Asymptotic behavior of the martingale type integral functionals for unstable solutions to stochastic differential equations
Authors:
G. L. Kulinich, S. V. Kushnirenko and Yu. S. Mishura
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 90 (2015), 115-126
MSC (2010):
Primary 60H10; Secondary 60F17
DOI:
https://doi.org/10.1090/tpms/953
Published electronically:
August 7, 2015
MathSciNet review:
3242024
Full-text PDF Free Access
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Abstract: We consider functionals of the type $\int _ {0} ^ {t} g (\xi (s)) dW (s)$, $t \ge 0$. Here $g$ is a real valued and locally square integrable function, $\xi$ is a unique strong solution of the Itô stochastic differential equation $d \xi (t) = a (\xi (t)) dt + dW (t)$, $a$ is a measurable real valued bounded function such that $| xa (x) | \le C$. The behavior of these functionals is studied as $t \to \infty$. The appropriate normalizing factor and the explicit form of the limit random variable are established.
References
- A. Ju. Veretennikov, Strong solutions of stochastic differential equations, Teor. Veroyatnost. i Primenen. 24 (1979), no. 2, 348–360 (Russian, with English summary). MR 532447
- G. L. Kulīnīch and Ē. P. Kas′kun, On the asymptotic behavior of solutions of a class of one-dimensional Itô stochastic differential equations, Teor. Ĭmovīr. Mat. Stat. 56 (1997), 96–104 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 56 (1998), 97–105. MR 1791858
- Tokuzo Shiga and Shinzo Watanabe, Bessel diffusions as a one-parameter family of diffusion processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 27 (1973), 37–46. MR 368192, DOI https://doi.org/10.1007/BF00736006
- G. L. Kulinich, S. V. Kushnirenko, and Y. S. Mishura, Asymptotic behavior of the integral functionals for unstable solutions of one-dimensional Itô stochastic differential equations, Teor. Ĭmovīr. Mat. Stat. 89 (2013), 91–103 (English, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 89 (2014), 101–114. MR 3235178, DOI https://doi.org/10.1090/s0094-9000-2015-00938-8
- G. L. Kulīnīč, Limit distributions for functionals of integral type of nonstable diffusion processes, Teor. Verojatnost. i Mat. Statist. Vyp. 11 (1974), 81–85, 180 (Russian, with English summary). MR 0400423
- G. L. Kulīnīč, Limit theorems for one-dimensional stochastic differential equations under irregular dependence of the coefficients of a parameter, Teor. Verojatnost. i Mat. Statist. Vyp. 15 (1976), 99–114, 156 (Russian, with English summary). MR 0415771
- G. L. Kulinich, On necessary and sufficient conditions for convergence of homogeneous additive functionals of diffusion processes, Proceedings of the Second Ukrainian–Hungarian Conference: New Trends in Probability and Mathematical Statistics (M. Aráto and M. Yadrenko, eds.), vol. 2, “TViMS”, Kyiv, 1995, pp. 381–390.
- N. I. Portenko, Some limit theorems for additive functionals of processes with independent increments, Teor. Verojatnost. i Mat. Statist. Vyp. 4 (1971), 130–136 (Russian, with English summary). MR 0287611
- J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, “Fizmatlit”, Moscow, 1994; English transl., Springer-Verlag, Berlin, 1987.
- G. L. Kulīnīč, The limit distribution behavior of the solution of a stochastic diffusion equation, Teor. Verojatnost. i Primenen 12 (1967), 548–551 (Russian, with English summary). MR 0215365
- A. V. Skorohod and N. P. Slobodenjuk, Predel′nye teoremy dlya sluchaĭ nykh bluzhdaniĭ, Izdat. “Naukova Dumka”, Kiev, 1970 (Russian). MR 0282419
- I. I. Gikhman and A. V. Skorokhod, Stokhasticheskie differentsial′nye uravneniya i ikh prilozheniya, “Naukova Dumka”, Kiev, 1982 (Russian). MR 678374
- I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, “Naukova Dumka”, Kiev, 1968; English transl., Springer-Verlag, Berlin, 1972.
