# American Institute of Mathematical Sciences

June  2022, 7(2): 101-118. doi: 10.3934/puqr.2022007

## Path independence of the additive functionals for stochastic differential equations driven by G-lévy processes

 1 School of Mathematics, Southeast University, Nanjing 211189, Jiangsu, China 2 Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 3 Department of Mathematics, Computational Foundry, Swansea University, Bay Campus, Swansea SA1 8EN, UK

hjqiaogean@seu.edu.cn

Received  January 30, 2022 Accepted  May 29, 2022 Published  June 2022 Early access  June 2022

Fund Project: The first author thanks Professor Renming Song for providing her an excellent environment to work in the University of Illinois at Urbana-Champaign. She also thanks Professor Hongjun Gao for valuable discussion. Both authors are grateful to the referees for their constructive suggestions and comments which led to improve the results and the presentation of this paper. This work was partly supported by NSF of China (Grant Nos. 11001051, 11371352, 12071071) and China Scholarship Council (Grant No. 201906095034).

In this study, we are interested in stochastic differential equations driven by G-Lévy processes. We illustrate that a certain class of additive functionals of the equations of interest exhibits the path-independent property, generalizing a few known findings in the literature. The study is ended with many examples.

Citation: Huijie Qiao, Jiang-Lun Wu. Path independence of the additive functionals for stochastic differential equations driven by G-lévy processes. Probability, Uncertainty and Quantitative Risk, 2022, 7 (2) : 101-118. doi: 10.3934/puqr.2022007
##### References:
 [1] Gao, F., Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian 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 Black-Scholes option pricing, Econom. J., 1993, 103(417): 395−405. [3] Hu, M. and Peng, S., G-Lévy 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 G-Brownian motion, Stochastic Proc. Appl., 2011, 121(7): 1492−1508. doi: 10.1016/j.spa.2011.03.009. [5] Osuka, E., Girsanov’s formula for G-Brownian 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 G-Lévy processes, arXiv: 1211.2973v3, 2014. [7] Paczka, K., G-martingale representation in the G-Lévy 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/s10255-004-0161-3. [9] Peng, S., Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 2005, 26(2): 159−184. doi: 10.1142/S0252959905000154. [10] Peng, S., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, 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 G-Brownian Motion, Probability Theory and Stochastic Modelling, Springer, Berlin, Heidelberg, 2019. [12] Qiao, H., Euler-Maruyama approximation for SDEs with jumps and non-Lipschitz coefficients, Osaka J. Math, 2014, 51(7): 47−66. [13] Qiao, H., The cocycle property of stochastic differential equations driven by G-Brownian motion, Chinese Annals of Mathematics, Series B, 2015, 36(1): 147−160. doi: 10.1007/s11401-014-0869-1. [14] Qiao, H. and Wu, J.-L., Characterising the path-independence of the Girsanov transformation for non-Lipschitz SDEs with jumps, Statistics & Probability Letters, 2016, 119: 326−333. [15] Qiao, H. and Wu, J.-L., 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. [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 G-framework, Front. Math. China, 2019, 14(1): 135−148. doi: 10.1007/s11464-019-0752-1. [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 G-martingales 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/s11425-012-4463-2. [21] Wang, B. and Gao, H., Exponential stability of solutions to stochastic differential equations driven by G-Lévy process, Applied Mathematics & Optimization, 2021, 83(3): 1191−1218. [22] Wang, B. and Yuan, M., Existence of solution for stochastic differential equations driven by G-Lévy process with discontinuous coefficients, Advances in Difference Equations, 2017, 2017: 188. doi: 10.1186/s13662-017-1242-y. [23] Xu, J., Shang, H. and Zhang, B., A Girsanov type theorem under G-framework, Stoch. Anal. Appl., 2011, 29(3): 386−406. doi: 10.1080/07362994.2011.548985.

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##### References:
 [1] Gao, F., Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian 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 Black-Scholes option pricing, Econom. J., 1993, 103(417): 395−405. [3] Hu, M. and Peng, S., G-Lévy 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 G-Brownian motion, Stochastic Proc. Appl., 2011, 121(7): 1492−1508. doi: 10.1016/j.spa.2011.03.009. [5] Osuka, E., Girsanov’s formula for G-Brownian 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 G-Lévy processes, arXiv: 1211.2973v3, 2014. [7] Paczka, K., G-martingale representation in the G-Lévy 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/s10255-004-0161-3. [9] Peng, S., Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 2005, 26(2): 159−184. doi: 10.1142/S0252959905000154. [10] Peng, S., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, 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 G-Brownian Motion, Probability Theory and Stochastic Modelling, Springer, Berlin, Heidelberg, 2019. [12] Qiao, H., Euler-Maruyama approximation for SDEs with jumps and non-Lipschitz coefficients, Osaka J. Math, 2014, 51(7): 47−66. [13] Qiao, H., The cocycle property of stochastic differential equations driven by G-Brownian motion, Chinese Annals of Mathematics, Series B, 2015, 36(1): 147−160. doi: 10.1007/s11401-014-0869-1. [14] Qiao, H. and Wu, J.-L., Characterising the path-independence of the Girsanov transformation for non-Lipschitz SDEs with jumps, Statistics & Probability Letters, 2016, 119: 326−333. [15] Qiao, H. and Wu, J.-L., 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. [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 G-framework, Front. Math. China, 2019, 14(1): 135−148. doi: 10.1007/s11464-019-0752-1. [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 G-martingales 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/s11425-012-4463-2. [21] Wang, B. and Gao, H., Exponential stability of solutions to stochastic differential equations driven by G-Lévy process, Applied Mathematics & Optimization, 2021, 83(3): 1191−1218. [22] Wang, B. and Yuan, M., Existence of solution for stochastic differential equations driven by G-Lévy process with discontinuous coefficients, Advances in Difference Equations, 2017, 2017: 188. doi: 10.1186/s13662-017-1242-y. [23] Xu, J., Shang, H. and Zhang, B., A Girsanov type theorem under G-framework, Stoch. Anal. Appl., 2011, 29(3): 386−406. doi: 10.1080/07362994.2011.548985.
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