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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 |
Based on a recent result in [
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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