applications [Russian translation], Vol. 2, Mir, Moscow (1984).
I. I. Gikhman and A. V. Skorokhod, Theory of Random Processes [in Russian], Vol. 2, Nauka, Moscow (1973).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 8, pp. 1019–1029, August, 2015.
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Bratiichuk, M.S. Potential Method in the Limit Problems for the Processes with Independent Increments. Ukr Math J 67, 1146–1158 (2016). https://doi.org/10.1007/s11253-016-1142-4
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DOI: https://doi.org/10.1007/s11253-016-1142-4