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2013 | OriginalPaper | Chapter

Approximation and Stability of Solutions of SDEs Driven by a Symmetric α Stable Process with Non-Lipschitz Coefficients

Author : Hiroya Hashimoto

Published in: Séminaire de Probabilités XLV

Publisher: Springer International Publishing

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Abstract

Firstly, we investigate Euler–Maruyama approximation for solutions of stochastic differential equations (SDEs) driven by a symmetric α stable process under Komatsu condition for coefficients. The approximation implies naturally the existence of strong solutions. Secondly, we study the stability of solutions under Komatsu condition, and also discuss it under Belfadli–Ouknine condition.

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D. Applebaum, Lévy Processes and Stochastic Calculus, 1st edn. Cambridge Stud. Adv. Math. 93 (2004)

R.F. Bass, Stochastic differential equations driven by symmetric stable processes. Séminaire de Probabilités, XXXVI. Lecture Notes in Mathematics, vol. 1801 (Springer, New York, 2004), pp. 302–313

R. Belfadli, Y. Ouknine, On the pathwise uniqueness of solutions of Stochastic differential equations driven by symmetric stable Lévy processes. Stochastics 80, 519–524 (2008)MathSciNetMATH

C. Dellacherie, P.A. Meyer, Probabilités et Potentiel (Théorie des Martingales, Hermann, 1980)MATH

M. Émery, Stabilité des solutions des équations différentielles stochastiques applications aux intégrales multiplicatives stochastiques. Z. Wahr. 41, 241–262 (1978)MATHCrossRef

E. Giné, M.B. Marcus, The central limit theorem for stochastic integrals with respect to Lévy processes. Ann. Prob. 11, 58–77 (1983)MATHCrossRef

H. Hashimoto, T. Tsuchiya, T. Yamada, On stochastic differential equations driven by symmetric stable processes of index α. Stochastic Processes and Applications to Mathematical Finance (World Scientific Publishing, Singapore, 2006), pp. 183–193

A. Janicki, Z. Michna, A. Weron, Approximation of stochastic differential equations driven by α-stable Lévy motion. Appl. Math. 24, 149–168 (1996)MathSciNetMATH

H. Kaneko, S. Nakao, A note on approximation for stochastic differential equations. Séminaire de Probabilités, XXII. Lecture Notes in Mathematics, vol. 1321 (Springer, New York, 1998), pp. 155–162

10.

Y. Kasahara, K. Yamada, Stability theorem for stochastic differential equations with jumps. Stoch. Process. Appl. 38, 13–32 (1991)MathSciNetMATHCrossRef

11.

S. Kawabata, T. Yamada, On some limit theorems for solutions of stochastic differential equations. Séminaire de Probabilités, XVI. Lecture Notes in Mathematics, vol. 920 (Springer, New York, 1982), pp. 412–441

12.

T. Komatsu, On the pathwise uniqueness of solutions of one-dimentional stochastic differential equations of jump type. Proc. Jp. Acad. Ser. A. Math. Sci. 58, 353–356 (1982)MathSciNetMATHCrossRef

13.

J.F. Le Gall, Applications des temps locaux aux équations différentielles stochastiques unidimensionnelles. Séminaire de Probabilités, XVII. Lecture Notes in Mathematics, vol. 986 (Springer, New York, 1983), pp. 15–31

14.

S. Nakao, On the pathwise uniqueness of solutions of stochastic differential equations. Osaka J. Math. 9, 513–518 (1972)MathSciNetMATH

15.

P. Protter, Stochastic Integration and Differential Equations (Springer, New York, 1992)

16.

D. Revuz, M. Yor, Continuous Martingales and Brownian Motion (Springer, New York, 1991)MATHCrossRef

17.

T. Tsuchiya, On the pathwise uniqueness of solutions of stochastic differential equations driven by multi-dimensional symmetric α stable class. J. Math. Kyoto Univ. 46, 107–121 (2006)MathSciNetMATH

18.

T. Yamada, Sur une Construction des Solutions d’Équations Différentielles Stochastiques dans le Cas Non-Lipschitzien. Séminaire de Probabilités. Lecture Notes in Mathematics, vol. 1771 (Springer, New York, 2004), pp. 536–553

19.

P.A. Zanzotto, On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion. Stoch. Process. Appl. 68, 209–228 (1997)MathSciNetMATHCrossRef

Title: Approximation and Stability of Solutions of SDEs Driven by a Symmetric α Stable Process with Non-Lipschitz Coefficients
Author: Hiroya Hashimoto
Publisher: Springer International Publishing
Book: Séminaire de Probabilités XLV
Print ISBN: 978-3-319-00320-7

Electronic ISBN: 978-3-319-00321-4

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-3-319-00321-4_7