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

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

* 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.

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. KurtzÉ. 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. KurtzÉ. 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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