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May  2021, 26(5): 2499-2508. doi: 10.3934/dcdsb.2020192

A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale

Xianming Liu 1,, and  Guangyue Han 2,

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

* Corresponding author: Xianming Liu

Received  August 2019 Revised  March 2020 Published  May 2021 Early access  June 2020

Fund Project: The first author is supported by NSF grants of China No. 11971186

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 [10,Theorem 6.5], we establish stronger convergence results under the Skorokhod $ M_1 $-topology, which, among other possible applications, implies the convergence of the first passage time of the solution to the approximating stochastic differential equation.

Keywords: Wong-Zakai approximation, stochastic differential equation, semimartingale, the Skorokhod $ M_1 $-topology, random time change.
Mathematics Subject Classification: Primary: 60H10; Secondary: 34F05.
Citation: Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192
References:
[1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511809781.  Google Scholar
[2] R. B. Ash, Probability and Measure Theory, Harcourt/Academic Press, Burlington, MA, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. I. Marcus, Modelling and approximation of stochastic differential equations driven by semimaringales, Stochastics, 4 (1980/81), 223-245.  doi: 10.1080/17442508108833165.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. V. Skorokhod, Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen., 1 (1956), 289-319.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

W. Whitt, Weak convergence of first passage time processes, J. Appl. Probability, 8 (1971), 417-422.  doi: 10.2307/3211913.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar
[2] R. B. Ash, Probability and Measure Theory, Harcourt/Academic Press, Burlington, MA, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. I. Marcus, Modelling and approximation of stochastic differential equations driven by semimaringales, Stochastics, 4 (1980/81), 223-245.  doi: 10.1080/17442508108833165.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. V. Skorokhod, Limit theorems for stochastic processes, Teor. Veroyatnost. i Primenen., 1 (1956), 289-319.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

W. Whitt, Weak convergence of first passage time processes, J. Appl. Probability, 8 (1971), 417-422.  doi: 10.2307/3211913.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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