- A. V. Skorokhod, Studies in the theory of random processes, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. Translated from the Russian by Scripta Technica, Inc. MR 0185620
- N. V. Krylov, Controlled diffusion processes, Applications of Mathematics, vol. 14, Springer-Verlag, New York-Berlin, 1980. Translated from the Russian by A. B. Aries. MR 601776
- I. I. Gikhman and A. V. Skorokhod, Introduction to the theory of random processes, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. Translated from the Russian by Scripta Technica, Inc. MR 0247660
- Michel Loève, Probability theory, 2nd ed. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-New York-London, 1960. MR 0123342
References
- A. Yu. Veretennikov, On the strong solutions of stochastic differential equations, Teor. Veroyatnost. Primenen. XXIV (1979), no. 2, 348–360; English transl. in Theory Probab. Appl. 24 (1979), no. 2, 354–366. MR 532447 (81b:60058)
- G. L. Kulinich and E. P. Kas’kun, On the asymptotic behavior of solutions of a class of one-dimensional Itô stochastic differential equations, Teor. Imovir. Mat. Stat. 56 (1997), 96–104; English transl. in Theory Probab. Math. Statist. 56 (1998), 97–105. MR 1791858 (2002m:60111)
- T. Shiga and S. Watanabe, Bessel diffusions as a one-parameter family of diffusion processes, Z. Wahrscheinlichkeitstheory und verw. Geb. 27 (1973), no. 1, 37–46. MR 0368192 (51:4433)
- G. L. Kulinich, S. V. Kushnirenko, and Y. S. Mishura, Asymptotic behavior of the integral functionals for unstable solutions of one-dimensional Itô stochastic differential equations, Theory Probab. Math. Statist. 89 (2013), 93–105. MR 3235178
- G. L. Kulinich, Limit distributions for integral type functionals of non-stable diffusion processes, Teor. Imovir. Mat. Stat. 11 (1974), 81–85; English transl. in Theory Probab. Math. Statist. 11 (1975), 82–86. MR 0400423 (53:4257)
- G. L. Kulinich, Limit theorems for one-dimensional stochastic differential equations under nonregular dependence of coefficients on a parameter, Teor. Imovir. Mat. Stat. 15 (1976), 99–114; English transl. in Theory Probab. Math. Statist. 15 (1978), 101–116. MR 0415771 (54:3850)
- G. L. Kulinich, On necessary and sufficient conditions for convergence of homogeneous additive functionals of diffusion processes, Proceedings of the Second Ukrainian–Hungarian Conference: New Trends in Probability and Mathematical Statistics (M. Aráto and M. Yadrenko, eds.), vol. 2, “TViMS”, Kyiv, 1995, pp. 381–390.
- N. I. Portenko, Some limit theorems for additive functionals of processes with independent increments, Teor. Imovir. Mat. Stat. 4 (1971), 130–136; English transl. in Theory Probab. Math. Statist. 4 (1972), 121–126. MR 0287611 (44:4814)
- J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, “Fizmatlit”, Moscow, 1994; English transl., Springer-Verlag, Berlin, 1987.
- G. L. Kulinich, On the limit behavior of the distribution of the solution of a stochastic diffusion equation, Teor. Veroyatnost. Primenen. XII (1967), no. 3, 348–360; English transl. in Theory Probab. Appl. (1967) 12, no. 3, 497–499. MR 0215365 (35:6206)
- A. V. Skorokhod and N. P. Slobodenyuk, Limit Theorems for Random Walks, “Naukova Dumka”, Kiev, 1970. (Russian) MR 0282419 (43:8130)
- I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and their Applications, “Naukova Dumka”, Kiev, 1982. (Russian) MR 678374 (84j:60003)
- I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, “Naukova Dumka”, Kiev, 1968; English transl., Springer-Verlag, Berlin, 1972.
- A. V. Skorokhod, Studies in the Theory of Random Processes, Kiev University Press, Kiev, 1961; English transl., Addison-Wesley, Reading, 1965. MR 0185620 (32:3082b)
- N. V. Krylov, Controlled Diffusion Processes, “Nauka”, Moscow, 1977; English transl., Springer, Berlin, 1980. MR 601776 (82a:60062)
- I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Stochastic Processes, “Nauka”, Moscow, 1965; English transl., W. B. Saunders, Philadelphia, PA, 1969. MR 0247660 (40:923)
- M. Loève, Probability Theory, 4th ed., Springer-Verlag, New York, 1977. MR 0123342 (23:A670)
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Additional Information
G. L. Kulinich
Affiliation:
Department of General Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email:
zag_mat@univ.kiev.ua
S. V. Kushnirenko
Affiliation:
Department of General Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email:
bksv@univ.kiev.ua
Yu. S. Mishura
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email:
myus@univ.kiev.ua
Keywords:
Itô stochastic differential equations,
unstable solutions,
asymptotic behavior of martingale type functionals
Received by editor(s):
March 14, 2014
Published electronically:
August 7, 2015
Article copyright:
© Copyright 2015
American Mathematical Society