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Path independence of the additive functionals for stochastic differential equations driven by Glévy processes
1.  School of Mathematics, Southeast University, Nanjing 211189, Jiangsu, China 
2.  Department of Mathematics, University of Illinois at UrbanaChampaign, Urbana, IL 61801, USA 
3.  Department of Mathematics, Computational Foundry, Swansea University, Bay Campus, Swansea SA1 8EN, UK 
In this study, we are interested in stochastic differential equations driven by GLévy processes. We illustrate that a certain class of additive functionals of the equations of interest exhibits the pathindependent property, generalizing a few known findings in the literature. The study is ended with many examples.
References:
[1] 
Gao, F., Pathwise properties and homeomorphic flows for stochastic differential equations driven by GBrownian motion, Stochastic Proc. Appl., 2009, 119(10): 3356−3382. doi: 10.1016/j.spa.2009.05.010. 
[2] 
Hodges, S. and Carverhill, A., Quasi mean reversion in an efficient stock market: The characterisation of economic equilibria which support BlackScholes option pricing, Econom. J., 1993, 103(417): 395−405. 
[3] 
Hu, M. and Peng, S., GLévy processes under sublinear expectations, Probab., Uncertain. Quant. Risk, 2021, 6(1): 1–22. 
[4] 
Li, X. and Peng, S., Stopping times and related Itô’s calculus with GBrownian motion, Stochastic Proc. Appl., 2011, 121(7): 1492−1508. doi: 10.1016/j.spa.2011.03.009. 
[5] 
Osuka, E., Girsanov’s formula for GBrownian motion, Stoch. Process. Appl., 2013, 123(4): 1301−1318. doi: 10.1016/j.spa.2012.12.009. 
[6] 
Paczka, K., Itô calculus and jump diffusions for GLévy processes, arXiv: 1211.2973v3, 2014. 
[7] 
Paczka, K., Gmartingale representation in the GLévy setting, arXiv: 1404.2121v1, 2014. 
[8] 
Peng, S., Filtration consistent nonlinear expectations and evaluations of contingent claims, Acta Mathematicae Applicatae Sinica, English Series, 2004, 20(2): 191−214. doi: 10.1007/s1025500401613. 
[9] 
Peng, S., Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 2005, 26(2): 159−184. doi: 10.1142/S0252959905000154. 
[10] 
Peng, S., Multidimensional GBrownian motion and related stochastic calculus under Gexpectation, Stoch. Process. Appl., 2008, 118(12): 2223−2253. doi: 10.1016/j.spa.2007.10.015. 
[11] 
Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and GBrownian Motion, Probability Theory and Stochastic Modelling, Springer, Berlin, Heidelberg, 2019. 
[12] 
Qiao, H., EulerMaruyama approximation for SDEs with jumps and nonLipschitz coefficients, Osaka J. Math, 2014, 51(7): 47−66. 
[13] 
Qiao, H., The cocycle property of stochastic differential equations driven by GBrownian motion, Chinese Annals of Mathematics, Series B, 2015, 36(1): 147−160. doi: 10.1007/s1140101408691. 
[14] 
Qiao, H. and Wu, J.L., Characterising the pathindependence of the Girsanov transformation for nonLipschitz SDEs with jumps, Statistics & Probability Letters, 2016, 119: 326−333. 
[15] 
Qiao, H. and Wu, J.L., On the pathindependence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces, Discrete & Continuous Dynamical SystemsB, 2019, 24(4): 1449−1467. 
[16] 
Qiao, H. and Wu, J.L., Path independence of the additive functionals for McKean–Vlasov stochastic differential equations with jumps, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2021, 24(1): 2150006. doi: 10.1142/S0219025721500065. 
[17] 
Ren, P. and Yang, F., Path independence of additive functionals for stochastic differential equations under Gframework, Front. Math. China, 2019, 14(1): 135−148. doi: 10.1007/s1146401907521. 
[18] 
Stein, E. M. and Stein, J. C., Stock price distributions with stochastic volatility: An analytic approach, Rev. Financ. Stud., 1991, 4(4): 727−752. doi: 10.1093/rfs/4.4.727. 
[19] 
Song, Y., Uniqueness of the representation for Gmartingales with finite variation, Electron. J. Probab., 2012, 17: 1−15. 
[20] 
Truman, A., Wang, F.Y., Wu, J.L., and Yang, W., A link of stochastic differential equations to nonlinear parabolic equations, Science China Mathematics, 2012, 55(10): 1971−1976. doi: 10.1007/s1142501244632. 
[21] 
Wang, B. and Gao, H., Exponential stability of solutions to stochastic differential equations driven by GLévy process, Applied Mathematics & Optimization, 2021, 83(3): 1191−1218. 
[22] 
Wang, B. and Yuan, M., Existence of solution for stochastic differential equations driven by GLévy process with discontinuous coefficients, Advances in Difference Equations, 2017, 2017: 188. doi: 10.1186/s136620171242y. 
