Properties of solutions of stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion
Authors:
Yu. S. Mishura, S. V. Posashkova and G. M. Shevchenko
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 79 (2009), 117-126
MSC (2000):
Primary 60H10; Secondary 91B28
DOI:
https://doi.org/10.1090/S0094-9000-09-00774-1
Published electronically:
December 28, 2009
MathSciNet review:
2494541
Full-text PDF Free Access
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Abstract: Properties of solutions of stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion are studied in the paper. Conditions on the coefficients of an equation are obtained ensuring that a solution does not vanish over a finite time interval in the case of the diffusion $\sigma (t)\sqrt {x}$. We prove a limit theorem that solutions continuously depend on the parameter $n$ in the space $L_1(\mathsf {P})$ for a sequence of stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion.
References
- Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. MR 637061
- Goran Peskir, A change-of-variable formula with local time on curves, J. Theoret. Probab. 18 (2005), no. 3, 499–535. MR 2167640, DOI https://doi.org/10.1007/s10959-005-3517-6
- Daniel W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math. 22 (1969), 345–400. MR 253426, DOI https://doi.org/10.1002/cpa.3160220304
- T. Yamada, Sur l’approximation des solutions d’équations différentielles stochastiques, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), no. 2, 153–164 (French). MR 413269, DOI https://doi.org/10.1007/BF00533998
- Toshio Yamada and Shinzo Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), 155–167. MR 278420, DOI https://doi.org/10.1215/kjm/1250523691
References
- N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co. and Kodansha, Ltd., Amsterdam–New York and Tokyo, 1981. MR 637061 (84b:60080)
- G. Peskir, A change-of-variable formula with local time on curves, J. Theoret. Probab. 18 (2005), no. 3, 499–535. MR 2167640 (2006k:60096)
- D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients I, Comm. Pure Appl. Math. 22 (1969), no. 3, 345–400. MR 0253426 (40:6641)
- T. Yamada, Sur l’approximation des solutions d’équations différentielles stochastiques, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), 153–164. MR 0413269 (54:1386)
- T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), 155–167. MR 0278420 (43:4150)
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Additional Information
Yu. S. Mishura
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
myus@univ.kiev.ua
S. V. Posashkova
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
revan1988@gmail.com
G. M. Shevchenko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
zhora@univ.kiev.ua
Keywords:
Stochastic differential equations,
non-Lipschitz diffusion,
Cox–Ingersoll–Ross model,
continuous dependence on a parameter
Received by editor(s):
March 12, 2008
Published electronically:
December 28, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
