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

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.

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. BaoA. 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. MarinelliC. 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. TrumanF.-Y. WangJ.-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. BaoA. 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. MarinelliC. 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. TrumanF.-Y. WangJ.-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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