[23] 
Xu, J., Shang, H. and Zhang, B., A Girsanov type theorem under Gframework, Stoch. Anal. Appl., 2011, 29(3): 386−406. doi: 10.1080/07362994.2011.548985. 
show all references
References:
[1] 
Gao, F., Pathwise properties and homeomorphic flows for stochastic differential equations driven by GBrownian motion, Stochastic Proc. Appl., 2009, 119(10): 3356−3382. doi: 10.1016/j.spa.2009.05.010. 
[2] 
Hodges, S. and Carverhill, A., Quasi mean reversion in an efficient stock market: The characterisation of economic equilibria which support BlackScholes option pricing, Econom. J., 1993, 103(417): 395−405. 
[3] 
Hu, M. and Peng, S., GLévy processes under sublinear expectations, Probab., Uncertain. Quant. Risk, 2021, 6(1): 1–22. 
[4] 
Li, X. and Peng, S., Stopping times and related Itô’s calculus with GBrownian motion, Stochastic Proc. Appl., 2011, 121(7): 1492−1508. doi: 10.1016/j.spa.2011.03.009. 
[5] 
Osuka, E., Girsanov’s formula for GBrownian motion, Stoch. Process. Appl., 2013, 123(4): 1301−1318. doi: 10.1016/j.spa.2012.12.009. 
[6] 
Paczka, K., Itô calculus and jump diffusions for GLévy processes, arXiv: 1211.2973v3, 2014. 
[7] 
Paczka, K., Gmartingale representation in the GLévy setting, arXiv: 1404.2121v1, 2014. 
[8] 
Peng, S., Filtration consistent nonlinear expectations and evaluations of contingent claims, Acta Mathematicae Applicatae Sinica, English Series, 2004, 20(2): 191−214. doi: 10.1007/s1025500401613. 
[9] 
Peng, S., Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 2005, 26(2): 159−184. doi: 10.1142/S0252959905000154. 
[10] 
Peng, S., Multidimensional GBrownian motion and related stochastic calculus under Gexpectation, Stoch. Process. Appl., 2008, 118(12): 2223−2253. doi: 10.1016/j.spa.2007.10.015. 
[11] 
Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and GBrownian Motion, Probability Theory and Stochastic Modelling, Springer, Berlin, Heidelberg, 2019. 
[12] 
Qiao, H., EulerMaruyama approximation for SDEs with jumps and nonLipschitz coefficients, Osaka J. Math, 2014, 51(7): 47−66. 
[13] 
Qiao, H., The cocycle property of stochastic differential equations driven by GBrownian motion, Chinese Annals of Mathematics, Series B, 2015, 36(1): 147−160. doi: 10.1007/s1140101408691. 
[14] 
Qiao, H. and Wu, J.L., Characterising the pathindependence of the Girsanov transformation for nonLipschitz SDEs with jumps, Statistics & Probability Letters, 2016, 119: 326−333. 
[15] 
Qiao, H. and Wu, J.L., On the pathindependence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces, Discrete & Continuous Dynamical SystemsB, 2019, 24(4): 1449−1467. 
[16] 
Qiao, H. and Wu, J.L., Path independence of the additive functionals for McKean–Vlasov stochastic differential equations with jumps, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2021, 24(1): 2150006. doi: 10.1142/S0219025721500065. 
[17] 
Ren, P. and Yang, F., Path independence of additive functionals for stochastic differential equations under Gframework, Front. Math. China, 2019, 14(1): 135−148. doi: 10.1007/s1146401907521. 
[18] 
Stein, E. M. and Stein, J. C., Stock price distributions with stochastic volatility: An analytic approach, Rev. Financ. Stud., 1991, 4(4): 727−752. doi: 10.1093/rfs/4.4.727. 
[19] 
Song, Y., Uniqueness of the representation for Gmartingales with finite variation, Electron. J. Probab., 2012, 17: 1−15. 
[20] 
Truman, A., Wang, F.Y., Wu, J.L., and Yang, W., A link of stochastic differential equations to nonlinear parabolic equations, Science China Mathematics, 2012, 55(10): 1971−1976. doi: 10.1007/s1142501244632. 
[21] 
Wang, B. and Gao, H., Exponential stability of solutions to stochastic differential equations driven by GLévy process, Applied Mathematics & Optimization, 2021, 83(3): 1191−1218. 
[22] 
Wang, B. and Yuan, M., Existence of solution for stochastic differential equations driven by GLévy process with discontinuous coefficients, Advances in Difference Equations, 2017, 2017: 188. doi: 10.1186/s136620171242y. 
[23] 
Xu, J., Shang, H. and Zhang, B., A Girsanov type theorem under Gframework, Stoch. Anal. Appl., 2011, 29(3): 386−406. doi: 10.1080/07362994.2011.548985. 
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