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A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale
1. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China |
2. | Department of Mathematics, The University of Hong Kong, Hong Kong, China |
We examine a Wong-Zakai type approximation of a family of stochastic differential equations driven by a general càdlàg semimartingale. For such an approximation, compared with the pointwise convergence result by Kurtz, Pardoux and Protter [
References:
[1] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511809781.![]() ![]() |
[2] |
R. B. Ash, Probability and Measure Theory, Harcourt/Academic Press, Burlington, MA, 2000.
![]() |
[3] |
T. Fujiwara and H. Kunita,
Canonical SDE's based on semimartingales with spatial parameters. I. Stochastic flows of diffeomorphisms, Kyushu J. Math., 53 (1999), 265-300.
doi: 10.2206/kyushujm.53.265. |
[4] |
M. Hairer and É. Pardoux,
A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan, 67 (2015), 1551-1604.
doi: 10.2969/jmsj/06741551. |
[5] |
R. Hintze and I. Pavlyukevich,
Small noise asymptotics and first passage times of integrated Ornstein-Uhlenbeck processes driven by $\alpha $-stable Lévy processes, Bernoulli, 20 (2014), 265-281.
doi: 10.3150/12-BEJ485. |
[6] |
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. |
[7] |
H. Kunita, Stochastic differential equations with jumps and stochastic flows of diffeomorphisms, in Itô's Stochastic Calculus and Probability Theory, Springer, Tokyo, 1996,197–211.
doi: 10.1007/978-4-431-68532-6_13. |
[8] |
H. Kunita, Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms, in Real and Stochastic Analysis, Trends Math., Birkhüser, Boston, MA, 2004,305–373.
doi: 10.1007/978-1-4612-2054-1_6. |
[9] |
T. G. Kurtz,
Random time changes and convergence in distribution under the Meyer-Zheng conditions, Ann. Probab., 19 (1991), 1010-1034.
doi: 10.1214/aop/1176990333. |
[10] |
T. G. Kurtz, É. Pardoux and P. Protter,
Stratonovich stochastic differential equations driven by general semimartingales, Ann. Inst. H. Poincaré Probab. Statist., 31 (1995), 351-377.
|
[11] |
T. G. Kurtz and P. Protter,
Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab., 19 (1991), 1035-1070.
doi: 10.1214/aop/1176990334. |
[12] |
T. G. Kurtz and P. E. Protter, Weak convergence of stochastic integrals and differential equations, in Probabilistic Models for Nonlinear Partial Differential Equations, Lecture Notes in Math., 1627, Fond. CIME/CIME Found. Subser., Springer, Berlin, 1996, 1–41.
doi: 10.1007/BFb0093176. |
[13] |
S. I. Marcus,
Modelling and approximation of stochastic differential equations driven by semimaringales, Stochastics, 4 (1980/81), 223-245.
doi: 10.1080/17442508108833165. |
[14] |
I. Pavlyukevich and M. Riedle,
Non-standard Skorokhod convergence of Lévy-driven convolution integrals in Hilbert spaces, Stoch. Anal. Appl., 33 (2015), 271-305.
doi: 10.1080/07362994.2014.988358. |
[15] |
P. Protter, Stochastic Integration and Differential Equations, Stochastic Modelling and Applied Probability, 21, Springer-Verlag, Berlin, 2005.
doi: 10.1007/978-3-662-10061-5. |
[16] |
A. A. Puhalskii and W. Whitt,
Functional large deviation principles for first-passage-time processes, Ann. Appl. Probab., 7 (1997), 362-381.
doi: 10.1214/aoap/1034625336. |
[17] |
A. V. Skorokhod,
Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen., 1 (1956), 289-319.
|
[18] |
G. Tessitore and J. Zabczyk,
Wong-Zakai approximation of stochastic evolution equations, J. Evol. Equ., 6 (2006), 621-655.
doi: 10.1007/s00028-006-0280-9. |
[19] |
W. Whitt, Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer Series in Operations Research, Springer-Verlag, New York, 2002.
doi: 10.1007/b97479. |
[20] |
W. Whitt,
Weak convergence of first passage time processes, J. Appl. Probability, 8 (1971), 417-422.
doi: 10.2307/3211913. |
[21] |
E. Wong and M. Zakai,
On the relation between ordinary and stochastic differential equations, Internat. J. Engrg. Sci., 3 (1965), 213-229.
doi: 10.1016/0020-7225(65)90045-5. |
[22] |
E. Wong and M. Zakai,
On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965), 1560-1564.
doi: 10.1214/aoms/1177699916. |
[23] |
X. Zhang,
Derivative formulas and gradient estimates for SDEs driven by $\alpha$-stable processes, Stochastic Process. Appl., 123 (2013), 1213-1228.
