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April  2019, 24(4): 1449-1467. doi: 10.3934/dcdsb.2018215

On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces

Huijie Qiao 1,, and  Jiang-Lun Wu 2,

1. 

School of Mathematics, Southeast University, Nanjing, Jiangsu 211189, China
2. 

Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK

* Corresponding author: Huijie Qiao

Received  November 2017 Published  April 2019 Early access  June 2018

Fund Project: The first author is supported by NSF of China (No. 11001051, 11371352).

Based on a recent result in [13], in this paper, we extend it to stochastic evolution equations with jumps in Hilbert spaces. This is done via Galerkin type finite-dimensional approximations of the infinite-dimensional stochastic evolution equations with jumps in a manner that one could then link the characterisation of the path-independence for finite-dimensional jump type SDEs to that for the infinite-dimensional settings. Our result provides an intrinsic link of infinite-dimensional stochastic evolution equations with jumps to infinite-dimensional (nonlinear) integro-differential equations.

Keywords: An Itô formula, a Girsanov transformation, path-independence, characterization theorems.
Mathematics Subject Classification: Primary: 60H15, 60H30; Secondary: 35R60.
Citation: Huijie Qiao, Jiang-Lun Wu. On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1449-1467. doi: 10.3934/dcdsb.2018215
References:
[1]

J. Bao, A. Truman and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 2111-2134.  doi: 10.1098/rspa.2008.0486.

[2]

R. Cont and E. Voltchkova, Integro-differential equations for option prices in exponential Lèvy models, Finance Stochast, 9 (2005), 299-325.  doi: 10.1007/s00780-005-0153-z.

[3]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications. Cambriddge: Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223.

[4]

C. Dellacherie and P. A. Meyer, Probabilities and Potential B: Theory of Martingales, NorthHolland, Amsterdam/New York/Oxford, 1982.

[5]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland/Kodanska, Amsterdam/Tokyo, 1989.

[6] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987.  doi: 10.1007/978-3-662-02514-7.
[7]

C. Marinelli, C. Prévôt and M. Röckner, Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise, Journal of Functional Analysis, 258 (2010), 616-649.  doi: 10.1016/j.jfa.2009.04.015.

[8]

M. Metivier, Semimartingales: A Course on Stochastic Processes, De Gruyer, Berlin, 1982.

[9]

K. R. Parthasarathy, Probability Measures on Metric Spaces, AMS Chelsea Publishing, 2005. doi: 10.1090/chel/352.

[10] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511721373.
[11]

P. E. Protter and K. Shimbo, No arbitrage and general semimartingales, Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, 4 (2008), 267-283.  doi: 10.1214/074921708000000426.

[12]

H. J. Qiao, Exponential ergodicity for SDEs with jumps and non-Lipschitz coefficients, J. Theor. Probab., 27 (2014), 137-152.  doi: 10.1007/s10959-012-0440-5.

[13]

H. J. Qiao and J.-L. Wu, Characterising the path-independence of the Girsanov transformation for non-Lipschnitz SDEs with jumps, Statistics and Probability Letters, 119 (2016), 326-333.  doi: 10.1016/j.spl.2016.09.001.

[14]

A. Truman, F.-Y. Wang, J.-L. Wu and W. Yang, A link of stochastic differential equations to nonlinear parabolic equations, SCIENCE CHINA Mathematics, 55 (2012), 1971-1976.  doi: 10.1007/s11425-012-4463-2.

[15]

M. Wang and J.-L. Wu, Necessary and sufficient conditions for path-independence of Girsanov transformation for infinite-dimensional stochastic evolution equations, Front. Math. China, 9 (2014), 601-622.  doi: 10.1007/s11464-014-0364-8.

[16]

F.-Y. Wang, Harnack Inequalities for Stochastic Partial Differential Equations, Springer Briefs in Mathematics. New York: Springer, 2013. doi: 10.1007/978-1-4614-7934-5.

[17]

J.-L. Wu and W. Yang, On stochastic differential equations and a generalised Burgers equation, In Stochastic Analysis and Its Applications to Finance- Festschrift in Honor of Prof. Jia-An Yan (eds T. S. Zhang, X. Y. Zhou), Interdisciplinary Mathematical Sciences, Vol. 13, World Scientific, Singapore, 2012,425-435. doi: 10.1142/9789814383585_0021.

show all references

References:
[1]

J. Bao, A. Truman and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 2111-2134.  doi: 10.1098/rspa.2008.0486.

[2]

R. Cont and E. Voltchkova, Integro-differential equations for option prices in exponential Lèvy models, Finance Stochast, 9 (2005), 299-325.  doi: 10.1007/s00780-005-0153-z.

[3]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications. Cambriddge: Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223.

[4]

C. Dellacherie and P. A. Meyer, Probabilities and Potential B: Theory of Martingales, NorthHolland, Amsterdam/New York/Oxford, 1982.

[5]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland/Kodanska, Amsterdam/Tokyo, 1989.

[6] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987.  doi: 10.1007/978-3-662-02514-7.
[7]

C. Marinelli, C. Prévôt and M. Röckner, Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise, Journal of Functional Analysis, 258 (2010), 616-649.  doi: 10.1016/j.jfa.2009.04.015.

[8]

M. Metivier, Semimartingales: A Course on Stochastic Processes, De Gruyer, Berlin, 1982.

[9]

K. R. Parthasarathy, Probability Measures on Metric Spaces, AMS Chelsea Publishing, 2005. doi: 10.1090/chel/352.

[10] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511721373.
[11]

P. E. Protter and K. Shimbo, No arbitrage and general semimartingales, Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, 4 (2008), 267-283.  doi: 10.1214/074921708000000426.

[12]

H. J. Qiao, Exponential ergodicity for SDEs with jumps and non-Lipschitz coefficients, J. Theor. Probab., 27 (2014), 137-152.  doi: 10.1007/s10959-012-0440-5.

[13]

H. J. Qiao and J.-L. Wu, Characterising the path-independence of the Girsanov transformation for non-Lipschnitz SDEs with jumps, Statistics and Probability Letters, 119 (2016), 326-333.  doi: 10.1016/j.spl.2016.09.001.

[14]

A. Truman, F.-Y. Wang, J.-L. Wu and W. Yang, A link of stochastic differential equations to nonlinear parabolic equations, SCIENCE CHINA Mathematics, 55 (2012), 1971-1976.  doi: 10.1007/s11425-012-4463-2.

[15]

M. Wang and J.-L. Wu, Necessary and sufficient conditions for path-independence of Girsanov transformation for infinite-dimensional stochastic evolution equations, Front. Math. China, 9 (2014), 601-622.  doi: 10.1007/s11464-014-0364-8.

[16]

F.-Y. Wang, Harnack Inequalities for Stochastic Partial Differential Equations, Springer Briefs in Mathematics. New York: Springer, 2013. doi: 10.1007/978-1-4614-7934-5.

[17]

J.-L. Wu and W. Yang, On stochastic differential equations and a generalised Burgers equation, In Stochastic Analysis and Its Applications to Finance- Festschrift in Honor of Prof. Jia-An Yan (eds T. S. Zhang, X. Y. Zhou), Interdisciplinary Mathematical Sciences, Vol. 13, World Scientific, Singapore, 2012,425-435. doi: 10.1142/9789814383585_0021.

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