American Institute of Mathematical Sciences

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  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 & Continuous Dynamical Systems - B, 2019, 24 (4) : 1449-1467. doi: 10.3934/dcdsb.2018215
References:

show all references

References:
 [1] Hassan Emamirad, Arnaud Rougirel. Feynman path formula for the time fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020246 [2] Jianhai Bao, Feng-Yu Wang, Chenggui Yuan. Limit theorems for additive functionals of path-dependent SDEs. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5173-5188. doi: 10.3934/dcds.2020224 [3] Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123 [4] Jörg Schmeling. A notion of independence via moving targets. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 269-280. doi: 10.3934/dcds.2006.15.269 [5] Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487 [6] Lyndsey Clark. The $\beta$-transformation with a hole. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1249-1269. doi: 10.3934/dcds.2016.36.1249 [7] Martin Frank, Armin Fügenschuh, Michael Herty, Lars Schewe. The coolest path problem. Networks & Heterogeneous Media, 2010, 5 (1) : 143-162. doi: 10.3934/nhm.2010.5.143 [8] Marc Chamberland, Victor H. Moll. Dynamics of the degree six Landen transformation. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 905-919. doi: 10.3934/dcds.2006.15.905 [9] Oğul Esen, Partha Guha. On the geometry of the Schmidt-Legendre transformation. Journal of Geometric Mechanics, 2018, 10 (3) : 251-291. doi: 10.3934/jgm.2018010 [10] Hong-Kun Zhang. Free path of billiards with flat points. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4445-4466. doi: 10.3934/dcds.2012.32.4445 [11] Matthias Gerdts, René Henrion, Dietmar Hömberg, Chantal Landry. Path planning and collision avoidance for robots. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 437-463. doi: 10.3934/naco.2012.2.437 [12] Peter Seibt. A period formula for torus automorphisms. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1029-1048. doi: 10.3934/dcds.2003.9.1029 [13] Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187 [14] Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267 [15] Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232 [16] Ivan Landjev, Assia Rousseva. Characterization of some optimal arcs. Advances in Mathematics of Communications, 2011, 5 (2) : 317-331. doi: 10.3934/amc.2011.5.317 [17] Shiri Artstein-Avidan and Vitali Milman. A characterization of the concept of duality. Electronic Research Announcements, 2007, 14: 42-59. doi: 10.3934/era.2007.14.42 [18] Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437 [19] N. Kamran, K. Tenenblat. Periodic systems for the higher-dimensional Laplace transformation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 359-378. doi: 10.3934/dcds.1998.4.359 [20] Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433

2019 Impact Factor: 1.27