doi: 10.1016/j.spa.2012.11.012. |
show all references
References:
[1] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511809781.![]() ![]() |
[2] |
R. B. Ash, Probability and Measure Theory, Harcourt/Academic Press, Burlington, MA, 2000.
![]() |
[3] |
T. Fujiwara and H. Kunita,
Canonical SDE's based on semimartingales with spatial parameters. I. Stochastic flows of diffeomorphisms, Kyushu J. Math., 53 (1999), 265-300.
doi: 10.2206/kyushujm.53.265. |
[4] |
M. Hairer and É. Pardoux,
A Wong-Zakai theorem for stochastic PDEs, J. Math. Soc. Japan, 67 (2015), 1551-1604.
doi: 10.2969/jmsj/06741551. |
[5] |
R. Hintze and I. Pavlyukevich,
Small noise asymptotics and first passage times of integrated Ornstein-Uhlenbeck processes driven by $\alpha $-stable Lévy processes, Bernoulli, 20 (2014), 265-281.
doi: 10.3150/12-BEJ485. |
[6] |
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. |
[7] |
H. Kunita, Stochastic differential equations with jumps and stochastic flows of diffeomorphisms, in Itô's Stochastic Calculus and Probability Theory, Springer, Tokyo, 1996,197–211.
doi: 10.1007/978-4-431-68532-6_13. |
[8] |
H. Kunita, Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms, in Real and Stochastic Analysis, Trends Math., Birkhüser, Boston, MA, 2004,305–373.
doi: 10.1007/978-1-4612-2054-1_6. |
[9] |
T. G. Kurtz,
Random time changes and convergence in distribution under the Meyer-Zheng conditions, Ann. Probab., 19 (1991), 1010-1034.
doi: 10.1214/aop/1176990333. |
[10] |
T. G. Kurtz, É. Pardoux and P. Protter,
Stratonovich stochastic differential equations driven by general semimartingales, Ann. Inst. H. Poincaré Probab. Statist., 31 (1995), 351-377.
|
[11] |
T. G. Kurtz and P. Protter,
Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab., 19 (1991), 1035-1070.
doi: 10.1214/aop/1176990334. |
[12] |
T. G. Kurtz and P. E. Protter, Weak convergence of stochastic integrals and differential equations, in Probabilistic Models for Nonlinear Partial Differential Equations, Lecture Notes in Math., 1627, Fond. CIME/CIME Found. Subser., Springer, Berlin, 1996, 1–41.
doi: 10.1007/BFb0093176. |
[13] |
S. I. Marcus,
Modelling and approximation of stochastic differential equations driven by semimaringales, Stochastics, 4 (1980/81), 223-245.
doi: 10.1080/17442508108833165. |
[14] |
I. Pavlyukevich and M. Riedle,
Non-standard Skorokhod convergence of Lévy-driven convolution integrals in Hilbert spaces, Stoch. Anal. Appl., 33 (2015), 271-305.
doi: 10.1080/07362994.2014.988358. |
[15] |
P. Protter, Stochastic Integration and Differential Equations, Stochastic Modelling and Applied Probability, 21, Springer-Verlag, Berlin, 2005.
doi: 10.1007/978-3-662-10061-5. |
[16] |
A. A. Puhalskii and W. Whitt,
Functional large deviation principles for first-passage-time processes, Ann. Appl. Probab., 7 (1997), 362-381.
doi: 10.1214/aoap/1034625336. |
[17] |
A. V. Skorokhod,
Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen., 1 (1956), 289-319.
|
[18] |
G. Tessitore and J. Zabczyk,
Wong-Zakai approximation of stochastic evolution equations, J. Evol. Equ., 6 (2006), 621-655.
doi: 10.1007/s00028-006-0280-9. |
[19] |
W. Whitt, Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer Series in Operations Research, Springer-Verlag, New York, 2002.
doi: 10.1007/b97479. |
[20] |
W. Whitt,
Weak convergence of first passage time processes, J. Appl. Probability, 8 (1971), 417-422.
doi: 10.2307/3211913. |
[21] |
E. Wong and M. Zakai,
On the relation between ordinary and stochastic differential equations, Internat. J. Engrg. Sci., 3 (1965), 213-229.
doi: 10.1016/0020-7225(65)90045-5. |
[22] |
E. Wong and M. Zakai,
On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965), 1560-1564.
doi: 10.1214/aoms/1177699916. |
[23] |
X. Zhang,
Derivative formulas and gradient estimates for SDEs driven by $\alpha$-stable processes, Stochastic Process. Appl., 123 (2013), 1213-1228.
doi: 10.1016/j.spa.2012.11.012. |
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